Lectures on Symplectic Field Theory Chris Wendl

Lectures on Symplectic Field Theory

Chris Wendl

Institut f

ur Mathematik, Humboldt-Universit

at zu Berlin, Unter

den Linden 6, 10099 Berlin, Germany

E-mail address: wendl@math.hu-berlin.de

Contents

Preface vii

About the current version ix

Lecture 1. Introduction 1

1.1. In the beginning, Gromov wrote a paper 1

1.2. Hamiltonian Floer homology 4

1.3. Contact manifolds and the Weinstein conjecture 9

1.4. Symplectic cobordisms and their completions 16

1.5. Contact homology and SFT 20

1.6. Two applications 23

Lecture 2. Basics on holomorphic curves 25

2.1. Linearized Cauchy-Riemann operators 25

2.2. Some useful So bolev inequalities 28

2.3. The fundamental elliptic estimate 30

2.4. Regularity 32

2.5. Linear local existence and applications 38

2.6. Simple curves and multiple covers 41

Lecture 3. Asymptotic operators 43

3.1. The linearization in Morse homology 43

3.2. Spectral ﬂow 46

3.3. The Hessian of the contact action functional 57

3.4. The Conley-Zehnder index 61

Lecture 4. Fredholm theory with cylindrical ends 67

4.1. Cauchy-Riemann operators with punctures 67

4.2. A global weak regularity result 70

4.3. Elliptic estimates on cylindrical ends 71

4.4. The semi-Fredholm property 73

4.5. Formal adjoints and proof of the Fredholm property 74

Lecture 5. The index formula 81

5.1. Riemann-Roch with punctures 81

5.2. Some remarks o n the formal adjoint 86

5.3. The index zero case on a torus 90

5.4. A Weitzenb¨ock formula for Cauchy-Riemann operators 92

5.5. Large antilinear perturbations and energy concentration 94

iii

iv Chris Wendl

5.6. Two Cauchy-Riemann type problems on t he plane 96

5.7. A linear g luing argument 97

5.8. Antilinear deformations of asymptotic operators 102

Lecture 6. Symplectic cobordisms and moduli spaces 107

6.1. Stable Hamiltonian structures and their symplectizations 107

6.2. Symplectic cobordisms with stable boundary 113

6.3. Moduli spaces of unparametrized holomorphic curves 117

6.4. Simple curves and multiple covers 118

6.5. A local structure result 120

Lecture 7. Smoothness of the moduli space 121

7.1. Transversality theorems in cobordisms 121

7.2. Functional analytic setup 126

7.3. Teichm¨uller slices 131

7.4. Fredholm regula r ity and the implicit function theorem 132

7.5. A universal moduli space 134

7.6. Applying the Sard-Smale theorem 137

7.7. From C

to C

∞

138

Lecture 8. Transversality in symplectizations 141

8.1. Statement of the theorem a nd discussion 141

8.2. Injective points of the pr ojected curve 144

8.3. Smoothness of the universal moduli space 147

Lecture 9. Asymptotics and compactness 151

9.1. Removal of singularities 152

9.2. Finite energy and asymptotics 155

9.3. Degenerations of holomo r phic curves 170

9.4. The SFT compactness theorem 182

Lecture 10. Cylindrical contact homology and the tight 3-t ori 191

10.1. Contact structures on T

and Giroux torsion 191

10.2. Deﬁnition of cylindrical contact homology 194

10.3. Computing HC

∗

, ξ

) 208

Lecture 11. Coherent orientations 225

11.1. Gluing maps and coherence 225

11.2. Permutations of punctures and bad orbits 230

11.3. Orienting moduli spaces in general 232

11.4. The determinant line bundle 234

11.5. Determinant bundles of moduli spaces 237

11.6. An algorithm f or coherent orientations 238

11.7. Permutations and bad orbits revisited 240

Lecture 12. The generating function of SFT 243

12.1. Some important caveats on transversality 243

12.2. Auxiliary data, grading and supercommutativity 244

Lectures on Symplectic Field Theory v

12.3. The deﬁnition of H and commutators 247

12.4. Interlude: How to count points in an orbifold 251

12.5. Cylindrical contact homology revisited 256

12.6. Combinatorics of gluing 259

12.7. Some remarks on torsion, coeﬃcients, and conventions 263

Lecture 13. Contact invariants 267

13.1. The Eliashberg-Givent al-Hofer package 268

13.2. SFT generating functions for cobordisms 278

13.3. Full SFT as a BV

∞

-algebra 288

Lecture 14. Transversality and embedding controls in dimension four 295

Lecture 15. Intersection theory for punctured holomorphic curves 297

Lecture 16. Torsion computations and applications 299

Appendix A. Sobo lev spaces 301

A.1. Approximation, extension and embedding theorems 301

A.2. Products, compositions, and r escaling 305

A.3. Spaces of sections of vector bundles 311

A.4. Some remarks on domains with cylindrical ends 316

Appendix B. The Floer C

space 319

Appendix C. Genericity in the space of asymptotic operators 323

Bibliography 329

Preface

This book is a slight ly expanded version of the lecture notes I produced for a

two-semester course taught at University College London in 2015–16, for Ph.D. stu-

dents with a background in basic symplectic geometry and int erest in symplectic

topology and/or geometric analysis. I say “slightly expanded,” although the reader

will quickly notice that most individual chapters contain far more material than can

reasonably ﬁt into a two-ho ur lecture. In reality, much of that material was only

sketched or mentioned in passing during lectures, and I ended up using the notes

to discuss everything that I would like to have explained if I’d had unlimited time.

This includes relatively detailed discussions of several importa nt technical po ints

(e.g. the deﬁnition of spectral ﬂow, generic transversality in symplectizations, the

punctured Riemann-Roch formula, ﬁnite energy and asymptotics with arbitrary sta-

ble Hamiltonian structures) which a re either incompletely covered by the existing

literature or, in my opinion, simply more diﬃcult to learn from o t her sources than

they should be. For topics that are on the other hand well covered elsewhere, I have

usually not felt obliged to explain every detail, but have tried always to provide

adequate references.

One of the interesting features of SFT is that its foundations ar e— at the time of

this writing—not yet complete. When the original “propaganda paper” [

EGH00]

appeared in 2000, it was widely believed that the technical details would be ﬁlled in

within a few years, and several papers introducing important applications of SFT

to contact topology were written under this assumption. Since then, a certain re-

alization has set in that the results in those papers cannot truly be regarded a s

“theorems” in the sense o f mathematics, and it has become less socially acceptable

to preface statements of results with caveats of the form, “t his theorem is dependent

on the foundations of SFT”. At the same time, the need f or a ro bust perturbation

scheme to achieve transversality in SFT spawned the development of a whole new

approach to inﬁnite-dimensional diﬀerential geometry, the po lyfol d project [

Hof06],

which is intended for much more general applications but is not yet ﬁnished. Opin-

ions vary among symplectic topologists as to how unsatisﬁed we should all be with

this state of aﬀairs, and what could be done about it—among other things, one could

make an entire course out of the discussion of such issues, but I have not chosen to

do that. My approach is instead to develop the classical

analysis of pseudoholo-

morphic curves in symplectizations and symplectic cobordisms, to explain how this

would lead to a t heory of algebraic contact invariants if tr ansversality for multiple

covers were not an issue, and then to use the tools and insights gained fro m this

For the purposes of this discussion, the word “classical” may be deﬁned as “not involving the

words polyfold, virtual o r Kuranishi”.

vii

viii Chris Wendl

discussion to prove rigorous mathematical theorems about contact manifolds. Typi-

cally, such theorems can be r egarded informally as consequences of computations in

a (not yet well-deﬁned) theory called SFT, but in a rigorous sense, they are actually

consequences of the methods used in those computations. Examples covered in these

notes include distinguishing t ight conta ct structures on the 3-torus that are homo-

topic but not isomorphic (Lecture

10), a nd the nonexistence of symplectic ﬁllings

or symplectic cobordisms between certain pairs of contact manifolds (Lecture

16).

The choice of applications is o f course biased somewhat toward my own research

interests.

Prerequisites. The stated target audience f or the lecture course was “Ph.D. stu-

dents in diﬀerential g eometry or r elat ed ﬁelds who are not afraid of analysis”. More

precisely, the notes assume some knowledge of the following topics:

• Diﬀerential geometry: manifolds and vector bundles, diﬀerent ial forms and

Stokes’ theorem, connections, basic familiarity with symplectic manifolds

• Functional analysis: linear operators on Banach spaces, basics of Sobolev

spaces, Fredholm operators

• Diﬀerential topology: smooth mapping degree, int ersection numbers, Sard’s

theorem

• Algebraic topology: fundamental group, homology and cohomo logy of man-

ifolds, Poincar´e duality, ﬁrst Chern class, homological intersection numbers

The following topics a re not considered formal prerequisit es, but some knowledge of

them is likely in any case t o be helpful to the reader, who may want to have a good

reference for them (as suggested below) within arm’s reach:

• Contact manifolds (e.g. Geiges [

Gei08])

• Diﬀerential calculus on Banach spaces and Banach manifolds (e.g. these

two books by Lang: [

Lan93] and [Lan99])

• Closed pseudoholomorphic curves (e.g. McDuﬀ-Salamon [

MS04] or my

other book in preparation [

Wend])

• Floer homology (e.g. Salamon [

Sal99] or Audin-Damian [AD14])

Acknowledgements. I wo uld like to thank the students who sat through the

course that gave rise to t hese notes, and in particular Alexandru Cioba and Agust´ın

Moreno for their assistance in editing the ﬁrst several lectures. My understanding

of Taubes’s approach to the Riemann-Roch formula (explained in Lecture

5) and its

generalization to the punctured case emerged in part from discussions with Chris

Gerig, and I am grateful also to Tim Perutz for helpful hints about Weitzenb¨ock

formula s, and Patrick Massot for patient discussions o f singular integral operators

and elliptic regularity. Thanks also to Michael Hutchings and Janko Latschev for

helping me understand the combinatorial factors in Lecture

12, to Jo Nelson for

helpful comments on coeﬃcients and orbifold singularities, and to Sam Lisi and

Barney Bramham for advice on the Floer C

space.

About the current version

At the time of posting this on the arXiv, Lectures

14, 15 and 16 each consist

of messy handwritten notes t hat have not yet been typed up, but will eventually

appear in the published version of the book. The main goal for those lectures

is to carry out some explicit computations of the torsion invariant introduced at

the end of Lecture

13, and to explain the consequences for ﬁlling and cobordism

obstructions, including for instance the classic result that overtwistedness implies

vanishing contact homology and thus obstructs ﬁllability. In keeping with the spirit

of the book, the theorems about torsion in Lecture 16 will need to be understo od

with the usual caveat that they depend on the unﬁnished f oundations of SFT, but

part of the point is also to extract complete and rigorous proofs of the importa nt

consequences regarding symplectic ﬁllings. Lectures

14 and 15 are more technical

in nature, in t he spirit of Lectures

2 through 9 except that they deal with topics

that are only relevant in low-dimensional settings (and thus signiﬁcantly increase

the power of the theory in those settings). Aside from dealing with topics that

are valuable in their own rig ht, they speciﬁcally precede Lecture

16 because they

introduce techniques that will be used in the computations in that lecture.

As far as the rest of the manuscript is concerned, I have tried to produce some-

thing that is relatively well polished, but I admit I have not tried quite as diligently

for that as I do with most of my research papers. Trying to produce another one

of these lectures every week while teaching the course was a formidable task, and I

had more time to be careful with it in some weeks than in others. I have since gone

back and reworked some portions, but not all, so I apo logize for any sloppiness that

I may have failed so fa r to expunge. All comments and corr ections are welcome,

and may be sent to [email protected]in.de. Updates on the publication of the

boo k will be posted periodically on my website at

https://www.mathematik.hu-berlin.de/

wendl/publications.html#notes

especially if those corrections are received before the book goe s to press

LECTURE 1

Introduction

Contents

1.1. In the beginning, Gromov wrote a pa per 1

1.2. Hamiltonian Floer homology 4

1.3. Cont act manifolds and the Weinstein conjecture 9

1.4. Symplectic cobordisms and t he ir completions 16

1.5. Cont act homology and SFT 20

1.6. Two applications 23

1.6.1. Tight contact structures on T

1.6.2. Filling and cobordism obstructions 23

Symplectic ﬁeld theory is a general framewor k for deﬁning invariants of contact

manifolds and symplectic cobordisms between them via counts of “asymptotically

cylindrical” pseudoholomorphic curves. In this ﬁrst lecture, we’ll summarize some

of the historical background of the subject, and then sketch the basic algebraic

formalism of SFT.

1.1. In the beginning, Gr omov wrote a paper

Pseudoholomorphic curves ﬁrst appeared in symplectic geometry in a 1985 paper

of Gromov [

Gro85]. The development was revolutionary for the ﬁeld of symplectic

topology, but it was not unprecedented: a few years before this, Donaldson had

demonstrated the power of using elliptic PDEs in geometric contexts to deﬁne in-

varia nts of smooth 4-manifolds (see [

DK90]). The PDE that Gro mov used was a

slight generalization of one that was already familiar from complex geometry.

Recall that if M is a smooth 2n-dimensional manifold, an almost complex

structure on M is a smooth linear bundle map J : T M → T M such that J

= −1.

This makes the ta ngent spaces of M into complex vector spaces and thus induces an

orientation on M; the pair (M, J) is called an almost complex manifold. In this

context, a Riemann surface is an almost complex manifold of real dimension 2

(hence complex dimension 1), and a pseudoholomorphic curve (also called J-

holomorphic) is a smooth map

u : Σ → M

satisfying the nonlinear Cauchy-Riemann equation

(1.1) T u ◦ j = J ◦T u,

2 Chris Wendl

where (Σ, j) is a Riemann surface and (M, J) is an almo st complex manifold ( of

arbitrary dimension). The almo st complex structure J is called integrable if M

is admits the structure of a complex manifold such that J is multiplication by i

in holomorphic coordinate charts. By a basic theorem of the subject, every almost

complex structure in real dimension two is integrable, hence one can always ﬁnd

local coordinates (s, t) on neighorhoods in Σ such that

j∂

= ∂

, j∂

= −∂

In these coordinates, (

1.1) takes the form

∂

u + J(u)∂

u = 0.

The fundamental insight of [Gro85] was that solutions to the equation (1.1)

capture information ab out symplectic structures on M whenever they are related to

J in the fo llowing way.

Definition 1 .1. Suppose (M, ω) is a symplectic manifold. An almost complex

structure J on M is said to be tamed by ω if

ω(X, JX) > 0 for all X ∈ T M with X 6= 0.

Additionally, J is compatible with ω if the pairing

g(X, Y ) := ω(X, JY )

deﬁnes a Riemannian metric on M.

We shall denote by J(M) the space of all smooth almost complex structures on

M, with the C

∞

lo c

-topology, and if ω is a symplectic form on M, let

(M, ω), J(M, ω) ⊂ J(M)

denote the subsets consisting of almost complex structures that are tamed by or

compatible with ω respectively. Notice that J

(M, ω) is an open subset of J(M),

but J(M, ω) is not. A proof of the following may be found in [

Wend, §2.2], among

other places.

Proposition 1.2. On any symplectic manif old (M, ω), the spa ces J

(M, ω) and

J(M, ω) are ea c h nonempty and contractible. 

Tameness implies that the energy of a J-holomorphic curve u : Σ → M,

E(u) :=

∗

ω,

is always nonnegative, and it is strictly po sitive unless u is constant. Notice moreover

that if the domain Σ is closed, then E(u) depends only on the cohomology class

[ω] ∈ H

(M) and the homology class

[u] := u

∗

[Σ] ∈ H

(M),

so in par ticular, any family of J-holomorphic curves in a ﬁxed homolo gy class sat-

isﬁes a uniform energy bound. This basic observa t ion is one of the key facts behind

Gromov’s compactness theorem, which states that moduli spaces of closed curves in

a ﬁxed homology class are compact up to “nodal” degenerations.

Lectures on Symplectic Field Theory 3

The most famous application of pseudoholomorphic curves presented in [Gro85]

is Gromov’s nonsqueezing theorem, which wa s t he ﬁrst known example of an obstruc-

tion for embedding symplectic domains that is subtler than the obvious obstruction

deﬁned by volume. The technolo gy introduced in [

Gro85] also led directly to the

development of the Gromov-Witten inva ri ants (see [

MS04, RT95, RT97]), which

follow the same pattern as Donaldson’s earlier smo oth 4-manifold invariants; they

use counts of J-holomorphic curves to deﬁne invariants of symplectic manifolds up

to symplectic deformation equivalence.

Here is another sample application from [

Gro85]. We denote by

A · B ∈ Z

the intersection number between two homolog y classes A, B ∈ H

(M) in a closed

oriented 4-ma nif old M.

Theorem 1.3. Suppose (M, ω) is a closed an d co nnected symplectic 4-man ifold

with the following properties:

(i) (M, ω) does not contain any symplectic submanif old S ⊂ M that is diﬀeo-

morphic to S

and satisﬁes [S] · [S] = −1.

(ii) (M, ω) contain s two s ymp l ectic submanifolds S

, S

⊂ M which are both

diﬀeomorphic to S

, satisfy

] · [S

] = [S

] · [S

] = 0 ,

and have exac tly one i ntersec tion point with each other, which is transv erse

and positive.

Then (M, ω) is symplectomorphic to (S

× S

, σ

⊕ σ

), where fo r i = 1, 2, the σ

are area forms on S

satisfying

= h[ω], [S

]i.

Sketch of the proof. Since S

and S

are both symplectic submanifolds,

one can choose a compat ible almost complex structure J on M for which both of

them are the images of embedded J-holomorphic curves. One then considers the

moduli spaces M

(J) a nd M

(J) o f equiva lence classes of J-holomorphic spheres

homologous to S

and S

respectively, where any two such curves are considered

equivalent if one is a reparametrization of the other (in the present setting this just

means they have the same imag e). These spaces are both manifestly no nempty,

and one can argue via Gromov’s compactness theorem for J-holomorphic curves

that both are compact. Moreover, an inﬁnte-dimensional version of the implicit

function theorem implies that both are smooth 2-dimensional manifolds, carrying

canonical orientations, hence both are diﬀeomorphic to closed surfaces. Fina lly, one

uses positivity of intersections to show that every curve in M

(J) intersects every

curve in M

(J) exactly once, and this intersection is always transverse and positive;

moreover, any two curves in the same space M

(J) or M

(J) a r e either ident ical

or disjoint. It follows tha t b oth moduli spaces a r e diﬀeomorphic to S

, and b oth

consist of smooth families of J-holomorphic spheres that foliate M, hence deﬁning

4 Chris Wendl

a diﬀeomorphism

(J) × M

(J) → M

that sends (u

, u

) to the unique point in the intersection im u

∩im u

. This identiﬁes

M with S

× S

such that each of the submanifolds S

× {∗} and {∗} × S

are

symplectic. The la tter observation can be used to determine the symplectic for m

up to deformation, so that by the Moser stability theorem, ω is determined up to

isotopy by its cohomology class [ω] ∈ H

× S

), which depends only on the

evaluation of ω on [S

×{∗}] and [{∗} ×S

] ∈ H

× S

). 

For a detailed expo sition of the above proof of Theorem

1.3, see [Wene, Theo-

rem E].

1.2. Hamiltonian Floer homology

Throughout the following, we write

:= R/Z,

so maps on S

are the same as 1-periodic maps on R. One popula r version of the

Arnold conjecture on symplectic ﬁxed points can be stated as follows. Suppose

(M, ω) is a closed symplectic manifo ld and H : S

× M → R is a smooth f unc-

tion. Writing H

:= H(t, ·) : M → R, H determines a 1-periodic time-dependent

Hamiltonian vector ﬁeld X

via the relation

(1.2) ω(X

, ·) = −dH

Conjecture 1.4 (Arnold conjecture). If all 1-periodic orbits of X

are nonde-

generate, then the number of these orbits is at least the sum of the Betti numbers

of M.

Here a 1-periodic orbit γ : S

→ M of X

is called nondegenerate if, denoting

the ﬂow of X

by ϕ

, the linearized time 1 ﬂow

dϕ

(γ(0)) : T

γ(0)

M → T

γ(0)

does not have 1 as an eigenvalue. This can be thought of as a Morse condition for

an action functional on the loop space whose critical p oints are periodic orbits; like

Morse critical points, no ndegenerate periodic orbits occur in isolation. To simplify

our lives, let’s restrict attention to contractible orbits and also assume that (M, ω)

is symplectically aspherical, which means

[ω]|

(M)

= 0.

Then if C

∞

contr

, M) denotes the space of all smoothly contractible smooth loops

in M, the symplectic action functional can be deﬁned by

: C

∞

contr

, M) → R : γ 7→ −

¯γ

∗

ω +

(γ(t)) dt,

Elsewhere in the literature, you will sometimes see (

1.2) without the minus s ign on the right

hand side. If you want to know why I s trongly believe that the minus sign be longs there, see

[

Wenc], but to some e xtent this is just a pers onal opinion.

Lectures on Symplectic Field Theory 5

where ¯γ : D → M is any smooth map on the closed unit disk D ⊂ C satisfying

¯γ(e

2πit

) = γ(t),

and the symplectic asphericity condition guarantees that A

(γ) does not depend

on the choice of ¯γ.

Exercise 1.5. Regarding C

∞

contr

, M) as a Fr´echet manifold with tangent

spaces T

∞

contr

, M) = Γ(γ

∗

T M), show that the ﬁrst var iation of the action func-

tional A

(γ)η =

[ω( ˙γ, η) + dH

(η)] dt =

ω( ˙γ − X

(γ), η) dt

for η ∈ Γ(γ

∗

T M). In particular, the critical points of A

are precisely the con-

tractible 1-periodic orbits of X

A few years after Gromov’s int r oduction o f pseudoholomorphic curves, Floer

proved the most important cases of the Arnold conjecture by developing a novel

version of inﬁnite-dimensional Morse theory for the functional A

. This approach

mimicked the homological approach to Morse theory which has since been popular-

ized in books such as [

AD14,Sch93], but was apparently only known to experts a t

the time. In Morse homology, one considers a smooth Riemannian manifold (M, g)

with a Morse function f : M → R, and deﬁnes a chain complex whose generators

are the critical points of f, graded according to their Morse index. If we denote the

generator corresponding to a given critical point x ∈ Crit(f) by hxi, the boundary

map on this complex is deﬁned by

∂hxi =

ind(y)=ind(x)−1



M(x, y)





hyi,

where M(x, y) denotes the moduli space of negative gradient ﬂow lines u : R → M,

satisfying ∂

u = −∇f(u(s)), lim

s→−∞

u(s) = x and lim

s→+∞

u(s) = y. This space

admits a natural R-action by shifting the variable in the domain, and one can show

that for generic choices of f and the metric g, M(x, y)/R is a ﬁnite set whenever

ind(x) −ind(y) = 1. The real magic however is contained in the following statement

about the case ind(x) − ind(y) = 2:

Proposition 1.6. For generic ch oices of f and g and any two critical poi nts

x, y ∈ Crit(f ) with ind(x) − ind(y) = 2, M(x, y)/R is homeomorphic to a ﬁnite

co llection of circles and open intervals whose end points are canonically i dentiﬁed

with the ﬁnite set

∂

M(x, y) :=

[

ind(z)=ind(x)−1

M(x, z) ×M(z, y).

We say that M(x, y) has a natural compatiﬁcation

M(x, y), which has the

topology of a compact 1-manifold with bo undary, and its boundary is the set of

all broken ﬂow lines from x to y, cf. Figure

1.1. This set of broken ﬂow lines

is precisely what is counted if one computes the hyi coeﬃcient of ∂

hxi, hence we

deduce

∂

= 0

6 Chris Wendl

Figure 1.1. One-parameter families of gradient ﬂow lines on a

Riemannian ma nif old degenerate to broken ﬂow lines.

as a consequence of the fact that compact 1-manifolds always have zero boundary

points when counted with appropriate signs.

The homology of the resulting chain

complex can be denoted by HM

∗

(M ; g, f) and is called the Morse homology

of M. The well-known Morse inequalities can then be deduced from a fundamen-

tal theorem stating that HM

∗

(M ; g, f ) is, fo r generic f and g, isomorphic to the

singular homology of M.

With the above notion of Morse homology understood, Floer’s approach to the

Arnold conjecture can now be summarized a s follows:

Step 1: Under suitable technical assumptions, construct a homology theory

∗

(M, ω ; H, {J

}),

depending a priori on the choices of a Hamiltonian H : S

×M → R with

all 1-periodic orbits nondegenerate, and a generic S

-parametrized family

of ω-compatible almost complex structures {J

}

t∈S

. The generators of the

chain complex a r e the critical points of the symplectic action functional

, i.e. 1-periodic orbits of the Hamiltonian ﬂow, and t he boundary map

is deﬁned by counting a suitable notio n of gradient ﬂow lines connecting

pairs of orbits (more on this below).

Step 2: Prove that HF

∗

(M, ω) := HF

∗

(M, ω ; H, {J

}) is a symplectic invariant,

i.e. it depends on ω, but not on the auxiliary choices H and {J

Step 3: Show that if H and {J

} are chosen to be time-independent and H is

also C

-small, then the chain complex for HF

∗

(M, ω ; H, {J

}) is isomor -

phic (with a suitable grading shift) to the chain complex for Morse ho-

mology HM

∗

(M ; g, H) with g := ω(·, J

·). The isomorphism between

∗

(M ; g, H) and singular ho mo logy thus implies that the Floer com-

plex must have at least as many generators (i.e. periodic orbits) as there

are generators of H

∗

(M), proving the Arnold conjecture.

Counting with signs presumes that we have chosen suitable orientations for the mo duli spaces

M(x, y), a nd this can always be done. Alternatively, one can avoid this issue by counting modulo 2

and thus deﬁne a homology theory with Z

coeﬃcients.

Lectures on Symplectic Field Theory 7

The implementatio n of Floer’s idea required a diﬀerent type of analysis than

what is needed for Morse homology. The moduli space M(x, y) in Morse homol-

ogy is simple to understand as the (generically transverse) intersection between the

unstable manifold o f x and t he stable manifold of y with respect to the negative

gradient ﬂow. Conveniently, both of those are ﬁnite-dimensional manifolds, with

their dimensions determined by the Morse indices of x and y. We will see in Lec-

ture

3 that no such thing is true for the symplectic action functional: to the extent

that A

can be thought of as a Morse function on an inﬁnite-dimensional manifold,

its Morse index and its Morse “co-index” at every critical point are both inﬁnite,

hence the stable and unstable manif olds are not nearly as nice as ﬁnite-dimensional

manifolds, providing no reason to expect that their intersection should be. There

are additional problems since C

∞

contr

, M) does not have a Banach space topology:

in order to view the negative gradient ﬂow of A

as an ODE and make use of the

usual local existence/uniqueness theorems (as in [

Lan99, Chapter IV]), one would

have t o extend to A

to a smoot h function on a suitable Hilbert manifold with a

Riemannian metric. There is a very limited range of situations in which o ne can do

this and obt ain a reasonable formula for ∇A

, e.g. [

HZ94, §6.2] explains the case

M = T

, in which A

can be deﬁned on the Sobolev space H

1/2

, R

) and then

studied using Fourier series. This approach is very dependent on the fact that the

torus T

is a quotient of R

; for general symplectic manifo lds (M, ω), one cannot

even deﬁne H

1/2

, M) since functions of class H

1/2

on S

need not be continuous

1/2

is a “Sobolev borderline case” in dimension one).

One of the novelties in F loer’s approach was to refrain from viewing the gradient

ﬂow as an ODE in a Banach space setting, but instead to write down a formal

version of the gradient ﬂow equation and regard it as an elliptic PDE. To this end,

let us regard C

∞

contr

, M) formally as a manifold with tangent spaces

∞

contr

, M) := Γ(γ

∗

T M),

choose a formal Riemannian metric on this manifold (i.e. a smoothly varying family

of L

inner pr oducts on the spaces Γ(γ

∗

T M)) and write down the resulting equation

for the negative gradient ﬂow. A suitable R iemannian metric can be deﬁned by

choosing a smoo t h S

-parametrized family of compatible a lmo st complex structures

∈ J(M, ω)}

t∈S

abbreviated in the following as {J

}, and setting

hξ, ηi

ω(ξ(t), J

η(t)) dt

for ξ, η ∈ Γ(γ

∗

T M). Exercise

1.5 then yields the formula

(γ)η = hJ

( ˙γ − X

(γ)), ηi

so that it seems reasonable to deﬁne t he so-called unregularized gradient of A

(1.3) ∇A

(γ) := J

( ˙γ − X

(γ)) ∈ Γ(γ

∗

T M).

Let us a lso think of a pa t h u : R → C

∞

contr

, M) as a map u : R ×S

→ M, writing

u(s, t) := u(s)(t). The negative gradient ﬂow equation ∂

u + ∇A

(u(s)) = 0 then

8 Chris Wendl

Figure 1.2. A f amily of smooth Floer trajectories can degenerate

into a broken Floer trajectory.

becomes the elliptic PDE

(1.4) ∂

u + J

(u) (∂

u −X

(u)) = 0.

This is called the Floer equation, and its solutions are often called Floer tra-

jectories. The relevance of Floer homology to our previous discussion of pseudo-

holomorphic curves should now be obvious. Indeed, the resemblance of the Floer

equation to the nonlinear Cauchy-Riemann equation is not merely superﬁcial—we

will see in Lecture 6 that the former can always be viewed as a special case of the

latter. In any case, one can use the same set of ana lytical techniques for both: el-

liptic regularity theory implies that Floer trajectories a r e always smooth, Fredholm

theory and the implicit function theorem imply that (under appropriate assump-

tions) they form smooth ﬁnite-dimensional moduli spaces. Most importa ntly, the

same “bubbling oﬀ” analysis that underlies Gromov’s compactness theorem can be

used to prove that spaces o f Floer trajectories are compact up to “breaking”, just as

in Morse homology (see Figure 1.2)—this is the main reason for the relation ∂

= 0

in Floer homology.

We should mention one complication that does not arise either in the study of

closed holomorphic curves or in ﬁnite-dimensional Morse theory. Since the gradient

ﬂow in Morse homolog y takes place on a closed manifold, it is obvious that every

gradient ﬂow line asymptotically approaches critical points at both −∞ and +∞.

The following example shows that in the inﬁnite-dimensional setting of Floer theory,

this is no longer tr ue.

Example 1.7. Consider the Floer equation on M := S

= C ∪{∞} with H := 0

and J

deﬁned as the standard complex structure i for every t. Then the orbits of X

are all constant, a nd a map u : R ×S

→ S

satisﬁes the Floer equation if and only

if it is holomorphic. Identifying R × S

with C

∗

:= C \ {0} via the biholomorphic

map (s, t) 7→ e

2π(s+it)

, a solution u approaches periodic orbits as s → ±∞ if and

only if the corresponding holomo r phic map C

∗

→ S

extends continuously ( and

therefore holomorphically) over 0 and ∞. But this is not true for every holomorphic

map C

∗

→ S

, e.g. take any entire function C → C that has an essentia l singularity

at ∞.

Lectures on Symplectic Field Theory 9

Exercise 1.8. Show that in the above example with an essential singularity

at ∞, the symplectic action A

(u(s, ·)) is unbounded as s → ∞.

Exercise 1.9. Suppose u : R ×S

→ M is a solution to the Floer equation with

lim

s→±∞

u(s, ·) = γ

uniformly f or a pair of 1-periodic orbits γ

∈ Crit(A

). Show

that

(1.5)

A(γ

−

) − A(γ

) =

R×S

ω(∂

u, ∂

u −X

(u)) ds dt =

R×S

ω(∂

u, J

(u)∂

u) ds dt.

The right hand side of (

1.5) is manifestly nonnegative since J

is compatible

with ω, and it is strictly po sitive unless γ

−

= γ

. It is therefore sensible to call

this expression the energy E(u) of a Floer trajectory. The following converse of

Exercise

1.9 plays a crucial role in the compactness theory for Floer trajectories, a s it

guarantees that all the “levels” in a broken Floer trajectory are asymptotically well

behaved. We will prove a variant of this r esult in t he SFT context (see Prop.

1.23

below) in Lecture 9.

Proposition 1.10. If u : R ×S

→ M is a Floer trajectory with E(u) < ∞ and

all 1-periodic orbits of X

are non egenera te, then there exist orbits γ

−

, γ

∈ Crit(A

)

such that lim

s→±∞

u(s, ·) = γ

uniformly. 

Remark 1.11. It should be emphasized again that we have assumed [ω]|

(M)

0 throughout this discussion; Floer homolog y can also be deﬁned under more general

assumptions, but several details become mor e complicated.

For nice comprehensive treatments of Hamiltonian Floer homology—unfortunat ely

not always with the same sign conventions a s used here—see [

Sal99, AD14]. Note

that this is only one of a few “Floer homologies” that were introduced by Floer in

the late 80’s: the others include Lagrangian intersection Fl oer h o mology [

Flo88a]

(which has since evolved into the Fukaya category, see [

Sei08]), and instanton ho-

mology [

Flo88c], an extension of Donaldson’s gauge-theoretic smooth 4-manifold

invariants to dimension three. The development of new Floer-type theories has

since become a major industry.

1.3. Contact manifolds and t he Weinstein conjecture

A Hamiltonian system on a symplectic manifold (W, ω) is called autonomous if

the Ha miltonian H : W → R does not depend o n time. In this case, the Hamiltonian

vector ﬁeld X

deﬁned by

ω(X

, ·) = −dH

is time-independent and its orbits are conﬁned to level sets of H. The images of

these orbits on a given regular level set H

−1

(c)

but not on H itself, as they are the integral curves (also known as character istics)

of the characteristic line ﬁeld on H

−1

(c), deﬁned as the unique direction spanned

by a vector X such that ω(X, Y ) = 0 for a ll Y tangent to H

−1

(c). In 1978, Weinstein

[

Wei78] and Rabinowitz [Rab78] proved that certain kinds of regular level sets in

symplectic manifolds are gua ranteed to admit closed characteristics, hence implying

10 Chris Wendl

Figure 1.3. A star-shaped hypersurface in Euclidean space

the existence of periodic Hamiltonian orbits. In particular, this is true whenever

−1

(see Figure

1.3).

The following symplectic interpretation of the star-shaped condition provides

both an intuitive reason to believe Rabinowitz’s existence result and motivation for

the more general conjecture of Weinstein. In any symplectic manifold (W, ω), a

Liouville vector ﬁeld is a smooth vector ﬁeld V that satisﬁes

ω = ω.

By Cart an’s formula for the Lie derivative, the dual 1-form λ deﬁned by λ := ω(V, ·)

satisﬁes dλ = ω if and only if V is a L iouville vector ﬁeld; moreover, λ then also

satisﬁes L

λ = λ, and it is referred to as a Liouville form. A hypersurface

M ⊂ (W, ω) is said to be of contact type if it is tr ansverse to a Liouville vector

ﬁeld deﬁned on a neighborhood of M.

Example 1.12. Using coordinates (q

, p

, . . . , q

, p

) on R

, the standard sym-

plectic form is written as

std

j=1

∧ dq

Lectures on Symplectic Field Theory 11

and the Liouville form λ

std

j=1

−q

) is dual to the radial Liouville

vector ﬁeld

std

j=1



∂

∂p

+ q

∂

∂q



Any star-shaped hypersurface is therefore of contact type.

Exercise 1.13. Suppose (W, ω) is a symplectic manifold of dimension 2n, M ⊂

W is a smoothly embedded and oriented hypersurface, V is a Liouville vector ﬁeld

deﬁned near M and λ := ω(V, ·) is the dual Liouville form. Deﬁne a 1-form on M

by α := λ|

T M

(a) Show that V is positively transverse to M if and only if α satisﬁes

(1.6) α ∧ (dα)

n−1

> 0.

(b) If V is positively transverse to M, choose ǫ > 0 suﬃciently small and

consider the embedding

Φ : (−ǫ, ǫ) × M ֒→ W : ( r, x) 7→ ϕ

(x),

where ϕ

denotes the time t ﬂow of V . Show that

∗

λ = e

α,

hence Φ

∗

ω = d(e

α).

The above exercise presents any contact-type hypersurface M ⊂ (W, ω) as

one member o f a smooth 1-parameter family of contact-type hypersurfaces M

(M) ⊂ W , each canonically identiﬁed with M such that ω|

T M

= e

dα. In

particular, the characteristic line ﬁelds on M

are the same for all r, thus the ex-

istence of a closed characteristic on any of these implies that there also exists one

on M. This observation has sometimes been used to prove such existence theorems,

e.g. it is used in [

HZ94, Chapter 4] to reduce Rabinowitz’s result to an “almost

existence” theorem based o n symplectic capacities. This discussion hopefully makes

the following conjecture seem believable.

Conjecture 1.14 (Weinstein conjecture, symplectic version). Any cl osed contact-

type hypersurface in a symplectic manifold admits a clos ed characteristic.

Weinstein’s conjecture admits a natural rephrasing in t he language of contact

geometry. A 1-form α on an o r iented (2n − 1)-dimensional manifold M is called a

(positive) contact form if it satisﬁes (

1.6), and the resulting co-oriented hyperplane

ﬁeld

ξ := ker α ⊂ T M

is then called a (positive and co-oriented) contact structure.

We call the pair

(M, ξ) a contact manifold, and r efer to a diﬀeomorphism ϕ : M → M

′

as a

The adjective “positive” refers to the fact that the orie ntation of M agrees with the one deter-

mined by the volume form α ∧(dα)

n−1

; we call α a negative c ontact form if these two orientations

disagree. It is also possible in general to deﬁne contact structures without co-orientations, but con-

tact structures of this type will never appear in these notes; for our purp oses, the co-orientation is

always considered to be par t of the data of a contact s tructure.

12 Chris Wendl

contact omorphism from (M, ξ) to (M

′

, ξ

′

) if ϕ

∗

maps ξ to ξ

′

and also preserves

the respective co-orientations. Equivalently, if ξ and ξ

′

are deﬁned via contact forms

α and α

′

respectively, this means

∗

′

= f α for some f ∈ C

∞

(M, (0, ∞)).

Contact topology studies the cat egory of contact manifolds (M, ξ) up to con-

tactomorphism. The following basic result provides one good reason to regard ξ

rather than α as the geometrically meaningful data, as the result holds for contact

structures, but not for contact forms.

Theorem 1.15 (Gray’s stability theorem). If M is a closed (2n−1)-dimensional

manifold and {ξ

}

t∈[0,1]

is a smooth 1-parameter family of contact structures on M,

then there exists a sm ooth 1-parameter family of diﬀeomorph i s ms {ϕ

}

t∈[0,1]

such

that ϕ

= Id and (ϕ

)

∗

= ξ

Proof. See [

Gei08, §2.2 ] or [Wend, Theorem 1.6.12]. 

A corollary is that while the cont act fo rm α induced on a contact- type hyper-

surface M ⊂ (W, ω) via Exercise

1.13 is not unique, its induced cont act structure is

unique up to isotopy. Indeed, the space of all Liouville vector ﬁelds transverse to M

is very large (e.g. one can add to V any suﬃciently small Hamiltonian vector ﬁeld),

but it is conv ex, hence any two choices of the induced conta ct form α on M are

connected by a smoot h 1-parameter family of contact forms, implying an isotopy of

contact structures via Gray’s theorem.

Exercise 1.16. If α is a nowhere zero 1-form on M and ξ = ker α, show tha t α

is contact if and only if dα|

deﬁnes a symplectic vector bundle structure on ξ → M.

Moreover, the orientation of ξ determined by this symplectic bundle structure is

compatible with the co-or ientation determined by α and the orientation of M for

which α ∧ (dα)

n−1

> 0.

The following deﬁnition is based on the fact that since dα|

is nondegenerate

when α is contact, ker dα ⊂ T M is always 1-dimensional and transverse to ξ.

Definition 1.17. Given a contact f orm α on M, the Reeb vector ﬁeld is the

unique vector ﬁeld R

that satisﬁes

dα(R

, ·) ≡ 0, and α(R

) ≡ 1.

Exercise 1.18. Show that the ﬂow o f any Reeb vector ﬁeld R

preserves both

ξ = ker α and the symplectic vector bundle structure dα|

Conjecture 1.19 (Weinstein conjecture, contact version). On any closed con-

tact manifold (M, ξ) with contact form α, the Reeb vector ﬁeld R

admits a periodic

orbit.

To see that this is equivalent to the symplectic version of the conjecture, ob-

serve that any contact manifold (M, ξ = ker α) can be viewed as the contact-type

hypersurface {0}× M in the open symplectic manifold

(R × M, d(e

α)) ,

called the symplectization of (M, ξ).

Lectures on Symplectic Field Theory 13

Exercise 1.20. Recall that on any smooth manifold M, there is a tautological

1-form λ that locally takes the form λ =

j=1

in any cho ice of local coo r di-

nates (q

, . . . , q

) on a neighbood U ⊂ M, with (p

, . . . , p

) denoting the induced

coordinates on t he cotangent ﬁbers over U. This is a Liouville form, with dλ deﬁn-

ing the canonical symplectic structure of T

∗

M. Now if ξ ⊂ T M is a co-oriented

hyperplane ﬁeld on M, consider the submanifold

M :=



p ∈ T

∗



ker p = ξ and p(X) > 0 for any X ∈ T M pos. transverse to ξ



Show that ξ is contact if and only if S

M is a symplectic submanifold o f (T

∗

M, dλ),

and the Liouville vector ﬁeld on T

∗

M dual to λ is tangent to S

M. Moreover, if ξ is

contact, then any choice of contact f orm for ξ determines a diﬀeomorphism of S

to R × M identifying the L iouville form λ along S

M with e

α.

Remark 1.21. Exercise

1.20 shows that up to symplectomorphism, our deﬁ-

nition of the symplectization of (M, ξ) above actually depends only on ξ and not

on α.

In 1993, Hofer [

Hof93] introduced a new approach to the Weinstein conjecture

that was based in part on ideas of Gromov and Floer. Fix a contact manifold (M, ξ)

with contact form α, and let

J(α) ⊂ J(R × M)

denote the nonempty and cont ractible space o f all almost complex structures J on

R × M satisfying the following conditions:

(1) The natur al translation action on R × M preserves J;

(2) J∂

= R

and JR

= −∂

, where r denotes the canonical coordinate on

the R-factor in R × M;

(3) Jξ = ξ and dα(·, J·) |

deﬁnes a bundle metric on ξ.

It is easy to check that any J ∈ J(α) is compatible with the symplectic structure

d(e

α) on R × M. Moreover, if γ : R → M is any periodic orbit of R

with period

T > 0, then for any J ∈ J(α), the so-called trivial cylinder

u : R × S

→ R × M : (s, t) 7→ (T s, γ(T t))

is a J-holomorphic curve. Following Floer, one version of Hofer’s idea would be to

look for J-holomorphic cylinders that satisfy a ﬁnite energy condition as in Prop.

1.10

forcing them to approach trivial cylinders asymptotically—the existence of such a

cylinder would then imply the existence of a closed Reeb orbit and thus prove the

Weinstein conjecture. The ﬁrst hindrance is that the “o bvious” deﬁnition of energy

in t his context,

R×S

∗

d(e

α),

is not t he right one: this integral is inﬁnite if u is a trivial cylinder. To circumvent

this, notice that every J ∈ J(α) is also compatible with any symplectic structure

of the form

:= d( e

ϕ(r)

α),

14 Chris Wendl

where ϕ is a function chosen freely from the set

(1.7) T :=



ϕ ∈ C

∞

(R, (−1, 1))



′

> 0



Essentially, choosing ω

means identifying R × M with a subset of the bounded

region (−1, 1) × M, in which trivial cylinders have ﬁnite symplectic area. Since

there is no preferred choice for the function ϕ, we deﬁne the Hofer energy

of a

J-holomorphic curve u : Σ → R × M by

(1.8) E(u) := sup

ϕ∈T

∗

This has the desired property of being ﬁnite for trivial cylinders, and it is also

nonnegative, with strict positivity whenever u is not constant.

Another useful observation from [

Hof93] was that if the goal is t o ﬁnd periodic

orbits, then we need not restrict our attention to J-holomorphic cylinders in par-

ticular. One can more generally consider curves deﬁned on an arbitrary punctured

Riemann surface

Σ := Σ \ Γ,

where (Σ, j) is a closed connected Riemann surface and Γ ⊂ Σ is a ﬁnite set of

punctures. For any ζ ∈ Γ, one can ﬁnd coordinates identifying some punctured

neighborhood o f ζ biholomorphically with the closed punctured disk

D := D \ {0} ⊂ C,

and then identif y this with either the positive or negative half- cylinder

:= [0, ∞) ×S

, Z

−

:= (−∞, 0] × S

via the biholomorphic maps

→

D : (s, t) 7→ e

−2π(s+it)

, Z

−

→

D : (s, t) 7→ e

2π(s+it)

We will refer to such a choice as a (positive or negative) holomorphic cylindrical

coordinate system near ζ, and in this way, we can present (

Σ, j) as a Riemann

surface with cylindrical ends, i.e. the union of some compact Riemann surface with

boundary with a ﬁnite collection of half-cylinders Z

on which j takes the standard

form j∂

= ∂

. Note that the standard cylinder R × S

is a special case of this, as

it can be identiﬁed biholomorphically with S

\ {0, ∞}. Another important special

case is the plane, C = S

\ {∞}.

If u : (

Σ, j) → (R × M, J) is a J-holomorphic curve and ζ ∈ Γ is o ne of its

punctures, we will say that u is positively/negatively asymptotic to a T -periodic

Reeb orbit γ : R → M a t ζ if one can choose holomorphic cylindrical coordinates

(s, t) ∈ Z

near ζ such that

u(s, t) = exp

(T s,γ(T t))

h(s, t) for |s| suﬃciently la rge,

Strictly speaking, the energy deﬁned in (

1.8) is not identical to the notion introduced in

[Hof93] and used in many of Hofer’s papers, but it is equivale nt to it in the s ense that uniform

bounds on either notion of energy imply uniform bounds on the other.

Lectures on Symplectic Field Theory 15

{∞} × M

{−∞} × M

Figure 1.4. An asymptotically cylindrical holomorphic curve in a

symplectization, with g enus 1, one positive puncture and two negative

punctures.

where h(s, t) is a vector ﬁeld along the trivial cylinder satisfying h(s, ·) → 0 uni-

formly as |s| → ∞, and the exponential map is deﬁned with respect to any R-

invariant choice of Riemannian metric on R×M. We say that u : (

Σ, j) → (R×M, J)

is asymptotically cylindrical if it is (positively or negatively) asymptotic to some

closd Reeb orbit at each of its punctures. Not e that this partitions the ﬁnite set of

punctures Γ ⊂ Σ into two subsets,

Γ = Γ

∪ Γ

−

the positive and negative punctures respectively, see Figure 1.4.

Exercise 1.22. Suppose u : (

Σ, j) → (R×M, J) is an asymptotically cylindrical

J-holomorphic curve, with the asymptotic orbit at each puncture ζ ∈ Γ

denoted

by γ

, having period T

> 0. Show that

ζ∈Γ

−

ζ∈Γ

−

∗

dα ≥ 0,

with equality if and only if the image of u is contained in that of a trivial cylinder.

In particular, u must have at least one positive puncture unless it is constant. Show

also that E(u) is ﬁnite and satisﬁes an upper bound determined o nly by the periods

of the positive asymptotic orbits.

The following analog ue of Prop.

1.10 will be proved in Lecture 9. For simplicity,

we shall state a weakened version of what Hofer proved in [

Hof93], which did not

require any nondegeneracy assumption. A T -periodic Reeb orbit γ : R → M is

called nondegenerate if the Reeb ﬂow ϕ

has the property t ha t its linearization

along the contact bundle (cf. Exercise 1.18),

dϕ

(γ(0))|

γ(0)

: ξ

γ(0)

→ ξ

γ(0)

does not have 1 as an eigenvalue. Note that since R

is not time-dependent, closed

Reeb orbits are never completely isolated—they always exist in S

-parametrized

families—but t hese families are isolated in the nondegenerate case.

16 Chris Wendl

Proposition 1.23. Suppose (M, ξ) is a closed contact manifold, with a co ntact

form α such that all closed Reeb orbits are nond egenerate. If u : (

Σ, j) → (R×M, J)

is a J-hol omorphic curve with E( u) < ∞ on a punctured Riemann surface such that

none of the punctures are removable, then u is asymptotically cylindrical. 

The main r esults in [Hof93] state that under certain assumptions on a closed

contact 3-manifold (M, ξ), namely if either ξ is ove rtwis ted (as deﬁned in [

Eli89])

or π

(M) 6= 0, one can ﬁnd for a ny contact form α on (M, ξ) and any J ∈ J(α) a

ﬁnite-energy J-holomor phic plane. By Proposition

1.23, this implies t he existence

of a contractible periodic Reeb orbit and thus proves the Weinstein conjecture in

these settings.

1.4. Symplectic cobordisms and their completions

After the developments described in the previous three sections, it seemed nat-

ural that one might deﬁne invariants of contact manifolds via a Floer-type theory

generated by closed Reeb orbits and counting asymptotically cylindrical holomor-

phic curves in symplectizations. This theory is what is now called SFT, and its

basic structure was outlined in a paper by Eliashberg, Giventa l and Hofer [EGH00]

in 2000, t hough some of its analytical foundations remain unﬁnished in 2016. The

term “ﬁeld theory” is an allusion to “topological quantum ﬁeld theories,” which

associate vector spaces to certain geometric objects and morphisms to cobordisms

between those objects. Thus in order to place SFT in its proper setting, we need to

introduce symplectic cobordisms between contact manifolds.

Recall that if M

and M

−

are smooth oriented closed manifolds of the same

dimension, an oriented cobordism fr om M

−

to M

is a compact smooth oriented

manifold W with oriented boundary

∂W = −M

−

⊔ M

where −M

−

denotes M

−

with its orientation r eversed. Given positive contact struc-

tures ξ

on M

, we say that a symplectic manifold (W, ω) is a symplectic cobor-

dism from (M

−

, ξ

−

) to (M

, ξ

) if W is an oriented cobor dism

from M

−

to M

such that both components of ∂W are contact-type hypersurfaces with induced con-

tact structures isotopic to ξ

. Note that our chosen orientation conventions imply

in this case that the Liouville vector ﬁeld chosen near ∂W must point outward at

and inward at M

−

; we say in this case that M

is a symplectically convex

boundary component, while M

−

is symplectically concave. As important special

cases, (W, ω) is a symplectic ﬁlling of (M

, ξ

) if M

−

= ∅, and it is a symplectic

cap of (M

−

, ξ

−

) if M

= ∅. In the literature, ﬁllings and caps are sometimes also

referred to as conv ex ﬁllings or concave ﬁlli ngs respectively.

The contact-type condition implies the existence of a Liouville fo rm λ near ∂W

with dλ = ω, such that by Exercise

1.13, neighborhoods of M

and M

−

in W can

be identiﬁed with the collars (see Fig ure

1.5)

(−ǫ, 0] × M

or [0, ǫ) × M

−

We assume of course that W is assigned the orientation determined by its symplectic form.

Lectures on Symplectic Field Theory 17

((−ǫ, 0] × M

, d(e

))

([0, ǫ) ×M

−

, d(e

−

))

(W, ω)

Figure 1.5. A symplectic cobordism with concave bo undary

−

, ξ

−

) and convex boundary (M

, ξ

), with symplectic collar neigh-

borhoods deﬁned by ﬂowing along Liouville vector ﬁelds near the

boundary.

respectively f or suﬃciently small ǫ > 0, with λ taking the form

λ = e

where α

:= λ|

T M

are conta ct forms for ξ

. The symplectic completion of

(W, ω) is the noncompact symplectic manifold (

W , ˆω) deﬁned by attaching cylindri-

cal ends to these collar neighbo rhoods (Figure 1.6):

(

W , ˆω) = ((−∞, 0] × M

−

, d(e

−

)) ∪

−

(W, ω)

∪

([0, ∞) × M

, d(e

)) .

(1.9)

In this cont ext, the symplectization (R × M, d(e

α)) is symplectomorphic to the

completion of the trivial symplectic cobordism ([0, 1] ×M, d(e

α)) from (M, ξ =

ker α) to itself. More generally, the object in the following easy exercise can also

sensibly b e called a trivial symplectic cobordism:

Exercise 1.24. Suppose (M, ξ) is a closed contact manifold with contact form

α, and f

: M → R is a pair of functions with f

−

< f

everywhere. Show that the

domain



(r, x) ∈ R × M



−

(x) ≤ r ≤ f

(x)



⊂ R × M

deﬁnes a symplectic cobor dism from (M, ξ) to itself, with a global Liouville form

λ = e

α inducing contact forms e

−

α and e

α on its concave and convex boundaries

respectively.

We say that (W, ω) is a n exact symplectic cobordism or Liouville cobor-

dism if the Liouville f orm λ can be extended from a neighborhood of ∂W t o deﬁne

a global primitive of ω on W . Equivalently, this means that ω admits a global Li-

ouville vector ﬁeld that points inward at M

−

and outward at M

. An exact ﬁlling

of (M

, ξ

) is an exact cobordism whose concave boundary is empty. Observe that

if (W, ω) is exact, then its completion (

W , ˆω) also inherits a global Liouville form.

Exercise 1.25. Use Stokes’ theorem to show that there is no such thing as an

exact symplectic cap.

18 Chris Wendl

−

(W, ω)

((−ǫ, 0] × M

, d(e

))

([0, ǫ) ×M

−

, d(e

−

))

([0, ∞) × M

, d(e

))

((−∞, 0] ×M

−

, d(e

−

))

Figure 1.6. The completion of a symplectic cobordism

The above exercise hint s at an important diﬀerence between cobordisms in the

symplectic as opposed to the oriented s mooth category: symplectic cobordisms are

not generally reversible. If W is an oriented cobor dism from M

−

to M

, then

reversing the orientation of W produces an oriented cobordism from M

to M

−

But one cannot simply reverse orientations in the symplectic category, since t he

orientation is determined by the symplectic form. For example, many obstructions

to the existence of symplectic ﬁllings of given contact manifolds are known—some

of them deﬁned in terms of SFT—but we do not know any obstructions at all to

symplectic caps, in fact it is known that a ll contact 3-manifolds admit them.

The deﬁnitions for holomorphic curves in symplectizations in the previous sec-

tion generalize to completions of symplectic cobo r disms in a fairly straightforward

way since these completions look exactly like symplectizations outside of a compact

subset. Deﬁne

J(W, ω, α

, α

−

) ⊂ J(

W )

as the space of all almost complex structures J on

W such tha t

∈ J(W, ω), J|

[0,∞)×M

∈ J(α

) and J|

(−∞,0]×M

−

∈ J(α

−

Lectures on Symplectic Field Theory 19

Figure 1.7. An asymptotically cylindrical holomorphic curve in a

completed symplectic cobordism, with genus 2, one positive puncture

and two negative punctures.

Occasionally it is useful to relax the compatibility condition on W to tameness,

i.e. J|

∈ J

(W, ω), producing a space that we shall denote by

(W, ω, α

, α

−

) ⊂ J(

W ).

As in Prop.

1.2, both of these spaces are nonempty and contractible. We can then

consider asymptotically cylindrical J-ho lomorphic curves

u : (

Σ = Σ \ (Γ

∪ Γ

−

), j) → (

W , J),

which are proper maps asymptotic to closed orbits of R

in M

at punctures in Γ

see Figure

1.7.

One must aga in tinker with the symplectic form on

W in order to deﬁne a notio n

of energy that is ﬁnite when we need it to be. We generalize (

1.7) as

T :=



ϕ ∈ C

∞

(R, (−1, 1))



′

> 0 and ϕ(r) = r near r = 0



and associate to each ϕ ∈ T a symplectic form ˆω

W deﬁned by

ˆω











d(e

ϕ(r)

) on [0, ∞) × M

ω on W,

d(e

ϕ(r)

−

) on (−∞, 0] ×M

−

One can again check that every J ∈ J(W, ω, α

, α

−

) or J

(W, ω, α

, α

−

) is com-

patible with or, respectively, tamed by ˆω

for every ϕ ∈ T . Thus it makes sense to

It seems natural to wonder whether one could not also relax the conditions on the cylindrical

ends and requir e J|

to be tamed by dα

instead o f compatible with it. I do not curr e ntly

know whether this works, but in later lectures we will see some reasons to worry that it might not.

20 Chris Wendl

deﬁne the energy of u : (

Σ, j) → (

W , J) by

E(u) := sup

ϕ∈T

∗

ˆω

It will be a straightforward matter to generalize Propo sition

1.23 a nd show that

ﬁnite energy implies asymptotically cylindrical behavior in completed cobordisms.

Exercise 1.26. Show tha t if (W, ω) is an exact cobo r dism, then every asymp-

totically cylindrical J-holo mo rphic curve in

W has at least one positive puncture.

1.5. Contact homology and SFT

We can now sketch the algebraic structure of SFT. We shall ignore or suppress

several pesky details that are best dealt with later, some of t hem algebraic, others

analytical. Due to analytical problems, some o f the “theorems” t hat we shall (often

imprecisely) state in this section are not yet provable at the current level of tech-

nology, though we exp ect that they will be soon. We shall use quotation marks to

indicate this caveat wherever appropriate.

The standard versions of SFT all deﬁne homology theories with varying levels of

algebraic structure which are meant to be invariants of a contact manifold (M, ξ).

The chain complexes always depend on certain auxiliary choices, including a nonde-

generate contact form α and a generic J ∈ J(α). The generators consist of formal

varia bles q

, one f or each

closed Reeb orbit γ. In the most straightforward gen-

eralization of Hamiltonian Floer homology, the chain complex is simply a graded

Q-vector space generated by the variables q

, and the boundary map is deﬁned by

∂

CCH

′



M(γ, γ

′

)





′

where M(γ, γ

′

) is the moduli space of J-holomorphic cylinders in R × M with a

positive puncture asymptotic to γ and a negative puncture asymptotic to γ

′

, and the

sum ranges over all orbits γ

′

for which this moduli space is 1-dimensional. The count

# (M(γ, γ

′

)/R) is ratio nal, as it includes rational weighting factors that depend on

combinatoria l informatio n and are best not discussed right now.

“Theorem” 1.27. If α admits no contractible Reeb orbits, then ∂

CCH

= 0, and

the resulting homology is independ ent of the choices of α with this property and

generic J ∈ J(α).

The invariant arising from this result is known as cylindrical contact homol-

ogy, and it is sometimes quite easy to work with when it is well deﬁned, though it

has the disadvantage of not always being deﬁned. Namely, the relation ∂

CCH

= 0

can f ail if α admits contra ctible Reeb orbits, because unlike in Floer homology, the

compactiﬁcation of the space of cylinders M(γ, γ

′

) generally includes objects that

are not broken cylinders. In fact, the objects arising in the “SFT compactiﬁcation”

Actually I should b e making a distinction here between “good” and “bad” Reeb orbits, but

let’s discuss that later; see Lecture

11.

Similar combinatorial fac tors ar e hidden behind the symbol “#” in our deﬁnitions of ∂

and H, and will be discussed in earnest in Lecture

12.

Lectures on Symplectic Field Theory 21

, ξ

)

−

, ξ

−

)

−

R × M

−

R × M

−

R × M

−

Figure 1.8. Degeneration of a sequence u

of ﬁnite energy punc-

tured holomorphic curves with genus 2, one positive puncture and two

negative punctures in a symplectic cobo rdism. The limiting holomor-

phic building (v

, v

−

, v

−

, v

−

) in this example has one upper level

living in the symplectization R × M

, a ma in level living in

W , and

three lower levels, each of which is a (possibly disconnected) ﬁnite-

energy punctured nodal holomorphic curve in R × M

−

. The building

has arithmetic genus 2 and the same numbers of positive and negative

punctures a s u

of moduli spaces of ﬁnite-energy curves in completed cobordisms can be quite elab-

orate, see Fig ure

1.8. The combinatorics of the situation are not so bad however

if the cobordism is exact, as is the case for a symplectization: Exercise

1.26 then

prevents curves without positive ends from appear ing. The only possible degen-

erations for cylinders then consist of broken conﬁgurations whose levels each have

exactly one positive puncture and arbitrary negative punctures; moreover, all but

one of the negative punctures must eventually be capped oﬀ by planes, which is why

“Theorem”

1.27 holds in the absence of planes.

If planes do exist, then one can account for them by deﬁning the chain complex

as an algebra rather than a vector space, producing the theory known as contact

homology. For this, the chain complex is taken t o be a graded unital algebra over

22 Chris Wendl

Q, and we deﬁne

∂

(γ

,...,γ

)



M(γ; γ

, . . . , γ

)





. . . q

with M( γ; γ

, . . . , γ

) denoting the moduli space of punctured J-holomorphic spheres

in R × M with a positive puncture at γ and m negative punctures at the orbits

, . . . , γ

, and the sum ranges over all integers m ≥ 0 and all m-tuples of orbits for

which the moduli space is 1-dimensional. The action of ∂

is then extended to the

whole algebra via a graded Leibniz rule

∂

′

) := ( ∂

) q

′

+ (−1)

|γ|

(∂

′

) .

The general compactness and gluing theory for genus zero curves with one positive

puncture now implies:

“Theorem” 1.28. ∂

= 0, and the resulting ho mology is (as a graded unital

Q-algebra) independent of the choices α and J.

Maybe yo u’ve noticed the pattern: in order to accommodate more general classes

of holomorphic curves, we need to add more algebraic structure. The full SFT

algebra counts all rigid holomorphic curves in R ×M, including all combinations of

positive and negative punctures and all genera. Here is a brief picture of what it

looks like. Counting all the 1-dimensional mo duli spaces of J-holomo r phic curves

modulo R-translation in R × M produces a formal power series

H :=



(γ

, . . . , γ

; γ

−

, . . . , γ

−

)



−

. . . q

−

. . . p

g−1

where the sum ranges over all integers g, m

, m

−

≥ 0 and tuples of orbits, ~ and p

(one for each orbit γ) ar e additional formal variables, and

(γ

, . . . , γ

; γ

−

, . . . , γ

−

)

denotes the moduli space of J-holomorphic curves in R × M with genus g, m

positive punctures at the orbits γ

, . . . , γ

, a nd m

−

negative punctures at the

orbits γ

−

, . . . , γ

−

. We can regard H as an operator on a graded algebra W of

formal power series in the variables {p

}, {q

} and ~, equipped with a graded bracket

operation that satisﬁes the quantum mechanical commutation relation

, q

] = κ

where κ

is a combinatorial f actor tha t is best igno red for now. Note that due t o the

signs that accompany the grading, odd elements F ∈ W need not satisfy [F, F] = 0,

and H it self is an odd element, thus the following statement is nontrivial; in fact,

it is the algebraic manifestation of the general compactness and gluing theory for

punctured holomorphic curves in symplectizations.

“Theorem” 1.29. [H, H ] = 0, hence by the graded Jacobi id e ntity, H deter-

mines an operator

SFT

: W → W : F 7→ [H, F]

satisfying D

SFT

= 0. The re sulting homology depend s on (M, ξ) b ut not o n the

auxiliary choices α and J.

Lectures on Symplectic Field Theory 23

It takes some time to understand how pictures such as Figure 1.8 tr anslate

into algebraic relations like [H , H] = 0, but this is a subject we’ll come back to.

There is also an intermediate theory between contact homology and full SFT, called

rational SFT, which counts only g enus zero curves with arbitrary positive and

negative punctures. Algebraically, it is obtained fro m the full SFT alg ebra as a

“semiclassical approximation” by discarding higher-order factors of ~ so that the

commutation bracket in W becomes a gr aded Poisson bracket. We will discuss all

of this in Lecture

12.

1.6. Two applications

We brieﬂy mention two applications that we will b e able to establish rigorously

using the methods developed in t his book. Since SFT itself is not yet well deﬁned

in full generality, this sometimes means using SFT for inspiration while proving

corollaries via more direct methods.

1.6.1. Tight cont act structures on T

. The 3-torus T

= S

×S

with

coordinates (t, θ, φ) admits a sequence of contact structures

:= ker (cos(2πkt) dθ + sin( 2πkt) dφ) ,

one for each k ∈ N. These cannot be distinguished fr om each other by a ny classical

invariants, e.g. they all have the same Euler class, in fact they are all homotopic as

co-oriented 2-plane ﬁelds. Nonetheless:

Theorem 1.30. For k 6= ℓ, (T

, ξ

) and (T

, ξ

ℓ

) are not con tactomorphic.

We will be a ble to prove this in Lecture

10 by rigorously deﬁning and computing

cylindrical contact homology for a suitable choice of contact forms on (T

, ξ

1.6.2. Filling and cobordism obstructions. Consider a closed connected

and oriented surface Σ presented as Σ

∪

−

, where Σ

⊂ Σ are each (not neces-

sarily connected) compact surfaces with a common boundary Γ. By an old result of

Lutz [

Lut77], the 3-manifold S

× Σ admits a unique isotopy class of S

-invariant

contact structures ξ

such that the loops S

× {z} are positively/negatively trans-

verse to ξ

for z ∈

and tangent to ξ

for z ∈ Γ. Now for each k ∈ N, deﬁne

, ξ

) := ( S

× Σ, ξ

)

where Σ = Σ

∪

−

is chosen such that Γ has k connected compo nents, Σ

−

connected with genus zero, a nd Σ

is connected with positive genus (see Figure

1.9).

Theorem 1.31. The contact manifolds (V

, ξ

) d o not admit any symplectic

ﬁllings. Moreo ver, if k > ℓ, then there exists no ex act symplectic cobordism from

, ξ

) to (V

ℓ

, ξ

ℓ

For these examples, one can use explicit constructions from [

Wen13, Avd] to

show that non-exact cobordisms from (V

, ξ

) to (V

ℓ

, ξ

ℓ

) do exist, and so do exact

cobordisms from (V

ℓ

, ξ

ℓ

) to (V

, ξ

), thus both the directionality of the cobordism

relation and the distinction between exact and non-exact are crucial. The proof

of the theorem, due to the author with Latschev and Hutchings [

LW11], uses a

24 Chris Wendl

−

(W, dλ)

, ξ

)

, ξ

)

Figure 1.9. This exact symplectic cobordism does not exist.

numerical contact invariant based o n the full SFT algebra—in part icular, the curves

that cause this phenomenon have multiple positive ends and are thus not seen by

contact homology. We will introduce the relevant numerical invariant in Lecture

and compute it for these examples in Lecture 16.

LECTURE 2

Basics on holomorphic curves

Contents

2.1. Linearized Cauchy-Riemann operators 25

2.2. Some useful Sobolev inequalities 28

2.3. The fundamental elliptic estimate 30

2.4. Regularity 32

2.4.1. The linear case 33

2.4.2. The nonlinear case 35

2.5. Linear local existence and applications 38

2.6. Simple curves and multiple covers 41

In this lecture we begin studying the analysis of J-holomorphic curves. The

coverage will necessarily be a bit sparse in some places, but more detailed proofs of

everything in this lecture can be found in [

Wend].

2.1. Linearized Cauchy-Riemann operators

In order to motivate the study of linear Cauchy-Riemann type operators, we

begin with a formal discussion of the nonlinear Cauchy-R iemann equation and its

linearization.

Fix a Riemann surface (Σ, j) and almost complex manifold (W, J), and suppose

that we wish to understand the structure of some space of the form

(2.1) {u : Σ → W | T u ◦ j = J ◦ T u plus furt her conditions},

where the “further conditions” (which we will for now leave unspeciﬁed) may impose

constraints on e.g. the regularity of u, as well as its boundary and/or asymptotic

behavior. The standard approach in global analysis can be summarized as follows:

Step 1: Construct a smooth Ba nach manifold B of maps u : Σ → W such that all

the solutions we’re interested in will be elements of B. The tangent spaces

B ar e then Banach spaces of sections of u

∗

T W .

Step 2: Construct a smooth Banach space bundle E → B such that for each u ∈ B,

the ﬁber E

is a Banach space of sections of the vector bundle

Hom

(T Σ, u

∗

T W ) → Σ

of complex-antilinear bundle maps (T Σ, j) → (u

∗

T W, J). Since our purpose

is to study a ﬁrst-order PD E, we need the sections in E

to be “one step

less regular” than t he maps in B, e.g. if B consists of maps of Sobolev class

k,p

, then the sections in E

should be of class W

k−1,p

26 Chris Wendl

Step 3: Show that

∂

: B → E : u 7→ T u + J(u) ◦T u ◦ j

deﬁnes a smooth section of E → B, whose zero set is precisely the space o f

solutions (

2.1).

Step 4: Show that under suitable assumptions (e.g. on regular ity and asymptotic

behavior), one can arra nge such t hat for every u ∈

∂

−1

(0), the lineariza-

tion of

∂

(u) : T

B → E

is a Fredholm operator and is generically surjective. (In g eometric terms,

this would mean that

∂

is transverse to the zero section.)

Step 5: Using the implicit function theorem in Banach spaces (see [

Lan93]), the

surjectivity of D

∂

(u) implies that

∂

−1

(0) is a smooth ﬁnite-dimensional

manifold, with its tangent space at each u ∈

∂

−1

(0) canonically identiﬁed

with ker D

∂

(u), hence the dimension of

∂

−1

(0) near u equals the Fredholm

index of D

∂

(u).

Without worrying about the fact that these are actually not Banach spaces, a nd

some Sobolev completion is needed, let us assume, for simplicity, tha t the bundle

E → B has as base the space B = C

∞

(Σ, W ) and the ﬁber over u ∈ B is given by

Hom

(T Σ, u

∗

T W ). The linearization of the section ∂

at a point u ∈

∂

−1

(0)

should then take the form

: Γ(u

∗

T W ) → Ω

0,1

(Σ, u

∗

T W ),

where t he right hand side denotes the space of u

∗

T W -valued (0, 1)-forms on Σ, or

equivalently, smooth sections of

Hom

(T Σ, u

∗

T W ) = T

0,1

Σ ⊗

∗

T W , where T

0,1

denotes the (0, 1)-part of the complexiﬁed cotangent bundle.

In order to compute such a linearization, we need to make a choice of “connec-

tion” on the bundle E. Choose a connection ∇ on W , and recall the fact that this

naturally induces a connection on the bundles T

0,1

Σ ⊗

∗

T W and End(u

∗

T W )

by setting ∇(α ⊗ s) = α ⊗ ∇s and (∇J)s = ∇(Js) − J∇s, for s ∈ Γ(u

∗

T W ),

J ∈ End (u

∗

T W ) and α ∈ Γ(T

0,1

Σ). We shall make the ansatz tha t for any smooth

1-parameter family of maps u

: Σ → W for ρ ∈ (−ǫ, ǫ) and a section η

∈ E

along t he path (i.e a section of the pullback bundle of E under the map (−ǫ, ǫ) → B

mapping ρ to u

), the connection takes the f orm

(∇

)X = ∇

(η

(X)),

for X ∈ T Σ, where this expression should be interpreted as the pullback connection

under the map displayed above. The tensorial property of connections implies that

∇

does not depend on the connection at the values ρ for which η

= 0.

Given u ∈

∂

−1

(0) and η in T

B = Γ(u

∗

T W ), take a one-para meter family u

∈ B

with u

= u and ∂

ρ=0

= η. We then have that

η = ∇



∂

)





ρ=0

= ∇

(T u

+ J(u

) ◦ T u

◦ j)|

ρ=0

Since

∂

u = 0, this is independent of the connection, and we may therefore choose

∇ to be symmetric.

Lectures on Symplectic Field Theory 27

Fix a po int z ∈ Σ and choose local holomorphic coordinates s+it around it . The

symmetry of the connection implies ∇

∂

ρ=0

= ∇

∂

ρ=0

= ∇

η, and similarly

for the variable t. Observing also that ∇

J = ∇

∂

ρ=0

J = ∇

(J(u

))|

ρ=0

, and

using the above ansatz, we obtain

η)∂

= ∇

(∂

+ J(u

) ◦ ∂

ρ=0

= ∇

η + J(u)∇

η + (∇

J)∂

Since D

η is an antilinear map, and ∂

= j∂

, it is therefore determined by its

action on ∂

. One can check that the operator on the right hand side below is also

antilinear, and thus removing t he ∂

, we obt ain

(2.2) D

η = ∇η + J(u) ◦ ∇η ◦ j + (∇

J) ◦ T u ◦ j.

Definition 2.1. Fix a complex vector bundle E over a Riemann surface (Σ, j).

A (real) linear Cauchy-Riemann type operator on E is a real-linear ﬁrst-order

diﬀerential operator

D : Γ(E) → Ω

0,1

(Σ, E)

such that for every f ∈ C

∞

(Σ, R) and η ∈ Γ(E),

(2.3) D(fη) = (

∂f)η + f Dη,

where

∂f denotes the complex-valued (0, 1)-form df + i df ◦ j.

Observe that D is complex linear if and only if the Leibniz rule (

2.3) also holds

for all smooth complex-valued f unctions f, not just real-va lued. It is a standard

result in complex geometry that choo sing a complex-linear Cauchy-Riemann type

operator D on E is equivalent t o endowing it with the structure of a holomorphic

vector bundle, where local sections η are deﬁned to be holomorphic if and only

if Dη = 0. Indeed, every holomorphic bundle comes with a canonical Cauchy-

Riemann o perator that is expressed as

∂ in holomorphic trivializations, and in the

other direction, the equivalence follows from a local existence result for solutions to

the equation Dη = 0, proved in §

2.5 below.

Exercise 2.2. If D is a linear Cauchy-Riemann type operato r on E, prove that

every other such operator is of the form D + A where A : E →

Hom

(T Σ, E) is

a smoo t h linear bundle map. Using this, show that in suitable local trivializations

over a subset U ⊂ Σ identiﬁed biholomorphically with an open set in C, every

Cauchy-Riemann type operator D takes the form

D =

∂ + A : C

∞

(U, C

) → C

∞

(U, C

where

∂ = ∂

+ i∂

in complex coor dina t es z = s + it and A ∈ C

∞

(U, End

)).

Exercise 2.3. Verify that t he linearized operator D

of (

2.2) is a real-linear

Cauchy-Riemann type operator.

This statement about the existence of holomorphic vector bundle structure s is true when

the base is a Riemann surface, but not if it is a higher-dimensional complex manifold. In highe r

dimensions there are obstructions, see e.g. [

Kob87].

28 Chris Wendl

2.2. Some useful Sobolev inequalities

In this section, we review a few general properties o f Sobolev spaces that a re

essential for applications in no nlinear analysis. The results stated here are explained

in more detail in Appendix

Throughout this section we consider functions with values in C unless otherwise

speciﬁed, and deﬁned on an open domain U in either R

or a quotient of R

which the Lebesgue measure is well deﬁned. Certain regularity assumptions must

generally be placed on the boundary of U in order for all the results stated below

to hold; we will ignore this detail except to mention that the necessary assumptions

are satisﬁed for the two classes of domains that we are most interested in, which are

U =

D ⊂ C,

U = (0, L) × S

⊂ C/Z, 0 < L ≤ ∞.

Here D denotes the closed unit disk and

D is its interior. Certain results will be

speciﬁed to hold only for bounded domains, which means in practice that they hold

D and (0, L) ×S

for any L > 0, but not on (0, ∞) × S

Recall that f or p ∈ [1, ∞) we deﬁne the L

norm of a measurable function

f : U → R

to be

kfk



|f|



1/p

For the space L

∞

we deﬁne the norm to b e the essential supremum of f over U.

Denote by

∞

(U) ⊂ C

∞

(U)

the set of smooth functions with compact support in U. We say a function f has

a weak j-th partial derivative g if the integration by parts formula holds for all

ϕ ∈ C

∞

(U):

gϕ = −

f ∂

ϕ.

Equivalently, this means that g is a partial derivative of f in the sense of distribu-

tions (see e.g. [

LL01]). Higher order weak partial derivatives are deﬁned similarly:

recall tha t for a multiindex α = (i

, ...i

) we denote

∂

f =

∂

|α|

∂x

. . . ∂x

where |α| :=

. We then write ∂

f = g if for all ϕ ∈ C

∞

(U),

gϕ = (−1)

|α|

f ∂

ϕ.

Now we may deﬁne W

k,p

(U) to be the set of functions on U with weak par t ial

derivatives up to order k lying in L

, and deﬁne the norm of such a function by:

kfk

k,p =

|α|≤k

k∂

Lectures on Symplectic Field Theory 29

As W

k,p

(U) can be regarded as a subset of a k-fold product of L

(U), it is a Banach

space, and it is reﬂexive and separable for 1 < p < ∞.

While the Sobolev spaces W

k,p

(U) are generally deﬁned on open domains, we of-

ten consider the closure

U as the domain for spaces of diﬀerentiable functions C

(U)

and C

∞

(

U). For instance, C

(U) is the Banach space of k-times diﬀerentiable func-

tions on U whose derivatives up to order k are bounded and uniformly continuous

on U; note that uniform cont inuity implies the existence of continuous extensions

to the closure

U. Given suitable regularity assumptions for the boundary of U, one

can show (with some eﬀort) that C

(

U) is precisely t he set of functions which admit

k-times diﬀerentiable extensions to some open set containing

The following two results are special cases of the more general Theorems

A.6

and A.9 in Appendix A, proofs of which may be found e.g. in [AF03].

Proposition 2.4 (Sobolev embedding theorem). Assume 1 ≤ p < ∞, kp > n

and d ≥ 0 is an in teger. Then there exists a continuous inclusion

k+d,p

(U) ֒→ C

(U),

which is co mpact if U is bounded. 

Proposition 2.5 (Rellich- Kondrachov compactness theorem). If 1 ≤ p < ∞

and U is bounded, then the natural inclusion

k+1,p

(U) ֒→ W

k,p

(U)

is compact. 

Exercise 2.6. Show that Proposition

2.5 fails in general for unbounded do-

mains, e.g. for R.

The next three results for t he case kp > n are proved in §

A.2 as corollar ies of

the Sobolev embedding theorem.

Proposition 2.7 (Banach algebra property). Suppose 1 ≤ p < ∞, kp > n and

0 ≤ m ≤ k. Then the product pairin g (f, g) 7→ fg deﬁnes a continuous bilinear map

k,p

(U) × W

m,p

(U) → W

m,p

(U).

In particular, W

k,p

(U) is a Banach algebra. 

The continuity statements above tr anslate into inequalities between the norms

in the respective spaces. For example, continuous inclusions W

k+d,p

֒→ C

and

k+1,p

֒→ W

k,p

respectively imply that

kfk

≤ ckfk

k+d,p

kfk

k,p

≤ ckfk

k+1,p

for some constants c > 0 which may depend on d, k, p or U, but not f. Similarly,

the Banach algebra property implies

kfgk

m,p

≤ ckfk

k,p kgk

m,p

where again, the constant c is independent of g and f .

30 Chris Wendl

We state the next r esult only for the case of bounded domains; it does have

an extension to unbounded domains, but the statement becomes more complicated

(cf. Theorem

A.12). Given an open set Ω ⊂ R

, we denote

k,p

(U, Ω) :=

u ∈ W

k,p

(U, R

)



u(U) ⊂ Ω

Note that this is an open subset if kp > n, due to the Sobolev embedding theorem.

Proposition 2.8 (C

-continuity property). Assume 1 ≤ p < ∞, kp > n, U is

bo und ed a nd Ω ⊂ R

is an open s et. Then the map

(Ω, R

) ×W

k,p

(U, Ω) → W

k,p

(U, R

) : (f, u) 7→ f ◦ u

is well deﬁned and continuous. 

Remark 2.9 . Though we will not yet use it in this lecture, Propositions

2.4,

2.7 and 2.8 are the essential conditions needed in order to deﬁne smooth Banach

manifold structures on spaces of W

k,p

-smooth maps f rom one manifold to another,

cf. [

El˘ı67, Pal68]. This only works under the condition kp > n, as the smooth

category is not well equipped to deal with discontinuous ma ps!

The following rescaling result will be needed for nonlinear regularity arguments;

see Theorem

A.15 in Appendix A for a proof.

Proposition 2.10. Assume p ∈ [1, ∞) and k ∈ N satisfy kp > n, let

denote the open unit ball in R

, and for each f ∈ W

k,p

(

) and ǫ ∈ (0, 1], deﬁne

∈ W

k,p

(

) by

(x) := f(ǫx).

Then there exist constants C > 0 and r > 0 such that for eve ry f ∈ W

k,p

(

− f(0) k

k,p

(

)

≤ Cǫ

kf − f (0)k

k,p

(

)

for all ǫ ∈ (0, 1].



Exercise 2 .11. Working on a 2 -dimensional domain with kp > 2, prove directly

that for any multiindex α of positive degree k,

k∂

(

≤ ǫ

k−2/p

k∂

(

for f ∈ W

k,p

(

D). Find examples (e.g. in W

1,2

(

D)) to show that no estimate of the

form

k∂

(

≤ C

kf − f (0)k

k,p

(

with lim

ǫ→0

= 0 is possible when kp ≤ 2.

2.3. The fundamental elliptic estimate

We will make considerable use of the fact t hat the linear ﬁrst-order diﬀerential

operator

∂ := ∂

+ i∂

: C

∞

(C, C) → C

∞

(C, C)

is elliptic. There is no need to discuss here precisely what ellipticity means in full

generality (see [

Wend, §2.B] if you’re curious about this); in pra ctice, the main

consequence is the following pair of analytical results.

Lectures on Symplectic Field Theory 31

Theorem 2.12. If 1 < p < ∞, then

∂ : W

1,p

(

D) → L

(

D) admits a bounded

right inverse T : L

(

D) → W

1,p

(

D).

Theorem 2.13. If 1 < p < ∞ and k ∈ N, then there exists a constant c > 0

such that for all f ∈ W

k,p

(

D),

kfk

k,p ≤ ck

∂fk

k−1,p .

Here W

k,p

(

D) denotes the W

k,p

-closure of C

∞

(

D), the latter being space of

smooth functions on

D with compact suppo r t.

The complete proofs of the two theorems above are rather lengthy, and we shall

refer to [

Wend, §2.6 and 2.A] for the details, but we can at least explain why

they hold in the case p = 2. Fir st, it is straightforward to show that the function

K ∈ L

lo c

K(z) =

2πz

is a fundament al solution for the equation

∂u = f, meaning it satisﬁes

∂K = δ

in the sense o f distributions, where δ denotes the Dirac δ-function. Hence for any

f ∈ C

∞

(C), one ﬁnds a smooth solution u : C → C to the equation

∂u = f as the

convolution

u(z) = (K ∗ f)(z) :=

K(z − ζ) f (ζ) dµ(ζ),

where dµ(ζ) denotes the Lebesgue measure with respect to the variable ζ ∈ C. It

is not hard to show from this formula that whenever f ∈ C

∞

, K ∗ f has decaying

behavior at inﬁnity (see [

Wend, Lemma 2.6.13]) . Thus if u ∈ C

∞

and

∂u = f, it

follows that u −K ∗f is a holomorphic function on C that decays at inﬁnity, hence

u ≡ K ∗f. Since C

∞

(

D) is dense in L

(

D) for all p < ∞, Theorem

2.12 now follows

from the claim that for all f ∈ C

∞

(

D), there exist estimates of the f orm

(2.4) kK ∗ fk

(

≤ ckfk

(

, k∂

(K ∗f)k

(

≤ ckfk

(

with ∂

= ∂

or ∂

for j = 1, 2 r espectively, and the constant c > 0 independent of f.

Exercise 2.14. Use Theorem

2.12 and the remarks above to prove Theorem 2.13

for the case k = 1 with f ∈ C

∞

(

D), then extend it to f ∈ W

1,p

(

D) by a density

argument. Then extend it to the general case by diﬀerent iating both f and

∂f.

The ﬁrst estimate in (2.4) is not too hard if yo u remember your introductory

measure theory class: it follows fro m a general “potential inequality” for convolu-

tion operators (see [

Wend, Lemma 2.6.10]), similar to Young’s inequality, the key

points being that K is locally of class L

and

D has ﬁnite measure. For the second

inequality, observe that

∂(K ∗f ) = f, and the rest of the ﬁrst derivative o f K ∗f is

determined by ∂(K ∗f), where

∂ := ∂

− i∂

32 Chris Wendl

Diﬀerentiat ing K in the sense of distributions provides a formula for ∂(K ∗ f) as a

principal value integral, namely

∂(K ∗f)(z) = −

lim

ǫ→0

|ζ−z|≥ǫ

f(ζ)

(z − ζ)

dµ(ζ).

This is a so-called singular integral operat or: it is similar to our pr evious con-

volution operator, but more diﬃcult to handle because the kernel

is not of class

lo c

on C. The proof of the estimate k∂(K ∗ f) k

≤ ckf k

for all f ∈ C

∞

(

follows from a r ather diﬃcult general estimate on singular integral operato r s, known

as the Calder´on-Zygmund inequality, cf. [

Wend, §2.A] and the references therein.

The good news however is that the ﬁrst step in that proof is no t hard: that is the

case p = 2.

As is the case for all elliptic operat ors with constant coeﬃcients, the L

-estimate

on the fundament al solution of

∂ admits an easy proof using Fourier t ransforms:

Proposition 2.15 . For all f ∈ C

∞

(C), we have k∂(K ∗ f)k

= kfk

Proof. A suﬃcient ly nice function u : C → C is related to its Fourier transform

ˆu : C → C by

u(z) =

ˆu(ζ)e

2πi(z·ζ)

dµ(ζ)

and thus satisﬁes the identities

∂u(ζ) = 2πiζ ˆu(ζ),

∂u(ζ) = 2πi

ζ ˆu(ζ).

Since u = K ∗ f we have ˆu =

f, and since

∂K = δ, we have 2πiζ

K = 1. Hence

we may apply Plancharel’s t heorem to deduce

k∂(K ∗f )k

= k∂uk

= k

∂uk

= k2 πi

ζ ˆuk

= k2 πi



2πiζ



= k

= kfk



2.4. Regularity

We will now use the estimate kuk

k,p ≤ ck

∂uk

k−1,p f r om the previous section

to prove three types of results about solutions to Cauchy-Riemann type equations:

(1) All solutions of reasonable Sobolev-type regularity are smooth.

(2) Any collection of solutions satisfying uniform bounds in certain Sobo lev

norms also locally satisfy uniform C

∞

-bounds.

(3) All reasonable Sobolev-type topologies on spaces of solutions are (locally)

equivalent to the C

∞

-topology.

In the following,

⊂ C

Lectures on Symplectic Field Theory 33

denotes the closed disk of radius r > 0, and

denotes its interior. Note that func-

tions of class C

∞

) are assumed to be smooth up to the boundary (or equivalently,

on some open neighborhood of D

in C), not just on

2.4.1. The linear case. Recall from Exercise

2.2 that every linear Cauchy-

Riemann type operator on a vector bundle of complex rank n locally takes the form

∂ + A, where

∂ = ∂

+ i∂

, and A is a smooth function with values in End

Using the Sobolev embedding theorem, the following result implies by induction

that solutions u ∈ W

1,p

to the equation (

∂ + A)u = 0 are always smoot h.

Theorem 2.16 (Linear regularity). Assume 1 < p < ∞ and k, m ∈ N.

(1) If u ∈ W

k,p

(

D) satisﬁes

∂u ∈ W

m,p

(

D), then u is in W

m+1,p

on every

co mpact subset of

(2) Suppose f

∈ W

m,p

(

D) is a sequence converging in the W

m,p

-topo l ogy to

f ∈ W

m,p

(

D) as ν → ∞, and u

∈ W

k,p

(

D) is a sequence with

∂u

= f

(a) If there exist uniform bounds on ku

k,p

and kf

m,p

over

D as

ν → ∞, then ku

m+1,p

is also uniformly bounded on every com pact

subset of

(b) If the seq uen ce u

is W

k,p

-convergent on

D to a function u ∈ W

k,p

(

satisfying

∂u = f, then it is a l s o W

m+1,p

-convergent on every compact

subset of

Proof. We begin by proving statement (

2a), assuming t hat statement (1) is

already known, hence u

∈ W

m+1,p

lo c

(

D) since f

∈ W

m,p

(

D). Assume m = k,

since there is otherwise nothing to prove. Then by induction, it suﬃces t o show

that uniform bounds on ku

k,p

(

and kf

k,p

(

imply a uniform bound on

k+1,p

(

)

for any g iven r < 1; equivalently, t his would mean there is a uniform

bound on k∂

k,p

(

)

for j = 1, 2. In order to apply the elliptic estimate, we need

to work with functions with compact support in

D, thus choose a smooth bump

function

β ∈ C

∞

(

D, [0, 1])

that satisﬁes β|

≡ 1. We then have β ∂

∈ C

∞

(

D), so by Theorem

2.13,

k∂

k,p

(

)

≤ kβ ∂

k,p

(

≤ c



∂ (β ∂

)



k−1,p

(

≤ ck(

∂β)(∂

k−1,p + ckβ

∂(∂

k−1,p .

(2.5)

The ﬁrst term on the right hand side is uniformly bounded since

∂β is smo oth and

k,p

is uniformly bounded. To control the second term, we diﬀerentiate the

equation

∂u

= f

, giving

∂(∂

) = ∂

This also ha s a uniformly bounded W

k−1,p

-norm since kf

k,p is uniformly bounded.

Since β is smooth, this bounds the second term on the right hand side of (

2.5) as

ν → ∞, and we ar e done.

Statement (

2b) follows by a similar argument bounding k∂

(u − u

k,p

(

)

terms of ku − u

k,p

(

and kf − f

k,p

(

; we leave the details as an exercise.

34 Chris Wendl

Lastly, we prove stat ement (1), where again it suﬃces to assume

∂u = f ∈

k,p

(

D) and show that u|

∈ W

k+1,p

(

) for some r < 1. The idea is to use the

same argument that was used for statement (

2a), but with the partial derivatives

∂

u replaced by the diﬀerence quotients

u(z) :=

u(z + he

) −u(z)

, j = 1, 2,

where e

:= ∂

, e

:= ∂

, and the r ole of the index ν → ∞ is now played by the

parameter h ∈ R \ {0} approaching 0. Note that if u ∈ W

k,p

(

D), t hen β D

u is a

well-deﬁned function on

D for all |h| 6= 0 suﬃciently small and belongs to W

k,p

(

D).

The analogue of (

2.5) in this context is then

k,p

(

)

≤ kβD

k,p

(

≤ c



∂



β D





k−1,p

(

≤ ck(

∂β)(D

u)k

k−1,p + ckβ

∂(D

u)k

k−1,p .

The ﬁrst t erm is bounded independently o f h since ∂

u ∈ W

k−1,p

(

D), implying a

uniform W

k−1,p

-bound on D

u as h → 0. To control the second term, we can apply

the operator D

to the equation

∂u = f, giving

∂(D

u) = D

(

∂u) = D

This satisﬁes a W

k−1,p

-bound that is uniform in h since ∂

f ∈ W

k−1,p

(

D), so we

conclude t hat for all |h| suﬃciently small,

k,p

(

)

≤ c

for some constant c > 0 tha t does not change as h → 0. By a standard application

of the Banach-Alaoglu theorem (cf. [

Eva98, §5.8.2]), this implies the existence of a

sequence h

→ 0 for which D

u is W

k,p

-convergent on

, and its limit is necessarily

∂

u, which therefore belongs to W

k,p

. Indeed, if k = 0, the uniform L

-bound

on D

u over

for any sequence h

→ 0 gives rise to a weakly L

-convergent

subsequence via the Banach-Alaoglu theorem. The limit of this subsequence belongs

to L

(

), and it is straightforward to show using the deﬁnition of weak derivatives

that this limit is ∂

u. One ﬁnds the same result for a ny k ∈ N by applying this

argument to higher-order derivatives of ∂

u. The conclusion is that u is in W

k+1,p

, since u and both of its ﬁrst par t ial derivatives belong to W

k,p

. 

Exercise 2.17. Show that all three parts of Theorem

2.16 continue to hold if

the operator

∂ is replaced by

∂ + A or

∂ + A

, where A, A

∈ C

∞

(D, End

))

with A

→ A in C

∞

as ν → ∞.

Exercise 2.18. Use Theorem

2.16(1) to extend Theorem 2.12 to the existence

of a bounded right inverse for

∂ : W

k,p

(

D) → W

k−1,p

(

D).

Hint: For any R > 1, there exists a bounded linear extension o perator E : W

k,p

(

D) →

k,p

(

) with the property (Ef)|

= f for all f ∈ W

k,p

(

D); see Theorem

A.4 and

Corollary A.5.

Lectures on Symplectic Field Theory 35

The above exercise can be used to improve the ﬁrst part of Theorem 2.1 6 to

cover weak solutions of class L

lo c

. We start with a classical result about “weakly

holomorphic” functions:

Lemma 2.19. If u ∈ L

(

D) satisﬁes

∂u = 0 in the se nse of distributions, then u

is smooth and holomorphic.

Proof. Taking real and imaginary parts, it suﬃces to prove that the same

statement holds for the Laplace equation. By molliﬁcation, a ny weakly ha r mo nic

function can be approximated in L

with smooth harmonic functions. The lat-

ter satisfy the mean value property, which behaves well under L

-convergence, so

the result fo llows from the mean value characterization of harmonic functions; see

[

Wend, Lemma 2.6.26] for more details. 

Lemma 2.20. Suppose 1 < p < ∞, k ∈ N, and u ∈ L

(

D) is a weak solution to

∂u = f for some f ∈ W

k,p

(

D). Then u is of class W

k+1,p

on every compact subset

Proof. Let T : W

k,p

(

D) → W

k+1,p

(

D) denote a bounded right inverse of

∂ :

k+1,p

(

D) → W

k,p

(

D) as provided by Exercise

2.18. Then u − T f ∈ L

(

D) is a

weak solution to

∂(u − T f) = 0 and is thus smooth by Lemma

2.19. In particular,

u − T f restricts to

for every r < 1 as a function of class W

k+1,p

, implying that

u also has a restriction in W

k+1,p

(

). 

Corollary 2.21 (Weak linear regularity). Suppos e 1 < p < ∞. Then given

A ∈ C

∞

(D, End

)), eve ry w eak solution u ∈ L

(

D, C

) of (

∂ +A)u = 0 is smooth

D. 

2.4.2. The nonlinear case. L ocally, every J-holomorphic curve can be re-

garded as a map u :

D → C

satisfying u(0) = 0 and

∂

u := ∂

u + J(u)∂

u = 0,

where J is a smooth almost complex structure on C

satisfying J(0) = i. Theo-

rem

2.16 now has the following analogue.

Theorem 2.22 (Nonlinear regularity). Assume 1 < p < ∞ and k ∈ N s atisfy

kp > 2, and ﬁx a smooth almost compl e x structure J on C

with J(0) = i.

(1) Every map u ∈ W

k,p

(

D, C

) satisfying u(0) = 0 and

∂

u = 0 is smooth

(2) Suppose J

is a sequence of s mooth alm ost comple x structures on C

co n-

verging in C

∞

lo c

to J as ν → ∞, and u

∈ W

k,p

(

D, C

) is a sequence of

smooth maps s atisfying

∂

= 0.

(a) If the maps u

are uniformly W

k,p

-bounded on

D, then they are also

uniformly C

-bounded on compact subsets of

D for every m ∈ N.

(b) If the sequence u

is W

k,p

-convergent on

D to a smooth map u :

D →

, then it is also C

∞

-convergent on every compact subset of

36 Chris Wendl

Our proo f of this will follow much the same outline as the proof of Theorem 2.16,

and indeed, one could use exactly the same argument if J were identically equal to i

(in which case the theorem can also be deduced from complex analysis). The r eason

it works in the general case is tha t if we zoom in on a suﬃciently small neighborhood

of the origin in C

, then J can be viewed as a C

∞

-small p erturbation of i. To make

this precise, we shall use the following rescaling trick.

Associate to any smooth almost complex structure J on C

the function

Q := i − J ∈ C

∞

, End

)).

In terms of Q, the equation ∂

u + J(u)∂

u = 0 then becomes

(2.6)

∂u − (Q ◦ u)∂

u = 0,

where we are regarding Q ◦ u as a function

D → End

). Given constants R ≥ 1

and ǫ ∈ (0, 1], associate to J and u the functions

J : C

→ End

J(p) := J(p/R),

Q : C

→ End

Q(p) := Q(p/R) = i −

J(p),

ˆu :

D → C

, u(z) := Ru(ǫz).

(2.7)

Now u satisﬁes (

2.6) if and only if ˆu satisﬁes

(2.8)

∂ˆu − (

Q ◦ ˆu)∂

ˆu = 0.

The rescaled almost complex structure has the convenient feature that if J(0) = i,

then

J can be made arbitrarily C

∞

-close to i on the unit disk

⊂ C

by choosing R suﬃciently large, which means k

)

can be made arbitrarily

small for every m ∈ N. If u is also continuous and satisﬁes u(0) = 0, then after

ﬁxing some large value for R, we can also choose ǫ ∈ (0 , 1] suﬃciently small to ensure

D) ⊂

and make k

Q ◦ ˆuk

(D)

arbitrarily small. By Propositions 2 .8 and 2.10,

we can similarly arrang e for k

Q ◦ ˆuk

k,p to be arbitrarily small if u is of class W

k,p

with kp > 2, and the same will hold for k

◦ ˆu

k,p when ν is large if ku

k,p

is uniformly bounded and u

(0) → 0. Here of course we abbreviate Q

:= i − J

and

(p) := Q

(p/R). The eﬀect is to make equations such as (

2.8) W

k,p

-close

to the linear equation

∂ˆu = 0 if ǫ > 0 and R > 0 are suﬃciently small and large

respectively.

The price we pay for this r escaling is t hat if we are able to prove e.g. a uniform

bound on the norms kˆu

k,p

(

for some sequence u

, then the resulting W

k+1,p

bound for u

will be valid only on

, a very small ball about the origin. But this

is good enough for obtaining estimates over all compact subsets of

D: indeed, we

can always reparametrize u :

D → C

to put the origin at some other point and

prove suitable estimates near that point, appealing in the end to the fact that any

compact subset of

D is covered by a ﬁnite union of small disks about points.

The need to use this rescaling trick is one of a few reasons why the condition

kp > 2 is needed in Theorem

2.22, while it was irrelevant in the linear case.

Lectures on Symplectic Field Theory 37

Proof of Theorem 2.22. We will prove statement (2a) and leave the rest as

exercises.

By the remarks above, it suﬃces to prove that if u

D → C

are smooth J

holomorphic curves satisfying a unifo rm bound in W

k,p

(

D), then for some r < 1, the

rescaled

-holomorphic curves ˆu

D → C

deﬁned as in (2.7) satisfy a uniform

k+1,p

-bound on

. In fact, it suﬃces to prove that every subsequence of u

has

a further subsequence for which this is true. Indeed, if the bound for the whole

sequence did not exist, then we would be able to ﬁnd a subsequence with norms

blowing up to inﬁnity, and no further subsequence of this subsequence could satisfy

a uniform bound. With this understood, we can appeal to the f act that W

k,p

bounded sequences are also C

-bounded for kp > 2 and thus replace u

with a

subsequence ( still denoted by u

) such that, after a suitable change of coordinat es

on C

(0) → 0.

Our goal is then to show that for a suitable choice of the rescaling parameters ǫ and

R, this subsequence admits a uniform bound on k∂

ˆu

k,p

(

)

for j = 1, 2 .

The a rgument b egins exactly the same as in the linear case: choose a smooth

bump function

β ∈ C

∞

(

D, [0, 1])

that satisﬁes β|

≡ 1. We then have β ∂

ˆu

∈ C

∞

(

D), so by Theorem

2.13,

(2.9) k∂

ˆu

k,p

(

)

≤ kβ ∂

ˆu

k,p

(

≤ c



∂ (β ∂

ˆu

)



k−1,p

(

Instead of rewriting

∂(β ∂

ˆu

) as a sum of two terms, let us derive a PDE satisﬁed

by β ∂

ˆu

. Diﬀerentiating the equation

∂ˆu

− (

◦ ˆu

)∂

ˆu

= 0 gives

∂(∂

ˆu

) = ∂

(

∂ˆu

) = ( d

◦ ˆu

) (∂

ˆu

, ∂

ˆu

) + (

◦ ˆu

)∂

∂

ˆu

thus β ∂

ˆu

satisﬁes

∂(β ∂

ˆu

) − (

◦ˆu

)∂

(β ∂

ˆu

)

= β(d

◦ ˆu

)(∂

ˆu

, ∂

ˆu

) +



∂β − (

◦ ˆu

)∂



∂

ˆu

= (d

◦ ˆu

)(β ∂

ˆu

, ∂

ˆu

) +



∂β − (

◦ ˆu

)∂



∂

ˆu

(2.10)

and combining this with (

2.9) gives

(2.11)

kβ ∂

ˆu

k,p

≤ c



(

◦ ˆu

)∂

(β ∂

ˆu

)



k−1,p

+ c



◦ ˆu

)(β ∂

ˆu

, ∂

ˆu

)



k−1,p

+ c





∂β − (

◦ ˆu

)∂



∂

ˆu



k−1,p

In order to ﬁnd bounds for the three terms on the right, recall that using Proposi-

tions

2.8 and 2.10 and the assumption u

(0) → 0, we can supp ose



◦ ˆu



k,p

≤ δ

38 Chris Wendl

for suﬃciently la r ge ν, where δ > 0 is a constant that may be assumed arbitrarily

small via suitable choices of the rescaling par ameters ǫ and R. This provides a

uniform bound on the third term in (

2.11), as there is also a continuous product

pairing W

k,p

× W

k−1,p

→ W

k−1,p

by Prop.

2.7, giving an estimate of the form





∂β − (

◦ ˆu

)∂



∂

ˆu



k−1,p

≤ c





∂β − (

◦ ˆu

)∂





k,p

· k∂

ˆu

k−1,p

≤ c

′

kˆu

k,p ≤ c

′′

For the ﬁrst term on the right side of (

2.11), the product pa iring similarly gives



(

◦ ˆu

)∂

(β ∂

ˆu

)



k−1,p

≤ c



◦ ˆu



k,p

· k∂

(β ∂

ˆu

k−1,p

≤ cδkβ ∂

ˆu

k,p .

Finally, since J

→ J in C

k+1

on compact subsets, we ar e also free to assume after

adjusting the rescaling parameters that

◦ ˆu

k,p

≤ δ,

so we can apply the product pairing W

k,p

×W

k−1,p

→ W

k−1,p

twice to estimate



◦ ˆu

)(β ∂

ˆu

, ∂

ˆu

)



k−1,p

≤ c



◦ ˆu



k,p

· kβ ∂

ˆu

k,p · k∂

ˆu

k−1,p

≤ cδkβ ∂

ˆu

k,p · kˆu

k,p

≤ cc

′

δkβ ∂

ˆu

k,p =: c

′′

δkβ ∂

ˆu

k,p .

Combining the three estimates for the rig ht hand side of (

2.11) now gives

kβ ∂

ˆu

k,p ≤ c + cδkβ ∂

ˆu

k,p ,

so after adjusting the scaling parameters R and ǫ to ensure cδ < 1, we obtain the

uniform bound

kβ ∂

ˆu

k,p ≤

1 − cδ

This pr ovides the desired unifor m bound on k∂

ˆu

k,p

(

)

. 

Exercise 2.23. Use an analogo us arg ument via diﬀerence quotients to prove

statement (

1) in Theorem 2.22. Hint: If you’re anything like me, you might get

stuck trying to estimate t he second term in the diﬀerence quotient a nalogue of

(

2.11). The diﬃculty is that this expression was derived using the chain rule for

derivatives, and there is no similarly simple chain rule for diﬀerence quotients. The

trick is to remember that diﬀerence quotients only diﬀer from t he corresponding

derivatives by a remainder term. The remainder will produce an extra term in the

diﬀerence quotient version of (

2.11), but the extra term can be bounded.

2.5. Linear local existence and applications

The following lemma can be applied in the case A ∈ C

∞

(D, End

)) to prove

the aforementioned standard fact that complex-linear Cauchy-Riemann type oper-

ators induce holomorphic structures on vector bundles. The version with weakened

regularity will be applied below to prove a useful “unique continuation” result about

solutions to (

∂ + A)f = 0 in the real-linear case.

Lectures on Symplectic Field Theory 39

Lemma 2.24. Assume 2 < p < ∞ and A ∈ L

(

D, End

)). Then for suﬃ-

ciently small ǫ > 0, the proble m

∂u + Au = 0

u(0) = u

has a so l ution u ∈ W

1,p

(

, C

Remark 2.25. Note that u :

→ C

in the above statement is only a weak

solution to

∂u + Au = 0, as it is not necessarily diﬀerentiable, but by the Sobolev

embedding theorem, it is at least continuous.

Proof of Lemma

2.24. The main idea is that if we take ǫ > 0 suﬃciently

small, then t he restriction of

∂ + A to

can be regarded as a small perturbat ion

∂ in t he space of b ounded linear operator s W

1,p

→ L

. Since the latter has a

bounded right inverse by Theorem 2.12, the same will be true for the perturbation.

Since p > 2, the Sobolev embedding theorem implies that functions u ∈ W

1,p

are also continuous and bounded by kuk

1,p

, thus we can deﬁne a bounded linear

operator

Φ : W

1,p

(

D) → L

(

D) × C

: u 7→ (

∂u, u(0)).

Theorem

2.12 implies that this operator is also surjective and has a bounded right

inverse, namely

(

D) ×C

→ W

1,p

(

D) : (f, u

) 7→ T f − T f (0) + u

where T : L

(

D) → W

1,p

(

D) is a right inverse of

∂. Thus any operator suﬃciently

close to Φ in the norm topology also has a right inverse. Now deﬁne χ

: D → R to

be the function that equals 1 on D

and 0 outside of it, and let

: W

1,p

(

D) → L

(

D) × C

: u 7→ ((

∂ + χ

A)u, u(0)).

To see that this is a bo unded operator, it suﬃces to check that W

1,p

→ L

: u 7→ Au

is bounded if A ∈ L

; indeed,

kAuk

≤ kAk

kuk

≤ ckAk

kuk

1,p

again using the Sobolev embedding theorem. Now by this same trick, we ﬁnd

kΦ

u − Φuk = kχ

Auk

(

≤ ckAk

(

)

kuk

1,p

(

thus kΦ

− Φk is small if ǫ is small, and it follows that in this case Φ

is surjective.

Our desired solution is therefore the restriction of any u ∈ Φ

−1

(0, u

) to

. 

Here is a corollary, which says that every solution to a real-linear Cauchy-

Riemann type equation looks locally like a holomorphic function in some continuous

local trivialization.

Theorem 2.26 (Similarity principle). Suppose A : D → End

) is smooth an d

u :

D → C

satisﬁes the equation

∂u + Au = 0 with u(0) = 0 . Then for suﬃciently

small ǫ > 0, there exist map s Φ ∈ C

, End

)) and f ∈ C

∞

(

, C

) such that

u(z) = Φ(z)f (z),

∂f = 0, and Φ(0) = 1.

40 Chris Wendl

Proof. After shrinking the domain if necessary, we may assume without loss

of generality that the smooth solution u :

D → C

is bounded. Choose a map

C : D → End

) satisfying C(z)u(z) = A(z)u(z) and |C(z)| ≤ |A(z)| for almost

every z ∈ D. Then C ∈ L

∞

(

D, End

)) and u is a weak solution to (

∂ + C)u = 0.

Note that since we do not know anything about the zero set of u, we cannot assume

C is continuous, but we have no trouble assuming C ∈ L

(

D) for every p > 2.

Since

∂+C is now complex linear, we can use Lemma

2.24 to ﬁnd a complex basis

of W

1,p

-smooth weak solutions to (

∂+C)v = 0 on

that deﬁne the standard basis of

at 0, and these solutions are continuous by the Sobolev embedding theorem. This

gives rise to a map Φ ∈ C

(

, End

)) that satisﬁes (

∂ + C)Φ = 0 in the sense

of distributions and Φ(0) = 1. Since Φ is continuous, we can assume without loss of

generality that Φ(z) is invertible everywhere on

. Setting f := Φ

−1

u :

→ C

the Leibniz rule then implies

0 = (

∂ + C)u = (

∂ + C)(Φf) =



(

∂ + C)Φ



f + Φ(

∂f ) = Φ(

∂f),

thus

∂f = 0, and f is smooth by Lemma

2.19. 

Corollary 2.27 (Unique continuation). Suppose D is a linear Cauchy-Riemann

type operator on a vector bundle E over a connected Riemann surface, and η ∈ Γ(E)

satisﬁes Dη = 0. Then either η is identically zero or its zeroes are isolated.

The similarity principle also has many nice applications for the nonlinear Cauchy-

Riemann equation. Here is ano t her “unique continuation” type result for t he non-

linear case.

Proposition 2.28. Suppose J is a smooth almost complex structure on C

and

u, v :

D → C

are smooth J-holomorphic curves such that u(0) = v(0) = 0 and

u and v have matching partial derivatives of all orders at 0. Then u ≡ v on a

neighborhood of 0.

Proof. Let h = v − u :

D → C

. We have

(2.12) ∂

u + J(u(z))∂

u = 0

and

∂

v + J(u(z))∂

v = ∂

v + J(v(z))∂

v + [J(u(z)) − J(v(z))] ∂

= −[J(u(z) + h(z)) − J(u(z))] ∂

= −



J(u(z) + th(z)) dt



∂

= −



dJ(u(z) + th(z)) ·h(z) dt



∂

v =: −A(z)h(z),

(2.13)

where the last step deﬁnes a smooth family of linear maps A(z) ∈ End

). Sub-

tracting (

2.12) from (2.1 3) gives the linear equation

∂

h(z) +

J(z)∂

h(z) + A(z)h(z) = 0,

where

J(z) := J(u(z)). This is a linear Cauchy-Riemann type equation on a trivial

complex vector bundle over

D with complex structure

J(z) on the ﬁber at z. The

Lectures on Symplectic Field Theory 41

similarity principle thus implies h(z) = Φ(z)f(z) near 0 for some holomor phic func-

tion f(z) ∈ C

and some continuous map Φ(z) ∈ GL(2n, R) representing a change of

trivialization. Now if h has vanishing derivatives of a ll orders at 0, Taylor’s formula

implies

lim

z→0

|Φ(z)f (z)|

|z|

= 0

for all k ∈ N, so f must also have a zero of inﬁnite order and thus f ≡ 0. 

2.6. Simple curves and multiple covers

We now prove a global result about the structure of closed J-holomorphic curves.

In Lecture

6 we will be able to generalize it in a straightforward way for punctured

holomorphic curves with a symptotically cylindrical behavior.

Theorem 2.29. Assume (Σ, j) is a closed conn ected Riemann surface, (W, J)

is a smooth almost com plex manifold and u : (Σ, j) → (W, J) is a nonconstant

pseudoholomorphi c curve. Then there exists a fac toriz ation u = v ◦ ϕ, where

• ϕ : (Σ, j) → (Σ

′

, j

′

) is a hol omorphic map of positive degree to another

closed and con nected Riemann surface (Σ

′

, j

′

);

• v : (Σ

′

, j

′

) → (W, J) is a pse udo holomorphic curve which is embed ded except

at a ﬁnite set of critical points and self-intersections.

Note that holomorphic maps (Σ, j) → (Σ

′

, j

′

) of degree 1 are always diﬀeomor-

phisms, so the factorization u = v ◦ ϕ in this case is just a repara metrizatio n, and

u is then called a simple curve. In all other cases, k := deg(ϕ) ≥ 2 and ϕ is in

general a branched cover; we then call u a k-fold branched cover of the simple

curve v.

The main idea in the proof is to construct Σ

′

(minus some punctures) explicitly

as the image of u a fter removing ﬁnitely many singular points, so that we can take

v to be the inclusion Σ

′

֒→ W . The map ϕ : Σ → Σ

′

is then uniquely determined.

In order to carr y out this program, we need some information on what the image

of u can look like near each of its singularities. These come in two types, each type

corresponding to one of the lemmas below, both of which should seem immediately

plausible if your intuition comes from complex analysis.

Lemma 2.30 (Intersectio ns). Suppose u : (Σ, j) → (W, J) an d v : (Σ

′

, j

′

) →

(W, J) are two nonconstant pseudoh olomorphic curves with an intersection u(z) =

v(z

′

). Then there exist neighborhoods z ∈ U ⊂ Σ and z

′

∈ U

′

⊂ Σ

′

such that

either u(U) = v(U

′

) or u(U \ {z}) ∩ v(U

′

) = u(U) ∩ v( U

′

\ {z

′

}) = ∅.



Lemma 2.31 (Branching). Suppose u : (Σ, j) → (W, J) is a nonconstant pseudo-

holomorphic curve an d z

∈ Σ is a critical point of u. Then a neighborhood U ⊂ Σ

of z

ca n be biholom o rphically identiﬁed with the unit disk D ⊂ C such that

u(z) = v(z

) for z ∈ D = U,

where k ∈ N, and v : D → W is an injective J-holo morphic map w i th no critical

po ints except possibly at the origin. 

42 Chris Wendl

These two local results follow from a well-known formula of Micallef and White

[

MW95] describing the local behavior of J-holomorphic curves near critical points

and their intersections. The proof of that theorem is analytically quite involved, but

one can also use a n easier “approximate” version, which is proved in [

Wend, §2.14].

Since both ar e closely relat ed to the phenomenon of unique continuation, you will

not be surprised to learn that the similarity principle plays a role in the proof: the

main idea is again to exploit the fact that locally J is always a small perturbation

of i, hence the local behavior of J-holomorphic curves is also similar to the integrable

case.

Proof of Theorem 2.29. Let Crit(u) = {z ∈ Σ | du(z) = 0} denote t he set

of critical points, and deﬁne ∆ ⊂ Σ to be the set of all points z ∈ Σ such that there

exists z

′

∈ Σ and neighborhoods z ∈ U ⊂ Σ and z

′

∈ U

′

⊂ Σ with u(z) = u(z

′

) but

u(U \ {z}) ∩ u(U

′

\ {z

′

}) = ∅.

The lemmas quoted above imply that both of t hese sets are discrete. Both are

therefore ﬁnite, and the set

′

= u(Σ \ ( Crit(u) ∪ ∆)) ⊂ W is then a smooth

submanifold of W with J- invariant tangent spaces, so it inherits a natural complex

structure j

′

for which the inclusion (

′

, j

′

) ֒→ (W, J) is pseudoholomorphic. We

shall now construct a new Riemann surface (Σ

′

, j

′

) from which (

′

, j

′

) is obtained

by removing a ﬁnite set of points. Let

∆ = (Crit(u) ∪ ∆)/ ∼, where two points

in Crit(u) ∪ ∆ are deﬁned to be equivalent whenever they have neighbor hoods in

Σ with identical images under u. Then for each [z] ∈

∆, the branching lemma

provides an injective J-holomorphic map u

[z]

from the unit disk D onto the image

of a neighborhood of z under u. We deﬁne (Σ

′

, j

′

) by

′

∪





[z]∈

∆





where the gluing map Φ is the disjoint union of the maps u

[z]

: D \{0} →

′

for each

[z] ∈

∆; since this map is holomorphic, the complex structure j

′

extends from

′

. Combining the maps u

[z]

: D → W with the inclusion

′

֒→ W now deﬁnes a

pseudoholomorphic map v : (Σ

′

, j

′

) → (W, J) which restricts to

′

as an embedding

and otherwise has at most ﬁnitely many critical points and double points. Moreover,

the restriction of u to Σ \(Crit(u) ∪∆) deﬁnes a holomorphic map to (

′

, j

′

) which

extends by removal of singularities to a proper holomorphic map ϕ : (Σ, j) → (Σ

′

, j

′

)

such that u = v ◦ ϕ. Its holomorphicity implies that it has positive degree. 

LECTURE 3

Asymptotic operators

Contents

3.1. The linearization in Morse homology 43

3.2. Spectral ﬂow 46

3.2.1. Geometry in the space of Fredholm operators 47

3.2.2. Symmetric operators of index zero 49

3.2.3. Perturbation of eigenvalues 52

3.2.4. Homotopies of eigenvalues 55

3.3. The Hessian of the contact action functional 57

3.4. The Conley-Ze hnde r index 61

We now begin with the analysis of the particular class of J-holomorphic curves

that a r e important in SFT. The next three lectures will focus on the linearized

problem, the goal being to prove t hat this linearization is Fredholm and to compute

its index. Using this along with the implicit function theorem and the Sard-Smale

theorem (on genericity of smooth nonlinear Fredholm maps), we will lat er be able

to show that moduli spaces of asymptotically cylindrical J-holomorphic curves are

smooth ﬁnite-dimensional manifolds under suitable genericity assumptions.

3.1. The linearization in Morse homology

Since Morse homology is the prototype for all Floer-type theories, we can g ain

useful intuition by recalling how the analysis works for the linearization of the gradi-

ent ﬂow problem in Morse theory. The basic features of the problem were discussed

already in §

1.2.

Assume (M, g) is a closed n-dimensional Riemannian manifold, f : M → R is a

smooth function, and for two critical points x

, x

−

∈ Crit(f), consider the moduli

space of parametrized gradient ﬂow lines

M(x

−

, x

) :=



u ∈ C

∞

(R, M)



˙u + ∇f (u) = 0, lim

s→±∞

u(s) = x



The map M(x

−

, x

) → M : u 7→ u(0) gives a nat ural identiﬁcation of M(x

−

, x

)

with the intersection between the unstable manifold of x

−

and the stable manifold

of x

for the negative gradient ﬂow, and we say the pair (g, f) is Morse-Smale if

f is Morse and this intersection is transverse, in which case M(x

−

, x

) is a smooth

manifold with

dim M(x

−

, x

) = ind(x

−

) −ind(x

44 Chris Wendl

This can all be proved using ﬁnite-dimensional diﬀerent ial topology, but since that

approach does not work in the study of Floer trajectories or holomorphic curves

in symplectizations, let us instead see how o ne proves it using nonlinear functional

analysis. For more details on the following discussion, see [

Sch93].

Following the strategy laid out in §

2.1, M(x

−

, x

) can be identiﬁed with the

zero set of a smooth section

σ : B → E : u 7→ ˙u + ∇f(u),

where B is a Banach manifold of maps u : R → M satisfying lim

s→±∞

u(s) = x

and E → B is a smoot h Banach space bundle whose ﬁbers E

contain Γ(u

∗

T M).

The linearization Dσ(u) : T

B → E

of this section at a zero u ∈ σ

−1

(0) deﬁnes a

ﬁrst-order linear diﬀerential operato r

: Γ(u

∗

T M) → Γ(u

∗

T M)

which takes the form

η = ∇

η + ∇

∇f

for any choice of symmetric connection ∇ on M. Taking suitable Sobolev comple-

tions of Γ(u

∗

T M), we are therefore led to consider bounded linear operato r s

of the

form

(3.1) D

= ∇

+ ∇∇f : W

k,p

∗

T M) → W

k−1,p

∗

T M)

for k ∈ N and 1 < p < ∞, and the ﬁrst task is to prove that whenever x

and

−

satisfy the Morse condition, this is a Fredholm operator of index ind D

ind(x

−

) − ind(x

Choose coordinates near x

in which g looks like the standard Euclidean inner

product a t x

. This induces a trivialization of u

∗

T M over [T, ∞) for T > 0 suf-

ﬁciently large, and we are free to assume that the connection ∇ is t he standard

one determined by these coordinat es on [T, ∞) . Using the trivialization to identify

sections β ∈ Γ(u

∗

T M) over [T, ∞) with maps f : [T, ∞) → R

, D

now acts on f

(3.2) (D

f)(s) = ∂

f(s) + A(s)f(s),

where A(s) ∈ R

n×n

is the matrix of the linear transformation dX(s) : R

→ R

with X(s) ∈ R

being the coordinate representatio n of ∇f(u( s)) ∈ T

u(s)

M. As

s → ∞, the zeroth-order term in this expression converges to a symmetric matrix

:= lim

s→∞

A(s),

which is the coordinate representation of the Hessian ∇

f(x

). Any choice of coor-

dinates near x

−

produces a similar formula f or D

over (−∞, −T ], A(s) converging

as s → −∞ to another symmetric matrix A

−

representing ∇

f(x

−

). Both the

Morse condition and the dimension ind(x

−

)−ind(x

) can now be expressed entirely

We are ignoring an a nalytical subtlety: since u

∗

T M → R ha s no canonic al trivialization and R

is noncompact, it is not completely obvious what the deﬁnition of the Sobolev space W

k,p

∗

T M )

should be. We will retur n to this issue in a more gener al context in the next lecture.

Lectures on Symplectic Field Theory 45

in terms of these two matrices: x

is Morse if and only if A

is invertible, and the

Fredholm index of D

will then be

ind(x

−

) −ind(x

) = dim E

−

) −dim E

−

where for any symmetric mat rix A we denote by E

−

(A) the direct sum of all its

eigenspaces with negative eigenvalue. The main linear functional analytic result

underlying Morse homology can now be stated as follows (cf. [

Sch93]):

Proposition 3.1. Assume k ∈ N and 1 < p < ∞. Suppose E → R is a

smooth vector bundl e with trivializations ﬁxed in neighborhoods of −∞ and +∞, and

D : W

k,p

(E) → W

k−1,p

(E) is a ﬁrst-order diﬀe rential operator which asymptotically

takes the form (

3.2) near ±∞ wi th respect to the cho sen trivializations, where A(s)

is a smooth family of n-by-n matrices with well-deﬁned asymptotic limits A

lim

s→±∞

A(s) which are symme tric. If A

and A

−

are also invertible, then D is

Fredholm and

(3.3) ind(D) = dim E

−

) − dim E

−



Remark 3.2. The hypothesis that A

is invertible in Prop.

3.1 cannot be lifted:

indeed, suppose D is Fredholm but e.g. A

has 0 in its spectrum. Then one can

easily perturb A( s) and hence A

in two distinct ways producing two distinct values

of dim E

−

), pushing the zero eigenvalue either up or down. This produces two

perturbed Fredholm operators that have diﬀerent indices according to (3.3), but

they also belong to a continuo us family of Fredholm operato r s, and must therefore

have the same index, giving a contradiction.

The formula (

3.3) makes sense of course because E

−

) are both ﬁnite-dimen-

sional vector spaces, but in F loer-type theories we typically encounter critical points

with inﬁnite Morse index. With this in mind, it is useful to note that (

3.3) can

be rewritten without explicitly referencing E

−

) or E

−

). Indeed, choose

a continuous path of symmetric matrices {B

}

t∈[−1,1]

connecting B(−1) := A

−

B(1) := A

. The spectrum of B

varies continuously with t in the following sense:

one can choose a family of continuous functions

{λ

: [−1, 1] → R}

j∈I

for the index set I = {1, . . . , n} such that for every t ∈ [−1, 1], the set of eigenvalues

of B

counted with multiplicity is {λ

(t)}

j∈I

. The spectral ﬂow from A

−

to A

then deﬁned as a signed count of the number of paths of eigenvalues that cross from

one side of zero to the other, namely (cf. Theorem

3.3)

spec

−

, A

) := #



j ∈ I



(−1) < 0 < λ

(1)



−#



j ∈ I



(−1) > 0 > λ

(1)



The index formula (3.3) now becomes

ind(D) = µ

spec

−

, A

This description of the index has the advantag e that it could potentia lly make

sense and give a well-deﬁned integer even if A

were symmetric o perators on an

inﬁnite-dimensional Hilbert space: they might bot h have inﬁnitely many positive

46 Chris Wendl

and negative eigenvalues, but only ﬁnitely many that cha ng e sign along a path fro m

−

to A

. We will make this discussion precise in the next section.

3.2. Spectral ﬂow

We will see in §

3.3 that in Floer-type theories, the role of the symmetric linear

transformation T

M → T

M deﬁned by the Hessian ∇

f(x) of a Morse function

f : M → R at a critical point is played by a certain class of symmetric operators on

the space of loops η : S

→ R

, namely operators of the form

(3.4) (Aη)(t) := −J

∂

η(t) − S(t)η(t),

where J

denotes the standard complex structure on R

= C

, and S : S

→

End(R

) is a smooth loop of symmetric matrices. The goal of this section is to

deﬁne a no t ion of spectral ﬂow for operators of this type. Regarding A as an

unbounded linear operator on L

, R

) with dense domain H

, R

), we will

see that its spectrum consists of isolat ed real eigenvalues with ﬁnite multiplicity. We

shall prove:

Theorem 3.3. Assume {S

: S

→ End(R

)}

s∈[−1,1]

is a sm ooth family of loops

of symmetric matrices, and consider the corresponding 1-parameter fami l y of un-

bo und ed li near operators

= −J

∂

− S

(t) : L

, R

) ⊃ H

, R

) → L

, R

Then there exists a set of continuous functions

{λ

: [−1, 1] → R}

j∈Z

such that for every s ∈ [−1, 1], the spectrum of A

co nsists of the numbers {λ

(s)}

j∈Z

ea ch of which is an eigenv alue with ﬁnite multiplicity equal to the number of times

it is repeated as j varies in Z.

Moreover, if additionally A

−

:= A

−1

and A

:= A

bo th have trivial kernel,

then the number µ

spec

−

, A

) ∈ Z deﬁned by



j ∈ Z



(−1) < 0 < λ

(1)



− #



j ∈ Z



(−1) > 0 > λ

(1)



is well deﬁned and depends only on A

−

and A

We will start by giving a more abstract deﬁnition of spectral ﬂow as an int er-

section number between a path of symmetric index 0 Fredholm operators and the

subvariety of noninvertible operators. This relies on the general fact that spaces

of operators with kernel and cokernel of ﬁxed ﬁnite dimensions form smooth ﬁnite-

codimensional submanifolds in the Banach space of all bounded linear operators.

We explain t his fact in §

3.2.1, and then specialize to the case of symmetric index 0

operators to deﬁne the abstract version of spectral ﬂow in §

3.2.2. In §3.2.3, we show

that the spectra of such operators vary continuously under small perturbations, and

in §

3.2.4 we specialize further to operators of the form (3.4) and explain how to

interpret the abstract deﬁnition of spectral ﬂow in terms of eigenvalues crossing the

origin in R, leading to a proof o f Theorem

3.3.

Spectral ﬂow can be deﬁned more generally for certain classes of self-adjoint

elliptic partia l diﬀerentia l operators, see e.g. [

APS76, RS95], and standard proofs

Lectures on Symplectic Field Theory 47

of its existence typically rely on perturbation results as in [Kat95] for the spectra of

self-adjoint operators. In the following presentation, we have chosen to avoid making

explicit use of self-adjointness and instead focus on the Fredholm property; in this

way t he discussion is mostly self-contained and, in particular, does not require a ny

results from [

Kat95].

3.2.1. Geometry in the space of Fredholm operators. Fix a ﬁeld

F := R or C.

Given Banach spaces X and Y over F, denote by L

(X, Y ) the Banach space of

bounded F-linear ma ps from X to Y , with L

(X) := L

(X, X), and let

Fred

(X, Y ) ⊂ L

(X, Y )

denote the open subset consisting of Fredholm operators. Recall that an operator

T ∈ L

(X, Y ) is Fredholm if its image is closed,

and its kernel and cokernel

(i.e. the quotient coker T := Y / im T) are both ﬁnite dimensional. Its index is

deﬁned as

ind

(T) := dim

ker T − dim

coker T ∈ Z.

The index deﬁnes a continuous and thus locally constant function Fred

(X, Y ) → Z,

and for each i ∈ Z, we shall denote

Fred

(X, Y ) :=



T ∈ Fred

(X, Y )



ind(T) = i



We will often have occasion to use the following general construction. Given

∈ Fred

(X, Y ), one can choose splittings into closed linear subspaces

X = V ⊕ K, Y = W ⊕ C

such that K = ker T

, W = im T

, the quotient projection π

: Y → coker T

restricts to C ⊂ Y as a n isomorphism, and T

deﬁnes an isomorphism from V

to W . Using these splittings, any other T ∈ Fred

(X, Y ) can be written in block

form as

T =



A B

C D



with T

itself written in this way as



0 0



for some Banach space isomorphism

: V → W . Let O ⊂ Fred

(X, Y ) denote the open neighborhood of T

for which

the block A is invertible, and deﬁne a map

(3.5) Φ : O → Hom

(ker T

, coker T

) : T 7→ D − CA

−1

Lemma 3 .4. The map Φ in (

3.5) is smooth, and holomo rp hic in the cas e F = C,

and its derivative at T

deﬁnes a surjective bounded linear operator L

(X, Y ) →

Hom

(ker T

, coker T

) of the form

dΦ(T

)H = π

ker T

It is not strictly necessary to require that im T ⊂ Y be closed, as this follows from the

ﬁnite-dimensionality of the kernel and cokernel, cf. [

AA02, Cor. 2.17].

48 Chris Wendl

where π

denotes the natural projection Y → coker T

. Moreover, there exists a

smooth function Ψ : O → L

(X) such that for every T ∈ O, Ψ(T) : X → X maps

ker Φ(T) ⊂ ker T

isomorphically to ker T.

Proof. Smoothness, holomorphicity

and the formula for the derivative are

easily veriﬁed fro m the given fo r mula for Φ; in particular, since the blocks B and C

both vanish for T = T

, we have

dΦ(T

) : L

(X, Y ) → Ho m

(K, C)



′



7→ D

′

The map Ψ : O → L

(X) = L

(V ⊕K) is deﬁned by

Ψ(T) =



1 −A

−1

0 1



For each T, this is an isomorphism; indeed, its inverse is given by

Ψ(T)

−1



1 A

−1

0 1



Then TΨ(T) =



A 0

C Φ(T)



, and since A is invertible, ker TΨ(T) = {0}⊕ker Φ(T).



Proposition 3.5. For each i ∈ Z and each n onnegative integer k ≥ i, the subset

Fred

i,k

(X, Y ) :=



T ∈ Fred

(X, Y )



dim

ker T = k and dim

coker T = k − i



admits the structure of a smooth (and complex-analytic if F = C) ﬁnite-codimensiona l

Banach subman i f old of L

(X, Y ), with

codim

Fred

i,k

(X, Y ) = k(k − i).

Proof. Applying the implicit function theorem to the map Φ from Lemma

3.4

endows a neighborhood of T

in Φ

−1

(0) ⊂ Fred

(X, Y ) with the structure of a

smooth Banach submanifold with

codim

−1

(0) = dim

Hom

(ker T

, coker T

) = k(k − i).

If F = C, then Φ is also holomo rphic and Φ

−1

(0) is thus a complex-analytic sub-

manifold near T

. Now observe that for every T ∈ O,

dim

ker T = dim

ker Φ(T) ≤ dim

ker T

= k,

with equality if and only if Φ(T) = 0, hence, since the index is locally constant, we

get Φ

−1

(0) = Fred

i,k

(X, Y ) in a neighborhood of T

. 

Holomorphicity in this inﬁnite-dimensio nal setting means the same thing as usual: L

(X, Y )

and Hom

(ker T

, coker T

) bo th have natural complex structures if T

∈ Fred

(X, Y ), and we

require dΦ(T) to commute with them for all T ∈ O.

Lectures on Symplectic Field Theory 49

For real-linear operators of index 0, one can use Prop. 3.5 to deﬁne the following

“relative” invariant. Given two Banach space isomorphisms T

: X → Y that lie in

the same connected component of Fred

(X, Y ), deﬁne

spec

−

, T

) ∈ Z

as the parity of the number of times that a generic smooth path [−1, 1] → Fred

(X, Y )

from T

−

to T

passes through operators with nontrivial kernel. This is well

deﬁned due to the following consequences of standard tra nsversality theory (see

Exercise

3.6): ﬁrst, generic paths {T(t) ∈ Fred

(X, Y )}

t∈[−1,1]

are transverse to

Fred

0,k

(X, Y ) for every k ∈ N, which implies via the codimension formula in Prop.

3.5

that they never intersect Fred

0,k

(X, Y ) for k ≥ 2, and their intersections with

Fred

0,1

(X, Y ) are transverse and thus isolated. Second, transversality also holds

for generic homotopies

[0, 1] ×[−1, 1] → Fred

(X, Y ) : (s, t) 7→ T

(t)

with ﬁxed end points between any pair of generic paths T

(t) and T

(t), so that the

set of intersections with Fred

0,k

(X, Y ) is again empty for k ≥ 2 and forms a smoot h

1-dimensional submanifold in [0, 1] ×[−1, 1] for k = 1. This submanifold, moreover,

is disjoint from [0, 1] × {−1, 1} since T

(±1) = T

, and it is also compact since

the set of T ∈ Fred

(X, Y ) with nontrivial kernel is a closed subset. We therefore

obtain a compact 1-dimensional cobordism between the intersection sets of T

and

respectively with Fred

0,1

(X, Y ), implying that the count of intersections modulo 2

does not depend on the choice of generic path.

Exercise 3.6. Convince yourself that the standard results (as in e.g. [

Hir94,

§3.2] about generic transversality of intersections between smoo t h maps f : M → N

and submanifolds A ⊂ N continue to hold—with minimal modiﬁcations to the

proofs—when N is an inﬁnite-dimensional Banach manifold and A ⊂ N has ﬁnite

codimension.

Exercise 3.7. For matrices A

∈ GL(n, R), show that µ

spec

−

, A

) = 0 if and

only if det A

and det A

−

have the same sign.

Remark 3.8. The Z

-valued spectral ﬂow µ

spec

cannot be lifted to Z without

restricting the class of operators under consideration. See Exercise

3.19.

3.2.2. Symmetric operators of index zero. We now add the following as-

sumptions to the setup from the previous subsection:

• Y is a Hilbert space H over F, with inner product denoted by h , i

;

• X is an F-linear subspace D ⊂ H, carrying a Banach space structure for

which the inclusion D ֒→ H is a compact linear operator.

The notation D = X is motivated by the fact that if T ∈ L

(D, H), then we can

also regard T as an unbounded operator on H with domain D a nd thus consider

the spectrum of T, see §

3.2.3 below.

Since H is a Hilbert space, the space L

(H) of bounded linear o perators from

H to itself contains a distinguished closed linear subspace

sym

(H) ⊂ L

(H),

50 Chris Wendl

consisting of self-adjoint operators. For operators tha t are bounded f rom D to H

but not necessarily deﬁned or bounded on H, there is also the space of symmetric

operators

sym

(D, H) :=



T ∈ L

(D, H)



hx, Tyi

= hTx, yi

for all x, y ∈ D



Important examples of symmetric operators are those which are self-adjoint (see

Remark

3.12 below), though for our purposes, it will suﬃce to restrict attention to

symmetric operators that are a lso Fredholm with index 0. It turns out that the space

of symmetric operators in Fred

0,1

(D, H) is a canonically co-oriented hypersurface in

sym

(D, H), so t hat the invariant µ

spec

−

, T

) deﬁned above can be modiﬁed to

have integer values when T

are symmetric. We will need a slightly more specialized

version of this statement in order to give a general deﬁnition of spectral ﬂow.

Fix an operator T

ref

∈ L

sym

(D, H) that is Fredholm with index zero, and con-

sider the space

Fred

sym

(D, H, T

ref

) :=



ref

+ K : D → H



K ∈ L

sym

(H)



Note t hat the restriction o f each K ∈ L

(H) to D is a compact o perator D →

H, thus Fred

sym

(D, H, T

ref

) has a natural continuous inclusion into the space of

symmetric Fr edholm o perators with index 0. It is also an aﬃne space over L

sym

(H)

and can thus be regarded naturally as a smooth Banach manifold locally modeled

on L

sym

(H); in particular, its tangent spaces are

(Fred

sym

(D, H, T

ref

)) = L

sym

(H).

A remark about the case F = C is in order: L

sym

(D, H) is a real-linear and not a

complex subspace of L

(D, H), thus Fred

sym

(D, H, T

ref

) is a real Banach manifold

but does not carry a natural complex structure.

Lemma 3.9. For any T ∈ L

sym

(D, H) that is Fredholm with index 0, ker T i s the

orthog onal complem ent o f im T in H, hence there exist splittings into closed linear

subspace s

D = V ⊕ K, H = W ⊕ C

where K = C = ker T, W = im T and V = W ∩ D.

Proof. If x ∈ K := ker T, then symmetry implies hx, Tyi

= hTx, yi

= 0

for all y ∈ D, hence K ⊂ W

⊥

, where W := im T. But since ind T = 0 , the

dimension of ker T equals the codimension of im T, implying that K already has the

largest po ssible dimension for a subspace that intersects W trivially, a nd therefore

W ⊕K = H. Since K is also a subspace of D and the latter is a subspace of H, any

x ∈ D can be written uniquely as x = v + k where k ∈ K and v ∈ W ∩ D = : V .

The continuous inclusion of D into H and the f act that W is closed in H imply that

V is a closed subspace of D. 

We now have the following modiﬁcation of Prop.

3.5.

Proposition 3.10 . For each integer k ≥ 0, the subset

Fred

sym,k

(D, H, T

ref

) :=



T ∈ Fred

sym

(D, H, T

ref

)



dim

ker T = k



Lectures on Symplectic Field Theory 51

is a smooth ﬁnite-codimensional Banach submanifold o f Fred

sym

(D, H, T

ref

), with

codim

Fred

sym,k

(D, H, T

ref

) =

(

k(k + 1)/2 if F = R,

if F = C.

In particular, Fred

sym,1

(D, H, T

ref

) is a subma nifold of Fred

sym

(D, H, T

ref

) with codi-

mension 1, and moreover, it carries a canonical co-orientation.

Proof. Given T

∈ Fred

sym,k

(D, H, T

ref

), ﬁx the splittings D = V ⊕ K and

H = W ⊕ K as in Lemma

3.9. Using these in the construction of the map Φ

from (

3.5) produces a neighborhood O ⊂ Fred

sym

(D, H, T

ref

) of T

such that, by

Lemma

3.4, {T ∈ O | dim

ker T = k} = Φ

−1

(0), where

Φ : O → End

(K) :



A B

C D



7→ D − CA

−1

Since the splittings are orthogonal, an element T =



A B

C D



∈ O is symmetric if

and only if

hx, Ayi

= hAx, yi

for all x, y ∈ V ,

hx, Dyi

= hDx, yi

for all x, y ∈ K,

hx, Byi

= hCx, yi

for all x ∈ V , y ∈ K,

hx, Cyi

= hBx, yi

for all x ∈ K, y ∈ V ,

and it follows then that Φ(T) ∈ End

sym

(K), where End

sym

(K) ⊂ End

(K) is the real

vector space of symmetric (or Hermitian when F = C) linear maps on (K, h , i

We thus have O∩Fred

sym,k

(D, H, T

ref

) = Φ

−1

(0) with Φ regarded as a smo oth map

O ∩ Fred

sym

(D, H, T

ref

) → End

sym

(K). The derivative at T

again takes the form

dΦ(T

) : L

sym

(H) → End

sym

(K) :



′



7→ D

′

where now the block matrix represents an element of L

sym

(H) with resp ect to the

splitting H = W ⊕ K. This o perator is evidently surjective, hence by the implicit

function theorem, Φ

−1

(0) is a smooth Banach submanifold with codimension equal

to dim

End

sym

(K).

Finally, we observe that in the case k = 1, the above identiﬁes Fred

sym,1

(D, H, T

ref

)

locally with the zero set of a submersion to End

sym

(K), which is a real 1-dimensional

vector space since K is a 1-dimensional vector space over F. The canonical isomor-

phism

R → End

sym

(K) : a 7→ a1

thus determines a co-orientat ion on Fred

sym,1

(D, H, T

ref

). 

The co-orientation of Fred

sym,1

(D, H, T

ref

) means that one can now deﬁne signed

intersection numbers between Fred

sym,1

(D, H, T

ref

) and smooth paths in the ambient

space Fred

sym

(D, H, T

ref

). Moreover, the codimensions of Fred

sym,k

(D, H, T

ref

) f or

each k ≥ 2 are still at least 3, hence large enough to ensure that generic paths or

52 Chris Wendl

homotopies o f paths will never intersect them. The following notion is therefore

independent of choices.

Definition 3.11. Suppose T

: T

−

∈ Fred

sym

(D, H, T

ref

) are both Banach

space isomorphisms D → H. The spectral ﬂow

spec

−

, T

) ∈ Z

from T

−

to T

is then deﬁned as the signed count of intersections of T : [−1, 1] →

Fred

sym

(D, H, T

ref

) with Fred

sym,1

(D, H, T

ref

), where T : [−1, 1] → Fred

sym

(D, H, T

ref

)

is any smooth path that is transverse to Fred

sym,k

(D, H, T

ref

) for every k ≥ 1 and

satisﬁes T(±1) = T

3.2.3. Perturbation of eigenvalues. Continuing in t he setting of the previous

subsection, we shall now regard each T ∈ Fred

sym

(D, H, T

ref

) as an unbounded

operator on H with domain D, see e.g. [

RS80, Chapter VIII]. Notice that for each

scalar λ ∈ F, the operator T−λ also belongs to Fred

sym

(D, H, T

ref

). The spectrum

σ(T) ⊂ F

of T is deﬁned as the set of all λ ∈ F for which T − λ : D → H does not ad-

mit a bounded inverse. In particular, λ ∈ σ(T) is an eigenvalue of T whenever

T − λ : D → H has nontrivial kernel, and the dimension of this kernel is called the

multiplicity of the eigenvalue. We call λ a simple eigenvalue if it has multiplic-

ity 1. By a standard argument familiar to both mathematicians and physicists, the

eigenvalues of a symmetric complex-linear operator are always real.

Remark 3.12. If D ⊂ H is dense, then the adjoint of T is deﬁned as an

unbounded operator T

∗

with domain D

∗

satisfying

hx, Tyi

= hT

∗

x, yi

for all x ∈ D

∗

, y ∈ D,

where D

∗

is the set of all x ∈ H such that there exists z ∈ H satisfying hx, Tyi

hz, yi

for all y ∈ D. O ne says that T is self-adjoint if T = T

∗

, which means

both that T is symmetric and D = D

∗

. In many applications (e.g. in Exercise

3.28),

the latter amounts to a condition on “regularity of weak solutions”. This condition

implies that the inclusion ker T ֒→ (im T)

⊥

—va lid for all symmetric operators—is

also surjective, so if T : D → H is Fredholm, it is then automatic that ind(T) = 0.

Proposition 3.13 . Assume T

∈ Fred

sym

(D, H, T

ref

). Then:

(1) Every λ ∈ σ(T

) is an e i genvalue with ﬁ nite multiplicity.

(2) The spectrum σ(T

) is a discrete subset of R.

(3) Suppose λ

∈ σ(T

) is an eigenvalue with multiplicity m ∈ N a nd ǫ > 0 is

chosen such that no other eigenvalues li e in [λ

−ǫ, λ

+ ǫ]. Then T

has a

neighorhood O ⊂ Fred

sym

(D, H, T

ref

) such that for all T ∈ O,

λ∈σ(T)∩[λ

−ǫ,λ

+ǫ]

m(λ) = m,

where m(λ) ∈ N den otes the multiplicity of λ ∈ σ(T).

Lectures on Symplectic Field Theory 53

Proof. For every λ ∈ F, T

− λ is a Fredholm operato r with index 0, so it is

a Banach space isomorphism D → H and thus has a bounded inverse if and only

if its kernel is trivial. The Fredholm property also implies that the kernel is ﬁnite

dimensional whenever it is nontrivial, so this proves (

1).

For (

2) and ( 3), let us assume F = C, as the case F = R will follow by taking

complexiﬁcations of real vector spaces. We claim therefore that σ( T

) is a discrete

subset of C. To see t his, suppose λ

∈ R is an eigenvalue of T

with multiplicity m,

− λ

∈ Fred

sym,m

(D, H).

By Lemma 3.9, there are splittings D = V ⊕ K and H = W ⊕ K with K =

ker(T

− λ

), W = im(T

− λ

) and V = W ∩ D. Any scalar λ ∈ C appears in

block-diagonal form



λ 0

0 λ



with respect to these splittings, and the block form for

is thus



+ λ

0 λ



for some Banach space isomorphism A

: V → W . Writing nearby operators

T ∈ Fred

(D, H) as



A B

C D



, we can imitate the construction in (

3.5) to pro-

duce neighbo r hoods O(T

) ⊂ Fred

(D, H) of T

and D

(λ

) ⊂ C of λ

, admitting

a holomorphic map

Φ : O(T

) × D

(λ

) → End

(K) : (T, λ) 7→ (D − λ) − C (A − λ)

−1

such that ker(T −λ)

∼

ker Φ(T, λ). The set of eigenvalues of T

near λ

is then the

zero set of the holomorphic function

(3.6) D

(λ

) → C : λ 7→ det Φ(T

, λ).

This function cannot be identically zero since there are no eigenvalues outside of R,

thus the zero at λ

is isolated, proving (

2).

To prove (

3), note ﬁnally that if t he neighborhood O(T

) ⊂ Fred

(D, H) of T

is suﬃciently small, then for every T ∈ O(T

), the holomorphic function

: D

(λ

) → C : λ 7→ det Φ(T, λ)

has the same alg ebraic count of zeroes in D

(λ

), all of which lie in [λ

− ǫ, λ

+ ǫ]

if T is symmetric. Observe moreover that since

∂

Φ(T

, λ

) = −1 ∈ End

(K),

we are free to assume after possibly shrinking ǫ and O(T

) that ∂

Φ(T, λ) is always

a nonsingular transformation in End

(K). Since Φ(T, λ) is in End

sym

(K) and thus

diagonalizable whenever T is symmetric and λ ∈ R, it follows via Exercise

3.14

below that the order of any zero f

(λ) = 0 is precisely the multiplicity of λ as an

eigenvalue of T.



54 Chris Wendl

Exercise 3.14. Suppose U ⊂ C is an open subset, A : U → C

n×n

is a ho lomor-

phic map and z

∈ U is a point at which A(z

) is noninvertible but diagonalizable,

and A

′

) ∈ GL(n, C). Show that dim

ker A(z

) is the order of the zero of the

holomorphic function det A : U → C at z

The next result implies that for a generic path of symmetric index 0 operators

as appears in our deﬁnition of µ

spec

−

, T

), the spectral ﬂow is indeed a signed

count of eigenvalues crossing 0.

Proposition 3.15 . Suppose {T

∈ Fred

sym

(D, H, T

ref

)}

t∈(−1,1)

is a smooth path

and λ

∈ R is a simple eigen v alue of T

. Then:

(1) For suﬃciently sma ll ǫ > 0, there exists a unique smooth function λ :

(−ǫ, ǫ) → R such that λ(0) = λ

and λ(t) is a simple eigenva l ue of T

for

ea ch t ∈ (−ǫ, ǫ).

(2) The derivative λ

′

(0) is n onzero if and only if the intersection of the path

− λ

∈ Fred

sym

(D, H, T

ref

)}

t∈(−1,1)

with Fred

sym,1

(D, H, T

ref

) at t = 0

is transverse, and the sign of λ

′

(0) is then the si gn of the intersection.

Proof. Using the same construction as in the proof of Proposition

3.13, we can

ﬁnd small numbers ǫ > 0 and δ > 0 such that



(t, λ) ∈ (−ǫ, ǫ) ×(λ

−δ, λ

+ δ)



λ ∈ σ(T

)



= Φ

−1

(0),

where

Φ : (−ǫ, ǫ) × (λ

− δ, λ

+ δ) → End

sym

(K) : (t, λ) 7→ (D

−λ) − C

−λ)

−1

and we write T





with respect to splittings D = V ⊕K and H = W ⊕K

with K = ker(T

− λ

), W = im(T

− λ

) and V = W ∩ D. In saying this, we’ve

implicitly used the assumption that λ

is a simple eigenvalue, as it follows that

dim

ker(T − λ) cannot be larger than 1 for any T near T

and λ near λ

, so that

−1

(0) catches all nearby eigenva lues. Simplicity also means that End

sym

(K) is real

1-dimensional, and we have

∂

Φ(0, λ

) = ∂

t=0

, ∂

Φ(0, λ

) = −1.

The implicit function t heorem thus gives Φ

−1

(0) near (0, λ

) t he structure of a

smooth 1-manifold with tangent space at (0, λ

) spanned by the vector

∂

+ (∂

t=0

) ∂

where we ar e identifying ∂

t=0

∈ End

sym

(K) with a real number via the natural

isomorphism End

sym

(K) = R. Therefore Φ

−1

(0) can be written as the graph of

a uniquely determined smooth function λ, whose derivative at zero is a multiple

of ∂

t=0

. This proves both statements in the proposition, since by the proof of

Proposition

3.10, the intersection of {T

}

t∈(−1,1)

with Fred

sym,1

(D, H, T

ref

) is trans-

verse if and only if ∂

t=0

6= 0, and its sign is then the sign of ∂

t=0

. 

The purpose of t he next lemma is to prevent eigenvalues from escaping to ±∞

under smooth families of operators in Fred

sym

(D, H, T

ref

Lectures on Symplectic Field Theory 55

Lemma 3.16. Suppose {K

∈ L

sym

(H)}

t∈(a,b)

is a smooth path of s ymmetric

bo und ed linear operators, and λ : (a, b) → R is a smooth f unc tion such that for every

t ∈ (a, b), λ( t) is a simple eigenvalue of T

:= T

ref

∈ Fred

sym

(D, H, T

ref

). Then

λ(t)| ≤ k∂

L (H)

for all t ∈ (a, b).

Proof. Since {T

− λ(t) ∈ Fred

sym

(D, H, T

ref

)}

t∈(a,b)

is a smooth f amily of

operators in Fred

(D, H) with 1-dimensional kernel, one can use the local families

of isomorphisms Ψ(T

−λ(t)) ∈ L

(D) from Lemma

3.4 to ﬁnd a smoo t h family of

eigenvectors x(t) ∈ ker(T

− λ(t)) for t ∈ (a, b). Normalize these so that kx(t)k

1 fo r all t. Then 0 = ∂

hx(t), x(t)i

= h˙x(t), x(t)i

+ hx(t), ˙x(t)i

and λ(t) =

hx(t), T

x(t)i

, so writing

:= ∂

= ∂

, we have

λ(t) = ∂

hx(t), T

x(t)i

= hx(t),

x(t)i

+ h˙x(t), T

x(t)i

+ hx(t), T

˙x(t)i

= hx(t),

x(t)i

as the last two terms in the ﬁrst line become λ(t) [h˙x(t), x(t)i

+ hx(t), ˙x(t)i

] = 0

since T

is symmetric and T

x(t) = λ(t)x(t). We obtain

λ(t)| ≤ kx(t)k

L (H)

kx(t)k

= k

L (H)



3.2.4. Homotopies of eigenvalues. Specializing further, we now set H and

D equal to the speciﬁc real Hilb ert spaces

H := L

, R

), D := H

, R

and set T

ref

:= −J

∂

, where J

denotes the standard complex structure on R

. Observe that any bounded linear opera t or on L

determines a compact operator

→ L

via composition with the compact inclusion. In particular, we shall

consider compact perturbations of −J

∂

in the form

(3.7) A = −J

∂

−S(t)

with S : S

→ End

sym

) smooth. It is straightforward to check t hat this operator

is symmetric with respect to the L

-product since S(t) is symmetric for every t. The

following then implies that A ∈ Fred

sym

(D, H, T

ref

Lemma 3.17. T he operator −J

∂

: H

, R

) → L

, R

) is Fredholm with

index 0.

Proof. Since J

deﬁnes an isomorphism, it suﬃces actually to show that the

ordinary diﬀerential operato r

∂

: H

, R

) → L

, R

)

is Fredholm with index 0. The kernel of this operator is the space of constant

functions S

→ R

, which has dimension 2n. To compute the dimension of the

cokernel, we o bserve tha t if f = ∂

F lies in the image of this operator, we have

f(t) dt = 0 since F is periodic in t. Conversely, if

f(t) dt = 0 with f ∈

56 Chris Wendl

, R

), t hen the function F (s) =

f(t) dt is periodic in s and deﬁnes an

element of H

, R

) satisfying ∂

F = f. Hence the image of ∂

is exactly the set

im(∂

) =



f ∈ L

, R

)



f(t) dt = 0



which has codimension 2n. 

The proof of Theorem

3.3 requires only one more technical ingredient, whose

proof is given in Appendix

C and should probably be skipped on ﬁrst reading unless

you have already read Lecture

7 or seen similar applications of the Sard-Smale

theorem. You might however ﬁnd the result plausible in accordance with the notio n

that maps from 2-dimensional domains, such as a map of the form

(−1, 1) ×R → Fred

sym

(D, H, T

ref

) : (t, λ) 7→ T

− λ

should generically not intersect submanifolds that have codimension 3 or more, such

as Fred

sym,k

(D, H, T

ref

) when k ≥ 2.

Lemma 3.18. Fix a smooth map S : [−1, 1] × S

→ End

sym

) and consider

the 1-parameter family of unbounded linear operators

:= −J

∂

− S(s, ·) : L

, R

) ⊃ H

, R

) → L

, R

)

for s ∈ [−1, 1]. One can arrange after a C

∞

-small perturbation of S ﬁxed at s = ±1

that the following conditions hold:

(1) For each s ∈ (−1, 1), all eigenvalues of A

are simple.

(2) All intersections of the path

(−1, 1) → Fred

sym

(D, H, T

ref

) : s 7→ A

with Fred

sym,1

(D, H, T

ref

) are transverse.



Proof of Theorem

3.3. Given a smooth family {A

}

s∈[−1,1]

as stated in the

theorem, use Lemma

3.18 to obtain a C

∞

-small perturbation for which the eigen-

values are simple for s ∈ (−1, 1) and all intersections with Fred

sym,1

(D, H) are

transverse. Proposition

3.15 then implies that the eigenvalues depend smoothly

on s, and Lemma

3.16 imposes a uniform bound on their derivatives with respect

to s so that each one varies only in a bounded subset of R for s ∈ (−1, 1) . The

smooth families of eigenvalues f or s ∈ (−1, 1 ) therefore extend to continuous families

for s ∈ [−1, 1] since the space of noninvertible Fredholm operators with index 0 is

closed. Proposition

3.13 ensures mo r eover that these continuous families hit every

eigenvalue with the correct multiplicity at s = ±1, and by Proposition

3.15, the

formula for µ

spec

−

, A

) stated in the theorem is corr ect for the perturbed family

with simple eigenvalues and transverse crossings. To obtain the same result for the

original family, suppose we have a sequence of perturbations {A

}

s∈[−1,1]

converging

in C

∞

as ν → ∞ to {A

}

s∈[−1,1]

. L emma

3.16 then provides a uniform C

-bound

for each sequence of smooth families of eigenvalues, so they have C

-convergent sub-

sequences as ν → ∞, giving rise to the continuous families in the statement of the

theorem. 

Lectures on Symplectic Field Theory 57

Exercise 3.19. Show tha t the smooth hypersurface of ( no t necessarily sym-

metric!) rank 1 matrices in R

2×2

is not co- orientable. Hint: Consider a path from

1 ∈ R

2×2

to itself for which the two eigenvalues ﬁrst move from 1 to −1 along R but

then move back to 1 along the unit circle in C. What would be it s integer-valued

spectral ﬂow if such a thing were deﬁned for no n-symmetric matrices?

3.3. The Hessian of the contact action functional

Before returning to contact geometry, let’s quickly revisit the Floer homology for

a time-dependent Hamiltonian {H

: M → R}

t∈S

on a symplectic manifold (M, ω).

In Lecture

1, we introduced the symplectic a ctio n functional A

: C

∞

contr

, M) → R

and wrote down t he formula

∇A

(γ) = J

(γ) ( ˙γ − X

(γ)) ∈ Γ(γ

∗

T M) =: T

∞

contr

, M)

for the “unregularized” gradient of A

at a cont ractible loop γ ∈ C

∞

contr

, M).

Here X

denotes the Hamiltonian vector ﬁeld and J

is a time-dependent family of

compatible almost complex structures, which determines the L

-product

hη

, η

ω(η

(t), J

(t)) dt.

The critical po ints of A

are the loops γ such that ∇A

(γ) = 0. Formally, the

Hessian of A

at γ ∈ Crit(A

) is the “linearizatio n of A

at γ,” which gives a

linear operator

:= ∇

(γ) : Γ(γ

∗

T M) → Γ(γ

∗

T M).

To write it down, o ne can choose any connection ∇ o n M, and choose for η ∈

Γ(γ

∗

T M) a smooth family {γ

: S

→ M}

ρ∈(−ǫ,ǫ)

with γ

= γ and ∂

ρ=0

= η, and

then compute

η := ∇

[∇A

(γ

)]|

ρ=0

The result is independent of the choice of connection since ∇A

(γ) = 0.

Exercise 3.20. Show that if the connection ∇ on M is chosen to be symmetric,

then A

η = J

(∇

η − ∇

We now introduce the class of symmetric operators that appear in asymptotic

formula s in SFT. Fix a (2n − 1)-dimensional contact manifold (M, ξ) with contact

form α, induced Reeb vector ﬁeld R

, and a complex structure J : ξ → ξ compatible

with the symplectic structure dα|

. L et

: T M → ξ

denote the projection along R

. The contact action functional is deﬁned by

: C

∞

, M) → R : γ 7→

∗

α.

The ﬁrst variation of this functional fo r γ ∈ C

∞

, M) and η ∈ Γ(γ

∗

T M) is

(γ)η =

dα(η, ˙γ) dt = −

dα(π

˙γ, η) dt.

58 Chris Wendl

The functional has a built-in degeneracy since it is parametrization-invariant; in

particular, dA

(γ)η = 0 whenever η points in the direction of the Reeb vector ﬁeld,

a symptom of the fact that closed Reeb orbits always come in families related to

each other by repara metrizatio n. A loop γ : S

→ M is critical for A

if and

only if ˙γ is everywhere tangent to R

, allowing for an inﬁnite-dimensional family

of distinct perturbations—however, there exist preferred para metrizatio ns, namely

those fo r which ˙γ is a constant multiple of R

, meaning

(3.8) ˙γ = T · R

(γ), T := A

(γ).

Such a loop corresp onds to a T -periodic solution x : R → M to ˙x = R

(x), where

γ(t) = x(T t).

The discussion above indicates that we cannot derive a “Hessian” of A

in t he

same straightforward way as in Floer homology, a s t he resulting operator will always

have nontrivial kernel due to the degeneracy in the R

direction. To avoid this, we

shall consider only preferred parametrizations γ : S

→ M of the for m (

3.8), and

perturbations in directions tangent to ξ, which is transverse to every Reeb orbit.

For η ∈ Γ(γ

∗

ξ), we then have

(γ)η =

dα(−Jπ

˙γ, Jη) dt = h−Jπ

˙γ, ηi

where we deﬁne an L

-product for sections of γ

∗

ξ by

(3.9) hη, η

′

dα(η, Jη

′

) dt.

It t herefore seems sensible to write

∇A

(γ) := −Jπ

˙γ ∈ Γ(γ

∗

ξ),

and we shall deﬁne the Hessian at a critical point γ as the linearization in ξ directions,

i.e.

∇

(γ) : Γ(γ

∗

ξ) → Γ(γ

∗

ξ).

Given η ∈ Γ(γ

∗

ξ), choose a smooth family {γ

: S

→ M}

ρ∈(−ǫ,ǫ)

with γ

= γ a nd

∂

ρ=0

= η, and ﬁx a symmetric connection ∇ on M. Since π

˙γ = 0, the covariant

derivative of ∇A

(γ

) at ρ = 0 is then

∇

(−Jπ

˙γ

ρ=0

= −J ∇

(π

˙γ

ρ=0

= −J ∇

[ ˙γ

− α( ˙γ

(γ

)]|

ρ=0

= −J (∇

η − T ∇

− ∂

[α( ˙γ

)] |

ρ=0

· R

(γ)) .

In the last term, we can write ∂

[α( ˙γ

)] |

ρ=0

= dα(η, ˙γ) + ∂

[α(η)] = 0 since ˙γ =

T R

(γ) and α(η) = 0 for η ∈ Γ(γ

∗

ξ). O ne can now check that the remaining terms

deﬁne a section of γ

∗

ξ, thus we are led to the following deﬁnition.

Definition 3.21. Given a loop γ : S

→ M parametrizing a closed Reeb orbit

in (M, ξ = ker α) with period T ≡ α( ˙γ), the asymptotic operator associated to γ

is the ﬁrst-order diﬀerential operator on γ

∗

ξ deﬁned by

: Γ(γ

∗

ξ) → Γ ( γ

∗

ξ) : η 7→ −J(∇

η − T ∇

)

Lectures on Symplectic Field Theory 59

Exercise 3.22. Show t hat A

is symmetric with respect t o the L

inner product

(

3.9) on Γ(γ

∗

ξ). Moreover, γ is nondegenerate (see §1.3) if and only if ker A

trivial. Hint for nondegeneracy: Consider the pullback of γ

∗

ξ via the cover R →

= R/Z, and show that solutio ns to ∇

η −T ∇

= 0 on the pullback are given

by operating on ξ

γ(0)

with the linearized Reeb ﬂow. To see t his, try diﬀerentiating

families of solutions to the equation ˙x = T R

(x).

Remark 3.2 3. Another way of phrasing the hint in the the above exercise is

as follows: A

can a lso be written as −J

∇

, where

∇

is the unique symplectic

co nnection on (γ

∗

ξ, dα) for which para llel transport is given by the linearized Reeb

ﬂow.

You might be slightly concerned about the sign diﬀerence between the two for-

mulas we’ve derived for asymptotic operat ors in conta ct geometry and in Floer

homology. I also ﬁnd this troubling, but the discrepancy seems to originate from

the fact that our account o f Floer homology has referred always to the negative gra-

dient ﬂow of A

, while SFT is actually deﬁned via the posi tive g r adient ﬂow of A

The words “gradient ﬂow” in SFT must in any case be interpreted very loosely. If

u : [0, ∞) × S

→ R × M

is the cylindrical end of a ﬁnite-energy J-holomorphic curve for some J ∈ J(α) as

we described in Lecture

1, then u(s, t) does not satisfy anything so straightforward

as ∂

− ∇A

(u(s, ·)) = 0, but it does satisfy

∂

u + Jπ

∂

u = 0,

which can be interpreted as the projection of a positive gradient ﬂow equation to the

contact bundle. This observation is a local symptom of a mo r e important global fact

that follows from Sto kes’ theorem: any asymptotically cylindrical J-holomorphic

curve u :

Σ → R ×M with positive and negative punctures Γ

asymptotic to orbits

{γ

}

z∈Γ

satisﬁes

z∈Γ

(γ) −

z∈Γ

−

(γ) =

∗

dα ≥ 0.

This generalizes the basic fact in Floer ho mo logy that ﬂow lines decrease action and,

conversely, have their energy controlled by the a ctio n.

We would now like to develop some of the general properties of asymptotic

operators. Recall that on any symplectic vector bundle (E, ω), a compat ible complex

structure J determines a Hermitian inner product

hv, wi = ω(v, Jw) + iω(v, w),

and conversely, any Hermitian inner product on a complex vector bundle determines

a symplectic structure via the same relation. For this reason, we shall refer to any

vector bundle E with a compatible pair (J, ω) as a Hermitian vector bundle. A

unitary trivialization of such a bundle is a trivialization that identiﬁes ﬁbers with

such that J and ω become the standard complex structure J

and symplectic

structure ω

respectively.

60 Chris Wendl

Definition 3.24. Fix a Hermitian vector bundle (E, J, ω) over S

. An asymp-

totic operat or on (E, J, ω) is any real-linear diﬀerential operator A : Γ(E) → Γ(E)

that takes the form

(3.10) A : C

∞

, R

) → C

∞

, R

) : η 7→ −J

∂

η − S(t)η

in unitary trivializations, where S : S

→ End(R

) is a smooth loop of symmetric

matrices.

Equivalently, an asymptotic operator on (E, J, ω) is any operator of the form

−J∇ where ∇ is a symplectic connection on E.

Exercise 3.25. Show that any asymptotic operator on a Hermitian vector bun-

dle (E, J, ω) over S

is symmetric with respect to the real L

bundle metric

hη

, η

ω(η

(t), Jη

(t)) dt.

Exercise 3.26. Show that the asymptotic operator A

for a closed Reeb orbit

γ is also an asymptotic opera t or on (γ

∗

ξ, J, dα) in the sense of Deﬁnition

3.24.

For f unctiona l analytic purposes, we shall r egard asymptotic operators on Her-

mitian bundles (E, J, ω) as bounded real-linear operators

A : H

(E) → L

(E).

By Lemma

3.17, all asymptotic operators are then Fr edholm with index 0, and any

two such operators on the same bundle are compact perturbations of each other.

Regarding them a lt ernatively as unbounded symmetric operators on L

(E), the

spectral ﬂow

spec

−

, A

) ∈ Z

between two such operator s A

with tr ivial kernel is deﬁned by choosing any unitary

trivialization to write both in the form −J

∂

− S(t), and it is independent of this

choice. The following is what we mean when we say tha t critical points of the action

functional have “inﬁnite Morse index” and “inﬁnite Morse co-index”:

Proposition 3.27. Every asymptotic operator has inﬁnitely many eigenvalues

of both signs.

Proof. It is easy to verify that this is true for A

:= −J

∂

: H

, R

) →

, R

); see the pr oof of theorem

3.34 below. It is therefore also tr ue for A

+ ǫ

for any ǫ ∈ R, and this operator has tr ivial kernel whenever ǫ 6∈ 2πZ. For any

other trivialized asymptotic operator A with 0 6∈ σ(A), the result then follows from

Theorem 3.3 since µ

spec

+ ǫ, A) is ﬁnite, and this is precisely the signed count

of eigenva lues which change sign. The condition 0 6∈ σ(A) can then be lifted by

replacing A with A + ǫ. 

Exercise 3.28. Show that asymptotic operators are self-adjoint (as unbounded

operators on L

with domain H

) in the sense of Remark 3.12.

Lectures on Symplectic Field Theory 61

3.4. The Conley-Zehnder index

We are now in a position to deﬁne a suitable replacement for the Morse index in

the context of SFT. We shall say that an asymptotic operator A is nondegenerate

whenever 0 6∈ σ(A). We will begin by deﬁning the Conley-Zehnder index as an

integer-valued invariant of homotopy classes of nondegenerate asymptotic operators

on the trivial Hermitian bundle S

× R

; the deﬁnition o n arbitrary Hermitian

bundles will then depend on a choice of trivialization.

It is customary elsewhere in the literature (see e.g. [

SZ92]) to adopt a somewhat

diﬀerent perspective on the Conley-Zehnder index, in which it deﬁnes an integer-

valued invariant o f connected components of the space of “nondegenerate symplectic

arcs”



Ψ ∈ C

([0, 1], Sp(2n))



Ψ(0) = 1 a nd 1 6∈ σ(Ψ(1))



These are two diﬀerent perspectives on the same notion. A dictionary from ours

to the other perspective is provided by associating to any tr ivialized nondegenerate

asymptotic operator A = −J

∂

− S(t) the symplectic arc Ψ deﬁned by the initial

value problem

(−J

∂

− S(t))Ψ(t) = 0, Ψ(0) = 1.

Conversely, any smooth symplectic arc determines via this same formula a smooth

path of symmetric matrices S : [0, 1] → End(R

), producing a mild generalization

of our notion of an asymptotic operator.

Definition 3.29. The Conley-Zehnder index associates to every trivialized

nondegenerate asymptotic operator A : H

, R

) → L

, R

) as in (

3.10) an

integer

(A) ∈ Z

determined uniquely by the following properties:

(1) Set µ

(A) := 0 for the operator A = −J

∂

−



1 0

0 −1



(2) For any two nondegenerate operators A

, set

−

) − µ

) := µ

spec

−

, A

Definition 3.30. Given a no ndegenerate asymptotic operator A on a Hermitian

bundle (E, J, ω) over S

and a choice o f complex trivialization τ for (E, J), the

Conley-Zehnder index of A with respect to τ is the integer

(A) ∈ Z

deﬁned by choosing any unitary trivialization homotopic to τ to write A as an

operator H

, R

) → L

, R

) and then plug ging in Deﬁnition

3.29.

If S(t) is not continuous on S

but is continuous on [0, 1], then −J

∂

− S(t) cannot be

regarded as a linear operator on C

∞

, R

) but is still a very well-b e haved symmetric Fredholm

operator from H

) to L

). All of the impor tant functional analytic results in this lecture

can thus be generalized to allow this.

62 Chris Wendl

If γ is a nondegenerate Reeb orbit γ in a (2n −1)-dimensional conta ct manifold

(M, ξ = ker α), then for any complex trivialization τ of γ

∗

ξ → S

, t he Conley-

Zehnder index of γ relative to τ is deﬁned as

(γ) := µ

Remark 3.31. Fr om the perspective of [

SZ92], µ

(γ) is the Conley-Zehnder

index of the linearized Reeb ﬂow along γ restricted to ξ, expressed via a choice of

unitary t rivialization as a nondegenerate arc in Sp(2n − 2).

Exercise 3.32. Show tha t if A

and A

are nondegenerate asymptotic o perators

on Hermitian bundles E

and E

respectively, then A

⊕A

deﬁnes a nondegenerate

asymptotic operator on E

⊕ E

, and given trivializations τ

for j = 1 , 2,

⊕τ

⊕A

) = µ

) + µ

The following is a functional-analytic version of the well-known fact that the

Conley-Zehnder index classiﬁes homotopy classes of nondegenerate symplectic arcs.

Theorem 3 .33. On any Hermitian bundle ( E, J, ω) → S

with complex trivial-

ization τ , two nondegenerate asymptotic operators A

lie in the same connected com-

po nent of the space of no ndegenerate asymptotic operators if and only if µ

) =

−

Proof. Trivializing the bundle, we need to show that if A

= −J

∂

− S

(t)

satisfy µ

spec

−

, A

) = 0, then there exists a pa th of asymptotic operators between

them fo r which no eigenvalues cross 0. To see this, we can ﬁrst choose any path

}

t∈[−1,1]

of asymptotic operators with A

±1

= A

, and then use Lemma 3.18 to

add generic compact perturbations producing a family



′

∈ Fred

sym

, L

, A

)



t∈[−1,1]

whose intersections with Fred

sym,k

, L

, A

) are tr ansverse f or every k ≥ 1, hence

only simple eigenvalues cross 0 and they cross t r ansversely. Any neighboring pair

of crossings with opposite signs can then be eliminated by changing {A

′

}

t∈[−1,1]

′

+ c(t)}

t∈[−1,1]

for a suitable choice of smooth function c : [−1, 1] → R. Since

the spectral ﬂow is zero, one can repeat this modiﬁcation unt il one obtains a path

of perturbed operators with no crossings, and it is a small perturbation of the path

of asymptotic operators {A

+ c(t)}

t∈[−1,1]

. Since A

are both no ndegenerate, one

can assume moreover that all eigenvalues of A

+ c(t) stay a ﬁxed distance δ > 0

away from 0, where δ is independent of the perturbation. O ne can therefore “turn

oﬀ the p erturba tion” as in the proof o f Theorem

3.3, i.e. there exists a sequence of

perturbed paths {A

}

t∈[−1,1]

converging to {A

+c(t)} whose eigenvalues stay a ﬁxed

distance away from 0, and the same is therefore true fo r the continuous families of

eigenvalues of A

+ c(t) obtained as ν → ∞. 

To compute Conley-Zehnder indices, Exercise

3.32 shows that it suﬃces if we

know how to compute t hem for operators on Hermitian line bundles. The next two

theorems provide a tool fo r handling the latter.

Lectures on Symplectic Field Theory 63

Theorem 3.34. Let A = −J

∂

− S(t) : H

, R

) → L

, R

), where S( t)

is a smooth loop of symmetric 2-by-2 matrices. For each λ ∈ σ(A), denote the

co rresponding eigenspace by E

⊂ H

, R

(1) Every nontrivial eigenfunction e

∈ E

is nowhere zero and thus has a

well-deﬁned winding numbe r wind(e

) ∈ Z.

(2) Any two nontrivial eig enfunctions i n the same eigen space E

have the same

winding number.

(3) If λ, µ ∈ σ(A ) satisfy λ < µ, then any two nontrivial eigenfunctions e

∈ E

and e

∈ E

satisfy wind(e

) ≤ wind(e

(4) For every k ∈ Z, A has exactly two eigenvalues (co unting m ultiplic i ty) for

which the corresponding eigenfunctions have winding number eq ual to k.

Proof. We follow the proof given in [

HWZ95].

Observe ﬁrst that (

1) fo llows from the fact that nontrivial eigenfunctions are so-

lutions to an ODE, for which classical existence and uniqueness results are available.

Since the t rivial ma p is a solutio n, every eigenfunction which vanishes at a point

must be itself t r ivial, by uniqueness.

To prove (

2), let ν

and ν

be nontrivial eigenfunctions for the same eigenvalue λ.

If their winding numbers are diﬀerent, then there exists t

∈ S

at which ν

) is a

nonzero real mult iple of ν

), so after rescaling, we can assume ν

) = ν

). But

and ν

are both solutions t o the same linear ODE, so this implies ν

(t) = ν

(t)

for all t and thus contradicts the assumption on the winding numbers.

We ﬁrst prove the rest for the case S = 0 and the operator A

= −J

∂

. Given

ν ∈ H

, R

), written as ν(t) = (x(t), y(t)), we have that ν is an element of E

for the operator A

if and only ( ˙y, −˙x) = λ(x, y). This has solutions of the form



x(t) = A cos(λt) − B sin(λt)

y(t) = B cos(λt) + A sin(λt)

for some constants A, B ∈ R, which are deﬁned on S

as lo ng as λ ∈ 2πZ. In other

words, the sp ectrum of this operator is σ(A

) = 2πZ. Hence ν(t) = ν(0) e

iλt

, which

has winding number

wind(ν) =

2π

Statements (

2) and (3) are now o bvious, and (4) f ollows from the observation tha t E

is two-dimensional, so in this case each eigenvalue is to be counted with multiplicity

two.

For the general case, consider the path of asymptotic operators given by

= −J

∂

−τS( t)}

τ∈[0,1]

Theorem

3.3 gives continuous families {λ

: [0, 1] → R}

j∈Z

and {ν

: [0, 1] →

, R

)}

j∈Z

such that for every τ ∈ [0, 1], ν

(τ) is an eigenfunction for the

operator A

with eigenvalue λ

(τ), whose multiplicity is given by the number of i ∈ Z

for which λ

(τ) = λ

(τ), and such that λ

2n+k

(0) = 2πn, f or k = 0, 1 (this eigenva lue

has mult iplicity 2). Now, since the winding number is a homoto py invariant (hence

invariant under deformations), we have

wind(ν

2n+k

(τ)) = wind(ν

2n+k

(0)) = n,

64 Chris Wendl

for k = 0, 1. Moreover, since the winding only depends on the eigenvalue, the only

paths that can possibly meet are λ

and λ

2n+1

, which implies that the multiplicity

of every eigenvalue λ

(τ) is at most two, with equality where these two “ br anches”

meet. Hence (

3) and ( 4) follow, where equality in (3) holds if and only if the two

branches of paths of eigenvalues with same winding number end up at diﬀerent

points. 

The theorem implies the existence of a well-deﬁned and nondecreasing function

σ(A) → Z : λ 7→ wind(λ),

where wind(λ) is deﬁned as wind(e

) for a ny nontrivial e

∈ E

, and this function

attains every value exactly twice (counting multiplicity of eigenvalues). Since eigen-

values of A are isolated, we can therefore associate to any nondegenerate asymptotic

operator A on the trivial Hermitian line bundle its extremal winding numbers

and its parity,

(A) = min

λ∈σ(A)∩(0,∞)

wind(λ) ∈ Z,

−

(A) = max

λ∈σ(A)∩(−∞,0)

wind(λ) ∈ Z,

p(A) = α

(A) − α

−

(A) ∈ {0, 1}.

(3.11)

Theorem 3.35. If A is a non degenerate asymptotic operator on the trivial Her-

mitian line bundle S

× R

→ S

, then

(A) = 2α

−

(A) + p(A) = 2α

(A) − p(A).

Proof. The operator A

= −J

∂

−



1 0

0 −1



satisﬁes µ

) = 0 by def-

inition, and it has two constant eigenfunctions with eigenvalues of oppo site signs,

hence

−

) = α

) = 0,

consistent with the stated formula. The general case then follows by computing

the spectral ﬂow from A

to any other no ndegenerate operator A, and observing

that the winding number associated to any continuous family of eigenvalues (as in

Theorem

3.3) for a path {A

}

t∈[−1,1]

of asymptotic operators cannot change. 

For any Hermitian line bundle (E, J, ω) over S

with a nondegenerate asymptotic

operator A, we can similarly choose a complex trivialization τ to deﬁne the winding

numbers α

(A) ∈ Z and parity p(A) = α

(A) − α

−

(A) ∈ {0, 1}; note that the

dependence on τ cancels out in the last for mula, so that p(A) is independent of

choices. We then can a ssociate to any nondegenerate Reeb orbit γ in a cont act

3-manifold (M, ξ = ker α) with a trivialization τ of γ

∗

ξ the integers α

(γ) and p(γ),

such that

(γ) = 2α

−

(γ) + p(γ) = 2α

(γ) − p(γ)

holds.

Lectures on Symplectic Field Theory 65

Exercise 3.36. Given a Hermitian vector bundle (E, J, ω) → S

with two com-

plex trivializations τ

: E → S

× R

for j = 1 , 2, denote by

deg(τ

◦ τ

−1

) ∈ Z

the winding number of det g : S

→ C \ {0}, where g : S

→ GL(n, C) is the

transition map appearing in the formula τ

◦ τ

−1

(t, v) = (t, g(t)v). Show t hat for

any asymptotic operator A on (E, J, ω),

(A) = µ

(A) + 2 deg(τ

◦ τ

−1

Exercise

3.36 provides t he useful f ormula

(γ) = µ

(γ) + 2 deg(τ

◦ τ

−1

)

for any two trivializations τ

, τ

of ξ along a nondegenerate Reeb orbit γ. In partic-

ular, this shows that the parity

(γ) := [µ

(γ)] ∈ Z

of the orbit does not depend on a choice of trivialization. We sometimes refer to

even orbits and odd orbits accordingly.

Exercise 3.37. Show that if a Reeb orbit γ : S

→ M in a contact 3-manifold

(M, ξ = ker α) is no ndegenerate and has even parity, then the same is true for all of

its multiple covers

: S

→ M : t 7→ γ(kt), k ∈ N.

LECTURE 4

Fredholm theory with cylindrical ends

Contents

4.1. Cauchy-Riemann operators with punctures 67

4.2. A global weak regularity result 70

4.3. Elliptic estimate s on cylindrical ends 71

4.4. The semi-Fredholm property 73

4.5. Formal adjoints and proof of the Fredholm property 74

In this lecture we will study the class of linear Cauchy-R iemann type operators

that arise by linearizing the nonlinear equation for moduli spaces in SFT. We saw

in the previous lecture that linearizing PDEs over domains with cylindrical ends

naturally leads one to consider certain symmetric asymptotic operators (e.g. the

Hessian of a Morse function at its critical points), which have tr ivial kernel if and only

if a nondegeneracy (i.e. Mor se) condition is satisﬁed. Our goal in this lecture is to

write down the SFT version of this story and show that the linear Cauchy-Riemann

type operators are Fr edholm if their asymptotic operators are nondegenerate.

4.1. Cauchy-Riemann operators with punctures

The setup throughout this lecture will be as follows.

Assume (Σ, j) is a closed connected Riemann surfa ce of genus g ≥ 0, Γ ⊂ Σ is a

ﬁnite set partitioned into two subsets

Γ = Γ

∪ Γ

−

and

Σ := Σ \ Γ denotes the resulting punctured Riemann surface. We shall ﬁx

a choice of holomorphic cylindrical coordinate near each puncture z ∈ Γ

meaning the following. Given R ≥ 0, let ( Z

, i) denote the half-cylinders

:= [R, ∞) × S

, Z

−

:= (−∞, −R] × S

, Z

:= Z

with complex structure i∂

= ∂

, i∂

= −∂

in coordinates (s, t) ∈ R × S

. The

standard half-cylinders Z

are each biholomorphically equivalent to the punctured

disk

D := D \ {0} via the maps

: Z

→

D : (s, t) 7→ e

∓2π(s+it)

For z ∈ Γ

, we choo se a closed neighborhood U

⊂ Σ of z with a biho lomorphic

map

: (

, j) → (Z

, i),

68 Chris Wendl

where

:= U

\ {z}, such that ψ

◦ ϕ

→

D extends holomorphically to

→ D with z 7→ 0. One can always ﬁnd such coordinates by choosing holomorphic

coordinates near z. We can thus view the punctured neighborhoods

⊂

Σ as

cylindrical ends Z

Suppose (E, J) is a smooth complex vector bundle of rank m over (

Σ, j). An

asymptotically Hermitian structure on (E, J) is a choice of Hermitian vector

bundles (E

, J

, ω

) o f rank m associated t o each puncture z ∈ Γ

, together with

choices of complex bundle isomorphisms

→ pr

∗

covering ϕ

→ Z

, where pr

: Z

→ S

denotes the natural projection to the

factor. This isomorphism induces from any unitary trivialization τ of (E

, J

, ω

)

a complex trivialization

(4.1) τ : E|

→ Z

× R

over the cylindrical end, which we will call an asymptotic trivialization near z.

The bundle (E

, J

, ω

) will be referred to as the asymptotic bundle associated

to (E, J) near z.

Fixing asymptotic trivializations near every puncture, we can now deﬁne Sobolev

spaces of sections of E by

k,p

(E) :=

η ∈ W

k,p

lo c

(E)



∈ W

k,p

(

, R

) for every z ∈ Γ

where η

: Z

→ R

denotes the expression of η|

in terms of the asymptotic trivi-

alization, and we use the standard a r ea form ds∧dt on Z

to deﬁne the norm. Since

is compact, diﬀerent choices of asymptotic trivialization give rise to equivalent

norms, however:

Exercise 4.1. Convince yo ur self that diﬀerent choices of asymptotically Her-

mitian structure on E →

Σ can give rise to inequivalent W

k,p

-norms.

Any linear Cauchy-R iemann type operator on E ha s as its target the complex

vector bundle

F :=

Hom

Σ, E),

so sections of F are the same thing as E-valued (0, 1)-for ms. An asymptotic trivi-

alization τ as in (

4.1) then also induces a complex trivialization

F |

→ Z

×R

: λ 7→ τ(λ(∂

)),

where ∂

is the vector ﬁeld o n

arising from its identiﬁcation with Z

. This

trivialization yields a corresponding deﬁnition for the Sobolev spaces W

k,p

(F ), which

depend on the asymptotically Hermitian structure of E but not on the choices of

asymptotic trivializations. Having ma de these choices, a Cauchy-Riemann type

operator D : Γ(E) → Γ(F ) always appears over

as a linear map on C

∞

, R

)

of the form

(4.2) Dη(s, t) =

∂η(s, t) + S(s, t)η(s, t),

where

∂ := ∂

+ J

∂

and S ∈ C

∞

, End(R

)).

Lectures on Symplectic Field Theory 69

Definition 4.2. Suppose A

is an asymptotic operator on (E

, J

, ω

) and D is

a linear Cauchy-Riemann type operator on (E, J). We say that D is asymptotic to

at z if D appears in the form (

4.2) with respect to an asymptotic trivialization

near z, with

kS − S

∞

)

→ 0 as R → ∞

for all k ∈ N, where S

∞

(s, t) := S

∞

(t) is a smooth loop of symmetric matrices

such that A

appears in the correspo nding unitary trivialization of (E

, J

, ω

) a s

−J

∂

− S

∞

Recall that an asymptotic operator is called nondegenerate if 0 is not in its

spectrum, which means it deﬁnes an isomorphism H

→ L

. The objective o f this

lecture will be to prove the fo llowing:

Theorem 4.3. Suppose (E, J) is an asymptotically Hermitian vector bundle

over (

Σ, j), A

is a nondegenerate asymptotic operator on the assoc i ated asymptotic

bundle (E

, J

, ω

) for each z ∈ Γ, and D is a linear Cauchy-Riemann type operator

asymptotic to A

at each puncture z. Then for every k ∈ N and 1 < p < ∞,

D : W

k,p

(E) → W

k−1,p

(F )

is Fredholm. Moreover, ind D a nd ker D are each independent of k and p, the latter

be i ng a space o f smooth sections whose derivatives of all orders decay to 0 at inﬁnity.

Remark 4.4. The asymptotic decay conditions on S(s, t) in Deﬁnition

4.2 can

be relaxed at the cost of limiting the rang e of k ∈ N for which Theorem

4.3 is

valid. To prove that D : W

1,p

→ L

is Fr edholm, it suﬃces to assume S(s, ·) → S

∞

uniformly as |s| → ∞.

The index of D is determined by a generalization of the Riemann-Roch formula

involving the Conley-Zehnder indices µ

) that were introduced in the previous

lecture. We will postpone serious discussion of the index formula until the next

lecture, but here is the statement:

Theorem 4.5. In the setting of Theorem

4.3,

ind D = mχ(

Σ) + 2c

(E) +

z∈Γ

) −

z∈Γ

−

where τ is an arbitrary choice of asymptotic trivializations , c

(E) ∈ Z is the relative

ﬁrst Chern number of E w i th respect to τ, and the sum is independent of this choice.

For the rest of t his lecture, we maintain as standing assumptions that k ∈ N,

1 < p < ∞, and D is a linear Cauchy-Riemann type operat or on E asymptotic at

the punctures t o a ﬁxed set of asymptotic operators {A

}

z∈Γ

. We will not always

need to assume that the A

are nondegenerate, so this condition will be speciﬁed

whenever it is relevant. For subdomains Σ

⊂

Σ, we will sometimes denote the

k,p

-norm on sections of E restricted to Σ

kηk

k,p

(Σ

)

:= kηk

k,p

(E|

)

70 Chris Wendl

and we will use the same notation for sections of other bundles such as F =

Hom

Σ, E) over this domain when there is no danger of confusion. The space

k,p

(Σ

) ⊂ W

k,p

(E)

is deﬁned in this case as the W

k,p

-closure of the space of smooth sections of E with

compact support in Σ

\∂Σ

. For some background discussion on Sobolev spaces of

sections of vector bundles, see Appendix

4.2. A global weak regularity result

In Lecture 2 we proved that for 1 < p < ∞, weak solutions of class L

lo c

to linear

Cauchy-Riemann type equations are always smooth. Here is a g lobal version of that

result.

Proposition 4.6. Suppose 1 < p < ∞ an d k ∈ N. If η ∈ L

(E) weakly satisﬁes

Dη ∈ W

k−1,p

(F ),

then η ∈ W

k,p

(E).

Proof. By induction, it suﬃces to show that if η ∈ W

k−1,p

and Dη ∈ W

k−1,p

then η ∈ W

k,p

. We already know that this is true locally, so the task is to bound the

k,p

-norm of η on the cylindrical ends. Pick an asymptotic trivialization and write

D on one of the ends Z

∼

∂ +S(s, t). Let us assume for concreteness that the

puncture is a positive one, and now consider the W

k,p

-norm of η on (N, N +1)×S

⊂

for N ∈ N. Choo sing a smooth bump function β : R × S

→ [0, 1] supported in

(N − 1, N + 2) × S

with β = 1 on [N, N + 1] × S

, we can use the usual elliptic

estimate to write

kηk

k,p

((N,N+1)×S

)

≤ kβηk

k,p

((N−1,N +2)×S

)

≤ ck

∂(βη)k

k−1,p

((N−1,N +2)×S

)

≤ ckηk

k−1,p

((N−1,N +2)×S

)

+ ck

∂ηk

k−1,p

((N−1,N +2)×S

)

= ckηk

k−1,p

((N−1,N +2)×S

)

+ ckDη − Sηk

k−1,p

((N−1,N +2)×S

)

≤ c

′

kηk

k−1,p

((N−1,N +2)×S

)

+ c

′

kDηk

k−1,p

((N−1,N +2)×S

)

An important detail here is that the constants in these estimates can be assumed

independent of N: indeed, one can use shifts of the same cutoﬀ function for any N,

and the C

k−1

-norm of S on [N −1, N + 2] ×S

is also bo unded uniformly in N since

S(s, t) converges asymptotically to some S

∞

(t). We can therefore take the sum of

this estimate for all N ∈ N, producing

kηk

k,p

(

)

≤ ckηk

k−1,p

(

)

+ ckDηk

k−1,p

(

)



Corollary 4.7. For 1 < p < ∞, any weak s olution η ∈ L

(E) of D η = 0 is

smooth, with derivatives of all orders decaying to 0 at inﬁnity.

Proof. Proposition

4.6 implies η ∈ W

k,p

(E) for every k ∈ N, so smoothness

follows from the Sobolev embedding theorem. Moreover, suppose k and p are large

Lectures on Symplectic Field Theory 71

enough to have a continuous inclusion W

k,p

֒→ C

for some m ∈ N. Then the

ﬁniteness of the W

k,p

-norm also implies that for each end

= Z

kηk

)

≤ ckηk

k,p

(

)

→ 0 as R → ∞.



4.3. Elliptic estimates on cylindrical ends

The local elliptic estimates for

∂ = ∂

+ J

∂

in Lecture

2 applied to functions

D ⊂ C with compact support. Using a ﬁnite open covering with a subordinate

partition of unity, it is a straightfor ward matter to tur n these local estimates into

the following global result (cf. [Wend, Lemma 3.3.2]):

Proposition 4.8. If Σ

⊂

Σ is a compact 2-dimensional submanifold with

bo und ary, then there exi sts a constant c > 0 s uch that

kηk

k,p

(Σ

)

≤ ckDηk

k−1,p

(Σ

)

+ ckηk

k−1,p

(Σ

)

for all η ∈ W

k,p

(Σ

). 

This unfortunately is unsuﬃcient for the global problem under consideration,

since one has to chop oﬀ the cylindrical ends of

Σ in order to obtain a compact

domain. We therefore supplement the previous local estimates with an asymptotic

estimate.

Proposition 4.9. Suppose z ∈ Γ

is a puncture such that the asymptotic op-

era tor A

is nondegenerate. Then on Z

⊂

for suﬃciently large R ≥ 0, there

exists a constant c > 0 such that

kηk

k,p

(

)

≤ ckDηk

k−1,p

(

)

for all η ∈ W

k,p

(

Remark 4.10. Recall that W

k,p

(

) denotes the W

k,p

-closure of C

∞

(

), so

such functions remain in W

k,p

if they are extended as zero to larger domains con-

taining

. Note that functions of class W

k,p

need not actually have compact

support; in fa ct C

∞

is dense in W

k,p

(R × S

), see §

A.4.

The proof of this requires a ba sic result about translation-invariant Cauchy-

Riemann type operators on t he cylinder. Other than the elliptic estimates we dis-

cussed in Lecture

2, this is the main analytical ingredient that makes all Floer-type

theories in symplectic geometry work.

Theorem 4.11. Suppose k ∈ N, 1 < p < ∞, and A = −J

∂

− S( t) is a

nondegenerate as ymptotic operator on the trivial Hermitian vector bundle S

×R

→

. Then the operator

∂

− A = ∂

+ J

∂

+ S(t) : W

k,p

(R ×S

, R

) → W

k−1,p

(R × S

, R

)

is an isomorphism. 

A detailed proof of this result f or k = 1 can be found in [

Sal99, Lemma 2.4],

and the general result follows easily from t his using regularity (Proposition

4.6). I

will not attempt to reproduce the proof in Salamon’s notes here since it is somewhat

72 Chris Wendl

involved, but let us informally sketch the ﬁrst step, which is the interesting part.

The goal is to prove that D

:= ∂

− A is an invertible operator from H

(R × S

)

to L

(R × S

). To gain some intuition on this, consider the special case where t he

asymptotic operator is of the form A = −i∂

− C for some constant C ∈ R. O ne

can then write down an inverse o f D

explicitly by combining a Fourier transfor m

in t he s variable with a Fourier series in the t variable. That is, suﬃciently nice

functions u on R × S

can be expressed as

u(s, t) =

k∈Z

ˆu

(σ)e

2πiσs

2πikt

dσ,

where the hybrid Fourier transform/series ˆu depends on a continuous variable σ ∈ R

and a discrete variable k ∈ Z. One can then obtain ˆu from u by

ˆu

(σ) =

R×S

u(s, t)e

−2πiσs

−2πikt

ds dt,

and we have the usual derivative formulas

∂

(σ) = 2πiσˆu

(σ) and

∂

(σ) =

2πikˆu

(σ). The relation (∂

+i∂

+C)u = f therefore produces an inversion formula

of the form

ˆu

(σ) =

(σ)

2πiσ − 2πk + C

This is a nice formula and produces from any f ∈ L

an element u ∈ H

unless

C ∈ 2πZ, in which case the denominator has a singularity. This condition means C

must not be an eigenvalue o f −i∂

, or in other words, A = −i∂

−C is nondegenerate.

One can perhaps imagine carrying out a similar argument in the general case using

an orthonormal set of eigenfunctions

for A in place of the functions e

2πikt

; this is

presumably part of the idea behind the actual proof in [

Sal99], which uses strongly

continuous semigroups generated by the self-adjoint operato r A.

Proof of Proposition

4.9. Write D = ∂

+ J

∂

+ S(s, t) and D

= ∂

∂

+ S

∞

(t) in an asymptotic trivialization on

= Z

, where the nondegenerate

asymptotic operator is A = −J

∂

− S

∞

(t) and we a ssume

kS − S

∞

k−1

)

→ 0 as R → ∞.

For η ∈ W

k,p

(

), there is a canonical extension η ∈ W

k,p

(R ×S

) that equals zero

outside Z

, so by Theorem

4.11 we have

kηk

k,p

(

)

= kηk

k,p

(R×S

)

≤ ckD

ηk

k−1,p

(R×S

)

= ckD

ηk

k−1,p

(R×S

)

Rewriting t his in terms o f D gives

kηk

k,p

(

)

≤ ckDηk

k−1,p

)

+ ck(S

∞

− S)ηk

k−1,p

)

Recall from Lecture

3 that the spectrum σ(A) of an arbitr ary asymptotic operator A always

consists only of isolated real eigenvalues, thus one can ﬁnd λ ∈ R for which λ − A : H

) →

) is invertible. Its inverse, also known a s the resolvent, then deﬁnes a compact self-adjo int

operator (λ − A)

−1

: L

) → L

) due to the co mpact inclusion H

) ֒→ L

). The

spectral theorem for compa ct self-adjoint operators now provides an orthonormal basis of L

)

consisting of eigenfunctions of (λ − A)

−1

, which are also eigenfunctions of A.

Lectures on Symplectic Field Theory 73

where the constants c > 0 do not depend on R. For this reason, we are free to make

R ≥ 0 large enough to make the C

k−1

-norm of S

∞

−S on Z

less than an arbitrarily

small number δ > 0, in which case the above gives

kηk

k,p

(

)

≤ ckDηk

k−1,p

)

+ cδkηk

k−1,p

)

and thus by the inclusion W

k−1,p

֒→ W

k,p

kηk

k,p

(

)

≤

1 −cδ

kDηk

k−1,p

)



4.4. The semi-Fredholm property

The standard approach for proving that elliptic operators are Fredholm begins

by proving that they are semi-Fredholm, meaning dim ker D < ∞ and im D is

closed. In most settings, it is no t hard to show that local elliptic estimates give rise

to global estimates of the form kηk

k,p ≤ ckDηk

k−1,p + kηk

k−1,p . The step from

these estimates to the semi-Fredholm property is then provided by the following

lemma.

Lemma 4.12. Suppose X, Y and Z are Banach spaces, T ∈ L (X, Y ), K ∈

L (X, Z) is compact, and there is a constant c > 0 such that for all x ∈ X,

(4.3) kxk

≤ ckTxk

+ ckKxk

Then ker T is ﬁnite dimensional and im T is closed.

Proof. A vector space is ﬁnite dimensional if and only if the unit ball in that

space is a compact set, so we begin by proving the la t ter holds fo r ker T. Suppose

∈ ker T is a bounded sequence. Then since K is a compact operator, Kx

has

a convergent subsequence in Z, which is therefore Cauchy. But (

4.3) then implies

that the corresponding subsequence of x

in X is also Cauchy, and thus converges.

Since we now know ker T is ﬁnite dimensional, we also know there is a closed

complement V ⊂ X with ker T ⊕ V = X. Then the r estriction T|

has the same

image as T, thus if y ∈

im T, there is a sequence x

∈ V such that Tx

→ y.

We claim that x

is bounded. If not, t hen T(x

/kx

) → 0 and K(x

/kx

)

has a convergent subsequence, so (

4.3) implies that a subsequence o f x

/kx

also

converges to some x

∞

∈ V with kx

∞

k = 1 and Tx

∞

= 0, a contradiction. But now

since x

is bounded, Kx

also has a convergent subsequence and Tx

converges by

assumption, thus (4.3) yields also a convergent subsequence of x

, whose limit x

satisﬁes Tx = y. This completes the proof that im T is closed. 

In the analysis of closed J-holomorphic curves, one makes use of the above lemma

by placing the inclusion W

k−1,p

֒→ W

k,p

in the role of the compact operator K.

Unfortunately, W

k−1,p

֒→ W

k,p

is not compact when the domain

Σ has cylindrical

ends; in contrast to the case of a compact domain, there is no way t o write the norm

on the ends as a ﬁnite sum of nor ms for functions on domains of ﬁnite measure. To

circumvent this problem, let

⊂

74 Chris Wendl

denote the compact complement of t he ends

⊂

for all z ∈ Γ.

Lemma 4.13. Fix k ∈ N an d 1 < p < ∞, and assume all the A

are non-

degenerate. Then for suﬃciently large R > 0, there exists a constant c > 0 such

that

kηk

k,p

(

Σ)

≤ ckDηk

k−1,p

(

Σ)

+ ckηk

k−1,p

(Σ

)

for all η ∈ W

k,p

(E).

Proof. Fix a smooth cutoﬀ function β ∈ C

∞

(Σ

) such that β|

R−1 ≡ 1, and

write

⊂

for the union of all the ends

⊂

for z ∈ Γ

∪ Γ

−

. Then we can write any

η ∈ W

k,p

(E) as η = βη+(1−β)η so that βη ∈ W

k,p

(Σ

) and ( 1−β)η ∈ W

k,p

(

R−1

Choosing R large enough to make Proposition

4.9 valid, we can apply this together

with Proposition

4.8 to show

kηk

k,p

(

Σ)

≤ kβηk

k,p

(Σ

)

+ k(1 − β)ηk

k,p

(

R−1

)

≤ ckD(βη)k

k−1,p

(Σ

)

+ ckβηk

k−1,p

(Σ

)

+ kD [(1 − β)η] k

k−1,p

(

R−1

)

After applying the Leipbniz rule and absorbing the norms of β and

∂β into the

constants, this produces the stated inequality since the term involving the W

k−1,p

norm of η on the cylindrical ends includes

∂(1−β), which vanishes outside of Σ

. 

Lemma

4.12 is now applicable since the o perator

k,p

(

Σ) → W

k−1,p

(Σ

) : η 7→ η|

involves the compact inclusion W

k,p

(Σ

) ֒→ W

k−1,p

(Σ

) and is thus compact.

Corollary 4.14. If all the A

are nondegenerate, then

D : W

k,p

(E) → W

k−1,p

(F )

is semi- Fredholm. 

4.5. Formal adjoints and proof of the Fredholm property

In order to show that coker D is also ﬁnite dimensional, we will apply the above

arguments to the formal adjoint of D, an operator whose kernel is naturally isomor-

phic to t he cokernel of D. Let us choose Hermitian bundle metrics h , i

on E a nd

h , i

on F , and ﬁx an area form d vol on

Σ that takes the fo rm d vol = ds ∧ dt

on the cylindrical ends. The formal adjoint of D is then deﬁned as the unique

ﬁrst-order linear diﬀerential operato r

∗

: Γ(F ) → Γ(E)

that satisﬁes the relation

hλ, Dηi

(F )

= hD

∗

λ, ηi

(E)

for all η ∈ C

∞

(E), λ ∈ C

∞

(F ),

Lectures on Symplectic Field Theory 75

where we use the real-valued L

-pairings

hη, ξi

(E)

:= Re

hη, ξi

d vol, for η, ξ ∈ Γ(E),

hα, λi

(F )

:= Re

hα, λi

d vol, for α, λ ∈ Γ(F ).

The word “formal” refers to the fact that we are not viewing D

∗

as the adjoint of

an unbounded operator on a Hilbert space (cf. [

RS80]); that would be a stronger

condition.

Exercise 4.15. Show that D

∗

is well deﬁned and, for suitable choices of complex

local trivializations of E and F and holomorphic coordinates on open subsets U ⊂

Σ,

can be written locally as

∗

= −∂ + A : C

∞

(U, R

) → C

∞

(U, R

)

for some A ∈ C

∞

(U, End(R

)), where ∂ := ∂

−J

∂

The f ormula in the above exercise r eveals that D

∗

is also an elliptic operator

and thus has the same local properties as D; indeed, −∂ + A can be transformed

into

∂ + B for some zeroth-o r der term B if we conjugate it by a suitable complex-

antilinear change of trivialization. In particular, our local estimates for D and their

consequences, notably Proposition

4.8, are all equally valid for D

∗

To obtain suitable asymptotic estimates for D

∗

, let us ﬁx asymptotic trivializa-

tions τ of E, use the corresponding trivializations of F over the ends as described

in §

4.1, and choose the bundle metrics such that both appear standard in these

trivializations over the ends. We will say that the bundle metrics a re compatible

with the asymptotically Hermitian structure of E whenever they are chosen

in this way outside of a compact subset of

Σ. We can then express D as

∂ + S(s, t)

= Z

, and integrate by parts to o bt ain

∗

= −∂ + S(s, t)

To identify this expression with a Cauchy-Riemann type o perator, let C :=



1 0

0 −1



denote the R-linear transformation on R

= C

representing complex conjugat ion.

Then since C anticommutes with J

, we have

−1

∗

C)η = −C∂

(Cη) + CJ

∂

(Cη) + CS(s, t)

Cη

= −∂

η − J

∂

η + CS(s, t)

Cη = −(

∂η − CS(s, t)

Cη)

=: −(

∂ +

S(s, t))η,

where we’ve deﬁned

S(s, t) := −CS(s, t)

C. Now if the asymptotic operator A

z ∈ Γ

is written in the chosen trivialization as A := −J

∂

−S

∞

(t), the asymptotic

Technically, this property of the formal adjoint is part of the deﬁnition of ellipticity: we call a

diﬀerential operator elliptic whenever (1) it has the properties necessary for proving fundamental

estimates using Fourier transforms a s we did with

∂ in §

2.3, and (2) its formal adjoint also has this

property. The former requires the principal symbol of the operator to be everywhere injective, and

the latter requires it to be surjective.

76 Chris Wendl

convergence of S(s, t) implies that similarly

S −

∞

)

→ 0 as R → ∞

for all k ∈ N, where

∞

(t) := −CS

∞

(t)C.

This deﬁnes a t rivialized asymptotic operator

A = −J

∂

−

∞

(t) to which −D

∗

(after a suitable change of trivialization) asymptotic at the puncture z; in particular,

our proof of the global r egularity result, Proposition

4.6, now also works for D

∗

Finally, notice t hat A and −

A are conjuga t e: indeed,

−1

AC)η = −CJ

∂

(Cη) + CCS

∞

(t)C(Cη) = J

∂

η + S

∞

(t)η = −Aη.

This implies that A is nondegenerate if and only if

A is; applying this assumption

for all of the A

, the proofs of Proposition

4.9 and Lemma 4.13 now also go through

for D

∗

We’ve proved:

Proposition 4.16. Suppose D

∗

is deﬁned with respect Hermitian bundle metrics

on E and F =

Hom

Σ, E) that are compatible with the asymptotically Hermitian

structure of E. If additionally all the as ymptotic operators A

are nondegenerate,

then

∗

: W

k,p

(F ) → W

k−1,p

(E)

is semi-Fredholm, and its kernel is a space of smooth sections contained in W

m,q

(F )

for all m ∈ N and q ∈ (1, ∞) . 

Since ker D

∗

is now known to be ﬁnite dimensional, the next result completes

the proof of the Fredholm property for D by showing that its imag e has ﬁnite

codimension:

Lemma 4.17. Under the sam e assumptions as in Proposition

4.16,

k−1,p

(F ) = im D + ker D

∗

Proof. Consider ﬁrst the case k = 1. Since D : W

1,p

(E) → L

(F ) is semi-

Fredholm, its image is closed, hence im D + ker D

∗

is a closed subspace of L

(F ).

Then if im D + ker D

∗

6= L

(F ), the Hahn-Banach theorem

provides a nontrivial

element α ∈ (L

(F ))

∗

∼

(F ) for

= 1 such that

(4.4) hDη + λ, αi

(F )

= 0 for all η ∈ W

1,p

(E), λ ∈ ker D

∗

Choosing λ = 0, this implies in particular

hDη, αi

(F )

= 0 for a ll η ∈ C

∞

(E),

which means that α is a weak solution of class L

to the formal adjoint equation

∗

α = 0. By Proposiiton

4.6, α is therefore smooth and belong s to ker D

∗

. But

this contradicts (

4.4) if we plug in η = 0 and λ = α, so t his completes the proof for

k = 1.

In the case p = 2, one can forego the Hahn-Banach theorem and simply take an L

-orthogonal

complement.

Lectures on Symplectic Field Theory 77

For k ≥ 2, suppose α ∈ W

k−1,p

(F ) ⊂ L

(F ) is given: then the case k = 1

provides elements η ∈ W

1,p

(E) and λ ∈ ker D

∗

such that Dη + λ = α. Since

Proposition

4.6 implies λ ∈ W

m,q

(F ) for all m ∈ N and q ∈ (1, ∞), we have

Dη = α − λ ∈ W

k−1,p

(F ) and thus, by Prop.

4.6 aga in, η ∈ W

k,p

(E), completing

the proof f or all k ∈ N. 

The proof of Theorem 4.3 is now complete, but as long as we’re talking about

the formal adjoint , let us take note of a few more properties t hat will be useful in

the future. Assume from now on tha t all the assumptions of Proposition

4.16 are

satisﬁed. We can now strengthen Lemma

4.17 as fo llows.

Proposition 4.18. W

k−1,p

(F ) = im D⊕ker D

∗

and W

k−1,p

(E) = im D

∗

⊕ker D.

In particular, the projections deﬁned by these splittings give i s omorphisms

coker D

∼

ker D

∗

and coker D

∗

∼

ker D,

thus D

∗

: W

k,p

(F ) → W

k−1,p

(E) is a Fredholm operator with

ind D

∗

= −ind D .

Proof. By Lemma

4.17, the ﬁrst splitting follows if we can show that im D ∩

ker D

∗

= {0}. Recall ﬁrst (see §

A.4) that C

∞

(

Σ) is dense in W

k,p

(

Σ) for every

k ≥ 0 and p ∈ [1, ∞), so the deﬁnition of the formal adjoint implies via density and

H¨older’s inequality that if 1 < p, q < ∞ and

= 1,

(4.5) hλ, D ηi

(F )

= hD

∗

λ, ηi

(E)

for all η ∈ W

1,p

(E), λ ∈ W

1,q

(F ).

Now suppose λ ∈ im D ∩ ker D

∗

and write λ = Dη, assuming η ∈ W

k,p

(E). Reg-

ularity implies that since D

∗

λ = 0, λ ∈ W

1,q

(F ), where q can be chosen to satisfy

= 1. We can therefore apply (

4.5) and obtain

hλ, λi

(F )

= hλ, Dηi

(F )

= hD

∗

λ, ηi

(E)

= 0,

hence λ = 0.

The proof that W

k−1,p

(E) = im D

∗

⊕ ker D is analogous. 

This result hints at t he fact that D

∗

is in fact—under some natural extra

assumptions—globally equivalent to another Cauchy-Riemann type operato r. To

see this, let us impose a further constraint on the relation between the Hermitian

bundle metrics h , i

and h , i

. Note that since the area form d vol is necessarily

j-invariant, it induces a Hermitian structure on T

Σ, namely

hX, Y i

:= d vol(X, jY ) + i d vol(X, Y ),

which matches the standard bundle metric in the trivializations over the ends deﬁned

via the cylindrical coordinates. This induces real-linear isomorphisms from T

Σ to

the complex-linear and -antilinear parts of the complexiﬁed cotangent bundle,

Σ → Λ

1,0

∗

Σ : X 7→ X

1,0

:= hX, ·i

Σ → Λ

0,1

∗

Σ : X 7→ X

0,1

:= h·, Xi

where the ﬁrst isomorphism is complex antilinear and the second is complex linear.

We use these to deﬁne Hermitian bundle metrics on Λ

1,0

∗

Σ and Λ

0,1

∗

Σ in terms

78 Chris Wendl

of the metric on T

Σ; note that this is a straig htforward deﬁnition for Λ

0,1

∗

Σ, but

since the isomorphism to Λ

1,0

∗

Σ is complex antilinear, we really mean

1,0

, Y

1,0

:= hY, Xi

for X, Y ∈ T

Σ.

Now observe that as a vector bundle with complex structure λ 7→ J ◦ λ, F =

Hom

Σ, E) is naturally isomorphic to the complex tensor product

F = Λ

0,1

∗

Σ ⊗E.

We can therefore make a natural choice for h , i

as the tensor product metric

determined by h , i

and h , i

. It is easy to check that this choice is compatible

with the asymptotically Hermitian structure o f E.

Next, we notice that the area form d vol also induces a natural complex bundle

isomorphism

E → Hom

Σ, F ).

Indeed, the right hand side is canonically isomorphic to the complex tensor product

Hom

Σ, F ) = Λ

1,0

∗

Σ ⊗ F = Λ

1,0

∗

Σ ⊗Λ

0,1

∗

Σ ⊗ E,

and Λ

1,0

∗

Σ⊗Λ

0,1

∗

Σ is isomorphic to the trivial complex line bundle ǫ

Σ×C →

Σ via

1,0

∗

Σ ⊗ Λ

0,1

∗

Σ → ǫ

: X

1,0

⊗Y

0,1

7→ X

1,0

(Y ) = hX, Y i

Exercise 4.19. Assuming h , i

is cho sen as the tensor product metric described

above, show that under the natural identiﬁcation of E with Hom

Σ, F ),

−D

∗

: Γ(F ) → Ω

1,0

(

Σ, F )

satisﬁes the Leibniz rule

−D

∗

(fλ) = (∂f)λ + f(−D

∗

λ)

for all f ∈ C

∞

(

Σ, R), where ∂f ∈ Ω

1,0

(

Σ) denotes the complex-valued (1, 0)- form

df − i df ◦ j.

We might summarize this exercise by saying that −D

∗

is an “anti-Cauchy-

Riemann type” operator on F . But such an object is easily transformed into an

honest Cauchy-Riemann type operator: let

F denote the conjugate bundle to F ,

which we deﬁne as the same real vector bundle F but with the sign of its complex

structure reversed, so λ 7→ −J ◦ λ. Now there is a canonical isomorphism

Hom

Σ, F ) =

Hom

Σ,

F ),

and the same operator deﬁnes a real-linear map

−D

∗

: Γ(

F ) → Ω

0,1

(

Σ,

F )

which satisﬁes our usual Leibniz rule for Cauchy-Riemann type operators.

Its asymptotic behavior also ﬁts into the scheme we’ve been describing: we

have already seen this by computing D

∗

on the ends with respect to asymptotic

trivializations. To express this in trivialization-invariant language, observe that each

of the Hermitian bundles (E

, J

, ω

) over S

for z ∈ Γ has a conjugate bundle

with complex structure −J

and symplectic structure −ω

; it s natural Hermitian

Lectures on Symplectic Field Theory 79

inner product is then the complex conjugate of the one on E

. The asymptotic

operator A

on E

can be expressed as −J

∇

, where

∇

is a symplectic connection

on (E

, ω

). Then

∇

is also a symplectic connection on (

, −ω

), so we naturally

obtain an asymptotic operator on

in t he form

(4.6)

:= −A

: Γ(

) → Γ(

where t he sign reversal arises from the reversal of the complex structure. One can

check that if we choose a unitary trivialization of E

and the conjugate trivialization

, this relatio nship between A

and

produces precisely the relat ionship

between A = −J

∂

− S

∞

(t) and

A = −J

∂

−

∞

(t) that we saw previously, with

∞

(t) = −CS

∞

(t)C. Let us summarize all this with a theorem.

Theorem 4.20. Assume h , i

is chosen to be the tensor product metric on

F = Λ

0,1

∗

Σ ⊗ E induced by h , i

and the area form d vol. Then under the

isomorphism induced b y d vol from E to Hom

Σ, F ) and the natural identiﬁcation

of the latter with its conj uga te

Hom

Σ,

F ), the operator −D

∗

: Γ(F ) → Γ(E)

deﬁnes a linear Cauchy-Riemann type o perator on the conjugate bundl e

F ,

−D

∗

: Γ(

F ) → Ω

0,1

(

Σ,

F ),

and it is asymp totic at each puncture z ∈ Γ to the co njugate asymptotic operator

(4.6). 

LECTURE 5

The index formula

Contents

5.1. Riemann-Roch with punctures 81

5.2. Some remarks on the formal adjoint 86

5.3. The index zero case on a torus 90

5.4. A Weitzenb¨ock formula for Cauchy-Riemann operators 92

5.5. Large antilinear p e rturbations and energy concentration 94

5.6. Two Cauchy-Riemann type problems on the plane 96

5.7. A linear gluing argument 97

5.8. Antilinea r deformations of asymptotic operators 102

5.1. Riemann-Roch with punctures

As in the previous lecture, let D denote a linear Cauchy-Riemann type operator

on an asymptotically Hermitian vector bundle E of complex ra nk m over a punctured

Riemann surface (

Σ = Σ \ (Γ

∪ Γ

−

), j), and assume that D is asymptotic at each

puncture z ∈ Γ to a nondegenerate asymptotic operator A

on the asymptotic

bundle (E

, J

, ω

) over S

. Writ ing

F :=

Hom

Σ, E)

for the bundle o f complex-antilinear homomorphisms T

Σ → E, the main result of

the previous lecture was that

D : W

k,p

(E) → W

k−1,p

(F )

is Fr edholm fo r any k ∈ N and p ∈ (1, ∞), and its kernel and index do not depend

on k or p. The main goal of this lecture is to compute ind(D) ∈ Z.

The index will dep end on the Conley-Zehnder indices µ

) ∈ Z introduced

in Lecture

3, but since these depend on arbitr ary choices of unitary trivializations τ,

we need a way of selecting preferred trivializations. The most natur al condition is to

require that every (E

, J

, ω

) be endowed with a unitary trivialization such that t he

corresponding asymptotic trivializations of (E, J) extend to a global trivialization

;

if there is only one puncture z, for instance, then this condition determines µ

)

uniquely. This convention has been used to state the formula for ind(D) in several

of the standard references, e.g. in [HWZ99]. We would prefer however to state a

formula which is also valid when Γ = ∅ and E → Σ is nontrivial. One way to do

Note that (E, J) is always globally trivializable unless Γ = ∅, as a punctured s urface can be

retracted to its 1-skeleton.

82 Chris Wendl

this is by allowing completely arbitrary asymptotic trivializations, but int r oducing

a topological invariant to measure their failure to extend globally over E.

Definition 5.1. Fix a compact oriented surface S with bo undary. The relative

ﬁrst Chern number associates to every complex vector bundle (E, J) over S and

trivialization τ of E|

∂S

an integer

(E) ∈ Z

satisfying the following properties:

(1) If (E, J) → S is a line bundle, then c

(E) is the signed count of zeroes for

a generic smooth section η ∈ Γ(E) that appears as a nonzero constant at

∂S with respect to τ.

(2) For any two bundles (E

, J

) and (E

, J

) with trivializations τ

and τ

respectively over ∂S,

⊕τ

⊕E

) = c

) + c

These two conditio ns uniquely determine c

(E) for all complex vecto r bundles

since bundles of higher rank can always be split into direct sums of line bundles.

The deﬁnition clearly matches the usual ﬁrst Chern number c

(E) when ∂S = ∅,

and it extends in an obvious way to the category of asymptotically Hermitian vector

bundles with asymptotic trivializations.

Exercise 5.2. Given two distinct choices of asymptotic trivializations τ

and τ

for an asymptotically Hermitian bundle E of rank m, show that

(E) = c

(E) − deg(τ

◦ τ

−1

where deg(τ

◦τ

−1

) ∈ Z denotes the sum over all punctures of the winding numbers

of the determinants of the transition maps S

→ U(m).

Exercise 5.3. Combining Exercise 5.2 above with Exercise 3.36, show that for

our asymptotically Hermitian vector bundle E with Cauchy-Riemann type operator

D and asymptotic operators A

, the number

(E) +

z∈Γ

) −

z∈Γ

−

)

is independent of the choice of asymptotic tr ivializations τ.

The above exercise shows that the right hand side of the following index formula

is independent of all choices.

Theorem 5.4. The Fredho l m index of D is given by

ind D = mχ(

Σ) + 2c

(E) +

z∈Γ

) −

z∈Γ

−

where m = rank

E a nd τ is an arbitrary choice of asymptotic trivializations.

Caution: to c ompute this winding number at a neg ative puncture using cylindrical coordinates

(s, t) ∈ (−∞, 0] × S

, one must traverse {−s} × S

for s ≫ 1 in the wrong direction, as this is

consistent with the or ie ntation induced on {−s}×S

as a boundary component of a large compact

sub domain of

Σ.

Lectures on Symplectic Field Theory 83

Notation. Throughout this lecture, we shall denote the integer on the right

hand side in Theorem

5.4 by

I(D) := mχ(

Σ) + 2c

(E) +

z∈Γ

) −

z∈Γ

−

) ∈ Z.

Our goal is thus to prove that ind(D) = I(D).

When Γ = ∅, Theorem

5.4 is equivalent to the classical Riemann-Roch f ormula,

which is more often stated f or holomorphic vector bundles over a closed Riemann

surface (Σ, j) with genus g as

(5.1) ind

) = m(1 − g) + c

(E).

This formula assumes that the Cauchy-Riemann type operator D

is complex linear,

but an arbitrary real-linear Cauchy-Riemann operator is then of the form D = D

B, where the zeroth-order term B ∈ Γ(Hom

(E, F )) deﬁnes a compact perturbation

since the inclusion W

k,p

(Σ) ֒→ W

k−1,p

(Σ) is compact. It follows that D has the same

real Fredholm index as D

, namely twice the complex index shown on the right hand

side of (

5.1), which matches what we see in Theorem 5.4.

Remark 5.5. Now seems a good moment to clarify explicitly that all dimensions

(and therefore also Fredholm indices) in this lecture ar e real dimensions, not complex

dimensions, unless otherwise stated.

Reduction t o the complex-linear case does not wo r k in general if there a r e punc-

tures: it remains true that arbit r ary Cauchy-Riemann type operators can be written

as D = D

+ B where D

is complex linear, but the perturbation introduced by

the zeroth-order term B is not compact since W

k,p

(

Σ) ֒→ W

k−1,p

(

Σ) is not compact

when Γ 6= ∅. Another indication that this idea cannot work is the f act t hat while

the formula in Theorem

5.4 always gives an even integer when Γ = ∅, it can be odd

when there are punctures, in which case D clearly cannot have the same index is

any complex-linear operator . Our proof will therefore have to deal with more than

just the complex category.

The punctured version of Theorem

5.4 was ﬁrst proved by Schwarz in his the-

sis [

Sch95], its main purpose at the time being to help deﬁne algebraic operations

(notably the pair-of-pan ts product) in Hamiltonian Floer homology. Schwarz’s proof

used a “linear gluing” construction that gives a relation between indices of opera-

tors on bundles over surfaces obtained by gluing to gether constituent surfaces along

matching cylindrical ends. Since any surface with ends can be “capped oﬀ” to form

a closed surface, one obtains the general index formula if one already knows how to

compute it for closed surfaces and for planes (i.e. caps). For the latter, it is simple

enough to write down model Cauchy-Riemann operators on planes and compute

their kernels and cokernels explicitly, so in this way the general case is reduced to

the classical Riemann-Roch formula. An analogous linear gluing argument for com-

pact surfaces with boundary is used in [MS04, Appendix C] to reduce the general

Riemann-Roch formula to an explicit computatio n for Cauchy-Riemann operators

on the disk with a totally real boundary condition.

84 Chris Wendl

In this lecture, we will follow a diﬀerent path and use an argument that was

ﬁrst sketched by Taubes for the closed case in [

Tau96a, §7], with an additional

argument fo r the punctured case suggested by Chris Gerig [

Ger]. The argument is

(in my opinion) analytically somewhat easier than the more standard a pproaches,

and in addition to proving the formula we need for punctured surfaces, it produces a

new proof in the closed case without assuming the classical Riemann-Roch formula .

It also provides a gentle preview of two analytical phenomena that will later assume

prominent roles in o ur discussion of SFT: bubbling and gluing.

To see the idea behind Taubes’s argument, we can start by noticing an apparent

numerical coincidence in the closed case. Assume (E, J) is a complex line bundle over

a closed Riemann surface (Σ, j), and D : Γ(E) → Γ(F ) = Ω

0,1

(Σ, E) is a Cauchy-

Riemann type operator. We know that ind(D) = ind(D + B) for any zeroth-order

term B ∈ Γ(Hom

(E, F )). But E and F are both complex vecto r bundles, so B

can always be split uniquely into its complex-linear and complex-antilinear parts,

i.e. there is a natural splitting of Hom

(E, F ) into a direct sum of complex line

bundles

Hom

(E, F ) = Ho m

(E, F ) ⊕ Hom

(E, F ).

Out of curiosity, let’s compute the ﬁrst Chern number of the second factor; this will

be the signed count of zeroes of a generic complex-antilinear zeroth-order perturba-

tion. To start with, note that

Hom

(E, F ) = Hom

(E, C) ⊗ F,

and then observe that Hom

(E, C) and E ar e isomorphic: indeed, any Hermitian

bundle metric h , i

on E gives rise to a bundle isomorphism

E → Hom

(E, C) : η 7→ h·, ηi

We thus have

Hom

(E, F )

∼

E ⊗ F , so c

(

Hom

(E, F )) = c

(E) + c

(F ). We can

compute c

(F ) by the same trick since

F =

Hom

(T Σ, E) = Hom

(T Σ, C) ⊗ E

∼

T Σ ⊗ E,

so c

(F ) = c

(T Σ) + c

(E) = χ(Σ) + c

(E), and thus

(

Hom

(E, F )) = χ( Σ) + 2c

(E).

Since we’re looking at a line bundle over a surface without punctures, this number

is the same as I(D). This coincidence is too improbable to ignore, and indeed, it

turns out not to be coincidental. Here is an informal statement of a r esult that we

will later prove a more precise version of in order to deduce Theorem 5.4.

“Theorem”. Given a Cauchy-Riemann type operator D : H

(E) → L

(F ) on a

line bundle (E, J) over a closed Riemann surface ( Σ, j), choose a complex-antilin ear

zeroth-ord er perturbation B ∈ Γ(Hom

(E, F )) whos e zeroes are all nondegenerate.

Here the complex structure on Hom

(E, F ) and its subbundles is deﬁned in terms of the

complex structure of F , i.e. it sends B ∈ Hom

(E, F ) to J ◦ B ∈ Hom

(E, F ).

We are assuming as usual that Hermitian inner products are complex antilinear in the ﬁrst

argument and linear in the second.

Lectures on Symplectic Field Theory 85

Then for suﬃciently large σ > 0, ker(D + σB) is approx i mately spanned by 1-

dimensional space s of sections with support localized near the positive zeroes of B.

In particular, dim ker(D + σB) equals the number of positive ze roes of B.

To deduce ind(D) = I(D) from this, we need to apply the same trick to the

formal adjo int D

∗

. As we will review in §

5.2, −D

∗

can be rega r ded under certain

natural assumptions as a Cauchy-Riemann type operat or on the bundle

F conjugate

to F , a nd the formal adjoint of D + σB then gives rise to a Cauchy-Riemann type

operator of the form

−D

∗

+ σB

′

: Γ(

F ) → Γ(

E) = Ω

0,1

(Σ,

F ),

where B

′

F →

E is a lso complex antilinear and has the same zeroes as B, but with

opposite signs. Applying the above “theorem” to −D

∗

thus identiﬁes ker(D + σB)

∗

for suﬃciently large σ > 0 with a space whose dimension equals the number of

negative zeroes of B. This gives

ind(D) = ind(D + σB) = dim ker(D + σB) − dim ker(D + σB)

∗

= c

(

Hom

(E, F )) = I(D).

It’s worth mentioning t hat the “larg e perturbation” argument we’ve just sketched

is only o ne simple example of an idea with a long and illustrious history: another

simple example is the observation by Witten [

Wit82] that a fter choosing a Morse

function on a Riemannian manifold, certain large deformations of the de Rha m

complex lead to an approximation of the Morse complex, with generators of the de

Rham complex having support concent r ated near the critical points of the Morse

function—this yields a somewhat novel proof of de Rham’s theorem. A much deeper

example is Taubes’s isomorphism [

Tau96b] between the Seiberg- Witt en invariant s

of symplectic 4-manifolds and certain ho lomorphic curve invariants: here also, the

idea is to consider a large compact perturbation of the Seiberg-Witten equations and

show tha t , in the limit where t he perturbation becomes inﬁnitely large, solutions of

the Seiberg-Witten equations localize near J- holomorphic curves. For a more recent

exploration of this idea in the context of Dirac operators, see [

Mar].

Before proceeding with the details, let us ﬁx two simplifying assumptions that

can be imposed without loss of generality:

Assumption 5.6. (E, J) has com plex rank 1.

Indeed, an asymptotically Hermitian bundle E of complex rank m ∈ N always

admits a decomposition into asymptotically Hermitian line bundles E = E

⊕. . . ⊕

, producing a cor resp onding splitting of the targ et bundle F = F

⊕ . . . ⊕ F

The operator D need not respect these splittings, but it is always homotopic through

Fredholm operators to one that does: we saw in Theorem

3.33 that the asymptotic

operators A

are homotopic through nondegenerate asymptotic operators to any

other operators A

′

that have the same Conley-Zehnder indices, so one can choose

′

to respect the splitting. Any homotopy of Cauchy-Riemann operators following

such a homotopy of nondegenerate asymptotic operat ors then produces a continuous

family of Fredholm operators by the main result of Lecture

4, implying that their

indices do not change. The general index formula then follows from the line bundle

86 Chris Wendl

case since any two Cauchy-Riemann type Fredholm operators D

and D

over the

same Riemann surface satisfy

ind(D

⊕D

) = ind(D

) + ind(D

) and I(D

⊕ D

) = I(D

) + I(D

Assumption 5.7. k = 1 and p = 2.

This means we will concretely be considering the operator

D : H

(E) → L

(F ),

where H

as usual is an abbreviation for W

1,2

. This assumption is clearly harmless

since we know tha t ind D does not depend on the choice of k and p.

5.2. Some remarks on the formal adjoint

For the beginning of this section we can drop the assumption that (E, J) is a

line bundle and assume rank

E = m ∈ N, though later we will again set m = 1.

Recall from the end of Lecture

4 that if we ﬁx global Hermitian structures h , i

and h , i

on (E, J) a nd (F, J) respectively a nd an ar ea form d vol on

Σ tha t matches

ds ∧dt on the cylindrical ends, then D has a formal adjoint

∗

: Γ(F ) → Γ(E)

satisfying

hλ, Dηi

(F )

= hD

∗

λ, ηi

(E)

for all η ∈ H

(E), λ ∈ H

(F ).

Here the real-valued L

pairings are deﬁned by

hη, ξi

(E)

:= Re

hη, ξi

d vol for η, ξ ∈ Γ(E),

and similarly for sections of F . The essential features of the formal adjoint are

that ker D

∗

∼

coker D and coker D

∗

∼

ker D, hence ind(D

∗

) = −ind(D). R ecall

moreover that d vol induces a natural Hermitian bundle metric on

Σ by

h·, ·i

= d vo l( ·, j·) + i d vol(·, ·),

which determines a bundle isomorphism

Σ → Λ

0,1

∗

Σ : X 7→ X

0,1

:= h·, Xi

as well as a complex-antilinear isomorphism

Σ → Λ

1,0

∗

Σ : X 7→ X

1,0

:= hX, ·i

If h , i

is then chosen to be the tensor product metric determined via the natural

isomorphism

F =

Hom

Σ, E) = Λ

0,1

∗

Σ ⊗ E = T

Σ ⊗ E,

then E admits a natural isomorphism to Λ

1,0

∗

Σ ⊗ F such that

−D

∗

: Γ(F ) → Γ(E) = Ω

1,0

(

Σ, F )

becomes an anti-C auchy-Riemann type operator, i.e. it satisﬁes the Leibniz rule

−D

∗

(fλ) = (∂f)λ + f(−D

∗

λ)

Lectures on Symplectic Field Theory 87

for all f ∈ C

∞

(

Σ, R), with ∂f := df − i df ◦ j ∈ Ω

1,0

(

Σ). Equivalently, −D

∗

deﬁnes

a Cauchy-Riemann type operator on the conjugate bundle

F →

Σ, deﬁned as the

real bundle F →

Σ but with the sign of its complex structure reversed; we shall

distinguish this Cauchy-Riemann operator from −D

∗

by writing it as

−D

∗

: Γ(

F ) → Ω

0,1

(

Σ,

F ),

though it is technically the same operator. Recall that t he identity map deﬁnes a

natural complex-antilinear isomorphism between any complex vector bundle and its

conjugate bundle; we shall denote this isomorphism generally by

E →

E : v 7→ ¯v,

so in particular it satisﬁes

cv = ¯c¯v for all scalars c ∈ C, and similarly

∗

λ = D

∗

for λ ∈ Γ( F ). The asymptotic operators f or −

∗

are

= −A

: Γ(

) → Γ(

Lemma 5.8. If τ is a choice of asymptotic trivialization on E and ¯τ denotes the

conjugate asymptotic trivialization

, then

¯τ

(

E) = −c

(E), and µ

¯τ

(

) = −µ

) for all z ∈ Γ.

Proof. Assuming E is a line bundle, suppose η is a generic section of E that

matches a nonzero constant with respect to τ on the cylindrical ends, so c

(E) is

the signed count of zeroes of η. Then ¯η ∈ Γ(

E) is similarly a nonzero constant on

the ends with respect t o ¯τ, but the signs of its zeroes are opposite those of η because

they are deﬁned as winding numbers with resp ect to conjugate local trivializations.

This pr oves c

¯τ

(

E) = −c

(E).

The Conley-Zehnder indices can be computed from the formula

) = α

) + α

−

see Theorem

3.35. Here α

−

) is the largest possible winding number relative to

τ of an eigenfunction for A

with negative eigenvalue, and α

) is the smallest

possible winding number with positive eigenvalue. The eigenfunctions of A

= −A

are the same, but the signs of their eigenvalues are reversed, and the signs of their

winding numbers are also reversed because they must be measured relative to the

conjugate trivialization, thus

¯τ

(

) = −α

∓

implying

¯τ

(

) = α

¯τ

) + α

¯τ

−

) = −α

−

) − α

) = −µ

The above calculations are all valid for line bundles, but the general case f ollows

by taking direct sums. 

If τ : E|

→ U×C

is a local trivialization of E with τ(v) = (z, w), the conjugate trivialization

¯τ :

→ U × C

is deﬁned by ¯τ (¯v) = (z, ¯w).

88 Chris Wendl

We are now able to show that Theorem 5.4 is consistent with what we already

know abo ut the f ormal adjo int.

Proposition 5.9. I(−

∗

) = −I(D).

Proof. Under the isomorphism F = Λ

0,1

∗

Σ ⊗ E = T

Σ ⊗ E, an asymptotic

trivialization τ on E induces an asymptotic trivialization ∂

⊗ τ on F , where ∂

denotes the asymptotic trivialization of T

Σ deﬁned via an outward pointing vector

ﬁeld on the cylindrical ends. Counting zeroes of vector ﬁelds then proves c

∂

Σ) =

χ(

Σ), so

∂

⊗τ

(F ) = c

∂

⊗τ

Σ ⊗ E) = mc

∂

Σ) + c

(E) = mχ(

Σ) + c

(E).

Applying Lemma

5.8 to the conjug ate bundle then gives

∂

⊗τ

(

F ) = −mχ(

Σ) − c

(E).

The unitary trivializations of the asymptotic bundles

corresponding to

∂

⊗ τ

are simply ¯τ, thus using Lemma

5.8 again for the Conley-Zehnder terms,

I(−

∗

) = mχ(

Σ) + 2c

∂

⊗τ

(

F ) +

z∈Γ

¯τ

(

) −

z∈Γ

−

¯τ

(

)

= −mχ(

Σ) − 2c

(E) −

z∈Γ

) +

z∈Γ

−

)

= −I(D).



We next consider the eﬀect of an antilinear zeroth-order perturbation on the

formal adjoint. By “antilinear zeroth-order perturbation,” we generally mean a

smooth section

B ∈ Γ(

Hom

(E, F )).

It is perhaps easier to understand B in terms of the conjugat e bundle

E: indeed,

there exists a unique

β ∈ Γ (Hom

(

E, F ))

such that

Bη = β ¯η,

and this correspondence deﬁnes a bundle isomorphism

Hom

(E, F ) = Hom

(

E, F ).

Exercise 5.10. Assume X and Y are complex vector bundles over the same

base.

(a) Show that

X ⊗

Y is canonically isomorphic to the conjugate bundle of

X ⊗ Y .

(b) Show that Hom

(

Y ) is canonically isomorphic to the conjugate bundle of

Hom

(X, Y ), and

Hom

(

Y ) is canonically isomorphic to the conj ug ate

bundle of

Hom

(X, Y ).

0,1

X :=

Hom

(X, C) is canonically isomorphic to the conjugate

bundle of Λ

1,0

X := Hom

(X, C).

Lectures on Symplectic Field Theory 89

Deﬁne the Cauchy-Riemann type operator

:= D + B : Γ(E) → Γ(F ) = Ω

0,1

(

Σ, E),

so D

η = Dη + β¯η. To write down D

∗

, observe that since β :

E → F is a complex-

linear bundle map between Hermitian bundles, it has a complex-linear adjoint

†

: F →

E such that hβ

†

λ, ¯ηi

= hλ, β ¯ηi

for λ ∈ F , ¯η ∈

Here the bundle metric on

E is deﬁned by h¯η,

ξi

:= hξ, ηi

. We then have

Rehλ, Bηi

= Rehλ, β ¯ηi

= Rehβ

†

λ, ¯ηi

= Rehη,

†

λi

= Rehβ

†

λ, ηi

= Reh

†

λ, ηi

where

†

∈ Γ(Ho m

(

F , E)) denotes the image of β

†

∈ Γ(Ho m

(F,

E)) under the

complex-antilinear identity map from Hom

(F,

E) to its conjugate bundle (see Ex-

ercise

5.10). The formal adjoint of D

is thus

∗

= D

∗

+ B

∗

: Γ(F ) → Γ(E),

where B

∗

: F → E is deﬁned by

∗

λ :=

†

λ.

To write down the resulting Cauchy-Riemann type operator o n

F , we replace B

∗

F → E with

∗

F →

E, deﬁned by

∗

λ := B

∗

λ = β

†

λ,

giving a Cauchy-Riemann operator

−

∗

= −

∗

+ (−B

∗

) : Γ(

F ) → Γ(

E) = Ω

0,1

(

Σ,

F ).

The point of writing down this formula is to make the following observat ions:

Lemma 5.11. Th e zeroth-ord er perturbation −

∗

F →

E appearing in −D

∗

has the following properties:

(1) −

∗

F →

E i s complex antilinear;

(2) There is a natural complex bundle isomorphism

Hom

(

F ,

E) = Hom

(F,

that ide ntiﬁes −

∗

with −β

†

;

(3) If m = 1 and B ∈ Γ(

Hom

(E, F )) has only nondegenerate zeroe s, then

−B

∗

∈ Γ(Hom

(

F ,

E)) has the sa me zeroes but with o p posi te si gns.

Proof. The ﬁrst two statements fo llow immediately from the fact that −

∗

is the composition of the canonical conjugation map

F → F with the complex-

linear bundle map −β

†

: F →

E. For the third, it suﬃces to compare what β ∈

Γ(Hom

(

E, F )) and −β

†

: Γ(Hom

(F,

E)) look like in local trivializatio ns near a

zero: o ne is minus the complex conjugate o f the other, hence their zeroes count with

opposite signs. 

90 Chris Wendl

5.3. The index zero case on a torus

As a wa r mup for the general case, we now ﬁll in the details of Taubes’s proof of

Theorem 5.4 in the case

Σ = T

:= C \ (Z ⊕ iZ)

and E = T

×C, i.e. a trivial line bundle. In t his case I(D) = χ(T

) + 2c

(E) = 0,

so our a im is to prove ind(D) = 0 . What we will show in fact is that D is homotopic

through a continuous f amily of Fredholm operators to one that is an isomorphism.

Since E and F are now both trivial, it will suﬃce to consider the operator

D :=

∂ = ∂

+ i∂

: H

, C) → L

, C),

whose formal adjoint is D

∗

:= −∂ = −∂

+ i∂

. An antilinear zeroth-or der pertur-

bation is then equivalent to a choice of function β : T

→ C, giving rise to a family

of operators

η :=

∂η + σβ¯η

for σ ∈ R, where ¯η : T

→ C now denotes the straightforward complex conjugate

of η. Let us assume t hat β : T

→ C is nowhere zero; note that this would not be

possible in more general situations, but is possible here b ecause Hom

(

E, F ) is a

trivial bundle.

Lemma 5.12. D

is injective for a ll σ > 0 suﬃciently large.

Proof. Elliptic regularity implies any η ∈ ker D

is smooth, so we shall restrict

our attention to smooth functions η : T

→ C. We start by comparing the two

second-order diﬀerential operators

∗

D and D

∗

: C

∞

, C) → C

∞

, C).

Both are nonnegative L

-symmetric operators, and in fact the ﬁrst is simply the

Laplacian

∗

D = −∂

∂ = (−∂

+ i∂

)(∂

+ i∂

) = −∂

−∂

= −∆.

The formal adjoint of D

takes the form

∗

η = D

∗

η + σB

∗

η = D

∗

η + σβ ¯η,

thus for any η ∈ C

∞

, C),

∗

η = (D

∗

+ σB

∗

)(D + σB)η

= D

∗

Dη + σ



∂η − ∂(β¯η)



+ σ

∗

Bη

= D

∗

Dη + σ (β∂¯η − (∂β)¯η − β∂¯η) + σ

∗

Bη

= D

∗

Dη + σ

∗

Bη − σ( ∂β)¯η.

(5.2)

This is a Weitzenb¨ock formula: its main message is that the Laplacian D

∗

D and

the related operator D

∗

diﬀer from each other only by a zeroth-order term t hat

Lectures on Symplectic Field Theory 91

will be positive deﬁnite if σ is suﬃciently large. Indeed, since β is nowhere zero, we

have |Bη| ≥ c|η| for some constant c > 0, thus

ηk

= hη, D

∗

ηi

= hη, D

∗

Dηi

+ σ

hη, B

∗

Bηi

− σhη, (∂β)¯ηi

= kDηk

+ σ

kBηk

−σhη, (∂β)¯ηi

≥



− σk∂βk



kηk

We conclude that as soon as σ > 0 is larg e enough to make the quantity in paren-

theses positive, D

η cannot vanish unless kηk

= 0. 

Proof of Theorem

5.4 for E = T

× C. The lemma above shows that one

can add a large antilinear perturbation to D =

∂ making the deformed operator

injective. By Lemma

5.11, the same argument applies to the formal adjoint

∗

, implying that for suﬃciently large σ > 0, D

∗

is injective and thus D

is also

surjective, and therefore an isomorphism. This proves ind(D) = ind(D

) = 0. 

Let’s consider which par ticular details o f the setup made the proof above possible.

First, the zeroth-order perturbation is complex antilinear. We used t his, if only

implicitly, in deriving the Weitzenb¨o ck formula (

5.2): the key step is in the third

line, where the two terms involving ∂ ¯η cancel each other out and leave nothing but

zeroth-order terms remaining. This would not have happened if e.g. B : E → F

had been complex linear—we would then have seen terms depending on the ﬁrst

derivative of η in D

∗

η − D

∗

Dη, and this would have killed the whole argument .

The fact that this cancellation happens when the perturba t ion is antilinear probably

looks like magic at this point, but t here is a principle behind it; we will discuss it

further in §

5.4 below, see Remark 5.1 7.

The second crucial fact we used was that β : T

→ C is nowhere zero, in

order to obtain the lower bound on kBηk

in terms of kηk

. This cannot always

be achieved—it is possible in this special case only because E and F are both

trivial bundles and thus so is Hom

(

E, F ). On more general bundles, the best we

could hope for would be to pick β ∈ Γ(Hom

(

E, F )) with ﬁnitely many zeroes, all

nondegenerate. In this case the above argument fa ils, but it still tells us something.

Suppose Σ

⊂ T

is a region disjoint fro m the isola ted zeroes of β. Then there exists

a constant c

> 0, dependent on the region Σ

, such that

kβ¯ηk

)

≥ kβ ¯ηk

(Σ

)

≥ c

kηk

(Σ

)

so instead of the estimate at the end of the proof above implying D

is injective, we

obtain one of the form

ηk

)

≥ c

kηk

(Σ

)

− cσkηk

)

To see what this means, imagine we have sequences σ

→ ∞ and η

∈ ker D

normalized so that kη

= 1 for all ν. The estimate above then implies

kη

(Σ

)

≤

→ 0 as ν → ∞,

so while all sections η

have the same amount of “energy” (as measured via their L

norms), the energy is escaping from Σ

as σ

increases. This is true for any domain

disjoint fro m the zeroes, so we conclude that in the limit as σ → ∞, sections in

92 Chris Wendl

ker D

have their energy concentrated in inﬁnitesimally small neighborhoods of the

zeroes of β. We will see in the f ollowing how to extract useful information from this

concentration of energy.

5.4. A Weitzenb¨ock formula for Cauchy-Riemann operators

The Weitzenb¨ock formula (

5.2) can be generalized to a useful relation between

any two Cauchy-Riemann type opera t ors that diﬀer by an antilinear zeroth-order

term. To see t his, we start with a short digression on holomorphic and antiholo mo r -

phic vector bundles.

A smooth function f : C ⊃ U → C is called antiholomorphic if it satisﬁes

(∂

−i∂

)f = 0, which means its diﬀerential anticommutes with the complex structure

on C. The class of antiholomorphic functions is not closed under composition, but

it is closed under products, hence one can deﬁne an antiholomorphic structure

on a complex vector bundle to be a system of local trivializations for which all

transition maps are antiholomorphic. Given the standard correspondence between

holomorphic structures and Cauchy-Riemann type operators, it is easy to establish

a similar correspondence between aniholomorphic structures and (complex-linear)

anti-Cauchy-Riemann type operators, i.e. those which satisfy

D(fη) = (∂f)η + fDη

for all f ∈ C

∞

(

Σ, C), where ∂f := df −i df ◦j ∈ Ω

1,0

(

Σ). We’ve seen one important

example of such an operator already: if D : Γ(E) → Γ(F ) is complex linear, then

−D

∗

is a complex-linear anti-Cauchy-Riemann operator on F and thus endows F

with an antiholo mo r phic structure. Another natural example occurs naturally on

conjugate bundles: if E has a holomorphic structure, then

E inherits from this an

antiholomorphic structure. This is immediate from the fact that f : C ⊃ U → C is

holomorphic if and only if

f : U → C is antiholomorphic. If D : Γ(E) → Γ(F ) =

Ω

0,1

(

Σ, E) is the corresponding complex-linear Cauchy-Riemann type o perator on

E, we shall denote the resulting anti-Cauchy-Riemann operator by

D : Γ(

E) → Γ(

F ) = Ω

1,0

(

Σ,

E),

where by deﬁnition

D¯η = Dη.

Exercise 5.13. Show that if X and Y are antiholomorphic vector bundles over

the same base, then X ⊗Y and Hom

(X, Y ) both naturally inherit antiholomorphic

bundle structures such that the obvious Leibniz rules are satisﬁed. Remark: the

proof of this is exactly the same as for ho lomorphic bundles, one only needs t o

change some signs.

Exercise 5.14. Suppose X and Y are complex vector bundles over the same

base, carrying real- linear anti-Cauchy-Riemann operators ∂

and ∂

respectively.

Show that H := Hom

(X, Y ) then admits a real-linear anti-Cauchy-Riemann oper-

ator ∂

such that for all Φ ∈ Γ(H) and η ∈ Γ(X),

∂

(Φη) = (∂

Φ)η + Φ(∂

η).

Lectures on Symplectic Field Theory 93

Hint: write ∂

and ∂

as complex-linear operator s with real-linear zeroth-order

perturbations, and apply Exercise

5.13. Show moreover that any C

-bounds satisﬁed

by the zeroth-o r der terms in ∂

and ∂

are inherited by the zeroth-order term in ∂

The setup for the next result is as follows. We assume again m = 1, so E and F

are line bundles. Fix β ∈ Γ(Hom

(

E, F )), deﬁne B ∈ Γ(Hom

(E, F )) by Bη := β ¯η,

and use this to deﬁne the p erturbed Cauchy- Riemann type operator

:= D + B : Γ(E) → Γ(F ),

whose formal adjoint is D

∗

= D

∗

+ B

∗

with B

∗

λ :=

†

λ.

Proposition 5.15. The second - order diﬀerential ope rators D

∗

D and D

∗

E a re related by

∗

η = D

∗

η + B

∗

Bη − (∂

β)¯η,

where ∂

is a real- l i near anti-Cauchy-Riemann type operator on Hom

(

E, F ). More-

over, if β is C

-bounded on

Σ, then ∂

β is C

-bounded.

Proof. We have real-linear ant i-Cauchy-Riemann operators

D and −D

∗

and F respectively, so Exercise

5.14 produces an operator ∂

on Hom

(

E, F ) for

which the Leibniz rule is satisﬁed. We can then write

∗

η = (D

∗

+ B

∗

)(D + B)η

= D

∗

Dη +

†

Dη − ( −D

∗

)(β¯η) + B

∗

Bη

= D

∗

Dη +

†

D¯η − (∂

β)¯η − βD¯η + B

∗

Bη

= D

∗

Dη + B

∗

Bη − (∂

β)¯η +



†

−β



D¯η.

Here β and

†

are both viewed as complex-linear bundle maps

F → E, the latter in

the obvious way, and the former acting as 1 ⊗ β on

F = Λ

1,0

∗

Σ ⊗

E with tar get

1,0

∗

Σ ⊗F = Λ

1,0

∗

Σ ⊗Λ

0,1

∗

Σ ⊗E = E. Choosing unitary local trivializations,

β and β

†

are represented by the same complex-valued function: indeed, the latter

is the transpose of the former as m-by-m complex matrices, but since m = 1 , this

means they are identical.

Finally, we observe that the asymptotic convergence conditions satisﬁed by D

on the cylindrical ends imply similar conditions for a ll other Cauchy-Riemann and

anti-Cauchy-Riemann operators in this picture, yielding an estimate of the form

k∂

βk

≤ ckβk

globally on

Σ. 

Remark 5.16. The above proof used the assumption m = 1 in order to conclude

†

−β ≡ 0. For higher rank bundles, this impo ses a nontrivial condition that must

be satisﬁed in order fo r the Weitzenb¨ock formula to hold, cf. [

GW].

Remark 5.17. We can now pick out a geometric reason for the miraculous can-

cellation in the Weitzenb¨ock f ormula: the perturbation B is described by a complex

bundle ma p

E → F , where

E and F both have natural antiholomorphic bun-

dle structures deﬁned via the complex-linear parts of

D and −D

∗

respectively. A

complex-linear perturbation B : E → F would not work because E is holomorphic

rather than antiholomorphic: while

D can be ﬁt into the same Leibniz rule with

−D

∗

, the same is not true o f D.

94 Chris Wendl

5.5. Large antilinear perturbations and energy concent ration

We continue in the setting of Proposition

5.15 and set

:= D + σB : Γ(E) → Γ(F )

for σ > 0. After a compact perturbation of D, we can without loss of generality

also impose the following assumptions on D, β ∈ Γ(Hom

(

E, F )) and the area

form d vol:

(i) All zeroes of β are nondegenerate.

(ii) Both |β| and 1/|β| are bounded outside of a compact subset of

Σ.

(iii) Near each point ζ ∈

Σ with β(ζ) = 0, t here exists a neighborhood D(ζ) ⊂

of ζ, a holomorphic coordinate chart identifying (D(ζ), j, ζ) with the unit

disk (D, i, 0), and a local trivialization of E over D(ζ) that ident iﬁes D with

∂ = ∂

+ i∂

: C

∞

(D, C) → C

∞

(D, C) and β with one of the functions

β(z) = z or β( z) = ¯z,

the former if ζ is a positive zero and the latter if it is negative.

(iv) In the holomorphic coordinate on D(ζ) describ ed above, d vol is the stan-

dard Lebesgue measure.

As in the torus case discussed in §

5.3, we will see that the Weitzenb¨ock formula

implies a concentr ation of energy near the zeroes of β for sections η ∈ ker D

σ → ∞. To understand what really happens in this limit, we will use a rescaling

trick. Denote the zero set of β by

Z(β) = Z

(β) ∪ Z

−

(β) ⊂

Σ,

partitioned into the positive and negative zeroes. For any η ∈ Γ(E), ζ ∈ Z

(β) and

σ > 0, we then deﬁne a rescaled function

(ζ,σ)

: D

√

→ C : z 7→

√

η(z/

√

σ),

where the right hand side denotes the local representation of η on D(ζ) in the chosen

coordinate and trivialization. Notice that the equation D

η = 0 appears in this local

representation as either

∂η + σz ¯η = 0 or

∂η + σ¯z¯η = 0 depending on the sign of ζ,

and the function f := η

(ζ,σ)

then satisﬁes

∂f + z

f = 0 or

∂f + ¯z

f = 0 on D

√

We will take a closer look at these two PDEs in §

5.6 below. But ﬁrst, observe tha t

by change of variables,



(ζ,σ)



√

)

= kηk

(D(ζ))

Lemma 5.18. Assume σ

→ ∞, and η

∈ ker D

is a sequence s atisfying a

uniform L

-bound. Then after passing to a subsequence, the rescaled functions η

(ζ,σ

)

: D

√

→ C for each ζ ∈ Z

(β) converge in C

∞

lo c

Lectures on Symplectic Field Theory 95

∞

∈ L

∂η

∞

+ z

∞

= 0 if ζ ∈ Z

(β),

∂η

∞

+ ¯z

∞

= 0 if ζ ∈ Z

−

(β).

Moreover, i f ξ

∈ ker D

is another sequence with these same properties and con-

vergence ξ

→ ξ

∞

, then

lim

ν→∞

hη

, ξ

(E)

ζ∈Z(β)

hη

∞

, ξ

∞

(C)

Proof. The uniform L

-bound implies uniform bounds on kη

)

for every

R > 0, where ν here is assumed suﬃciently large so that R <

√

. Since η

satisﬁes

a Cauchy-Riemann type equation on D

, the usual elliptic estimates (see Lecture

then imply uniform H

-bounds for every k ∈ N on every compact subset in the

interior of D

, hence η

has a C

∞

lo c

-convergent subsequence on C, and the limit η

∞

clearly satisﬁes the stated PDE. The uniform L

-bound also implies a uniform bound

on kη

√

)

and thus an R-independent uniform bound on kη

)

as ν → ∞,

implying that η

∞

is in L

(C).

The limit of hη

, ξ

(E)

is now proved using the Weitzenb¨ock formula. Let

Σ \

[

ζ∈Z(β)

D(ζ),

so there exists a constant c > 0 such that β satisﬁes |β(z)¯v| ≥ c|v| for all v ∈ E

z ∈

. ( Note that this depends on the assumption of 1/|β| being bounded outside

of a compact subset.) Now by Proposition

5.15,

0 = kD

(

Σ)

= hη

, D

∗

(

Σ)

= hη

, D

∗

Dη

(

Σ)

+ σ

hη

, B

∗

Bη

(

Σ)

− σ

hη

, (∂

β)¯η

(

Σ)

≥ kDη

(

Σ)

+ σ

kη

(

)

−σ

′

kη

(

Σ)

≥ σ

kη

(

)

− σ

′

kη

(

Σ)

for some constant c

′

> 0 independent of ν. This implies

kη

(

)

≤

′

kη

(

Σ)

→ 0 as ν → ∞

since kη

(

Σ)

is uniformly bounded. The same estimate applies to ξ

, so that

hη

, ξ

(

)

→ 0 and thus by change of varia bles,

lim

ν→∞

hη

, ξ

(

Σ)

= lim

ν→∞

ζ∈Z(β)

hη

, ξ

(D(ζ))

= lim

ν→∞

ζ∈Z(β)

hη

, ξ

√

)

ζ∈Z(β)

hη

∞

, ξ

∞

(C)



96 Chris Wendl

5.6. Two Cauchy-Riemann type problems on the plane

The rescaling trick in the previous section produced smooth solutions f : C → C

of class L

∂f + z

f = 0,

∂f + ¯z

f = 0.

It turns out tha t we can say pr ecisely what all such solutions are. Write D

f :=

∂f +

f and D

−

f :=

∂f + ¯z

f. Both operators diﬀer from

∂ by antilinear perturbations,

so they satisfy Weitzenb¨ock formulas relating D

∗

to the Laplacian −∆ =

∂

∗

∂ =

−∂

− ∂

. Indeed, repeating Proposition

5.15 in these special cases gives

∗

f = −∆f + |z|

f − 2

f and D

∗

−

f = −∆f + |z|

To make use of this, recall that a smooth function u : U → R on an open subset

U ⊂ C is called subharmonic if it satisﬁes

−∆u ≤ 0.

Subharmonic functions satisfy a mean value property:

−∆u ≤ 0 on U ⇒ u(z

) ≤

πr

)

u(z) dµ(z) for all D

) ⊂ U,

where D

) ⊂ C denotes the disk of radius r > 0 about a point z

∈ U, and dµ(z)

is the Lebesgue measure on C; see e.g. [

Eva98, p. 85].

Exercise 5.19. Show that for any smooth complex-valued function f o n an

open subset of C,

∆|f|

= 2 Rehf, ∆fi + 2|∇f |

where h , i denotes the standard Hermitian inner product o n C and |∇f|

|∂

+ |∂

Proposition 5.20 . The equation

∂f + ¯z

f = 0 does not admit any nontrivial

smooth solutions f ∈ L

(C, C).

Proof. If f : C → C is smooth with D

−

f = 0, then the Weitzenb¨ock formula

for D

−

implies ∆f = |z|

f. Then by Exercise

5.19,

∆|f|

= 2 Rehf, |z|

fi + 2|∇f|

= 2|z|

|f|

+ 2|∇f|

implying that |f|

: C → R is subharmonic. Now if f(z

) 6= 0 f or some z

∈ C, the

mean value property implies

)

|f(z)|

dµ(z) ≥ πr

|f(z

→ ∞ as r → ∞,

so f 6∈ L

(C). 

Proposition 5.21. Every smooth so l ution f ∈ L

(C, C) to the equation

∂f +

f = 0 is a constant real multiple of f

(z) := e

−

|z|

Lectures on Symplectic Field Theory 97

Proof. We claim ﬁrst that every smooth solution in L

(C, C) of D

f = 0 is

purely real valued. The Weitzenb¨ock formula for this case gives ∆f = |z|

f −2

f, and

taking the diﬀerence between t his equation and its complex conjugate then implies

that u := Im f : C → R satisﬁes

∆u = (|z|

+ 2)u.

Now by Exercise

5.19,

∆(u

) = 2|∇u|

+ 2(|z|

+ 2)u

≥ 0,

so u

: C → R is subharmonic, and the mean value property implies as in the proof

of Prop.

5.20 that u 6∈ L

claim.

It is easy to check however that f

is a solution and is in L

(C). Since it is also

nowhere zero, every other solution f must then take the form f(z) = v(z)f

(z) for

some real-valued function v : C → R. Since D

is a Cauchy-Riemann type operator,

the Leibniz rule then implies

∂v ≡ 0. But the only globally holomo rphic functions

with trivial imaginary parts are constant. 

5.7. A linear gluing argument

Now we’re getting somewhere.

Lemma 5.22. Suppose the assumptions of §

5.5 ho ld and β ∈ Γ(Hom

(

E, F )) has

≥ 0 positive and I

−

≥ 0 negative zeroes. Then for all σ > 0 suﬃciently large,

dim ker D

≤ I

and dim coker D

≤ I

−

In particular, for suﬃciently l arge σ, D

is injective i f all zeroes of β are n egative

and surjective if all zeroes are positive.

Proof. Arguing by contradiction, suppose there exists a sequence σ

→ ∞ such

that dim ker D

> I

, and pick (I

+1) sequences of sections η

, . . . , η

∈ ker D

which form L

-orthonormal sets for each ν. By Lemma

5.18, we can then extract

a subsequence such that rescaling near the zeroes of β produces C

∞

lo c

-convergent

sequences whose limits f orm an (I

+ 1)-dimensional or thonormal set in

ζ∈Z(β)

(C, C),

where the component functions f ∈ L

(C, C) for ζ ∈ Z

(ζ) satisfy

∂f + z

f = 0,

while those for ζ ∈ Z

−

(ζ) satisfy

∂f + ¯z

f = 0. Proposition

5.20 now implies that

the component functions fo r ζ ∈ Z

−

(ζ) are all trivial, and by Proposition

5.21,

the components for ζ ∈ Z

(ζ) belong to 1-dimensional subspaces ker D

⊂ L

(C)

generated by the function e

−

|z|

. We conclude that the limiting ort honormal set

lives in a precisely I

-dimensional subspace

ζ∈Z

(β)

ker D

⊂

ζ∈Z(β)

(C, C),

and this is a cont radiction since there are I

+ 1 elements in the set.

98 Chris Wendl

Applying the same argument to the formal adjoint implies similarly dim ker D

∗

≤

−

for σ suﬃciently large. 

We would next like to turn the two inequalities in the above lemma into equal-

ities, which means showing that the I

-dimensional subspace of

ζ∈Z

(β)

(C, C)

generated by solutions of

∂f +z

f = 0 is isomorphic to ker D

for σ suﬃcient ly large.

This requires a simple example of a linear gl uing argument, the point of which is

to reverse the “convergence after rescaling” process that we saw in Lemma

5.18.

The ﬁrst step is a pregluing construction which turns elements of

ζ∈Z

(β)

ker D

into approximate solutions to D

η = 0 for large σ. To this end, ﬁx a smooth bump

function

ρ ∈ C

∞

(

D, [0, 1]), ρ|

1/2

≡ 1

and deﬁne for each ζ ∈ Z

(β) and σ > 0 a linear map

: ker D

→ Γ(E)

such that Φ

(f) is a section with support in D(ζ) whose expression in our ﬁxed

coordinate and trivialization on that neighborhood is the function

(z) = ρ(z)

√

σf(

√

σz).

Adding up the Φ

for all ζ ∈ Z

(β) then produces a linear map

ζ∈Z

(β)

ker D

→ Γ(E)

whose ima ge consists of sections supported near Z

(β), each a linear combination

of cut-o ﬀ Gaussians with energy concentrated in smaller neighbor hoods of Z

(β)

for larger σ. These sections are manifestly not in ker D

since they vanish on open

subsets and thus violate unique continuation, but they are close, in a quantit ative

sense:

Lemma 5.23. For each σ > 0, there e xists a constant c

> 0 such that

(f)k

≤ c

kfk

for all f ∈

ζ∈Z

(β)

ker D

and c

→ 0 as σ → ∞. Moreover, for every pair f, g ∈

ζ∈Z

(β)

ker D

hΦ

(f), Φ

(g)i

→ hf, gi

as σ → ∞.

Proof. First, observe that any f ∈

ζ∈Z

(β)

ker D

is described by a collection

of functions {f

∈ L

(C)}

ζ∈β

(Z)

which take the form

(z) = K

−

|z|

Lectures on Symplectic Field Theory 99

for some constants K

∈ R. Since each f

is in ker D

, we ﬁnd



(f)|

D(ζ)



(z) =

∂ρ(z)

√

σf

(

√

σz) + ρ(z)σ∂f

(

√

σz)

+ σzρ(z)

√

σf

(

√

σz)

∂ρ(z)

√

σf

(

√

σz) + ρ(z)σ(D

)(

√

σz)

∂ρ(z)

√

σK

−

σ| z|

(5.3)

Now since

∂ρ = 0 in D

1/2

, we obt ain

(f)k

ζ∈Z

(β)

D(ζ)

(f)( z)|

dµ(z)

ζ∈Z

(β)

D\D

1/2

∂ρ(z)|

σK

−σ|z|

dµ(z)

≤ Iσe

−σ/4

ζ∈Z

(β)

where we abbreviate I :=

D\D

1/2



∂ρ(z)



dµ(z). The norm of f is given by

kfk

ζ∈Z

(β)

−|z|

dµ(z) =



−|z|

dµ(z)



ζ∈Z

(β)

We conclude that there is a bound of the form

(f)k

≤ C

√

σe

−σ/2

kfk

which proves the ﬁrst stat ement since

√

σe

−σ/2

→ 0 as σ → ∞.

The second statement follows by a change of variable, since

hΦ

(f), Φ

(g)i

ζ∈Z

(β)

hΦ

(f)|

D(ζ)

, Φ

(g)|

D(ζ)

(D(ζ))

ζ∈Z

(β)

(z)σf

(

√

σz)g

(

√

σz) dµ(z)

ζ∈Z

(β)

√



√



(z)g

(z) dµ(z)

The functions f

and g

are both real multiples of e

−

|z|

, so this last integral for each

ζ ∈ Z

(β) is bounded between

√

σ/2

(z)g

(z) dµ(z) and

√

(z)g

(z) dµ(z),

both of which converge to

(z)g

(z) dµ(z) a s σ → ∞, thus

lim

σ→ ∞

hΦ

(f), Φ

(g)i

= hf, gi



To t ur n approximate solutions into actual solutions, let

: L

(E) → ker D

denote the orthogonal projection. We will prove:

100 Chris Wendl

Proposition 5.24 . If a ll zeroes of β are positive, then the linear map

◦ Φ

ζ∈Z

(β)

ker D

→ ker D

is injective for a ll σ > 0 suﬃciently large.

This statement says in eﬀect that whenever σ > 0 is large enough and η :=

(f) ∈ Γ(E) is in the image of the pregluing map, with f normalized by kf k

= 1,

we can ﬁnd a “correction” ξ ∈ (ker D

)

⊥

such that

η + ξ 6= 0 but D

(η + ξ) = 0.

An element ξ ∈ (ker D

)

⊥

with the second property certainly exists, and in fact

it’s unique: indeed, the assumption Z

−

(β) = ∅ implies via Lemma

5.22 that D

surjective and thus restricts to an isomorphism from (ker D )

⊥

∩ H

(E) to L

(F ),

with a bounded right inverse

: L

(F ) → H

(E) ∩ (ker D)

⊥

hence ξ := −Q

η). We know moreover from Lemma

5.23 t ha t kηk

is close

to kfk

= 1, so to prove η + ξ 6= 0, it would suﬃce to show kξk

is small, which

sounds likely since we also know kD

ηk

is small and Q

is a bounded operator. To

make this reasoning precise, we just need to have some control over kQ

k as σ → ∞,

or equivalently, a quantitat ive measure of the injectivity of D

(ker D

)

⊥

∩H

(E)

. This

requires one last appeal to the Weitzenb¨ock formula.

Lemma 5.25. Assume all ze roes of β are positive. Then there e xist co nstants

c > 0 and σ

such that for all σ > σ

kηk

≤ ckD

ηk

for all η ∈ H

(E) ∩ (ker D

)

⊥

Proof. Let us instead prove that if zeroes of β are all nega tive, then the same

bound holds for all η ∈ H

(E). The stated result follows from this by considering

the formal adjoint and using Exercise

5.26 below. Note that by density, it suﬃces

to prove the estimate holds for all η ∈ C

∞

(E).

Assume therefore that Z

(β) = ∅ and, arguing by contradiction, suppose there

exist sequences σ

→ ∞ and η

∈ C

∞

(E) with kη

= 1 and

→ 0.

The usual rescaling trick and application of the Weitzenb¨ock formula then produces

for each ζ ∈ Z

−

(β) a sequence of functions η

:= η

(ζ,σ

)

: D

√

→ C which satisfy

ζ∈Z

−

(β)

kη

√

)

→ 1 and kD

−

√

)

→ 0

as ν → ∞. Indeed, deﬁning

as in the proof of Lemma

5.18, a similar application

of the Weitzenb¨ock formula yields

(

Σ)

≥ σ

kη

(

)

− σ

′

kη

(

Σ)

= σ

kη

(

)

− σ

′

Lectures on Symplectic Field Theory 101

for some c

′

> 0. Thus we obtain

kη

(

)

≤

(

Σ)

′

→ 0 as ν → ∞,

so there is again concentrat ion of energy near t he zeroes of t he antilinear perturba-

tion: in particular,

1 = lim

ν→∞

kη

(

Σ)

= lim

ν→∞

kη

(

)

+ lim

ν→∞

ζ∈Z

−

(β)

kη

(D(ζ))

= lim

ν→∞

ζ∈Z

−

(β)

kη

√

)

Moreover, we have

−

(z) =

∂η



√



¯z

√

¯η



√





√



Taking the square of the norms on each side, we may integrate and use change of

varia bles to obtain

−

√

)

√

(D(ζ))

→ 0 as ν → ∞.

The elliptic estimates from Lecture

2 now provide uniform H

-bounds for each η

on compact subsets of C for every k ∈ N, so tha t a subsequence converges in C

∞

lo c

(C)

to a smooth map η

∞

∈ L

(C, C) satisfying D

−

∞

= 0. But

ζ∈Z

−

(β)

kη

∞

(C)

= 1,

so at least one of these solutions is nontrivial and thus contra dicts Proposition

5.20.



Exercise 5.26. Show that for any Fredholm Cauchy-Riemann type operator D

on E, the following two estimates are equivalent, with the same constant c > 0 in

both:

(i) kηk

(E)

≤ ckDηk

(F )

for all η ∈ H

(E) ∩ (ker D)

⊥

;

(ii) kλk

(F )

≤ ckD

∗

λk

(E)

for all λ ∈ H

(F ) ∩ (ker D

∗

)

⊥

Hint: elliptic regularity implies that for D and D

∗

as bounded linear operators

→ L

, (ker D)

⊥

= im D

∗

and (ker D

∗

)

⊥

= im D.

Proof of Proposition 5.24. If the statement is not true, then there exist

sequences σ

→ ∞ and

∈

ζ∈Z

(β)

ker D

such that kf

= 1 and η

:= Φ

) ∈ (ker D

)

⊥

for all ν. Lemmas

5.23

and 5.25 then provide estimates of the form

• kη

→ 1 ,

• kD

→ 0 , and

• kη

≤ ckD

102 Chris Wendl

as ν → ∞, with c > 0 indep endent of ν. These imply:

1 = lim

ν→∞

kη

≤ lim

ν→∞

ckD

= 0.



We’ve proved:

Proposition 5.27. Suppose the ass umptions of §

5.5 hold and that the section

β ∈ Γ( Ho m

(

E, F )) has I

≥ 0 positive and I

−

≥ 0 negative zeroes. If I

−

= 0, then

is surjective with dim ker D

= I

for all σ > 0 suﬃciently large. If I

= 0,

then D

is injective with dim coker D

= I

−

for all σ > 0 suﬃci e ntly large. In either

ca se,

ind(D

) = I

−I

−

for all σ > 0 suﬃciently large. 

5.8. Antilinear deformations of asymptotic oper at ors

Proposition 5.27 suﬃces to prove the index formula in the closed case, but there

is an additio na l snag if Γ 6= ∅: since H

(

Σ) ֒→ L

(

Σ) is not a compact inclusion,

we have no guarantee that D and D

:= D + σB will have t he same index, and

generally they will not. A solution to this problem has been pointed out by Chris

Gerig [

Ger], using a special class of asymptotic operators that also originate in the

work of Taubes (see [

Tau10, Lemma 2.3]).

In general, the only obvious way to guarantee ind(D) = ind(D

) for large σ > 0

is if we can arrange f or ev e ry operator in the family {D

}

σ≥0

to be Fredholm, which

is not automatic since the zeroth-order perturbation B : E → F is required to be

bounded away from zero near ∞ and must therefore change the asymptotic operators

at the punctures. We are therefore led to ask:

Question. For w hat nondegenerate asymptotic operators A : H

(E) → L

(E)

on a Hermitian line bundle (E, J, ω) → S

ca n one ﬁnd complex-antilin ear bundle

maps B : E → E such that

:= A − σB : H

(E) → L

(E)

is an isomorphism for ev e ry σ ≥ 0?

It turns out that it will suﬃce to ﬁnd, for each unitary trivialization σ and every

k ∈ Z, a particular pair (A

, B

) such that A

−σB

is nondegenerate for all σ ≥ 0

and µ

) = k. To see why, let us proceed under the assumption that such pairs

can be found, and use them to compute the index:

Lemma 5.28. Given D as in Theorem 5.4, ﬁx asymptotic trivializations τ and

suppose that for each puncture z ∈ Γ there exists an asymptotic operator A

′

, J

, ω

) with µ

′

) = µ

), such that if A

′

is written with respect to τ as

−J

∂

− S

(t), then the deformed asymptotic operator

(5.4) C

∞

, R

) → C

∞

, R

) : η 7→ −J

∂

η − S

(t)η − σβ

(t)¯η

Lectures on Symplectic Field Theory 103

is nond egen erate for some loop β

: S

→ C \ {0} and every σ ≥ 0. Then

ind(D) = χ(

Σ) + 2c

(E) +

z∈Γ

wind(β

) −

z∈Γ

−

wind(β

Proof. Since µ

) = µ

′

), we can deform A

to A

′

continuously

through a family of nondegenerate asymptotic operators. It f ollows that we can

deform D throug h a continuous f amily of Fredholm Cauchy-Riemann type oper-

ators to a new operator D

′

whose asymptotic operators are A

′

for z ∈ Γ, and

ind(D

′

) = ind(D). We are fr ee to assume in fa ct that D

′

is written with respect to

the trivialization τ on the cylindrical end near z ∈ Γ

∂

+ J

∂

+ S

(t).

Now choose β ∈ Γ(Hom

(

E, F )) with nondegenerate zeroes such that the deformed

operators D

η := D

′

η + σβ¯η appear in trivialized form o n the cylindrical end near

z ∈ Γ

η = ∂

η + J

∂

η + S

(t)η + σβ

(t)¯η.

This means D

is asymptotic at z to (

5.4), which is nondegenerate for every σ ≥ 0,

implying D

is Fredholm for every σ ≥ 0 and thus

ind(D) = ind(D

The trivializations τ induce trivializations over the cylindrical ends for

E and

F = Λ

0,1

∗

Σ ⊗E, and the expression f or β in the resulting asymptotic trivialization

of Hom

(

E, F ) near z ∈ Γ is β

(t). It follows that the signed count of zeroes of β is

i(D) := c

(Hom

(

E, F )) +

z∈Γ

wind(β

) −

z∈Γ

−

wind(β

)

= χ(

Σ) + 2c

(E) +

z∈Γ

wind(β

) −

z∈Γ

−

wind(β

where the computation c

(Hom

(

E, F )) = χ(

Σ) + 2c

(E) follows from the natural

isomorphism

Hom

(

E, F ) =

∗

⊗ F = E ⊗ F = E ⊗ Λ

0,1

∗

Σ ⊗ E = Λ

0,1

∗

Σ ⊗ E ⊗ E

= T

Σ ⊗ E ⊗ E.

We are f r ee to assume that all zeroes of β are either positive or negative, depending

on the sign of i(D). Proposition

5.27 then implies ind(D

) = i(D) for la r ge σ. 

Notice that instead of nondegenerate families A−σB para metrized by σ ∈ [0, ∞),

it is just as well to ﬁnd such families which are nondegenerate and have the right

Conley-Zehnder index for all σ > 0, as the σ ≥ 1 portion of this fa mily can be

rewritten as (A −B) −σB for σ ≥ 0. The following lemma thus completes the proof

of Theorem

5.4.

Lemma 5.29. For every k ∈ Z, the trivial Hermitian line bundle over S

admits

an asymptotic operator A

and a loop β

: S

→ C \ {0} such that the deformed

asymptotic operators

k,σ

η := A

η − σβ

¯η

104 Chris Wendl

are nondegenerate for every σ > 0 and satisfy

k,σ

) = wind(β

) = k.

Proof. We claim that the choices

η := −J

∂

η − πkη and β

(t) := e

2πikt

do the trick. We prove this in three steps.

Step 1: k = 0. The a bove formula gives A

0,σ

= −J

∂

η − σ¯η, in which the

σ = 1 case is precisely the operator that we used in Lecture

3 to norma lize the

Conley-Zehnder index, hence µ

0,1

) = 0 by deﬁnition. More generally, all of

these operators can be expressed in the form A := −J

∂

− S where S ∈ End

)

is a constant nonsingular 2-by-2 symmetric matrix that anticommutes with J

. We

claim that all asymptotic operators of this f orm are nondegenerate. Indeed, the

conditions S

= S and SJ

= −J

S for J



0 −1

1 0



imply that S takes the form



a b

b −a



with det S = −a

− b

6= 0, and moreover S is of this form if and only if

S also is. In particular, J

S is traceless, symmetric, and nonsingular. Solutions

of Aη = 0 then satisfy ˙η = J

Sη, which has no periodic solutions since J

S has one

positive and one negative eigenvalue, hence ker A = {0}.

Step 2 : even k. There is a cheap trick to deduce the case k = 2m for any m ∈ N

from the k = 0 case. Recall that by Exercise

3.36 in Lecture 3, conjuga t ing A

0,σ

a change of trivialization changes it s Conley-Zehnder index by twice the degree of

that change. In particular, the operator

0,σ

η := e

2πimt

0,σ

−2πimt

η)

is also a nondegenerate asymptotic operator, but with µ

(

0,σ

) = µ

0,σ

)+2m =

k. Explicitly, we compute

0,σ

η = −J

∂

η − πkη − σke

2πikt

¯η,

so A

k,σ

0,σ/k

is also nondegenerate for every σ > 0.

Step 3: odd k. Another cheap trick relates each A

k,σ

to A

2k,σ

after an adjustment

in σ. Given an arbitrar y asymptotic operator A = −J

∂

−S(t) and m ∈ N, deﬁne

:= −J

∂

− mS(mt).

Geometrically, if A is a trivialized representation for the asymptotic operator of

a Reeb orbit γ : S

→ M, then A

is the operator fo r the m-fold covered orbit

: S

→ M : t 7→ γ(mt). It is easy to check in particular that if we deﬁne

(t) := η(mt) for any given loop η : S

→ R

, then

= m(Aη)

so this gives an embedding of ker A into ker A

, implying that whenever A

nondegenerate for some m ∈ N, so is A. To make use of this, o bserve that

k,σ

η = −J

∂

η − π2kη − 2σe

4πikt

¯η = A

2k,2σ

η,

so A

k,σ

is nondegenerate for all σ > 0 by Step 2, and therefore so is A

k,σ

. 

Lectures on Symplectic Field Theory 105

The proof of Theorem 5.4 is now complete.

Exercise 5 .30. Derive a Weitzenb¨ock formula for asymptotic operat ors and use

it to show that for any asymptotic operator A on the trivial Hermitian line bundle

and any smooth β : S

→ C \ {0}, the deformed operators A

η := Aη − σβ¯η are

all nondegenerate for σ > 0 suﬃciently large. Deduce from this that µ

) =

wind(β) for large σ > 0.

LECTURE 6

Symplectic cobordisms and moduli spaces

Contents

6.1. Stable Hamiltonian structures and their symplectizations 107

6.2. Symplectic cobordisms w ith stable boundary 113

6.3. Moduli spaces of unparametrized holomorphic curves 117

6.4. Simple curves and multiple covers 118

6.5. A local structure result 120

In this lecture we introduce the moduli spaces of holomorphic curves that a re

used to deﬁne SFT.

6.1. St able Hamiltonian structures and their symplectizations

In Lecture

1, we motivated the notion of a conta ct manifold by considering

hypersurfaces M in a symplectic manifold (W, ω) that satisfy a convexity (also known

as “contact type”) condition. The point of that condition was that it presents M

as one member of a smooth 1-parameter family of hypersurfaces that all have the

same Hamiltonian dynamics; that 1-parameter family furnishes the basic model of

what we call the s ymplectization of M with its induced contact structure. A useful

generalization of this notion was introduced in [

HZ94] and was later recognized to

be the most natural geometric setting for punctured holomorphic curves. It has the

advantage of allowing us to view seemingly distinct theories such as Hamiltonian

Floer homolo gy as special cases of SFT—and even if we are only interested in conta ct

manifolds, the generalization sometimes makes computations easier than they might

be in a purely contact setting.

Recall that every smooth hypersurface M in a 2n-dimensional symplectic mani-

fold (W, ω) has a characteristic line ﬁeld

ker (ω|

T M

) ⊂ T M,

whose integral curves are the orbits on M of any Hamiltonian vector ﬁeld generated

by a function H : W → R that has M as a regular level set. We say that M ⊂ (W, ω)

is stable if a neighbor hood of M admits a stabilizing vector ﬁeld V : this means

that V is transverse to M and the 1-parameter family of hypersurfaces

:= ϕ

(M), −ǫ < t < ǫ

generated by the ﬂow ϕ

of V has the property that each of the diﬀeomorphisms

M → M

deﬁned by ﬂowing along V preserves characteristic line ﬁelds.

107

108 Chris Wendl

Exercise 6.1. Show that if V is a stabilizing vector ﬁeld for M ⊂ (W, ω), then

the 2-form and 1-form pair (Ω, Λ) deﬁned on M by

Ω := ω|

T M

, Λ := ι

ω|

T M

has the following properties:

(i) Ω|

ker Λ

is nondegenerate;

(ii) ker Ω ⊂ ker dΛ.

Show moreover that if M is assigned the o rientation for which V is positively trans-

verse to M and ξ := ker Λ ⊂ T M is assigned the natural co-orienta tion determined

by Λ, then the induced orientation of ξ matches the or ientation determined by the

symplectic vector bundle structure Ω|

, hence condition

(i) can equivalently be writ-

ten as

(iii) Λ ∧ Ω

n−1

> 0

where dim W = 2n.

A stable Hamiltonian structure (or “SHS” for short) on an arbitrary oriented

(2n − 1)-dimensional manifold M is a pair (Ω, Λ) consisting of a closed 2-form Ω

and 1-form Λ such that properties

(ii) and (iii) in Exercise 6.1 are satisﬁed.

Exercise 6.2. Show that if (Ω, Λ) is a stable Hamiltonian structure, then

ω := d(rΛ) + Ω

is a symplectic form on (−ǫ, ǫ) ×M for ǫ > 0 suﬃciently small, where r denotes the

coordinate on (−ǫ, ǫ); moreover, {0}×M is a stable hypersurface in ((−ǫ, ǫ)×M, ω).

Example 6.3. If M ⊂ (W, ω) is a contact type hypersurface, then a Liouville

vector ﬁeld V t ransverse to M is a stabilizing vector ﬁeld, and the induced stable

Hamiltonian structure is (dα, α), where α := λ|

T M

with λ := ω(V, ·). We will refer

to this example henceforward as the contact case.

Proposition 6.4. Suppose M ⊂ (W, ω) is a closed stable hypersurface with

stabilizing vector ﬁeld V and induced stable Hamiltonian structure (Ω, Λ) where

Ω = ω|

T M

and Λ = ι

ω|

T M

. Th e n a neighborhood of M in (W, ω) admits a sym-

plectomorphism to ((−ǫ, ǫ) ×M, d(rΛ)+ Ω) for some ǫ > 0, identifying M ⊂ W with

{0} ×M ⊂ (−ǫ, ǫ) × M.

Proof. By the smoo th tubular neighbourhood theorem and the preceeding ex-

ercise, we can view ω

= d(rΛ) + Ω as a symplectic form in some neighbourhood

∼

((−ǫ, ǫ) ×M) of M. In this neighbourhood,

(ω

−ω)|

= 0

by deﬁnition of ω

and thus

−ω = dµ

for some 1-form µ such that µ|

= 0. Now deﬁne

= ω + t dµ

Lectures on Symplectic Field Theory 109

and observe that it is a closed 2-form which can be assumed to be non-degenerate

for a small enough choice of U

. Solving the Moser equation

= −µ

yields a well-deﬁned, time-dependent vector ﬁeld v

with the property that v

= 0.

Working back we produce an isotopy a s follows:

dι

= −dµ ⇒

= dι

+ ι

dω

= dι

= −dµ = −

dω

⇒

(ρ

∗

) = L

dω

= 0

where ρ

∗

is the ﬂow of v

. Then

∗

= ρ

∗

ω = ω

since ρ

is the identity. The required symplectomorphism is then

: ρ

−1

→ U

and the fact that M is ﬁxed under the isotopy follows from v

= 0. 

Example 6.5. In the contact case (Ω, Λ) = (dα, α), the symplectic form on the

collar neighborhood in Propo sition

6.4 can be rewritten as d(e

α) by deﬁning the

coordinate t := ln(r + 1). The proposition is easier to prove in this case: one can

construct the collar neighborhood simply by ﬂowing along V , with no need for the

Moser isotopy trick.

A stable Ha miltonian structure H = (Ω, Λ) gives rise to two importa nt additional

objects: a co-o riented hyperplane distribution

ξ := ker Λ,

and a positively transverse vector ﬁeld R determined by the conditions

Ω(R, ·) ≡ 0 and Λ(R) ≡ 1.

By analogy with the cont act case, we will refer to R as the Reeb vector ﬁeld of H.

The condition ker Ω ⊂ ker dΛ implies that it reduces to the usual contact notio n of

the Reeb vector ﬁeld for Λ whenever the latter happens also to be a contact form.

The symplectization of (M, H) for any stable Hamiltonian structure H =

(Ω, Λ) can be deﬁned by choosing suitable diﬀeomorphisms of (−ǫ, ǫ) × M with

R × M: equivalently, this means we consider R × M with the family of symplectic

forms ω

deﬁned by

(6.1) ω

:= d ( ϕ(r)Λ) + Ω

where ϕ is chosen arbitrar ily from the set

(6.2) T :=



ϕ ∈ C

∞

(R, (−ǫ, ǫ))



′

> 0



110 Chris Wendl

Example 6.6. The fo llowing stable Hamiltonian structure places Hamiltonian

Floer homology into the setting of SFT. Suppose (W, ω) is a closed symplectic man-

ifold and H : S

×W → R is a smoo th function, and denote H

:= H(t, ·) : W → R.

The time-dependent Hamiltonian vector ﬁeld X

deﬁned by dH

= −ω(X

, ·) can

then be viewed as deﬁning a symplec tic connection on the trivial symplectic ﬁber

bundle

M := S

× W

−→ S

i.e. t he ﬂow of R(t, x) := ∂

+ X

(x) deﬁnes symplectic parallel transport maps

between ﬁbers. The horizontal subbundle fo r this connection is the “symplectic

complement” of the vertical subbundle with respect to the closed 2-form

Ω = ω + dt ∧ dH.

In other words, Ω restricts to the ﬁbers o f M → S

as ω and the subbundle

{X ∈ T M | ω(X, ·)|

T ({const}×W )

} is generated by R, so Ω is the connection 2-

form deﬁning the connection, cf. [

MS98]. Setting Λ := dt then makes H := (Ω, Λ)

a stable Hamiltonian structure with Reeb vector ﬁeld R, and its closed orbits in

homotopy classes that project to S

with degree one are in 1-to-1 correspondence

with the 1-periodic Hamiltonian orbits on W . Notice that this is very diﬀerent from

the contact case: ξ = ker dt is as far as possible from being a cont act structure, it

is instead an integrable distribution whose integral submanifolds are the ﬁbers of

M → S

Exercise 6.7. Show that for any stable Hamiltonian structure H = (Ω, Λ), the

ﬂow of R preserves ξ = ker Λ along with its symplectic bundle structure Ω|

Definition 6 .8. A T -periodic orbit x : R → M of R is called nondegenerate

if 1 is not an eigenvalue of dϕ

x(0)

: ξ

x(0)

→ ξ

x(0)

, where ϕ

denotes the ﬂow of R.

Exercise 6.9. Show that in Example 6.6, the notions of nondegeneracy for

closed Reeb orbits on M and for 1-periodic Hamiltonian orbits on W (see Lecture

coincide.

If γ : S

→ M parametrizes a T -periodic orbit of R with ˙γ = T · R(γ), then the

formula of Lecture 3 for the asymptotic operator

η = −J(∇

η − T ∇

still makes sense in this more general context, and it deﬁnes an L

-symmetric oper-

ator on the Hermitian vector bundle (γ

∗

ξ, J, Ω) over S

. It can also be interpreted

as a Hessian at a critical point, though for an action functional that is only lo-

cally deﬁned: indeed, while Ω need not be globally exact, it is necessarily exact on

a neighborhood of γ

) for any given loop γ

: S

→ M, so one can pick any

primitive λ of Ω on this neighborhood and, for a suﬃciently small neighborhood

U(γ

) ⊂ C

∞

, M) of γ

, consider the action functional

(6.3) A

: U(γ

) → R : γ 7→

∗

λ.

Lectures on Symplectic Field Theory 111

Its ﬁrst var iation at γ ∈ U(γ

) in the direction η ∈ Γ(γ

∗

ξ) is then

(γ)η = −

Ω( ˙γ, η) dt = h−Jπ

˙γ, ηi

where π

: T M → ξ denotes t he projection along R and the L

-pairing on γ

∗

is deﬁned via the bundle metric Ω(·, J·)|

. This leads us to interpret −Jπ

˙γ as a

“gradient” ∇A

(γ), and if ˙γ = T · R(γ), then diﬀerentiat ing this gradient in the

direction of η ∈ Γ(γ

∗

ξ) gives A

η. As one would exp ect, nondegeneracy of γ is

then equivalent to the condition ker A

= {0}, and one can in this case deﬁne the

Conley-Zehnder index µ

(γ) ∈ Z as in Lecture

3, relative to a choice of unitary

trivialization τ for (ξ, J, Ω).

Exercise 6.10. In the setting of Example

6.6, work out the relationship b etween

and the symplectic action functional for Hamiltonian systems that we discussed

in L ecture

1. (Try not to worry too much about signs.)

Definition 6.11. Given a stable Hamiltonian structure H = (Ω, Λ), denote by

J(H) ⊂ J(R × M)

the space of smooth almost complex structures J on R × M with the following

properties:

• J is invariant under the R-a ctio n on R×M by translation of the ﬁrst factor;

• J∂

= R and JR = −∂

, where r denotes the natural coo rdinate on the

ﬁrst factor;

• J(ξ) = ξ and J|

is compatible with the symplectic vector bundle structure

Ω|

Notice that if H = (dα, α) for a cont act form α, then J(H) matches the space

J(α) deﬁned in Lecture

Exercise 6.12. Show that every J ∈ J(H) is tamed by all of the symplectic

structures ω

as deﬁned in (

6.1) for ϕ ∈ T .

Given J ∈ J(H), we deﬁne the energy of a J-holomorphic curve u : (Σ, j) →

(R × M, J) by

E(u) := sup

ϕ∈T

∗

Exercise

6.12 above implies tha t E(u) ≥ 0, with equality if and only if u is constant.

In t he contact case, this notion o f energy is not identical to the “Hofer energy” that

we deﬁned in Lecture

1, nor to Hofer’s origina l deﬁnition from [Hof93], but all three

are equivalent for our purposes since uniform bounds on any of them imply uniform

bounds on the others.

Just as in the contact case, the simplest example of a ﬁnite-energy J-holomorphic

curve is a tr ivial cylinder

: R × S

→ R × M : (s, t) 7→ (T s, γ(t)),

where γ : S

→ M is a “constant velocity” parametrization of a T -periodic orbit of

R, i.e. ˙γ = T ·R(γ). More generally, given a punctured Riemann surface (

Σ = Σ\Γ, j)

112 Chris Wendl

with Γ = Γ

∪Γ

−

, we consider asymptotically cylindrical J-holomorphic curves

u : (

Σ, j) → (R × M, J), which are assumed to have the property that for each

z ∈ Γ

, there exist holomorphic cylindrical coordinates identifying a punctured

neighborhood

⊂

Σ of z with Z

= [0, ∞)×S

or Z

−

= (−∞, 0]×S

respectively,

and a trivial cylinder u

: R ×S

→ R ×M such that

u(s, t) = exp

(s,t)

(s, t) for |s| suﬃciently large,

where h

(s, t) is a vector ﬁeld along u

satisfying |h

(s, ·)| → 0 uniformly as s →

±∞. As usual, both the norm |h

(s, t)| and the exponential map here are assumed

to be deﬁned with respect to a translation-invariant choice of Riemannian metric

on R × M. The vector ﬁelds h

along u

for each z ∈ Γ are sometimes called

asymptotic representatives of u near z.

Asymptotic representatives satisfy a regularity estimate that will be important to

know about, though its proof (given originally in [

HWZ96]) would be too lengthy to

present here. The methods behind the following statement involve a combination of

nonlinear regularity arguments as in Lecture

2 with the asymptotic elliptic estimates

from Lecture

4. To prepare for the statement, note that H induces a splitting of

complex vector bundles

(6.4) T(R × M) = ǫ ⊕ ξ,

where ǫ denotes the trivial complex line bundle generated by the vector ﬁeld ∂

, or

equivalently, the Reeb vector ﬁeld. It follows that if γ : S

→ M is a Reeb orbit

and u

: R × S

→ R × M is the corresponding trivial cylinder, then any unitary

trivialization τ of the Hermitian bundle (γ

∗

ξ, J, Ω) natura lly induces a trivialization

of u

∗

T (R × M) .

Proposition 6.13 ([

HWZ96]). Assume J ∈ J(H), u : (

Σ, j) → (R × M, J)

is J-holomorphic and asymptotically cylindrical, and its as ymp totic orbit γ

at z ∈

is nondegenerate. Let h(s, t) ∈ C

denote the asymptotic repre sentative of u

near z expressed via the trivialization induced b y a c hoice of unitary trivialization

for (γ

∗

ξ, J, Ω). If δ > 0 is small enough so that the asymptotic operator A

has no

eigenvalues in the closed interval between 0 and ∓δ, then

h(s, t) = e

∓δs

g(s, t)

for some bounded f unc tion g(s, t) ∈ C

whose deri v atives of all orders are bounded

as s → ±∞.

Remark 6.14 . The range of δ > 0 for which Prop.

6.13 holds is o pen, thus by

adjusting δ slight ly, o ne can equivalently say that h( s, t) = e

∓δs

g(s, t) where the

derivatives of all orders of g(s, t) decay to zero as s → ±∞.

Exercise 6.15. Convince yourself that the analog ue of Proposition 6.13 in Morse

theory is true. Namely, suppose (M, g) is a Riemannian manif old, f : M → R is

smooth a nd u : R → M is a solution to ˙u + ∇f(u) = 0 with lim

s→±∞

u(s) = x

∈

Crit(f), where x

are nondegenerate critical points. We can write u(s) asymptoti-

cally as

u(s) = exp

(s)

Lectures on Symplectic Field Theory 113

for some functions h

(s) ∈ T

M that are deﬁned for s close to ±∞ and satisfy

(s)| → 0 as s → ±∞. Show that if δ > 0 is small enough so that ∇

f(x

) has

no eigenvalue in the closed interval between 0 and ±δ, then

(s) = e

∓δs

(s)

for some functions g

(s) with bo unded derivatives o f all orders as s → ±∞.

Hint:

ﬁx local coordinates identifying x

with 0 ∈ R

and ﬁrst consider the case where

∇f(x) in these coordinates depends linearly on x. Then try to compare u(s) with

solutions of this idealized equation.

Example 6.16. In the setting of Example

6.6, a choice of J ∈ J(H) is equiv-

alent to a choice of smooth S

-parametrized family of compatible almost complex

structures {J

}

t∈S

on (W, ω), and J- holomorphic curves u : (

Σ, j) → (R × M, J)

can be written as

u = (f, v) :

Σ →



R × S



× W,

where f : (

Σ, j) → (R × S

, i) is holomorphic. In particular, if (

Σ, j) = (R × S

, i)

and f is taken to have an extension to S

→ S

of degree one, then u can be

reparametrized so that f is the identity map, hence u = (Id, v) : R × S

→ (R ×

) ×W is a section of the trivial ﬁber bundle (R ×S

) ×W → R ×S

, and one can

check that t he equation satisﬁed by v : R ×S

→ W is precisely the Floer equation

∂

v + J

(v)( ∂

v − X

(v)) = 0.

6.2. Symplectic cobordisms with stable boundary

We discussed symplectic cobor disms between contact manifolds in Lecture

Let us now generalize this notion in the context of stable Hamiltonian structures.

A symplectic cobordism with stable boundary is a compact symplectic

manifold (W, ω) with boundary ∂W = −M

−

⊔ M

, equipped with a stabilizing

vector ﬁeld V that points tr ansversely inward at M

−

and outward at M

. This

induces stable Hamiltonian structures H

= (ω

, λ

) on M

, where

:= ω|

T M

, λ

:= (ι

ω)|

T M

and observe that the orientation conventions for M

and M

−

(with the latter car-

rying the opposite of the natural boundary orientation) have been chosen such that

if dim W = 2n,

∧ ω

n−1

> 0 on M

We can now identify neighborhoods o f M

in ( W, ω) symplectically with collars of

the form

([0, ǫ) ×M

, d (rλ

) + ω

) ,

((−ǫ, 0] × M

−

, d (rλ

−

) + ω

−

) ,

see Figure

6.1.

The apparent discrepancy in s igns between this and Proposition

6.13 is due to the fact that

u(s) satisﬁes a negative gradient ﬂow equation, whereas the nonlinear Cauchy-Riemann equa tion

in symplectizations is interpreted loosely as a positive gradient ﬂow equation.

114 Chris Wendl

((−ǫ, 0] × M

, d(rλ

) + ω

)

([0, ǫ) × M

−

, d(rλ

−

) + ω

−

)

(W, ω)

Figure 6.1. A symplectic cobordism with stable boundary compo-

nents ∂W = −M

−

⊔M

and symplectic collar neighborhoods induced

by the stable Hamiltonian structures H

= (ω

, λ

) on M

Modifying (6.2) by

(6.5) T



ϕ ∈ C

∞

(R, (−ǫ, ǫ))



′

> 0 and ϕ(r) = r for r near 0



we can use any ϕ ∈ T

to deﬁne a symplectic completion (

W , ω

) of (W, ω) by

W :=



(−∞, 0] × M

−



∪

−

W ∪



[0, ∞) × M



where the above collar neighborhoods are used to g lue the pieces together smoo t hly

and the symplectic form is deﬁned by











d (ϕ(r)λ

−

) + ω

−

on (−∞, 0] × M

−

ω on W,

d (ϕ(r)λ

) + ω

on [0, ∞) × M

see Figure

6.2. For each r

≥ 0, we deﬁne the compact submanifold

:= ([−r

, 0] × M

−

) ∪

−

W ∪

([0, r

] × M

) ,

and observe that (W

, ω

) is also a symplectic cobordism with stable boundary fo r

every ϕ ∈ T

Since

W is noncompact, almost complex structures J on

W will need to satisfy

conditions near inﬁnity in order for moduli spaces of J-holomor phic curves to be well

behaved, but we would like to preserve the f r eedom of choosing arbitrary compatible

or tame almost complex structures in compact subsets.

Definition 6.17. Given ψ ∈ T

and r

≥ 0, let

(ω

, r

, H

−

) ⊂ J(

W )

denote the space of smooth almost complex structures J on

W such tha t:

• J on [r

, ∞) ×M

matches an element of J(H

);

• J on (−∞, −r

] × M

−

matches an element of J(H

−

);

• J on W

is tamed by ω

Lectures on Symplectic Field Theory 115

(W, ω)

((−ǫ, 0] × M

, d(rλ

) + ω

)

([0, ǫ) × M

−

, d(rλ

−

) + ω

−

)

([0, ∞) × M

, d(ϕ(r)λ

) + ω

)

((−∞, 0] ×M

−

, d(ϕ(r)λ

−

) + ω

−

)

Figure 6.2. The completion (

W , ω

) of a symplectic cobordism

with stable boundary.

Let

J(ω

, r

, H

−

) ⊂ J

(ω

, r

, H

−

)

denote the subset for which J is additio nally compatible with ω

on W

Setting

(6.6) T (ψ, r

) :=



ϕ ∈ T



ϕ ≡ ψ on [−r

, r

]



Exercise

6.12 implies that every J ∈ J(ω

, r

, H

−

) is tamed by ω

for every

ϕ ∈ T (ψ, r

). It is therefore sensible to deﬁne the energy of a J-holomorphic curve

u : (Σ, j) → (

W , J) by

E(u) := sup

ϕ∈T (ψ,r

)

∗

The notion of asymptotically cylindrical J-holomorphic curves extends in a straight-

forwar d way to the setting of (

W , J): such curves are proper maps whose posi-

tive/negative punctures are asymptotic to closed or bits of the Reeb vector ﬁeld R

induced by H

on {±∞} × M

, see Figure 6.3. The exponential decay estimate

in Proposition

6.13 is also immediately applicable in this more general setting since

asymptotically cylindrical curves in

W are indistinguishable near their punctures

from curves in the symplectizations R × M

It is easy to check that asymptotically cylindrical J-holomorphic curves always

have ﬁnite energy. We will prove in Lecture

8 that the converse is also true whenever

the Reeb orbits are nondegenerate.

116 Chris Wendl

Figure 6.3. An asymptotically cylindrical holomorphic curve in

(

W , J) with genus 2, one positive puncture and two negative punc-

tures.

Remark 6.18. Strictly speaking, the “trivial stable cobordism”

([0, 1] ×M, d(ϕ(r)Λ, Ω))

induces diﬀerent stable Hamiltonian structures at M

−

:= {0} × M and M

{1} × M, thus o ne cannot technically regard J(H) as contained in any space of

the for m J(ω

, r

, H

−

) without inventing questionable new notions such as

the “inﬁnitesimal trivial cobordism” [0, 0] × M (whose completion would be the

symplectization of (M, H)). It is nonetheless true for fairly trivial reasons that most

results about J(ω, r

, H

−

) apply equally well to J(H), and we shall use this

fact in the following without always mentio ning it.

Every asymptotically cylindrical curve u :

Σ →

W has a well-deﬁned relative

homology class, meaning the following. Denote the asymptotic orbits of u at

its punctures z ∈ Γ

by γ

, and let

⊂ M

denote the closed 1- dimensional

submanifold deﬁned as the union over z ∈ Γ

of the images of the orbits γ

. L et

denote the compact oriented topological surface with bo undar y obtained from

Σ by

appending {±∞} × S

to each o f its cylindrical ends, and let π :

W → W denote

the retraction deﬁned a s the ident ity o n W and π( r, x) = x ∈ M

⊂ ∂W for (r, x)

in [0, ∞) × M

or (−∞, 0] × M

−

. Then π ◦ u :

Σ → W has a natural continuous

extension

¯u : (

Σ, ∂Σ) → (W,

∪

−

)

and thus represents a relative homology class

[u] ∈ H

(W,

∪

−

Lectures on Symplectic Field Theory 117

6.3. Moduli spaces of unparametrized holomorphic curves

We continue in the setting of a completed symplectic cobordism

W with ﬁxed

choices of ψ ∈ T

, r

≥ 0 and J ∈ J(ω

, r

, H

−

). We shall denote by ξ

and

the hyperplane distribution and Reeb vector ﬁeld respectively determined by

the stable Hamiltonian structure H

= (ω

, λ

Fix integers g, m, k

, k

−

≥ 0 along with o r dered sets of Reeb orbits

= (γ

, . . . , γ

where each γ

is a closed orbit of R

in M

. Denote the union of the images of the

⊂ M

, and choose a relative homology class

A ∈ H

(W,

∪

−

)

whose image under the boundary map H

(W,

∪

−

)

∂

−→ H

(

∪

−

) deﬁned

via the long exact sequence of the pair (W,

∪

−

) is

∂A =

i=1

[γ

] −

−

i=1

[γ

−

] ∈ H

(

∪

−

The moduli space of unparametrized J-holomorphic curves of genus g with m

marked points, homologous to A and asymptotic to (γ

, γ

−

) is then deﬁned

as a set of equivalence classes of tuples

g,m

(J, A, γ

, γ

−

) =



(Σ, j, Γ

, Γ

−

, Θ, u)



∼,

where:

(1) (Σ, j) is a closed connected Riemann surface of genus g;

(2) Γ

= (z

, . . . , z

), Γ

−

= (z

−

, . . . , z

−

) and Θ = (ζ

, . . . , ζ

) are disjoint

ordered sets of distinct points in Σ;

(3) u : (

Σ := Σ \ (Γ

∪ Γ

−

), j) → (

W , J) is an asymptotically cylindrical J-

holomorphic map with [u] = A, asymptotic at z

∈ Γ

to γ

for i =

1, . . . , k

;

(4) Equivalence

(Σ

, j

, Γ

−

, Θ

, u

) ∼ (Σ

, j

, Γ

−

, Θ

, u

)

means the existence o f a biholomo r phic map ψ : (Σ

, j

) → (Σ

, j

), taking

to Γ

and Θ

to Θ

with the ordering preserved, such that

◦ ψ = u

We shall usually abuse notation by a bbreviating elements [( Σ, j, Γ

, Γ

−

, Θ, u)] in

this moduli space by

u ∈ M

g,m

(J, A, γ

, γ

−

The automorphism group

Aut(u) = Aut(Σ, j, Γ

, Γ

−

, Θ, u)

of u is deﬁned as the group of biholomorphic maps ψ : (Σ, j) → (Σ, j) which act

as the identity on Γ

∪ Γ

−

∪ Θ and satisfy u = u ◦ ψ. Clearly the isomorphism

class of this group dep ends only on the equivalence class [(Σ, j, Γ

, Γ

−

, Θ, u)] ∈

118 Chris Wendl

g,m

(J, A, γ

, γ

−

), and we will see in §6.4 below that it is always ﬁnite unless

u :

Σ →

W is constant. The signiﬁcance of the marked points is that they determine

an evaluation map

ev : M

g,m

(J, A, γ

, γ

−

) →

: [(Σ, j, Γ

, Γ

−

, Θ, u)] 7→ (u(ζ

), . . . , u(ζ

))

where Θ = (ζ

, . . . , ζ

). For most of our applications we will be free to assume

m = 0, as marked points are not needed for deﬁning the most basic versions of

SFT; the evaluation map does play a prominent role however in more algebraically

elaborate versions of the theory, and especially in the Gromov-Witten invariants

(the “closed case” of SFT).

We will assign a topology to M

g,m

(J, A, γ

, γ

−

) in the next lecture by locally

identifying it with subsets of certain manifolds of maps

Σ →

W with Sobolev-type

regularity and exponential decay conditions at the ends. In reality, this topology

admits a simpler description: one can deﬁne convergence o f a sequence

[(Σ

, j

, Γ

−

, Θ

, u

)] → [(Σ, j, Γ

, Γ

−

, Θ, u)]

to mean that for suﬃciently large ν, the equivalence classes in the sequence admit

representatives of the form (Σ, j

′

, Γ

−

, Θ, u

′

) such that

(1) j

′

→ j in C

∞

;

(2) u

′

→ u in C

∞

lo c

(

Σ,

W );

(3) ¯u

′

→ ¯u in C

(

Σ, W ).

The proof that this topology matches what we will deﬁne in the next lecture in

terms of weighted Sobolev spaces requires a symptotic elliptic regularity arguments

along the lines of Proposition

6.13.

6.4. Simple curves and multiple covers

In Lecture 2, we pr oved that closed J-holomorphic curves ar e all either embedded

in the complement of a ﬁnite set or are multiple covers of curves with this property.

The same thing holds in the punctured case:

Theorem 6.1 9. Assume u : (

Σ, j) → (

W , J) is a nonconstant as ymptoti-

ca lly cylindrical J-h o l omorphic curve whose asymptotic orbits are all nondegenerate,

where

Σ = Σ \ Γ for some closed Riemann surface (Σ, j) and ﬁnite subset Γ ⊂ Σ.

Then there exists a factorization u = v ◦ ϕ, w here

• ϕ : (Σ, j) → (Σ

′

, j

′

) is a hol omorphic map of positive degree to another

closed and con nected Riemann surface (Σ

′

, j

′

);

• v : (

′

, j

′

) → (

W , J) is an asymptotically cylindrical J-holomorphic curve

which is embedd ed except at a ﬁnite s et o f critical points and se l f - i ntersec tion s,

where

′

:= Σ

′

\ Γ

′

with Γ

′

:= ϕ(Γ) and Γ = ϕ

−1

(Γ

′

As in the closed case, we call u a simple curve if the holomorphic map ϕ :

(Σ, j) → (Σ

′

, j

′

) is a diﬀeomorphism, and u is otherwise a k-fold multiple cover

of v with k := deg(ϕ) ≥ 2.

The proof of this theorem is an almost verbatim repeat of the proof of The-

orem

2.29 in Lecture 2, but with one new ingredient added. Recall that in the

Lectures on Symplectic Field Theory 119

closed case, our proof required two lemmas which described the local picture of a J-

holomorphic curve u :

Σ →

W near either a double po int u(z

) = u(z

) for z

6= z

a critical point du(z

) = 0. Both statements were completely local and thus equally

valid for no n-closed curves, but we now need similar statements to describe what

kinds of singularities can appear in the neighborhood of a puncture. The following

lemma is due to Siefring [

Sie08] and follows from a “ relative asymptotic formula”

analogous to Proposition

6.13.

Lemma 6.20 (Asymptotics). Assume u : (

Σ = Σ \ Γ, j) → (

W , J) i s a s ymptot-

ically cylindrical and is asymptotic at z

∈ Γ to a nondegenerate Reeb orbit. Then

a punctured neighborhood

⊂

Σ of z

ca n be identiﬁed biholomorphically with the

punctured disk

D = D \ {0} such that

u(z) = v(z

) for z ∈

D =

where k ∈ N and v : (

D, i) → (

W , J) is an embedded and asymptotically cylindrical

J-holomorphic curve. Moreover, if u

′

: (

′

= Σ

′

\ Γ

′

, j

′

) → (

W , J) is another

asymptotically cylindrical curve with a puncture z

′

∈ Γ

′

, then the ima g es of u near

and u

′

near z

′

are either identical or disjoint. 

Exercise 6.21. With Lemma

6.20 in hand, adapt the proof of Theorem 2.29 in

Lecture

2 to prove Theorem 6.19. If you get stuck, see [Nel15, §3.2].

Proposition 6.22. If [(Σ, j, Γ

, Γ

−

, Θ, u)] ∈ M

g,m

(J, A, γ

, γ

−

) is represented

by a simple curve, then Aut(u) is trivial. If it is represented by a k-fold cover of a

simple curve, then |Aut(u)| ≤ k. In particular, Aut(u) is a l ways ﬁnite unless u is

co nstant.

Proof. If u is simple, then it is a diﬀeomorphism onto its image in a small

neighbourhood of some point, and any map ϕ satisfying u = u ◦ ϕ would be the

identity on such a neighbourhood. By unique continuation, we conclude that Aut(u)

is trivial. In general if u = v ◦ ϕ for some simple

v : Σ

′

→ W

and

ϕ : Σ → Σ

′

a k-fold branched cover, we have

Aut(u) = {f : Σ → Σ | v ◦ ϕ ◦ f = v ◦ ϕ}.

By a similar argument as in the previous case, knowing that v is simple implies we

only need to look at solutions to

ϕ ◦f = ϕ.

Remove the set of branch points B from Σ

′

together with the set ϕ

−1

(B) f rom

Σ, so that ϕ becomes an ho nest covering map. Any ϕ ∈ Aut(u) then deﬁnes a

biholomorphic deck transformation of the cover, so it remains to argue that t here

are at most k of them. In fact, there is at most one transformation that takes w

for any two given points w

, w

∈ ϕ

−1

(x). If there were two such transformations

120 Chris Wendl

f and g, then f ◦ g

−1

would be the identity on an open neighbour hood and would

thus be globally the identity by unique continuation. 

6.5. A local structure result

The following statement, which we will prove in the next lecture, is the main

goal of most of the analysis we have discussed recently. It is essentially an appli-

cation of the implicit function t heorem for a smooth nonlinear Fredholm section

of a Banach space bundle. The implicit function theorem (see [

Lan93]) implies in

particular that if F is a smoo t h map between Banach spaces such tha t F (x

) = 0

and dF (x

) is a surjective Fredholm operat or, then F

−1

(0) is a smooth manifold

near x

with its dimension equal to the Fredholm index of dF (x

). Surjectivity is

an extra hypothesis, referred to in t he statement below as “Fredholm regularity,”

a notion that we will deﬁne precisely in t he next lecture. The dimension for mula

should look familiar, but is only an indirect consequence of the index formula for

Cauchy-Riemann type operators tha t we proved in Lecture 5; one also needs to ac-

count for the fact that in deﬁning our moduli space M

g,m

(J, A, γ

, γ

−

), we did not

ﬁx the complex structures on our domain curves, hence they ar e free to move about

in the moduli space of Riemann surfaces, whose dimension therefore plays a role in

determining the dimension of M

g,m

(J, A, γ

, γ

−

Theorem 6.23. The set of Fredholm regular curves forms an open subset

reg

g,m

(J, A, γ

, γ

−

) ⊂ M

g,m

(J, A, γ

, γ

−

)

which naturally admits the structure of a smoo th ﬁnite-dimensional orbifold of di-

mension

dim M

reg

g,m

(J, A, γ

, γ

−

) = ( n − 3)(2 − 2g − k

− k

−

) + 2c

(A)

i=1

(γ

) −

−

i=1

(γ

−

) + 2m,

where dim W = 2n, τ is a choice of unitary trivia l i zation for (ξ

, J, ω

) along each

of the asymptotic orbits γ

, and c

(A) denotes the normal ﬁrst Chern number of the

co mplex vector bundle (u

∗

W , J) →

Σ with respect to the asymptotic trivialization

determined by τ and the sp l i tting T (R ×M

) = ǫ ⊕ξ

(cf. (

6.4)). The local isotropy

gro up of M

reg

g,m

(J, A, γ

, γ

−

) at u is Aut(u), hence the moduli space is a manifold

near any regular element with trivial automorphism group.

Exercise 6.24. Verify that the number in the above index formula is inde-

pendent of the choice of trivializations τ, and that c

∗

W ) depends only on the

relative homology class A.

LECTURE 7

Smoothness of the moduli space

Contents

7.1. Transversality theorems in cobordisms 121

7.2. Functional analytic setup 126

7.3. Teichm¨uller slices 131

7.4. Fredholm regularity and the implicit function theorem 132

7.5. A universal moduli space 134

7.6. Applying the Sard-Smale theorem 137

7.7. From C

to C

∞

138

In this lecture, we continue the study of the moduli space

M(J) := M

g,m

(J, A, γ

, γ

−

We assume as before that (W, ω) is a 2n-dimensional symplectic cobordism with

stable boundary ∂W = −M

−

⊔ M

inheriting stable Hamiltonian structures H

(ω

, λ

) with induced Reeb vector ﬁelds R

and hyperplane distributions ξ

ker λ

, g, m, k

, k

−

≥ 0 are integers, γ

= (γ

, . . . , γ

) are ordered sets of peri-

odic R

-orbits in M

, and A ∈ H

(W,

∪

−

) is a relative homology class with

∂A =

[γ

] −

[γ

−

] ∈ H

(W,

∪

−

). The noncompact completion of (W, ω) is

denoted by (

W , ω

) for some ﬁxed function ψ : R → (−ǫ, ǫ) that scales the symplec-

tic form on the cylindrical ends, and r

≥ 0 is a ﬁxed constant which determines the

size of the ends [r

, ∞) ×M

and (−∞, −r

] ×M

−

on which we require our almost

complex structures J ∈ J(ω

, r

, H

−

) to be R-invariant. The complement of

these ends has closure

:= ([−r

, 0] × M

−

) ∪

−

W ∪

([0, r

] × M

) .

We will often ma ke use of the fact t hat since J matches translation-invariant a lmo st

complex structures in J(H

) outside of W

, there are natural complex vector

bundle splittings

T (R × M

) = ǫ ⊕ ξ

where ǫ denotes the canonically trivial line bundle spanned by ∂

and the Reeb

vector ﬁeld.

7.1. Transversality theorems in cobordisms

We concluded the previous lecture with the stat ement of the following theorem.

121

122 Chris Wendl

Theorem 7.1. If the orbi ts γ

are all nondegenerate and J ∈ J(ω

, r

, H

−

then the moduli space M(J) contains an open subset

reg

(J) ⊂ M(J)

co nsisting of so-called Fredholm regular curves, which naturally admits the structure

of a smooth ﬁnite-dimensi o nal orbifold of dimension

dim M

reg

(J) = (n − 3)(2 − 2g − k

−k

−

) + 2c

(A)

i=1

(γ

) −

−

i=1

(γ

−

) + 2m,

where dim W = 2n, τ is a choice of unitary trivia l i zation for (ξ

, J, ω

) along each

of the asymptotic orbits γ

, and c

(A) denotes the normal ﬁrst Chern number of the

co mplex vector bundle (u

∗

W , J) →

Σ with respect to the asymptotic trivialization

determined by τ and the splitting T (R × M

) = ǫ ⊕ ξ

. The local isotropy group

of M

reg

(J) at u is Aut(u), hence the moduli space is a manifold near any regular

element with trivia l automorphism g roup.

The integer in the above dimension formula is o ften called the virtual dimen-

sion of M(J) and denoted by

vir-dim M(J) := (n − 3)(2 −2g − k

− k

−

) + 2c

(A)

i=1

(γ

) −

−

i=1

(γ

−

) + 2m.

Ignoring the marked po ints, the virtual dimension of a space M

g,0

(J, A, γ

, γ

−

)

containing a curve u : (

Σ, j) → (

W , J) with punctures z ∈ Γ

and nondegenerate

asymptotic orbits {γ

}

z∈Γ

is sometimes also called the index of u,

ind(u) := (n − 3)χ(

Σ) + 2c

∗

W ) +

z∈Γ

(γ

) −

z∈Γ

−

(γ

) ∈ Z,

and we will see that it is in fact the Fredholm index of an operator closely related

to the linearized Cauchy- Riemann operator D

at u. The word “virtual” refers to

the f act that in general, the regularity condition may fail and thus M(J) might not

be smooth, or if it is, it might actually be of a diﬀerent dimension (see Example

7.5

below), but in an ideal world where transversality is always satisﬁed, its dimension

would be vir-dim M(J). This notion makes sense in ﬁnite-dimensional contexts as

well: if f : R

→ R

is a smooth map, then we would say that f

−1

(0) has virtual

dimension n −m, even though f

−1

(0) might in general be all sorts of strange things

other than a smoo t h (n − m)-dimensional manifo ld. In part icular, n − m could

be negative, in which case f

−1

(0) would be empty if transversality were satisﬁed,

but in general this need not be the case. It is true however that f can always be

pe rturbed to a map whose zero set is an (n − m)-dimensional manifold (or empty

if n − m < 0). The same is true in principle of the nonlinear Cauchy-Riemann

equation, but in general it is a formidably diﬃcult problem to ﬁnd perturbations

that respect all symmetries inherent in the setup as well as the extra structure

Lectures on Symplectic Field Theory 123

provided by the compatiﬁcation of M(J), which is usually crucial for meangingful

applications. Such issues require more sophisticated methods than we will discuss

here, but a good place to read a bout them is [

FFGW].

The ﬁrst goal of this lecture is to deﬁne the notion “Fredholm regular” and prove

Theorem

7.1. In practice, however, Fredholm regula rity is a t echnical condition that

can rarely be directly checked. To remedy this, we will also prove a genericity result

for somewhere injective J-holomorphic curves. A smooth map u :

Σ →

W is said to

have an injective point z ∈

Σ if

du(z) : T

Σ → T

u(z)

W is injective and u

−1

(u(z)) = {z}.

If u is a proper map, then it is easy to see that the set of injective points is open in

Σ,

though in general it could also be empty; this is the case e.g. for multiply covered J-

holomorphic curves. We say u is somewhere injective if its set of injective points is

nonempty; for asymptotically cylindrical J-holomorphic curves with no ndegenerate

asymptotic orbits, Theorem

6.19 implies that somewhere injectivity is equivalent to

being simple, i.e. not multiply covered.

Recall that if X is a topological space, a subset Y ⊂ X is called comeager if it

contains a countable intersection of open and dense sets.

If X is complete, then the

Baire category theorem implies that comeager subsets are always dense; moreover,

any countable intersection of comeager subsets is also comeager and therefore dense.

Comeager subsets often play the role in inﬁnite dimensions that the term “almost

everywhere” plays in ﬁnite dimensions. Informally, we often say that a given state-

ment dependent on a choice of auxiliary data (living in a complete metric space)

is true generically, or “for generic choices,” if it is true whenever the data are

chosen from some comeager subset o f the space of all possible data.

Theorem 7 .2. Fix the same d ata as in Theorem

7.1, an almost complex struc-

ture J

ﬁx

∈ J(ω

, r

, H

−

) and an open subset

U ⊂ W

Then there exists a comeager subset

reg

⊂

J ∈ J(ω

, r

, H

−

)



J = J

ﬁx

W \ U

such that for every J ∈ J

reg

, every curve u ∈ M(J) that has an inj ective point

mapped into U is Fredholm regular. In particular, the c urves with this property

deﬁne an open subset of M(J) that is a smooth manifold with dimen s i on equal to

its virtual di mension.

Remark 7.3. Since U ⊂

W has compact closure, the set

J ∈ J(ω

, r

, H

−

)



J = J

ﬁx

W \ U

Elsewhere in the symplectic literature, comeager subsets are sometimes r eferred to as “sets

of second category,” which is unfortunately slightly at odds with the standa rd mea ning of “second

category,” though it is accurate to say that the complement of a comea ger subset (a lso known as

a “meage r” subset) is a set of ﬁrs t category. The term Baire subset is also sometimes used as a

synonym for “comeager subset”.

124 Chris Wendl

has a natural C

∞

-topology that makes it a Fr´echet manifold and thus a complete

metric space, hence comeager subsets of it are dense.

Remark 7.4. Both of the above theorems admit easy extensions to the study

of moduli spaces dependent on ﬁnitely many parameters. Concretely, suppose P

is a smooth ﬁnite-dimensional manifold and {J

}

s∈P

is a smooth family of almost

complex structures satisfying the usual conditions. One can then deﬁne a parametric

moduli space

M({J

}

s∈P

) =



(s, u)



s ∈ P, u ∈ M(J

)



and a notion of parametric regularity for pairs (s, u) ∈ M({J

}), which is again an

open condition, such that the space M

reg

({J

}) of parametrically regular elements

will be an orbifold of dimension

dim M

reg

({J

}) = vir-dim M(J) + dim P.

Similarly, one can show that if the family {J

}

s∈P

is allowed to vary on a n open

subset U ⊂ W

for s lying in some precompact open subset V ⊂ P , then all

elements (s, u) for which s ∈ V and u has an injective point mapping to U will

be parametrically regular. See [

Wend, §4.5] for details in the closed case, which

is not fundamentally diﬀerent from the punctured case. The standard and most

important example is P = [0, 1] with V = (0, 1), so we consider generic h omotopies

of almost complex structures. Here it is important to observe that while regularity

in the sense of Theorem

7.1 always implies parametric regularity, the converse is

false: there can exist parametrically regular pairs (s, u) ∈ M({J

}) for which u is

not a Fredholm regular element of M(J

), hence M({J

}) may be smooth even if

M(J

) is not smooth for some s ∈ P . This can happen in particular whenever s is

a critical value of the projection map

M({J

}) → P : (s, u) 7→ s,

see Figure 7 .1. In general these cannot be excluded by making generic choices o f the

homotopy, though it is possible in certain cases using “automatic” transversality re-

sults, which guarantee regularity for all J

with no need for genericity (cf. [

Wen10]).

Example 7.5. It is not hard to imagine situations in which transversality must

fail generically for multiply covered curves. Suppose for instance that (W, ω) is an

8-dimensional symplectic manifold with compatible almost complex structure J

and u

: S

→ W is a simple J

-holomorphic sphere with no punctures and [u

] =

A ∈ H

(W ), where c

(A) = −1. This means u

represents an element of a moduli

space M

0,0

, A) with

vir-dim M

0,0

, A) = 2 − 2g + 2c

(A) = 0.

In par t icular if u

is regular and {J

∈ J(ω)}

s∈R

k is a smooth k-parameter family

of compatible almost complex structures including J

, then Remark

7.4 implies that

a neighborhood o f ( 0, u

) in the parametric moduli space M({J

}) = {(s, u) | s ∈

P, u ∈ M

0,0

, A)} is a smooth k-dimensional manifold, and this will be true no

matter how the family {J

} is chosen. But for each of the element s (s, u) ∈ M({J

})

Lectures on Symplectic Field Theory 125

M(J

)

M(J

)

Figure 7.1. The picture shows a smooth parametric moduli space

M({J

}

s∈[0,1]

) a nd its projection M({J

}) → [0, 1] : (s, u) 7→ s in a

case where vir-dim M(J

) = 0. The parametric moduli space is 1-

dimensional and the spaces M(J

) are regular and 0-dimensional for

almost every s ∈ [0, 1], but this need not hold when s is a critical

value of the projection; in the picture, one such space M(J

) conta ins

a 1-dimensional component consisting of non-regular curves, so its

dimension diﬀers fro m its virtual dimension.

parametrized by a J-holomorphic map u : (S

= C ∪ {∞}, i) → (W, J

), there is

also a double cover

′

: S

→ W : z 7→ u(z

with [u

′

] = 2A, so u

′

∈ M

0,0

, 2A) and

vir-dim M

0,0

, 2A) = 2 − 2g + 2c

(2A) = −2.

Negative virtual dimension means that M

0,0

, 2A) should be empty whenever

Fredholm regularity is achieved, but this is clearly impossible, even generically, since

elements of M

0,0

, A) always have double covers belonging to M

0,0

, 2A).

Remark 7.6. The most common way t o apply Theorem

7.2 is by setting U

equal to the interior of W

, so generic perturbations of J are allowed everywhere

except on the regions where it is required to be R-invariant. The theorem then

achieves transversality f or all simple curves that are not conﬁned to the R-invariant

regions. We will show in the next lecture that transversality for all curves of the

latter type can also be achieved by generic perturbations within the spaces J(H

) of

compatible R-invariant a lmo st complex structures on t he symplectizations R ×M

hence generic choices in J(ω

, r

, H

−

) do achieve transversality for all simple

curves.

Our proof s of Theorems 7.1 and 7.2 will mostly follow the same line of argument

that is carried out for the closed case in [

Wend, Chapter 4], thus we will not discuss

every detail but will instead emphasize aspects which are unique t o the punctured

case.

126 Chris Wendl

7.2. Functional analytic setup

Fix k ∈ N and p ∈ (1, ∞) with kp > 2, a small number δ ≥ 0, a nd a Riemannian

metric on

W that is translation-invariant in the cylindrical ends. Fix also a closed

connected surface Σ of genus g, and disjoint ﬁnite ordered sets of distinct points

= (z

, . . . , z

), Θ = (ζ

, . . . , ζ

)

in Σ, together with disjoint neighborhoods

⊂ Σ

of each z

∈ Γ

with complex structures j

and biholomorphic identiﬁcations of

, j

, z

) with (D, i, 0) for each j = 1, . . . , k

. This determines holomorphic cylin-

drical coordinates identifying each of the punctured neighborhoods

⊂

Σ := Σ \ (Γ

∪ Γ

−

)

biholomorphically with the half-cylinder Z

For reasons tha t will become clear when we study the linearized Cauchy-Riemann

operator in the punctured setting, we will need to consider exponentially weight ed

Sobolev spaces. Suppose E →

Σ is an asymptotically Hermitian vector bundle:

then the Banach space

k,p,δ

(E) ⊂ W

k,p

lo c

(E)

is deﬁned to consist of sections η ∈ W

k,p

lo c

(E) whose represent atives f : Z

→ C

cylindrical coordinates (s, t) ∈ Z

and asymptotic trivializations at the ends satisfy

(7.1) ke

±δs

k,p

)

< ∞.

The norm of a section η ∈ W

k,p,δ

(E) is deﬁned by adding the W

k,p

-norm o f η over a

large compact subdomain in

Σ to the weighted norms (

7.1) f or each cylindrical end.

If δ = 0, this just produces the usual W

k,p

(E), but for δ > 0, sections in W

k,p,δ

(E)

are guaranteed to have exponential decay at inﬁnity.

Remark 7.7. It is occasionally useful to observe that the deﬁnition of W

k,p,δ

(E)

also makes sense when δ < 0. In this case, sections in W

k,p,δ

(E) are of class W

k,p

lo c

but need not be glo bally in W

k,p

(E), as they are also allowed to have exp onential

gro wth at inﬁnity.

We now want to deﬁne a Banach manifold of maps u :

Σ →

W t ha t will contain

all the asymptotically cylindrical J-holomorphic curves with our particular choice

of asymptotic orbits. Recall that the asymptotically cylindrical condition means

(7.2) u(s, t) = exp

s,γ

(t))

h(s, t) for suﬃciently large |s|

in suitable cylindrical coordinates (s, t) ∈ Z

near each puncture z

∈ Γ

, where

> 0 is the period of the orbit γ

: S

→ M

and h(s, t) is a vector ﬁeld along the

trivial cylinder that decays as s → ±∞. The catch is that this deﬁnition was not

formula t ed with respect to a ﬁxed choice of the holomorphic cylindrical coordinates

(s, t); in general the coordinat es in which (

7.2) is valid may depend on u, a nd diﬀerent

choices of coordinates might be required fo r diﬀerent maps. One can show however

Lectures on Symplectic Field Theory 127

that any two distinct choices of ho lomorphic cylindrical coordinates are related to

each other by a transformation tha t converges asymptotically to a constant shift,

which implies that for our ﬁxed choice of coordinates (s, t), every asymptotically

cylindrical map can be assumed to satisfy

u(s, t) = exp

s+a,γ

(t+b))

h(s, t), lim

s→±∞

h(s, t) = 0

for some constants a ∈ R and b ∈ S

. We therefore deﬁne the space

k,p,δ

:= W

k,p,δ

(

Σ,

W ; γ

, γ

−

) ⊂ C

(

Σ,

W )

to consist of all continuous maps u :

Σ →

W of the fo r m

u = exp

where:

• f :

Σ →

W is smooth and, in our ﬁxed cylindrical coordinates (s, t) ∈ Z

on neighborhoods of the punctures z

∈ Γ

, takes the form

f(s, t) = (T

s + a, γ

(t + b)) fo r |s| suﬃciently large,

where a ∈ R and b ∈ S

are arbitrar y constants and T

> 0 is the period

of the Reeb orbit γ

: S

→ M

;

• h ∈ W

k,p,δ

∗

W ).

Though it is not immediate since

Σ is noncompact, one can generalize the ideas

in [

El˘ı67] to give B

k,p,δ

the structure of a smooth, separable and metrizable Banach

manifold. The key point is the condition kp > 2, which guarantees the continuous

inclusion W

k,p,δ

∗

W ) ֒→ C

∗

W ) as well as Banach algebra and C

-continuity

properties, cf. Propositions 2.4, 2.7 and 2.8 in Lecture

2. These pro perties are

needed in order to show that the transition maps between pairs of charts of the

form exp

h 7→ h ar e smooth.

The tangent space to B

k,p,δ

at u ∈ B

k,p,δ

can be written as

k,p,δ

= W

k,p,δ

∗

W ) ⊕ V

where V

⊂ Γ(u

∗

W ) is a non-canonical choice of a 2(k

+ k

−

)-dimensional vector

space of smooth sections asymptotic at the punctures to constant linear combina-

tions of the vector ﬁelds spanning the canonical trivialization of the ﬁrst factor in

T (R × M

) = ǫ ⊕ ξ

, i.e. they point in the R- and R

-directions. The space V

appears due to the fact that two distinct elements of B

k,p,δ

are generally asymptotic

to collections of trivial cylinders that diﬀer from each other by k

+ k

−

pairs of

constant shifts (a, b) ∈ R × S

Fix J ∈ J(ω

, r

, H

−

) and a smooth complex structure j on Σ that matches

in the neighborhoods U

of the punctures. The nonlinear Cauchy-Riemann op-

erator is then deﬁned as a smooth section

∂

j,J

: B

k,p,δ

→ E

k−1,p,δ

: u 7→ T u + J ◦ T u ◦ j

of a Banach space bundle

k−1,p,δ

→ B

k,p,δ

128 Chris Wendl

with ﬁbers

k−1,p,δ

= W

k−1,p,δ

(

Hom

Σ, u

∗

W )).

The zero set of

∂

j,J

is the set of all maps u ∈ B

k,p,δ

that are pseudoholomorphic

from (

Σ, j) to (

W , J). Note that the smoothness of

∂

j,J

depends mainly on the fact

that J is smoo t h. Indeed, in local coordinates

∂

j,J

looks like u 7→ ∂

u + ( J ◦ u)∂

in which the most o bviously nonlinear ingredient is u 7→ J ◦ u. If J were only of

class C

, then the C

-continuity property would imply that t he map u 7→ J ◦ u

sends maps of class W

k,p

continuously to maps of class W

k,p

, and one can use an

inductive argument to show that this map then becomes r-times diﬀerentiable if J

is of class C

k+r

, see [

Wend, Lemma 2.12.5]. Moreover, the fact that

∂

j,J

u satisﬁes

the same expo nential weighting condition as u at t he cylindrical ends depends on

the fact t hat J is R-invariant near inﬁnity.

For u ∈

∂

−1

j,J

(0), t he linearizat ion D

∂

j,J

(u) : T

k,p,δ

→ E

k−1,p,δ

deﬁnes a bounded

linear operator

: W

k,p,δ

∗

W ) ⊕ V

→ W

k−1,p,δ

(

Hom

Σ, u

∗

W )).

We derived a formula for this operator in Lecture

2 and showed that it is of Cauchy-

Riemann type. Since V

is ﬁnite dimensional, D

will be Fredholm if and o nly if its

restriction to the ﬁrst factor is Fredholm; denote this restriction by

: W

k,p,δ

∗

W ) → W

k−1,p,δ

(

Hom

Σ, u

∗

W )),

where we’ve chosen the notation to emphasize the dependence o f t his operato r on

the choice of exponential weight δ ≥ 0 in the deﬁnition of o ur Banach space. We

will see presently why it’s imp ortant to pay at tention to this detail.

To see whether D

is Fredholm, consider ﬁrst the special case where u is a trivial

cylinder

: R × S

→ R × M : (s, t) 7→ (T s, γ(t))

over some Reeb orbit γ : S

→ M with period T > 0 in M with stable Hamiltonian

structure H = (ω, λ) on M. In this case, there is a more convenient way to write

down D

than the f ormula from Lecture

2. To start with, we use the splitting

T (R × M) = ǫ ⊕ ξ to decompo se u

∗

T (R × M) = u

∗

ǫ ⊕ u

∗

ξ and thus write D

block form



ǫξ

ξǫ



Exercise 7.8. Suppose D : Γ(E) → Ω

0,1

(

Σ, E) is a linear Cauchy-Riemann

type operator on a vector bundle E with a complex-linear splitting E = E

⊕ E

and

D =





is the resulting block decomposition of D. Use the Leibniz rule satisﬁed by D

to show tha t D

and D

are also Cauchy- Riemann type operators on E

and

respectively, while the oﬀ-diagonal terms are tensorial, i.e. they commute with

multiplication by smooth real-valued functions and thus deﬁne bundle maps D

→ Λ

0,1

∗

Σ ⊗E

and D

: E

→ Λ

0,1

∗

Σ ⊗ E

Lectures on Symplectic Field Theory 129

Now observe that if u = (u

, u

) : R ×S

→ R ×M is ano t her cylinder near u

the nonlinear opera t or (

∂

j,J

u)∂

= ∂

u + J ∂

u ∈ Γ ( u

∗

T (R × M)) = Γ(u

∗

ǫ ⊕ u

∗

ξ)

takes the form

(

∂

j,J

u)∂



∂

− λ(∂

) + i (∂

+ λ(∂

))

∂

+ Jπ

∂



where we a re using the canonical trivialization of u

∗

ǫ via ∂

and R to express the top

block as a complex-valued function. As we observed in Lecture

3, the bottom block of

this expression can be interpreted in terms of the gradient ﬂow of an actio n functional

: C

∞

) → R, with ∇A

(γ) = −Jπ

∂

γ. Linearizing in the direction of a

section η

∈ Γ(u

∗

ξ) and taking the ξ component thus yields an expression involving

the Hessian of A

at the critical point γ, namely

)∂

= (∂

− A

)η

To compute the blocks D

and D

ξǫ

, notice that D

= 0 whenever η

is a constant

linear combination of ∂

and R, as η

is then the derivative of a smooth family of

J-holomorphic reparametrizations of u

. This is enough to prove D

ξǫ

= 0 since

the la t t er is tensorial by Exercise

7.8, and expressing arbitrary sections of u

∗

ǫ as

f∂

+ gR, we can apply the Leibniz rule for D

and conclude

)∂

= (∂

+ i ∂

)η

in the canonical trivialization. To compute the remaining oﬀ-diagonal term, one

needs to compute dr(D

) and λ(D

) for an arbitrary section η

∈ Γ(u

∗

ξ),

e.g. by picking a smooth family u

: R × S

→ R × M with ∂

ρ=0

= η

and a

connection ∇ and computing



∇

(

∂

j,J

)



ρ=0



and λ



∇

(

∂

j,J

)



ρ=0



This calculation is straightforward but unenlightening, so I will leave it as an exercise

for now—in the next lecture we’ll derive a general formula (see Lemma

8.10), which

implies that since π

∂

≡ π

∂

≡ 0 in the present setting, D

ǫξ

= 0. All t his

leads to the formula

η)∂



∂

−



−i∂

0 A



η.

Here the upper left block is the “trivial” asymptotic operator acting on the tr ivial

line bundle over S

. Since every asymptotically cylindrical curve approximates a

trivial cylinder near inﬁnity, one can deduce from this calculuation the following:

Proposition 7.9. The Cauchy-Riemann type operator D

on u

∗

W is as-

ymptotic at its punctures z

∈ Γ

for j = 1, . . . , k

to the asymptotic operators

(−i∂

) ⊕ A

on (γ

)

∗

(ǫ ⊕ ξ

Perhaps you can now see a problem: even if the orbit s γ

are all nondegenerate,

the asymptotic operators (−i∂

) ⊕A

are degenerate, as they have nontrivial kernel

consisting of constant sections in the ﬁrst (trivial) factor o f (γ

)

∗

(ǫ ⊕ ξ

). This

implies in particular that

: W

k,p

∗

W ) → W

k−1,p

(

Hom

Σ, u

∗

W ))

130 Chris Wendl

is not Fredholm, except of course in the sp ecial case where there are no punctures.

The situation is saved by the exponential weight:

Lemma 7.10. For every δ > 0 suﬃciently small, the operator D

is Fredholm

and has index

ind(D

) = nχ(Σ) − (n + 1)#Γ + 2c

∗

W ) +

j=1

(γ

) −

−

j=1

(γ

−

Moreover, every element of M(J) can be represen ted by a map u ∈ B

k,p,δ

Proof. The second claim fo llows from the exponential decay estimate of Hofer-

Wysocki-Zehnder [

HWZ96] mentioned in t he previous lecture, see Proposition 6.13.

To see that D

: W

k,p,δ

→ W

k−1,p,δ

is Fredholm and to compute its index, we

can identify it with a Cauchy-Riemann type operator fr om W

k,p

to W

k−1,p

. Indeed,

pick any smooth function f :

Σ → R with f (s, t) = ∓δs on the cylindrical ends

near Γ

, deﬁne Banach space isomorphisms

: W

k,p

→ W

k,p,δ

: η 7→ e

η,

: W

k−1,p

→ W

k−1,p,δ

: θ 7→ e

θ,

and consider the bounded linear map

′

:= Ψ

−1

: W

k,p

∗

W ) → W

k−1,p

(

Hom

Σ, u

∗

W )).

Using the Leibniz rule for D

, it is straightforward to show that D

′

is also a linear

Cauchy-Riemann type operator. Moreover, suppose D

takes the form

∂ + S(s, t) in

coordinates a nd trivialization on the cylindrical end near z

, where S(s, t) → S

∞

(t)

as s → ±∞ a nd A

= −i∂

−S

∞

(t). Then D

′

on this same end ta kes the form

′

η = e

±δs

(

∂ + S(s, t))(e

∓δs

η) =

∂η + (S(s, t) ∓ δ)η

and is therefore asymptotic t o the perturbed a symptotic operator



(−i∂

) ⊕ A



±δ.

The latter is the direct sum of two asymptotic operators −i∂

± δ on the trivial

line bundle and A

± δ o n (γ

)

∗

respectively. Since γ

is nondegenerate by

assumption and the spectrum of A

is discrete, we can assume ker(A

±δ) remains

trivial if δ > 0 is suﬃcient ly small, and the Conley-Zehnder index of this perturbed

operator will be the same as without the perturbation. O n the other ha nd, the

spectrum of −i∂

consists of the integer multiples of 2π, thus −i∂

±δ also becomes

nondegenerate for any δ > 0 small. Its Conley-Zehnder index can be deduced from

the winding numbers o f its eigenfunctions using Theorem

3.35 in Lecture 3: −i∂

has a 2-dimensional nullspace consisting of sections with winding number 0, and

this becomes an eigenspace for the smallest positive eigenvalue if the puncture is

positive or the largest negative eigenvalue if the puncture is negative. Theorem 3.35

thus gives

(−i∂

±δ) = ∓1,

Lectures on Symplectic Field Theory 131

and therefore,

(

) = ∓1 + µ

(γ

Plugging this into the general index formula from Lecture

5 t hen gives the stated

result. 

Putting back the missing 2(#Γ) dimensions in the domain o f D

, we have:

Corollary 7.11. For all δ > 0 suﬃciently small, the linearized Cauchy-Riemann

operator D

: T

k,p,δ

→ E

k−1,p,δ

is Fredholm with in dex

ind(D

) = nχ(Σ) − (n − 1)#Γ + 2c

∗

W ) +

j=1

(γ

) −

−

j=1

(γ

−

7.3. Teichm¨uller slices

Since the moduli space M(J) is not deﬁned with reference to any ﬁxed complex

structure on the domains

Σ, we must build t his freedom into the setup. Fo r a more

detailed version of the following discussion, see [

Wend, §4.2 .1].

For any integers g, ℓ ≥ 0, the moduli space of Riemann surfaces of genus g

with ℓ marked point s is a space of equivalence classes

g,ℓ

= {(Σ, j, Θ)}



∼

where (Σ, j) is a compact connected surface with genus g, Θ ⊂ Σ is an ordered

set of ℓ points and equivalence is deﬁned via biholomorphic maps that preserve

the marked points with their ordering. This space has been studied extensively in

algebraic geometry, though it can also be understood using the same global analytic

methods that we have been applying for M(J). It is known in particular that M

g,ℓ

is always a smooth orbifold, and for any [(Σ, j, Θ) ] ∈ M

g,ℓ

, it satisﬁes

(7.3) dim Aut(Σ, j, Θ) − dim M

g,ℓ

= 3χ(Σ) − 2ℓ,

where Aut(Σ, j, Θ) is the group of biholomorphic t ransformations of (Σ, j) that

ﬁx the points in Θ. This group is ﬁnite whenever (Σ, j, Θ ) is stable, meaning

χ(Σ \ Θ) < 0, and in that case (

7.3) turns into the well-known dimension formula

dim M

g,ℓ

= −3χ(Σ) + 2ℓ = 6g − 6 + 2ℓ.

This is also the dimension of the Teichm¨uller space

T (Σ, Θ) := J(Σ)/ Diﬀ

(Σ, Θ),

where J(Σ) denotes the space of all smo oth complex structures on Σ compatible

with its orientat ion, and Diﬀ

(Σ, Θ) is the identity component of the group of dif-

feomorphisms that ﬁx Θ. It is a classical result that T (Σ, Θ ) is a smooth manifold of

the same dimension as M

g,ℓ

, and indeed, the latter can be presented as the quotient

of the former by the discrete action of the mapping class group of (Σ, Θ).

Equation (

7.3) is actually a formula for a Fredholm index. To see how this works,

consider ﬁrst the case ℓ = 0. The right hand side is then χ(Σ) + 2c

(T Σ), which

is, according to Riemann-Roch, the index of the natur al Cauchy-Riemann operator

on T Σ tha t deﬁnes it s holomorphic structure. This o perator can also be interpreted

132 Chris Wendl

as the linearizatio n at the identity map of the nonlinear Cauchy-Riemann opera-

tor for holomorphic maps (Σ, j) → (Σ, j), so its kernel is naturally isomorphic to

Aut(Σ, j). Similarly, one can show that the cokernel of this operato r is naturally

isomorphic to T

[j]

T (Σ). This discussion r emains valid if marked points are included:

the main diﬀerence is then that the Cauchy-Riemann operator on T Σ should be

restricted to a space of vector ﬁelds that vanish at Θ, deﬁning a 2ℓ-codimensional

subspace as the domain and thus reducing the index by 2ℓ.

For a proof of t he following, see [

Wend, Chapter 4] and [Wen10, §3.1].

Proposition 7.12. Given a closed Ri e mann surface (Σ, j) with a ﬁnite ordered

set Θ ⊂ Σ, there exists a smooth ﬁnite-di mensional submanifold T ⊂ J(Σ) with the

following properties:

(1) The map T → T (Σ, Θ) : j

′

7→ [j

′

] i s b i j ective onto a neighborhood of [j] in

T (Σ, Θ);

(2) The subspace T

T ⊂ Γ(

End

(T Σ)) is complementary in W

k−1,p

(End

(T Σ))

to the i mage o f the standard Cauchy-Riemann operator of T Σ acting on the

domain {X ∈ W

k,p

(T Σ) | X|

= 0};

(3) Every j

′

∈ T equals j near Θ and is invariant under the action of Aut( Σ, j, Θ)

by diﬀeomorphisms on Σ.



We will r efer to the f amily T ⊂ J(Σ) in this proposition as a Teichm¨uller slice

through j.

7.4. Fredholm regularity and the implicit function theorem

We are now in a position to deﬁne the necessary regularity condition and prove

that a neighborhood of any given regular element [(Σ, j

, Γ

−

, Θ, u

)] in M(J) is

an orbifold of the stated dimension. Af t er reparametrizing, we can assume without

loss of generality that Σ, Γ

and Θ are precisely the dat a that were ﬁxed in §7.2,

and j

∈ J(Σ) matches j

on our ﬁxed coordinate neighbor hoods of Γ

. We can

then choose a Teichm¨uller slice

T ⊂ J(Σ)

through j

as provided by Prop.

7.12, but with j in that statement replaced by j

and Θ replaced by Γ

∪ Γ

−

∪ Θ. In particular, T is invariant under the action of

the group

:= Aut(Σ, j

, Γ

∪ Γ

−

∪ Θ),

and (

7.3) now becomes

(7.4) dim G

−dim T = 3χ( Σ) − 2(k

+ k

−

+ m).

There is a na t ur al extension of the nonlinear operator

∂

j,J

in §

7.2 to a smooth

section

∂

: T × B

k,p,δ

→ E

k−1,p,δ

: (j, u) 7→ T u + J ◦T u ◦ j

of a Banach space bundle E

k−1,p,δ

→ T × B

k,p,δ

with ﬁbers

k−1,p,δ

(j,u)

= W

k−1,p,δ



Hom

((T

Σ, j), (u

∗

W , J))



Lectures on Symplectic Field Theory 133

The zero set

∂

−1

(0) ⊂ T ×B

k,p,δ

consists of pairs (j, u) for which u : (

Σ, j) → (

W , J)

is pseudoholomorphic, and it contains (j

, u

) by construction. It also admits a

natural action of the automor phism group G

∂

−1

(0) →

∂

−1

(0) : (ϕ, (j, u)) 7→ (ϕ

∗

j, u ◦ ϕ),

whose stabilizer at (j

, u

) is Aut(u

), a ﬁnite group whenever u

is not constant.

Observe that any two elements in the same G

-orbit of

∂

−1

(0) deﬁne equivalent

elements of the moduli space M(J), as they are related to each o ther by a biholo-

morphic repara metrization that ﬁxes the punctures and marked points.

Lemma 7.13. The map

∂

−1

(0)



→ M(J) : [(j, u)] 7→ [(Σ, j, Γ

, Γ

−

, Θ, u)]

is a homeomo rp hism between open neighborhoods of [(j

, u

)] and [(Σ, j

, Γ

−

, Θ, u

)].

Proof. This depends fundamentally on the same fact underlying the smooth-

ness of Teichm¨uller space: the action of Diﬀ

(Σ, Γ

∪ Γ

−

∪ Θ) on J(Σ) is free and

proper.

See the proof of [Wend, Theorem 4.3.6]. 

Definition 7.14. We say that [(Σ, j

, Γ

−

, Θ, u

)] is Fredholm regular if

there exists a choice of Teichm¨uller slice T through j

such that the linearization

∂

, u

) : T

T ⊕ T

k,p,δ

→ E

k−1,p,δ

)

is surjective.

One can show that the surjectivity condition in this deﬁnition does not actually

depend on the choice of Teichm¨uller slice. This follows from the identiﬁcation of

T with the cokernel of the natural Cauchy-Riemann operator on T

Σ; see [

Wend,

Lemma 4.3.2].

Proof of Theorem

7.1. The fact that M(J) is an orbifold in a neighbor-

hood o f [(Σ, j

, Γ

−

, Θ, u

)] with isotropy group Aut(u

) follows from Lemma

7.13

and the implicit function theorem, which gives

∂

−1

(0) the structure of a ﬁnite-

dimensional manifold near (j

, u

) if Fredholm regularity is satisﬁed. There is a bit

of work to be done in showing that transition maps relating any two overlapping

charts that arise in this way from the implicit function t heorem ar e smooth; for this,

we refer aga in t o the proof of Theorem 4.3.6 in [

Wend] and merely comment that

the key ingredient is elliptic regularity.

The dimension of M(J) is

dim M(J) = dim

∂

−1

(0) − dim G

= ind D

∂

, u

) − dim G

The restriction o f D

∂

, u

) to T

k,p,δ

is the operator D

that we studied in

7.2, hence

ind D

∂

, u

) = dim T + ind D

This is true at least in the stable case, i.e. when χ(

Σ \Θ) < 0. There are ﬁnitely many cases

not satisfying this hypo thesis, for which the lemma can be proved by more direct arguments since

explicit descriptions of both Teichm¨uller space and the automorphism groups of Riemann surfaces

are available; see [

Wen10, §3 .1 and §3.2] for more details.

134 Chris Wendl

Using (7.4) to replace dim T − dim G

and combining this with Corollary 7.11 now

gives the stated formula for dim M(J). 

7.5. A universal moduli space

The remainder of this lecture is devoted to the proo f of Theorem 7.2. The main

tool fo r this purpose is the Sard-Smale t heorem [

Sma65], an inﬁnite-dimensional

version of Sard’s theorem stating that the regular values of a smooth nonlinear

Fredholm map between separable Banach spaces (i.e. a smooth map whose deriv-

ative at every point is a Fr edholm operator) form a comeager subset of the tar get

space. In order to incorporate perturbations of the almost complex structure int o

our functional analytic setup, we need to choose a suitable Banach manifo ld of al-

most complex structures. All known ways of doing this are in some sense non-ideal,

e.g. one could take almost complex structures of class C

or W

k,p

, but this neces-

sarily introduces non-smooth almost complex structures into the picture, with the

consequence that the nonlinear Cauchy-Riemann operat or has only ﬁnitely many

derivatives. That is not the end of the world, and indeed, this is the approach t aken

in [

MS04], but I will instead present an approach that was introduced by Floer in

[

Flo88b], in terms of what is now called the “ Floer C

space”. The idea is t o work

with a Banach manifold that continuously embeds into the space of smooth almost

complex structures, so that the nonlinear Cauchy-Riemann operator will always be

smooth. It’s a nice trick, but the catch is that we obtain a space that is strictly

smaller than the actual space of smooth almost complex structures we’re interested

in, and has a much stronger topology. The C

space should be viewed as a useful

tool but not a deeply meaningful o bject—you might notice that while some of the

intermediate results stated below depend on its (somewhat ad hoc) deﬁnition, The-

orem

7.2 does not. This is due to a general trick described in §7.7 below for turning

results about C

into results about C

∞

As in the statement of Theorem

7.2, a ssume U ⊂ W

is open and J

ﬁx

∈

J(ω

, r

, H

−

). Let

J ∈ J(ω

, r

, H

−

)



J = J

ﬁx

W \ U

and choose any almost complex structure

ref

∈ J

We can regard J

as a smooth Fr´echet manifold with tangent spaces

ref J

Y ∈ Γ



End

W , J

ref

)





Y |

W \U

≡ 0 and ω

(·, Y ·) + ω

(Y ·, ·) ≡ 0

where the antilinearity of Y ∈ T

ref

means that Y is tangent to the space almost

complex structures, and the condition relating it to ω

means that these structures

are compatible with ω

. One can check t hat the map

Y 7→ J



1 +

ref



ref



1 +

ref



−1

Lectures on Symplectic Field Theory 135

maps a neighborhood of 0 ∈ T

ref J

bijectively to a neighborhood of J

ref

in J

We thus ﬁx a suﬃciently small constant c > 0 and deﬁne the space of “C

-small

perturbations of J

ref

” by

(

∈ J



Y ∈ T

ref

with

∞

ℓ=0

ℓ

kY k

ℓ

(

< c

)

where ε := (ε

ℓ

)

∞

ℓ=0

is a ﬁxed sequence of positive numbers with ε

ℓ

→ 0 as ℓ → ∞.

The sum

kY k

∞

ℓ=0

ℓ

kY k

ℓ

(

deﬁnes a norm, and the space of smooth sections Y ∈ T

ref J

for which this norm is

ﬁnite is then a separable Banach space; see Appendix

B for a proof of this statement.

This makes J

a separable and metrizable Banach manifold, as the map J

7→ Y

can be viewed as a chart identifying it with an open subset of the aforementioned

Banach space. Not every J ∈ J

near J

ref

belongs to J

, but there is a continuous

inclusion

֒→ J

where the lat ter carries its usual C

∞

-topology and J

carries the topology induced

by the C

-norm. By a lemma due to Floer, choosing a sequence ε

ℓ

that decays

suﬃciently fast makes J

large enough to contain perturbations in arbitrary direc-

tions with arbitrarily small support near arbitrary points in U; see Theorem

B.6 in

Appendix

B for a precise version of this statement and its proof. We will assume

from now on that a suitably fast decaying sequence has b een ﬁxed.

We now deﬁne a universal moduli space

∗

) :=



(u, J)



J ∈ J

, u ∈ M(J) and

u has an injective point mapped into U



The terminology is somewhat unfort una t e, as M

∗

) depends on many auxiliary

choices such as J

ref

and (ε

ℓ

)

∞

ℓ=0

and thus should not really be tho ught of as a “uni-

versal” object. Nonetheless:

Lemma 7.15. The universal moduli space M

∗

) is a smooth separable Banach

manifold, and the projection M

∗

) → J

: (u, J) 7→ J is sm ooth.

Proof. As in the proof of Theorem

7.1, one can identify M

∗

) locally with

the zero set of a smooth section of a Banach space bundle. Suppo se J

∈ J

and

[(Σ, j

, Γ

−

, Θ, u

)] ∈ M(J

) where u

Σ →

W has an inj ective point z

with

) ∈ U. Choose a Teichm¨uller slice T through j

as in Proposition

7.12 and

consider the smooth section

∂ : T × B

k,p,δ

× J

→ E

k−1,p,δ

: (j, u, J) 7→ T u + J ◦ T u ◦ j,

where E

k−1,p,δ

is the obvious extension of our previous Banach space bundle to a

bundle over T × B

k,p,δ

× J

. We’re assuming as before that k ∈ N, 1 < p < ∞,

136 Chris Wendl

kp > 2, and δ > 0 is small. A neighbo rhood of (u

, J

) in M

∗

) can then be

identiﬁed with a neighborhood of [(j

, u

, J

)] in

∂

−1

(0)



where G

:= Aut(Σ, j

, Γ

∪Γ

−

∪Θ) acts on

∂

−1

(0) by ϕ ·(j, u, J) := (ϕ

∗

j, u ◦ϕ, J).

Since u

has an injective point, Aut(u

) is trivial and the G

-action at (j

, u

, J

)

is therefore free; hence it suﬃces to show t hat

∂

−1

(0) is a smooth Banach manifold

near (j

, u

, J

). This follows from the implicit function theorem if we can show that

∂(j

, u

, J

) : T

T ⊕ T

k,p,δ

⊕ T

→ E

k−1,p,δ

)

is surjective; indeed, the inﬁnite-dimensional implicit function theorem (see [

Lan93])

requires the a dditional hypothesis that D

∂(j

, u

, J

) has a bounded right inverse,

but this is immediate since the restriction of this operator to the f actor T

k,p,δ

Fredholm (see Exercise 7.17 b elow). We claim in fact that

k,p,δ

⊕T

→ E

k−1,p,δ

)

(η, Y ) 7→ D

∂(j

, u

, J

)(0, η, Y ) = D

η + Y ◦ T u

◦ j

is surjective. Consider ﬁrst the case k = 1,

so we are looking at a bounded linear

map

1,p,δ

∗

W ) ⊕ V

⊕ T

→ L

p,δ

(

Hom

Σ, u

∗

W )).

Note that the dual of any space of sections of class L

p,δ

can be identiﬁed with

sections of class L

q,−δ

for

= 1 (recall R emark 7.7). Indeed, choosing a suitable

-pairing deﬁnes a bounded bilinear map

(7.5) h , i

: L

p,δ

× L

q,−δ

→ R,

and one can use isomorphisms of the form L

→ L

p,δ

: η 7→ e

η as in the proof

of Lemma

7.10 to prove (L

p,δ

)

∗

∼

q,−δ

as a corollary of the standard fact that

)

∗

∼

. With this understood, observe that since D

: W

1,p,δ

⊕ V

→ L

p,δ

is Fredholm, we know by Exercise

7.16 below that the map under consideration

has closed r ange. Thus if it is not surjective, the Hahn-Banach theorem provides

a nontrivial element θ ∈ L

q,−δ

(

Hom

Σ, u

∗

W )) that annihilates its image under

the pair ing (

7.5), which amounts to the two conditions

η, θi

= 0 for all η ∈ W

1,p,δ

∗

W ) ⊕ V

hY ◦ T u

◦ j

, θi

= 0 for all Y ∈ T

(7.6)

The ﬁrst relation is valid in particular for all smooth sections η with compact support

and thus means that θ is a weak solution to the formal adjoint equation D

∗

θ = 0;

applying elliptic regularity and the similarity principle, θ is therefore smooth and has

only isolated zeroes. We will see however that this contradicts the second relation

as long as there exists an injective point z

∈

Σ with u

) ∈ U. Indeed, since the

set of injective point s with this property is open and zeroes of θ are isolated, let

us assume without loss of generality that θ(z

) 6= 0. Then by a standa r d lemma

in symplectic linear algebra (see [

Wend, Lemma 4.4.12]), one can ﬁnd a smooth

Since the present discussion is purely linear, it does not require the assumption kp > 2.

Lectures on Symplectic Field Theory 137

section Y ∈ T

whose value at u

) is chosen such that Y ◦ T u

◦ j

= θ at

, so their pointwise inner product is positive in some neighborhood of z

. But

by Theorem

B.6, one can multiply a small perturbation of Y by a bump function

to produce a section (still denoted by Y ) of class C

so that the pointwise inner

product of Y ◦T u

◦j

with θ is positive near z

but vanishes everywhere else; note

that this requires the assumption u

−1

)) = {z

}, so that the value of Y near

) aﬀects the value of Y ◦ T u

◦ j

near z

but nowhere else. This violates the

second condition in (

7.6) and t hus completes the pr oof for k = 1. In the general

case, suppose θ ∈ W

k−1,p,δ

(

Hom

Σ, u

∗

W )). Then θ is also of class L

p,δ

, so

surjectivity in the k = 1 case implies the existence of η ∈ W

1,p,δ

and Y ∈ T

with D

η + Y ◦ T u

◦ j

= θ. Since Y ◦ T u

◦ j

is smooth with compact support,

one can then use elliptic regularity to show η ∈ W

k,p,δ

, and this proves surjectivity

for ar bitrary k ∈ N and p ∈ (1, ∞).

The implicit function theorem now implies that whenever kp > 2 so that B

k,p,δ

is a well-deﬁned Banach manif old,

∂

−1

(0) is a smooth Banach submanifold of T ×

k,p,δ

×J

in a neighborhood of (j

, u

, J

). The projection map

∂

−1

(0) → J

: (j, u, J) 7→ J

is also smo oth since it is the restriction to a smoo t h submanifold of the obviously

smooth projection map T × B

k,p,δ

× J

→ J

. Since G

acts freely and properly

∂

−1

(0), the quotient

∂

−1

then inherits a smooth Banach manifold structure

for which the projection is still smooth, and this quotient is identiﬁed locally with

∗

). Smoothness of transition maps is shown via the same regularity arguments

as in the proof of Theorem

7.1. 

Exercise 7.16. Show tha t if X, Y and Z are Ba nach spaces, T : X → Y is a

Fredholm operator and A : Z → Y is a bounded linear operator, then the linear

map

L : X ⊕Z → Y : (x, z) 7→ Tx + Az

has closed range. Hint: it might help to write X = V ⊕ker T and Y = W ⊕coker C

so that C

∼

coker T and V

−→ W is an isomorphism.

Exercise 7.17. Under the same a ssumptions as in Exercise

7.16, show that if

T is surjective, then L has a bounded rig ht inverse.

7.6. Applying the Sard-Smale theorem

We claim now that the smooth map

(7.7) M

∗

) → J

: (u, J) 7→ J

is a nonlinear Fredholm ma p, i.e. its derivative at every point is a Fredholm oper-

ator. Using the local identiﬁcation of M

∗

) with

∂

−1

(0)/G

as in the proof of

Lemma 7.15 and lifting the projection to

∂

−1

(0), the derivative of

∂

−1

(0) → J

, u

, J

) takes the for m

ker D

∂(j

, u

, J

) → T

: (y, η, Y ) 7→ Y.

138 Chris Wendl

The Fredholm property for this projection is a consequence of the Fredholm property

for D

via the following general lemma, whose proof is a routine matter of linear

algebra (cf. [

Wend, Lemma 4.4.13]):

Lemma 7.18. Under the a s sumptions of Exercise

7.16, suppose L is surjective.

Then the projection

Π : ker L → Z : (x, z) 7→ z

has kernel and cokernel isomorphic to the kernel and cokernel respectively of T :

X → Y . 

By the Sard-Smale theorem, the set of regular values of the projection (

7.7) is a

comeager subset

ε,reg

⊂ J

and by Lemma

7.18, every (u

, J

) ∈ M

∗

) with J ∈ J

ε,reg

then has the pro perty

that

∂

, u

) : T

T ⊕ T

k,p,δ

→ E

k−1,p,δ

)

is surjective, which means u

represents a Fredholm regular element of M(J

7.7. From C

to C

∞

The arguments above would constitute a proof of Theorem

7.2 if we were allowed

to replace the space of smooth almost complex structures J

with the space J

-small perturba t ions of J

ref

. Let us deﬁne

reg

⊂ J

to be the space of all J ∈ J

with the property t hat all curves in M(J) tha t have

injective points mapping to U are Fredholm regular. The theorem claims that this

set is comeager in J

. We can already see at t his point that it is dense: indeed, the

Baire category theorem implies that J

ε,reg

is dense in J

, so in particular there exists

a sequence J

∈ J

ε,reg

that converges in to J

ref

in the C

-topology and therefore also

in the C

∞

-topology. The choice of J

ref

∈ J

in this discussion was arbitrary, so this

proves density.

To prove that J

reg

is not only dense but a lso contains a countable intersection of

open and dense sets in J

, we can adapt an argument originally due t o Taubes. The

idea is to present the sets of somewhere injective curves in M(J) as countable unions

of compact subsets M

∗

(J) for N ∈ N, and thus present J

reg

as a corr esponding

countable intersection of spaces J

reg,N

that achieve regularity only for the elements

in M

∗

(J). The compactness of M

∗

(J) will then permit us to prove that J

reg,N

not only dense but also open.

The deﬁnition of M

∗

(J) is motivat ed in part by the knowledge that spaces of

J-holomorphic curves have natural compactiﬁcations. We have not yet discussed

the compactiﬁcation

M(J) of M(J), but we have covered enough of the ana lytical

techniques behind this construction to suﬃce for the present discussion. Recall ﬁrst

that the moduli space of Riemann surfaces M

g,ℓ

of genus g with ℓ marked points

Lectures on Symplectic Field Theory 139

also has a natural compactiﬁcation whenever 2g + ℓ ≥ 3, known as t he Deligne-

Mumford compactiﬁcation

g,ℓ

⊃ M

g,ℓ

The space

g,ℓ

consists of “nodal” Riemann surfaces, which can be understood as

objects t hat arise from smooth Riemann surfaces with pair-of-pants decompositions

in the limit where some of the lengths of the circles separating two pairs of pants

from each other may degenerate to 0 (see e.g. [SS92]). We will discuss this in a

bit more detail in Lecture

9; for now, all you really need to know is that M

g,ℓ

a compact and metrizable topological space that contains M

g,ℓ

as a n open subset.

Let us ﬁx a metric on M

g,ℓ

and denote the distance function by dist( , ).

Similarly, ﬁx Riemannian metrics on

W and

Σ with translation-invariance on

the cylindrical ends and use dist( , ) to denote the distance functions. For N ∈ N

and J ∈ J

, we deﬁne

∗

(J) ⊂ M(J)

to be the set of equivalence classes admitting representatives (Σ, j, Γ

, Γ

−

, Θ, u) with

the following properties:

• The equivalence class in M

g,k

−

represented by ( Σ, j, Γ

∪Γ

−

∪Θ) lies

at a distance of at most 1 /N from

g,k

−

\ M

g,k

−

;

• sup

z∈

|du(z)| ≤ N;

• There exists z

∈

Σ such that

dist(u(z

W \ U) ≥

, |du(z

)| ≥

and

inf

z∈

Σ\{z

}

dist(u(z

), u(z))

dist(z

, z)

≥

We observe that every element o f M

∗

(J) has an injective point mapped into U, and

conversely, every asymptotically cylindrical J-holomor phic curve with that property

belongs to M

∗

(J) for N ∈ N suﬃciently large. It is crucial to observe that all three

conditions in this deﬁnition are closed conditions: morally, we are deﬁning M

∗

(J)

to be a closed subset in the compactiﬁcation of M(J), and it will therefore be

compact.

Deﬁne

reg,N

⊂ J

as the set of all J ∈ J

for which every element of M

∗

(J) is Fredholm regular.

Lemma 7.19. For e v ery N ∈ N, J

reg,N

is open and dense.

Proof. Density is immediate, since we’ve seen a lready that every J ∈ J

admits

a C

∞

-small perturbation that achieves regularity for all curves in

N∈N

∗

(J).

For openness, suppose the contrary: then there exists J

∞

∈ J

reg,N

and a sequence

If the stability condition 2g + k

+ k

−

+ m ≥ 3 is not satisﬁed, one should amend this by

asking for the dis tance condition to hold for some tuple (Σ, j, Γ

∪Γ

−

, Θ

′

), where Θ

′

is the unio n

of Θ with enough extra marked points to achieve stability.

140 Chris Wendl

∈ J

reg,N

with J

→ J

∞

in the C

∞

-topology. There must also exist a sequence

of curves u

∈ M

∗

) that are not Fredholm regular. By the deﬁnition of M

∗

they have domains that are uniformly bounded away fr om the singular part of the

Deligne-Mumford space of Riemann surfaces, so we can extract a subsequence for

which these domains converge. Similarly, the ﬁr st derivatives of u

are uniformly

bounded, implying in particular a uniform W

1,p

-bound locally for some p > 2, and

elliptic regularity (Theorem

2.22 in Lecture 2) turns this into uniform C

∞

-bound

and thus a C

∞

-convergent subsequence u

→ u

∞

∈ M

∗

∞

). But u

∞

must then

be Fr edholm regular, which is an open condition, implying that u

is also regular

for ν suﬃciently large, and this is a contra diction. 

Proof of Theorem 7.2. Since the space of all curves in M(J) with injective

points mapped into U is the union of the spaces M

∗

(J) for N ∈ N, we have

reg

N∈N

reg,N

which is a countable intersection o f open and dense sets. 

LECTURE 8

Transversality in symplectizations

Contents

8.1. Statement of the theorem and discussion 141

8.2. Injective points of the projected curve 144

8.3. Smoothness of the universal moduli space 147

This lecture is an addendum to the transversality discussion in Lecture 7: we

need to prove that Fredholm regularity can also be achieved for generic translation-

invariant almost complex structures on symplectizations.

8.1. St atement of the theorem and discussion

Theorem

7.2 in the previous lecture stated that generic perturbations of J in a

precompact open subset U o f a completed symplectic cobordism suﬃce to achieve

regularity for all simple holomorphic curves that pass through that subset. In the

more specialized setting of a symplectization R × M with an R-invariant almost

complex structure J ∈ J(H), we need a more specialized transversality result, as

the generic perturbation from Theorem

7.2 cannot be expected to stay in the space

J(H), in particular it will usually not be R-invariant. The f ollowing statement refers

to a stable Hamiltonian structure H = (ω, λ) with induced hyperplane distribution

ξ = ker λ and Reeb vector ﬁeld R, and we denote by

: T (R × M) → ξ

the projection along the trivial subbundle generated by ∂

and R. We assume as

usual that M(J) denotes a moduli space of asymptotically cylindrical J-holomorphic

curves with a ﬁxed genus a nd number of marked points, representing a ﬁxed relative

homology class a nd asymptotic to ﬁxed sets of nondegenerate Reeb orbits a t its

positive and negative punctures.

Theorem 8.1. Suppose M is a closed (2n −1)-dimensional manifold carrying a

stable Hamiltonia n structure H = (ω, λ), J

ﬁx

∈ J(H), and

U ⊂ M

is an open s ubset. Then there exi s ts a comeager subset

reg

⊂



J ∈ J(H)



J = J

ﬁx

on R × (M \ U)



such that for eve ry J ∈ J

reg

, every curve u ∈ M(J) with a representative u :

Σ →

R × M that has an injective point z ∈

Σ satisfying

141

142 Chris Wendl

(i) u(z) ∈ R × U,

(ii) π

◦ du(z) 6= 0, and

(iii) im (π

◦ du(z)) ∩ ker (dλ|

) = {0}

is Fredholm reg ular.

This result is applied most frequently with U = M, in which case the condition

u(z) ∈ R × U is vacuous. The second and third conditions on the injective point z

can be rephrased by asking for the linear map

dλ(π

T u(X), ·)|

u(z)

: ξ

u(z)

→ R

to be nontrivial fo r every nonzero X ∈ T

Σ. If λ is contact, then this is immediate

whenever π

T u(X) 6= 0 since dλ|

is nondegenerate, and the condition π

T u(X) 6= 0

is also easy to achieve:

Proposition 8.2. If J ∈ J(H), then for any connected J-holomorphic curve

u : (

Σ, j) → (R × M, J), the section

◦ du ∈ Γ(Ho m

Σ, u

∗

ξ))

either is identically zero or has only iso l ated zeroes.

As you might guess, this result is a consequence of the similarity principle; see

8.2 fo r a proof. Notice that if π

◦ du ≡ 0, then u is everywhere tangent to the

vector ﬁelds ∂

and R, so if it is asymptotically cylindrical, then it can only be a

trivial cylinder or a cover thereof.

Proposition 8.3. All trivial cylinders over nondegenerate Reeb orbits hav e in-

dex 0 a nd are Fredholm regular.

Proof. Let u

: R × S

→ R × M denote the trivial cylinder over an orbit

γ : S

→ M. The virtual dimension formula proved in Lecture

7 gives

ind(u

) = ( n −3)χ(R × S

) + 2c

∗

T (R × M)) + µ

(γ) − µ

(γ)

= 2c

∗

T (R × M) ) = 0

since the asymptotic trivialization τ has an obvious extension to a global trivial-

ization of u

∗

ξ, and u

∗

T (R × M) is globally the direct sum of the latter with the

trivial line bundle spanned by ∂

and R. Using this splitting, the linearized Cauchy-

Riemann o perator D

can be identiﬁed with

∂ ⊕ (∂

−A

), where

∂ = ∂

+ i∂

: W

k,p,δ

(R × S

, C) ⊕V

→ W

k−1,p,δ

(R × S

, C)

and

∂

− A

: W

k,p,δ

∗

ξ) → W

k−1,p,δ

∗

ξ).

Here we are assuming without loss of generality that V

is a complex 2-dimensional

space of smooth sections of the trivial line bundle spanned by ∂

and R that are

constant near inﬁnity, and we are identifying this with a space of smo oth complex-

valued f unctions on R × S

. Nondegeneracy implies that ∂

− A : W

k,p

→ W

k−1,p

is an isomorphism, recall Theorem

4.11 in Lecture 4. Using weight functions as

in the proof of Lemma

7.10 to deﬁne isomorphisms between W

k,p,δ

and W

k,p

, one

can identif y ∂

− A

: W

k,p,δ

→ W

k−1,p,δ

with a small perturbation of the same

Lectures on Symplectic Field Theory 143

operator W

k,p

→ W

k−1,p

, hence it is also an isomorphism for δ > 0 suﬃciently

small. To see that

∂ : W

k,p,δ

⊕ V

→ W

k−1,p,δ

is also surjective, observe ﬁrst that

its index is 2; this follows f r om our calculation of ind(u

) and corresponds to the

fact that dim Aut(R × S

, i) = 2. The kernel of this operator consists of bounded

holomorphic C-valued functions on R × S

, so it is precisely the real 2 -dimensional

space of constant functions, implying

dim

coker(

∂) = dim

ker(

∂) − ind

(

∂) = 2 − 2 = 0,

so D

is surjective. 

Corollary 8.4. For any contact form α o n a closed manifold M, there exists

a co meager subset J

reg

(α) ⊂ J(α) such that for every J ∈ J

reg

(α), all somewhere

injective asymptotically cylindrical J-holomorphic curves in R × M are Fredholm

regular. 

Note t hat in the setting of Corollary

8.4, a curve t hat is not a cover of a trivial

cylinder always belongs to a smoo th 1-parameter family of curves related to each

other by R-translation, so that the kernel of the linearized Cauchy-Riemann operator

automatically has kernel of dimension at least 1. This precludes Fredholm regularity

for curves of index 0, thus:

Corollary 8.5. If α is a con tact form and J ∈ J

reg

(α), then all simple asymp-

totically cylindrical J-holomorphic curves u : (

Σ, j) → (R ×M, J) other than trivial

cylinders satisfy

ind(u) ≥ 1.



The following example shows that the third condition on the injective point in

Theorem

8.1 cannot be fully removed in general.

Example 8.6 (cf. Examples 6.6 and 6.16 in Lecture

6). Assume (W, ω) is a closed

symplectic manifold of dimension 2 n−2 with a periodic time-dependent Hamiltonian

H : S

× W → R, and M := S

× W is assigned the stable Ha miltonian structure

(Ω, Λ) := (ω + dt ∧ dH, dt). A choice of J ∈ J(H) is then equivalent to a choice of

t-dependent family of ω-compatible almost complex structures {J

∈ J(W, ω)}

t∈S

and for any t ∈ S

and s ∈ R, J

-holomorphic curves u : (Σ, j) → (W, J

) give rise

to J-holomorphic curves

¯u : (Σ, j) → (R × M, J) : z 7→ (s, t, u(z)).

In particular, when n = 2 one can consider the example where W = Σ is a closed

surface, so curves of this form exist for any choice of J ∈ J(H), no matter how

generic (remember that the domain complex structure j is arbitrary, it is not ﬁxed

in advance). If Σ has genus g and the map u : Σ → Σ has degree 1, then since ¯u

has no punctures a nd satisﬁes c

([¯u]) = c

(¯u

∗

T (R ×S

×Σ)) = c

(T Σ) = χ(Σ), the

index of ¯u is

ind(¯u) = (n − 3)χ(Σ) + 2χ(Σ) = χ(Σ) = 2 − 2g.

This shows that ¯u cannot be Fredholm regular unless g = 0.

144 Chris Wendl

Theorem 8.1 appeared for the ﬁrst time in the contact case in [Dra04], and

alternative proofs have since appeared in t he appendix of [

Bou06] (for cylinders in

the contact case) and in [

Wena] (under slightly diﬀerent assumptions in the stable

Hamiltonian setting). What I will describe below is a generalization of Bourgeois’s

proof.

8.2. Injective points of the project ed curve

One point o f diﬃculty in proving transversality in R × M is that in cont r ast to

the setting of Theorem

7.2, generic perturbations within J(H) can never be truly

local, i.e. if you perturb J near a point (r, x) ∈ R×M, then you are also perturbing it

in a neighborhood of the entire line R ×{x}. We therefore need to know that we can

ﬁnd a point z ∈

Σ that is the only point where u :

Σ → R ×M pa sses through such

a line; put another way, we need to know that not only u = (u

, u

) :

Σ → R × M

but also t he projected map u

Σ → M is somewhere inj ective. The ﬁrst step in

showing this is Proposition

8.2 above, as the zeroes of the section

◦ du ∈ Γ(Ho m

Σ, u

∗

ξ))

are precisely the critical points of u

Σ → M; everywhere else, u

is an immersion

transverse to the Reeb vector ﬁeld. To prove Proposition

8.2, we shall use the fact

that the vector ﬁelds ∂

and R generate an integrable J- invariant distribution on

R×M. Indeed, the zeroes of π

◦du are the points of tangency with this distribution,

hence the result is an immediate consequence of the following statement:

Lemma 8.7. Suppo s e (W, J) is an almost complex manifold, Ξ ⊂ T W is a smooth

integrable J-invariant distribution and u : (Σ, j) → (W, J) is a connected pseudo-

holomorphic curve whose image is n ot contained in a leaf of the foliation generated

by Ξ. Then all points z ∈ Σ with im du(z) ⊂ Ξ are isolated in Σ.

Proof. Statement is local, so assume (Σ, j) = ( D, i) with coordinates s + it,

W = C

, and u(0) = 0. Let 2m denote the real dimension of Ξ , and observe that

since Ξ is integrable, we can change coordinates near 0 and assume without loss of

generality that at every point p ∈ C

near 0, Ξ

= C

⊕ {0} ⊂ C

= T

. The

J-invariance of Ξ then implies that in coordinates (w, ζ) ∈ C

×C

n−m

, J takes the

form

J(w, ζ) =



(w, ζ) Y (w, ζ)

0 J

(w, ζ)



where J

and J

are both −1, and J

Y + Y J

= 0. Writing u(z) = (f(z), v(z)) ∈

×C

n−m

, the Cauchy-Riemann equation ∂

u + J(u)∂

u = 0 is then equivalent to

the two equations

∂

f + J

(f, v) ∂

f + Y (f, v) ∂

v = 0,

∂

v + J

(f, v) ∂

v = 0.

(8.1)

We have im du(z) ⊂ Ξ wherever ∂

v = ∂

v = 0; notice that it suﬃces to consider

the condition ∂

v = 0 since ∂

v = J

(f, v) ∂

v. Diﬀerentiating the second equation

in (

8.1) with respect to s gives

∂

(∂

v) + J

(f, v) ∂

(∂

v) + ∂

(f, v)] J

(f, v) ∂

v = 0,

Lectures on Symplectic Field Theory 145

where in the la st t erm we’ve substituted J

(f, v) ∂

v for ∂

v. Setting

J(z) :=

(f(z), v(z)) and A(z) := ∂

(f(z), f(z)] J

(f(z), v(z)), this becomes a linear

Cauchy-Riemann type equation ∂

(∂

v) +

J ∂

(∂

v) + A(∂

v) = 0, so the similarity

principle implies that zeroes of ∂

v are isolated unless it is identically zero. The

latter would mean v is constant, so u is contained in a leaf of Ξ. 

Lemma 8.8. Suppo se J ∈ J(H), γ : S

→ M is a clos ed Ree b o rb i t, and

u = (u

, u

) : (

Σ, j) → (R × M, J) is an asymptotically cylindrical J-holomorph i c

curve that is not a cover of a trivial cylinder. Then all intersections of the map

Σ → M with the image of the orbit γ are isolated.

Proof. The trivial cylinder over γ is a J-holomorphic curve, so the statement

follows from the fact that two a symptotically cylindrical J-holomorphic curves can

only have isolated intersections unless bo t h are covers of the same simple curve. 

We can now prove the statement we need about somewhere injectivity for u

Σ → M. This result ﬁrst appear ed in [

HWZ99, Theorem 1.13].

Proposition 8.9. Suppose J ∈ J(H) and

u = (u

, u

) : (

Σ, j) → (R ×M, J)

is a simple asymptotically cylindrical J-holomorphic curve which is not a trivial

cylinder and has only nondegenerate asymptotic o rb i ts. Then the set of injective

po ints z ∈

Σ of the map u

Σ → M for which u

(z) i s not contained in any of

the asymptotic orbits of u is open and dense.

Proof. Openness is clear, so our main task is to prove density. The idea is

ﬁrst to show via elementary topological arguments that if the set of injective points

is not dense, then

Σ contains two disjoint open sets on which u

is an embedding

with identical images. We will then conclude from this that if u is simple, it must

be equivalent to one of its nontrivial R-translations, and the latter is impossible for

an asymptotically cylindrical curve.

Step 1: We begin by harmlessly removing some discrete sets of points in

Σ tha t

would make the subsequent arguments more complicated. Let

P ⊂ M

denote the union of the images of the asymptotic orbits of u, a ﬁnite disjoint union of

circles. Lemma

8.8 implies that u

−1

(P ) is a discrete subset of

Σ. By Proposition

8.2,

there is also a discrete set Z ⊂

Σ \ u

−1

(P ) containing all points z 6∈ u

−1

(P ) where

◦ du(z) = 0, and we claim that

′

:= u

−1

(Z))

is a discrete subset of

Σ \ u

−1

(P ). Indeed, u

(Z) is a discrete subset of M \ P

since the points in Z can only accumulate at inﬁnity,

hence accumulation points of

(Z) ⊂ M can occur o nly in P . For each individual point p ∈ u

(Z), the f act that

p 6∈ P implies u

−1

(p) is compact, and it consists of a discrete (and therefore ﬁnite)

Actually the asymptotic formula of [

HWZ96] implies that both Z and u

−1

(P ) are always

ﬁnite for curves that are not cove rs of trivial cylinders, but we do not need to use that here.

146 Chris Wendl

set of points with π

◦du(z) = 0, plus possibly some other points where π

◦du(z) 6= 0,

but u

is an embedding near each point of the latter type, so that these points of

−1

(p) must always be isolated and are therefore also ﬁnite in number. This proves

the claim, and we conclude that

Σ :=

Σ \



−1

(P ) ∪ Z

′



an open and dense subset of

Σ, as it is obta ined by removing a discrete subset from

the open and dense subset

Σ \u

−1

(P ). To prove the proposition, it will now suﬃce

to prove that the set of points z ∈

Σ which are injective points of u

Σ → M is

dense in

Σ. We shall argue by contradiction and assume from now o n t hat density

fails.

Step 2: We will ﬁnd two open subsets U, V ⊂

Σ such that u

restricts to a n

embedding on both, but

U ∩ V = ∅ and u

(U) = u

(V).

Indeed, assume the set of injective points of u

lying in

Σ is not dense in

Σ. Then

there exists a point z

∈

Σ with a closed neighborhood D(z

) ⊂

Σ such that no

z ∈ D(z

) is an inj ective point. Since z ∈

Σ implies π

◦ du(z) 6= 0, this means that

for every z ∈ D(z

), there exists ζ ∈

Σ \{z} with u

(z) = u

(ζ), and the deﬁnition

Σ implies ζ is also in

Σ, hence π

◦du(ζ) 6= 0 and u

is a local embedding near ζ.

Since u(z) 6∈ P and u

maps

Σ \ u

−1

(P ) properly to M \P , we also conclude that

−1

(z)) is ﬁnite. Now suppose u

−1

)) = {z

, ζ

, . . . , ζ

}, and let D(ζ

) ⊂

for j = 1, . . . , m denote closed neighborhoods on which u

is an embedding. We

claim tha t after possibly shrinking D(z

), we can assume

(D(z

)) ⊂

[

j=1

(D(ζ

Let us ﬁrst shrink D(z

) so that u

is an embedding o n D(z

), which is possible

since π

◦du(z

) 6= 0. Then if t he claim is false, there exists a sequence z

∈ D(z

) of

noninjective points with z

→ z

, hence there is also a sequence z

′

∈

Σ \D(z

) with

) = u

′

) but z

′

not converging to any of ζ

, . . . , ζ

. But since u

′

) →

) 6∈ P , the points z

′

are conﬁned to a compact subset of

Σ and therefore have

a subsequence z

′

→ z

′

∞

∈

Σ with u

′

∞

) = u

). The limit cannot be z

itself

since z

′

6∈ D(z

), thus z

′

∞

must be one of the ζ

, . . . , ζ

, and we have a contradiction.

We claim next that at least one of the sets u

(D(z

)) ∩ u

(D(ζ

)) has nonempty

interior. This is a simple exercise in metric space topology: it can be reduced to

the fact that if X is a metric space with closed subsets V, W ⊂ X that both have

empty interior (meaning no open subset of X is contained in V or W ), then V ∪W

also has empty interior. Since the subsets u

(D(z

)) ∩ u

(D(ζ

)) ⊂ u

(D(z

)) for

j = 1, . . . , m are all closed but their union is u

(D(z

)), they cannot all have empty

interior. This achieves the goal of Step 2.

Step 3: We show that u is biholomorphically equivalent to o ne of its R-translations

τ ·u := (u

+ τ, u

) :

Σ → R × M

Lectures on Symplectic Field Theory 147

for τ ∈ R \{0}. To see this, note that for J ∈ J(H), the nonlinear Cauchy-Riemann

equation T u ◦ j = J(u) ◦ T u is equivalent to the two equations

= u

∗

λ ◦j,

◦ T u

◦ j = J(u

) ◦ π

◦ T u

(8.2)

Since π

◦T u

Σ → u

∗

ξ is injective everywhere on the neighborhoods U and V, the

second equation determines j in terms of J on each of these regions; in particular,

the identiﬁcation of u

(U) with u

(V) provides a biholomorphic map of V to U

so that u|

and u|

may be regarded as two J-holomo r phic maps from the same

Riemann surface which diﬀer only in the R-factor. But with j and u

both ﬁxed,

the ﬁrst equation in (

8.2) determines du

and thus determines u

up to the addition

of a constant τ ∈ R. If τ = 0, t his means u has two disjoint regions on which its

images are identical, contradicting the assumption that u is simple. Thus τ 6= 0,

and since two distinct simple curves can only intersect each other at isolated points,

we conclude u = τ · u up to parametrization.

Step 4: We now derive a contradiction. The relation u = τ · u implies that in

fact u = kτ · u fo r every k ∈ Z, so we obtain a diverging sequence of R-tra nslatio ns

→ ∞ such that u and τ

· u always have identical images in R × M. It follows

that for some point z ∈

Σ with u(z) = ( r, x) where x is not contained in any of the

asymptotic orbits of u, the points (r − τ

, x) are all in the image o f u as τ

→ ∞.

But this contradicts the asymptotically cylindrical behavior of u. 

8.3. Smoothness of the universal moduli space

The overall outline of the proof o f Theorem 8.1 is the same as for Theorem 7.2:

one needs to deﬁne a suitable space J

of perturbed almost complex structures,

giving rise to a universal moduli space M

∗

) that is a smooth Banach manifold,

and then apply the Sard-Smale theorem to conclude that generic elements of J

are regula r values of the projection M

∗

) → J

: (u, J) 7→ J. If J

is a space

of C

-perturbed almost complex structures, then in the ﬁnal step one can use t he

Taubes tr ick as in §

7.7 to transform the genericity result in J

into a genericity

result within t he space J( H) of smooth almost complex structures. The only step

that diﬀers meaningfully fro m what we’ve already discussed is the smoothness of

the universal moduli space, so let us focus on this detail.

Assume J

ref

∈ J(H) with J

ref

= J

ﬁx

outside R×U, and J

is a Banach manifold

of C

-small perturbations of J

ref

in J(H) that are also ﬁxed outside of R ×U. The

relevant universal moduli space is then deﬁned by

∗

) :=



(u, J)



J ∈ J

, u ∈ M(J) and

u :

Σ → R × M has an injective point z ∈

Σ with

u(z) ∈ R × U and im (π

◦ du(z)) ∩ ker (dλ|

) = {0}



Notice that b oth of the constraints satisﬁed by u at the injective point ar e open.

The local structure of M

∗

) near an element (u

, J

) with representative u

(

Σ, j

) → (R × M, J

) can again be described via the zero set of a smooth section

∂ : T × B

k,p,δ

× J

→ E

k−1,p,δ

: (j, u, J) 7→ T u ◦ J ◦ T u ◦ j,

148 Chris Wendl

where T is a Teichm¨uller slice thro ugh j

, and it suﬃces to show that the lineariza-

tion

L : T

k,p,δ

⊕T

→ E

k−1,p,δ

)

: (η, Y ) 7→ D

η + Y ◦ T u

◦ j

is always surjective. As usual, here we’re assuming k ∈ N, 1 < p < ∞, and the

exp onential weight δ > 0 is small but positive so that D

is Fredholm. The image

of L is then closed, and focusing on the k = 1 case, if L is not surjective then there

exists a nontrivial element θ ∈ L

q,−δ

(

Hom

Σ, u

∗

T (R × M))) such t hat

η, θi

= 0 for all η ∈ W

1,p,δ

∗

T (R × M)) ⊕ V

hY ◦ T u

◦ j

, θi

= 0 for all Y ∈ T

(8.3)

The ﬁrst conditio n implies via elliptic regularity and the similarity principle that θ

is smooth and has only isolated zeroes. So far this is all the same as in the proof

of Theorem

7.2, but the next step is trickier: since perturbing J

within J(H)

only cha nges the action of the almo st complex structure on ξ but not on the trivial

subbundle generated by ∂

and R, it is not clear whether the range of values allowed

for Y is large enough to force hY ◦ T u

◦ j

, θi

> 0.

To overcome this, let us decompose everything in this picture with respect to

the nat ural splitting

T (R × M) = ǫ ⊕ ξ,

where ǫ denotes the trivial line bundle spanned by ∂

and R. In particular, the

domain and target bundles of the Cauchy-Riemann type operator D

now split as

∗

T (R × M) = u

∗

ǫ ⊕ u

∗

ξ,

Hom

Σ, u

∗

T (R × M) ) = Hom

Σ, u

∗

ǫ) ⊕ Hom

Σ, u

∗

ξ),

and we shall write η = (η

, η

) and θ = (θ

, θ

) accordingly. This gives a block

decomposition of D

η =



η)





ǫξ

ξǫ





It is easy to verify that D

and D

each satisfy suitable Leibniz rules and are thus

Cauchy-Riemann type operators on u

∗

ǫ and u

∗

ξ respectively, while the oﬀ-diagonal

terms are both tensorial, i.e. zeroth-order operators. Since perturbations of J

J(H) only change its action on ξ, Y ∈ T

now takes the block form

Y =



0 0

0 Y



where Y

is a C

-small section of the bundle

End

(ξ, J

) over M. Assuming the L

pairings are deﬁned so as to r espect these splittings, the second condition in (

8.3)

now becomes

◦ π

◦ T u

◦ j

, θ

= 0,

and given any injective point z

∈

Σ of (u

)

Σ → M satisfying u

) ∈ R×U, we

have enough freedom to choose Y

near R ×{u

)} such that t his pairing becomes

positive unless

= 0 near z

Lectures on Symplectic Field Theory 149

It remains to show that θ

also vanishes near z

, which will contradict the fa ct that θ

only has isolated zeroes. To this end, notice that the ﬁrst condition in (

8.3) implies

via separate choices o f the components η

and η

with support near z

that

, θ

= 0 for all η

supported near z

ǫξ

, θ

= 0 for all η

supported near z

(8.4)

The ﬁrst of these two conditions gives no new information, since we already know

that θ = (θ

, 0) solves an anti-Cauchy-Riemann equation. To get some information

out of the second condition, we will need an explicit formula for D

ǫξ

Lemma 8.10. T he tensorial operator D

ǫξ

: u

∗

ξ →

Hom

Σ, u

∗

ǫ) takes the form

ǫξ

−dλ



, J

◦π

◦ T u(·)



∂



dλ



, π

◦T u(·)



Proof. As a preliminary step, no tice that −dr ◦ J = λ for any J ∈ J(H);

indeed, the conditions J(ξ) = ξ ⊂ ker dr and J∂

= R imply t hat these two 1 -forms

have matching values on ∂

, R and ξ. As a consequence, λ◦J

= dr, so in particular

λ ◦J

is closed.

Choosing local holomorphic coordinates (s, t) in an arbitrary neighborhood in

Σ, we have

ǫξ

)∂

= dr



)∂



∂

+ λ



)∂



Extend u

Σ → R × M to a smooth 1-parameter family of maps {u

Σ →

R × M}

ρ∈R

with ∂

ρ=0

= η

∈ Γ(u

∗

ξ). Then by the deﬁnition of the linearized

Cauchy-Riemann opera tor,

)∂

= ∇

(∂

+ J

)∂

ρ=0

for any choice of connection ∇ on R × M. Since ∂

+ J

)∂

= 0, we ﬁnd



)∂



= λ



∇

(∂

+ J

)∂

ρ=0



= ∂

[λ(∂

+ J

)∂

)]|

ρ=0

= ∂

[λ(∂

)]|

ρ=0

+ ∂

[(λ ◦ J

)(∂

)]|

ρ=0

= dλ(η

, ∂

u) + d(λ ◦ J

)(η

, ∂

= dλ(η

, π

∂

u),

where we’ve used the formula

dλ(X, Y ) = L

[λ(Y )] − L

[λ(X)] − λ([X, Y ])

and eliminated severa l t erms using the fact t hat λ(η

) = λ(J

) = 0 since η

valued in ξ, plus d(λ ◦ J

) = 0. A similar computation gives



)∂



= −dλ(η

, π

∂

u) = −dλ(η

, J

◦ π

∂

u),

so removing the local coordinates from the picture produces the stated formula. 

The following exercise in symplectic linear algebra shows that this bundle map

∗

ξ →

Hom

Σ, u

∗

ǫ) is surjective on any ﬁber over a point z with π

◦ du

(z) 6=

0. (If you have no patience for the exercise, just convince yourself t hat it’s true

whenever dλ|

is nondegenerate and tames J|

, i.e. the contact case.)

150 Chris Wendl

Exercise 8.11. Assume V is a ﬁnite-dimensional vector space, X, Y ⊂ V are

linearly independent vectors, and Ω is an alternating bilinear form on V . Show that

the real-linear map

A : V → C : v 7→ Ω(v, X) + iΩ(v, Y )

is surjective if and only if Span(X, Y ) ∩ ker Ω = {0}.

Hint: Under the latter condition, one loses no generality by replacing V with a

subspace that is complementary to ker Ω a nd contains Span(X, Y ), in which case

(V, Ω) becomes a symplectic vector space. Now consider the restriction of A to a

2-dimensional subspace transverse t o the symplectic complement of Span(X, Y ).

The conclusion of this discussion is that unless θ

vanishes near z

, η

can be

chosen with supp ort near z

so that hD

, θ

> 0, violating the second condi-

tion in (

8.4). This proves that θ vanishes altogether near z

and thus, by unique

continuation, θ ≡ 0, a contradiction.

We’ve proved that the universal moduli space is smooth as claimed. Since the

rest of the proof of Theorem

8.1 is the same as in the non-R-invariant case, we leave

those details to the reader.

Remark 8.12. You may have noticed tha t in both this and the previous lecture,

our proof that the universal moduli space is smooth relied on a surjectivity result

that was actually stronger than needed: in both cases, we needed to prove that an

operator of the form

T ⊕ T

k,p,δ

⊕ T

−→ E

k−1,p,δ

)

was surjective, but we ended up proving that its restriction to the smaller domain

k,p,δ

⊕T

is a lr eady surjective. This technical detail hints at a stronger result

that can be proved using these methods: one can show that not only is M

∗

)

smooth but a lso the forget ful map

∗

) → M

g,k

−

([(Σ, j, Γ

, Γ

−

, Θ, u)], J) 7→ [(Σ, j, Γ

∪ Γ

−

∪ Θ)]

sending a J-holomorphic curve to its underlying domain in the moduli space of

Riemann surfaces is a submersion, cf. the blog post [

Wenb] and its sequel. One can

use this to prove generic t r ansversality results for spaces of J-holomorphic curves

whose domains are constrained within the moduli space of Riemann surfaces, which

can be used to deﬁne mor e elabor ate algebraic structures on SFT, e.g. this idea

plays a very prominent role in t he study of Gromov-Witten invariants.

LECTURE 9

Asymptotics and compactness

Contents

9.1. Removal of singularities 152

9.2. Finite energy and asymptotics 155

9.3. Degenerations of holomorphic curves 170

9.3.1. Bubbling 171

9.3.2. Breaking 172

9.3.3. The Deligne-Mumford space of Riemann surfaces 176

9.4. The SFT compactness theorem 182

9.4.1. Nodal curves 182

9.4.2. Holomorphic buildings 183

9.4.3. Convergence 186

9.4.4. Symplectizations, stretching and so forth 187

Moduli spaces of pseudoholomorphic curves are generally not compact, but they

have natural compactiﬁcations, obtained by allowing certain types of curves with

singular behavior. For closed holomorphic curves, this fact is known as Gromo v’s

co mpactness theorem, and our main goal in this lecture is to state its generalization

to punctured curves, which is usually called the SFT compactness theorem. The the-

orem was ﬁrst proved in [

BEH

03] (see also [CM05] for an alternative approach),

and we do not have space here to present a complete proof, but we can still describe

the main geometric and ana lytical ideas behind it.

The overar ching theme of this lecture is the notion of bubbling, of which we will

see several examples. Bubbling arises in a natural way from elliptic regularity: recall

that in Lecture

2, we proved that whenever kp > 2, any uniformly W

k,p

-bounded

sequence u

of holomorphic curves is also uniformly C

lo c

-bounded for every m ≥ N

(cf. Theorem

2.22). The Arzel`a-Ascoli theorem implies that such sequences have

∞

lo c

-convergent subsequences, and this is true in particular whenever u

is uniformly

-bounded, as a C

-bound implies a W

1,p

-bound with p > 2. Let us take note of

this fa ct for future use:

Proposition 9.1. If (W, J

) is a sequence of almost complex manifolds with

→ J in C

∞

, then any uniform l y C

-bounded sequence of J

-holomorphic maps

: D → W has a subsequence convergent in C

∞

lo c

If one wants to prove compactness for a moduli space of J-holomor phic curves,

it therefore suﬃces in general to establish a C

-bound. The catch is, of course,

that the ﬁrst derivatives of u

might not be uniformly bounded, and this is when

151

152 Chris Wendl

interesting things are seen to ha ppen: while the sequence u

is not compact, it

turns out that it becomes compact after removing ﬁnitely many points from its

domain, and near those points one can ta ke a sequence of reparametrizations to ﬁnd

additional nontrivial holomorphic curves in the limit, t he so-called “bubbles”. This

is one of the ways that the “nodal” curves in Gromov’s compactness theorem can

arise, and we will see the same phenomenon at work in several other contexts as

well.

9.1. Removal of singularities

As an important tool for use in the rest of this lecture, we begin with the following

result from [

Gro85]:

Theorem 9.2 (Gromov’s removable singularity theorem). Assume (W, ω) is a

symplectic manifold with a tame almost complex structure J, a nd u : D \ {0} → W

is a J-hol omorphic curve that has its image con tained in a com pact subset o f W and

satisﬁes

D\{0}

∗

ω < ∞.

Then u admits a s mooth extension to D.

We will prove the slightly weaker statement t hat u has a continuous extension.

If dim

W = 2, then the smooth extension follows from this by classical complex

analysis; in higher dimensions, one can instead apply results on local elliptic regu-

larity, see e.g. [

MS04]. We will use as a black box the following additional result

from [

Gro85], which is closely related to a standard result about minimal surfaces:

Theorem (Gromov’s monotonicity lemma [

Gro85]). Suppose (W, ω) is a com-

pa ct symplectic manifold (possibly with boundary), J is an ω-tame almost comp l ex

structure, and B

(p) ⊂ W denotes the open ball of radius r > 0 about p ∈ W with

respect to the Riemannian metric g(X, Y ) :=

ω(X, JY ) +

ω(Y, JX). T hen there

exist constants c, R > 0 such that for all r ∈ (0, R) and p ∈ W with B

(p) ⊂ W , ev-

ery prope r non-constant J-holomorphic curve u : (Σ, j) → (B

(p), J) passing through

p satisﬁes

∗

ω ≥ cr

In t he above statement , (Σ, j) is assumed to be an arbitrary (generally noncom-

pact) Riemann surface without boundary. In applications, one typically has a larger

(e.g. closed or punctured) domain Σ

′

in the picture, and Σ is deﬁned to be the con-

nected component of u

−1

(p)) ⊂ Σ

′

containing some point z ∈ u

−1

(p). The main

message of the theorem is tha t u must use up at least a certain amount of energy

for every ball whose center it passes through, so e.g. the portion of the curve passing

through B

(p) cannot become arbitrarily “thin” as in Figure

9.1.

Returning to the removable singularity theorem, we shall use the biholomorphic

map

:= [0, ∞) × S

→ D \ {0} : (s, t) 7→ e

−2π(s+it)

Lectures on Symplectic Field Theory 153

((−

(p)

u(Σ)

u(Σ

′

)

Figure 9.1. The intersection of a J-holomorphic curve u with an

open ball B

(p) deﬁnes a proper map Σ → B

(p). The monotonicity

lemma prevents this map from having arbitrarily small area if it passes

through p.

to transform J-holomorphic maps D \ {0} → W into maps Z

→ W , a nd the goal

will be to show that whenever such a map u has precompact image a nd satisﬁes

∗

ω < ∞, there exists a point p ∈ W such that

(9.1) u(s, ·) → p in C

∞

, W ) as s → ∞.

Fix t he obvious ﬂat metric on Z

and any Riemannian metric on W in order to

deﬁne norms such as |du(s, t)| for (s, t) ∈ Z

Lemma 9.3. There exists a constant C > 0 such that |du(s, t)| ≤ C for all

(s, t) ∈ Z

Proof, part 1. Arguing by contradiction, suppo se there exists a sequence z

, t

) ∈ Z

with |du(z

)| =: R

→ ∞. Choose a sequence of positive numbers

> 0 t hat converge to zero but not t oo fast, so that ǫ

→ ∞. We then consider

the sequence of reparametrized maps

: D

→ W : z 7→ u(z

+ z/R

These are also J-holomorphic since z 7→ z

+ z/R

is holomorphic, and the va lues

of v

depend only on the values of u over the ǫ

-disk about z

. Notice that since

→ ∞ and ǫ

→ 0, we are free to assume that all o f these ǫ

-disks a r e disjoint;

moreover, tameness of J implies u

∗

ω ≥ 0 and v

∗

ω ≥ 0, thus

∗

ω =

)

∗

ω ≤

∗

ω < ∞,

implying

(9.2)

∗

ω → 0 as k → ∞.

We would now like to say something a bout a limit of the maps v

as k → ∞, but this

will require a brief pause in the proof, as we don’t yet have quite enough inf ormation

154 Chris Wendl

to do so. We know that the v

are uniformly C

-bounded since u(Z

) is contained

in a compact subset. It would be ideal if we also had a uniform C

-bound, as then

elliptic regularity (Prop.

9.1) would give a C

∞

lo c

convergent subsequence on the union

of all the domains D

, i.e. on t he entire plane. We have

(z) =

du(z

+ z/R

hence |dv

(0)| = 1 , but we will need to know more about |du| on the rest of D

)

in order to deduce a C

-bound for v

on all of D

. We’ll come back to this in a

moment. proof to be continued. . .

Here is the auxiliary lemma that is needed to complete the above proo f:

Lemma 9.4 (Hofer). Suppose (X, d) is a com plete metric space, g : X → [0, ∞)

is continuous, x

∈ X and ǫ

> 0. Then there exist x ∈ X and ǫ > 0 such that,

(a) ǫ ≤ ǫ

(b) g(x)ǫ ≥ g(x

)ǫ

) ≤ 2ǫ

, and

(d) g(y) ≤ 2g(x) for all y ∈

(x).

Proof. If there is no x

∈

) such that g(x

) > 2g(x

), then we can

set x = x

and ǫ = ǫ

and are done. If such a point x

does exist, then we set

:= ǫ

/2 and repeat the above process for the pa ir (x

, ǫ

): that is, if there is

no x

∈

) with g(x

) > 2g(x

), we set (x, ǫ) = (x

, ǫ

) and are ﬁnished, and

otherwise deﬁne ǫ

= ǫ

/2 and repeat for (x

, ǫ

). This process must eventually

terminate, as otherwise we obtain a Cauchy sequence x

with g(x

) → ∞, which is

impossible if X is complete. 

Proof of Lemma

9.3, part 2. Applying Lemma 9.4 to X = Z

with g(z) =

|du(z)|, we can replace the original sequences ǫ

and z

with new sequences for which

all the previously stated properties still hold, but additionally,

|du(z)| ≤ 2|du(z

)| for all z ∈ D

Our sequence of repara metrizations v

then satisﬁes

|dv

(z)| ≤ 2 for all z ∈ D

so by elliptic regularity, v

has a subsequence convergent in C

∞

lo c

map

∞

: C → W

which is not constant since |dv

∞

(0)| = lim

k→∞

|dv

(0)| = 1. Informally, we say that

the blow-up of the derivatives at z

has caused a plane to “ bubble oﬀ”. However,

(

9.2) implies that for every R > 0, one can write ǫ

≥ R for k suﬃciently large

and thus

∗

∞

ω = lim

k→∞

∗

ω ≤ lim

k→∞

∗

ω = 0,

implying

∗

∞

ω = 0. It follows that v

∞

must be constant, so we have a contradic-

tion. 

Lectures on Symplectic Field Theory 155

To obtain the uniform limit of u(s, ·) as s → ∞, we now pick any sequence of

nonnegative numbers s

→ ∞ and consider the sequence of J-holomorphic half-

cylinders

: [−s

, ∞) × S

→ W : (s, t) 7→ u(s + s

, t).

By Lemma

9.3, these maps are uniformly C

-bounded, so elliptic regularity gives a

subsequence converging in C

∞

lo c

on R × S

to a J-holomorphic cylinder

∞

: R × S

→ W.

Observe that for any c > 0, we can write −s

/2 ≤ −c for suﬃciently large k and

thus compute

[−c,c]×S

∗

∞

ω = lim

k→∞

[−c,c,]×S

∗

ω ≤ lim

k→∞

[−s

/2,∞)×S

∗

= lim

k→∞

/2,∞)×S

∗

ω = 0

since

∗

ω < ∞. This implies

R×S

∗

∞

ω = 0, so u

∞

is a constant map to some

point p ∈ W , hence after replacing s

with a subsequence,

u(s

, ·) = u

(0, ·) → p in C

∞

, W ) as k → ∞.

To ﬁnish the proof of (

9.1), we need to show that one cannot ﬁnd two sequences

→ ∞ and s

′

→ ∞ such that u(s

, ·) → p and u(s

′

, ·) → p

′

for distinct points

p 6= p

′

∈ W . This is an easy consequence of the monotonicity lemma: indeed,

if two such sequences exist, then we can ﬁnd a sequence s

′′

→ ∞ for which the

loops u(s

′′

, ·) alternate between a r bitrarily small neighborhoods of p and p

′

. Since

u is continuous, it must then pass through ∂B

(p) inﬁnitely many times for r > 0

suﬃciently small, and in fact there exists an inﬁnite sequence of pairwise disjoint

neighborhoods U

⊂ Z

such that each

: U

→ B

)

is a proper map passing through some point q

∈ ∂B

(p). The monotonicity lemma

then implies

∗

ω ≥

∗

ω ≥

= ∞,

a contradiction.

Exercise 9.5. Given an area form ω on S

= C∪{∞} and a ﬁnite subset Γ ⊂ S

show that a holomorphic function f : S

\Γ → C has an essential singularity at one

of its punctures if and only if

∗

ω = ∞.

9.2. Finite energy and asymptotics

As further preparation for the compactness discussion, we now prove the long-

awaited converse of the fact that a symptotically cylindrical curves have ﬁnite en-

ergy. We work in the setting described in §

6.2: (W, ω) is a symplectic cobordism

with stable boundary ∂W = −M

−

⊔ M

carrying stable Hamiltonian structures

156 Chris Wendl

= (ω

, λ

) with induced hyperplane distributions ξ

= ker λ

and Reeb vector

ﬁelds R

. The completion (

W , ω

) carries the symplectic structure











d (h(r)λ

) + ω

on [0, ∞) × M

ω on W ,

d (h(r)λ

−

) + ω

−

on (−∞, 0] × M

−

for some C

-small smooth function h(r) with h

′

> 0 that is the identity near r = 0,

and for a ﬁxed constant r

, we deﬁne a compact subset

:= ([−r

, 0] × M

−

) ∪

−

W ∪

([0, r

] × M

) ⊂

W ,

outside of which our ω

-tame almost complex structures J ∈ J

(ω

, r

, H

−

)

are required to be translation-invariant and compatible with H

. The energy of a

J-holomorphic curve u : (

Σ, j) → (

W , J) is deﬁned by

E(u) := sup

f∈T (h,r

)

∗

where

T (h, r

) :=



f ∈ C

∞

(R, (−ǫ, ǫ))



′

> 0 and f ≡ h near [−r

, r

]



The constant ǫ > 0 should always be a ssumed suﬃciently small so that if J

∈

J(H

) and X ∈ ξ

(9.3) (ω

+ κ dλ

)(X, J

X) > 0 whenever X 6= 0 a nd κ ∈ (−2ǫ, 2ǫ).

This condition implies that every J ∈ J

(ω

, r

, H

−

) is tamed by every ω

for

every f ∈ T (h, r

), thus all J-holomorphic curves satisfy E(u) ≥ 0, with equality if

and only if u is constant.

Theorem 9.6. Assume all closed Reeb o rbits in (M

, H

) and (M

−

, H

−

) are

nondegenerate, J ∈ J

(ω

, r

, H

−

), ( Σ, j) is a closed Riemann surface with

Σ = Σ \Γ for some ﬁnite subset Γ ⊂ Σ, and u : (

Σ, j) → (

W , J) is a J-holomorphic

curve such that none of the singularities in Γ are removable a nd E(u) < ∞. Then

u is asymptotically cylindrical.

Remark 9.7. The theorem also holds in the setting of a symplectization (R ×

M, J) with J ∈ J(H) for a stable Hamiltonian structure H = (ω, λ) on M. The

only real diﬀerence in this case is the slightly simpler deﬁnition of energy,

E(u) = sup

f∈T

∗

where ω

:= d



f(r)λ



+ ω and

T =



f ∈ C

∞

(R, (−ǫ, ǫ))



′

> 0



This change necessitates a few tr ivial modiﬁcations to the proof of Theorem

9.6

given below.

Lectures on Symplectic Field Theory 157

Like removal of singularities, Theorem 9.6 is r eally a local result, so let us for-

mulate a more precise and more general statement in t hese terms. Let

D := D \ {0} ⊂ C

and deﬁne the two biholomorphic maps

: Z

:= [0, ∞) × S

→

D : (s, t) 7→ e

−2π(s+it)

−

: Z

−

:= (−∞, 0] × S

→

D : (s, t) 7→ e

2π(s+it)

(9.4)

Theorem 9.8 . Suppose J ∈ J

(ω

, r

, H

−

) and u :

D →

W is a J-

holomorphic map with E(u) < ∞. T hen either the s i ngularity at 0 ∈ D is removable

or u is a proper map. In the latter case the puncture is ei ther positive or negative,

meaning that u maps neigh borhoods of 0 to neighborhoods of {±∞} × M

, and the

puncture has a well-deﬁned charge, deﬁned as

Q = lim

ǫ→0

∂D

∗

which satisﬁes ±Q > 0. Moreover, the map

(s, t), u

(s, t)) := u ◦ ϕ

(s, t) ∈ R ×M

for (s, t) ∈ Z

near inﬁnity

satisﬁes

(s, ·) − T s → c in C

∞

) as s → ±∞

for T := |Q| and a constant c ∈ R, while for every sequence s

→ ±∞, one can

restrict to a subsequence such that

, ·) → γ(T ·) in C

∞

, M

) as k → ∞

for s ome T -periodic Reeb orbit γ : R/T Z → M

. If γ is nondegenerate or Morse-

Bott, then in fact

(s, ·) → γ(T ·) in C

∞

, M

) as s → ±∞

We will not prove this result in its full strength, as in particular the last step

(when γ is nondegenerate or Morse-Bott) requires some asymptotic elliptic regularity

results that we do not have space to explain here. No t e however that most of the

above statement does not require any nondegeneracy assumption at all. The price

for this level of generality is that if s

, s

′

→ ±∞ are two distinct sequences, then

we have no guarantee in general that the two Reeb orbits obtained a s limits of

subsequences of u

, ·) and u

′

, ·) will be t he same; at present, neither an

example of this rather unpleasant possibility no r any general argument to rule it

out is known. If one of these orbits is assumed to be isolated, however—which is

always true when the Reeb vector ﬁeld is nondegenerate—then we will be able to

show that both are the same up to parametrization, hence geometrically, u

(s, t)

lies in arbitrarily small neighborhoods of the orbit γ as s → ±∞. This turns out to

be also true in the more general Morse-Bott setting, though it is then much harder

to prove since γ need not be isolated. Once u

(s, ·) is localized near γ, one can

use the no ndegeneracy condition as we did in t he Fredholm theory of Lecture

4 to

develop asymptotic regularity results that give much ﬁner cont r ol over the behavior

of u

as s → ±∞, implying in particular that u

(s, ·) → γ(T ·) in C

∞

, M

158 Chris Wendl

For details o n this step, we refer to the original sources: [HWZ96,HWZ01] for the

nondegenerate case, and [

HWZ96, Bou02] when the Reeb vector ﬁeld is Morse-

Bott. Those pa pers deal exclusively with the contact case, but the setting of general

stable Hamiltonia n structures is also dealt with in [

Sie08].

Ignoring the ﬁnal step f or now, the proof of Theorem

9.8 will reuse most of the

techniques that we already saw in our proof of removal of singularities in §

9.1. The

main idea is to use a combination of the monotonicity lemma and bubbling analysis

to show that unless u has a removable singularity, it is a proper map, and fo r a ny

sequence s

→ ±∞, the holomorphic half-cylinders deﬁned by

(s, t) = u ◦ ϕ

(s + s

, t)

on a sequence of increasingly large half-cylinders must have a subsequence converging

in C

∞

lo c

(R ×S

) to either a constant map or a trivial cylinder. The ﬁrst case will turn

out to mean (a s in Theorem

9.2) that the puncture is removable, and the second

implies asymptotic convergence to a closed Reeb orbit.

One maj or diﬀerence between the proof of Theorem

9.8 and removal of singu-

larities is that since

W is noncompact, sequences of curves in

W with uniformly

bounded ﬁrst derivatives need not be locally C

-bounded. This issue will arise both

in the bubbling argument to prove |du

(s, t)| ≤ C and in the analysis of the sequence

itself. In such cases, one can use the R-translation action

(9.5) τ

: R × M

→ R × M

: (r, x) 7→ (r + c, x) for c ∈ R

on suitable subsets of the cylindrical ends t o replace unbounded sequences with

uniformly C

-bounded sequences of curves mapping into R ×M

or R ×M

−

. These

R-translations a r e the reason why our deﬁnition of energy needs to be something

slightly more complicated than just the symplectic area

∗

Ω for a single choice

of symplectic form. To understand bubbling in the presence of arbitrarily large

R-translations, we will need the following lemma.

Lemma 9.9. Suppose J ∈ J(H) for some stable Hamiltonian structure H =

(ω, λ) on an odd-dimensional manifold M, a nd u : (

Σ, j) → (R × M, J) is a J-

holomorphic curve s atisfying

E(u) < ∞ and

∗

ω = 0.

Σ = C, then u is constant. If

Σ = R × S

, then u either is constant or is

biholomorphically equivalent to a trivial cylinder over a closed Reeb orbi t.

Proof. Denote ξ = ker λ and let

: T (R × M) → ξ

denote the projection along the subbundle spanned by ∂

(the unit vector ﬁeld in

the R-direction) and the Reeb vector ﬁeld R. Then since ω annihilates both ∂

and

R, for any local holomo r phic coordinates (s, t) on a subset of

Σ, the compatibility

of J|

with ω|

implies

∗

ω(∂

, ∂

) = ω(∂

u, ∂

u) = ω(∂

u, J∂

u) = ω(π

∂

u, Jπ

∂

u) ≥ 0 ,

Lectures on Symplectic Field Theory 159

hence

∗

ω ≥ 0 for every J-ho lomorphic curve, and equality means that u is

everywhere tangent to the subbundle spanned by ∂

and R. This implies that im u

is contained in the image of some J-holomor phic plane of the form

: C → R × M : s + it 7→ (s, γ(t)),

where γ : R → M is a (not necessarily periodic) orbit of R. If γ is not periodic,

then u

is embedded, hence there exists a unique (and necessarily holomorphic) map

Φ : (

Σ, j) → (C, i) such that u = u

◦ Φ. If on the other hand γ is periodic with

minimal period T > 0, then u

descends to an embedding of the cylinder

ˆu

: C/iT Z → R × M,

and we can view u

as a covering map to this embedded cylinder. Now t here exists

a unique holomorphic map Φ :

Σ → C/iT Z such that u = ˆu

◦ Φ. If

Σ = C, then

since π

Φ : C → C with u

◦

Φ = u. Relabeling symbols, we conclude that in general if

Σ = C, then u = u

◦ Φ for a holomorphic map Φ : C → C.

Let us consider all cases in which the factorzat ion u = u

◦ Φ exists, where

Φ : (

Σ, j) → (C, i) is holomorphic and

Σ = Σ \ Γ fo r a closed Riemann surface

(Σ, j). We will now use the removable singularity theorem for Φ :

Σ → S

\ {0} to

show that unless Φ is constant,

∗

= ∞ for suitable choices of f ∈ T . This

integral can be rewritten as

(9.6)

∗

d (f(s) dt) =

∗

′

(s) ds ∧ dt)

since ω

= d



f(r) λ



+ω and u

(s, t) = ( s, γ(t)). Since f

′

> 0, f

′

(s) ds∧dt is an area

form on C with inﬁnite area. We claim now that for suitable choices of f ∈ T , one

can ﬁnd an area form Ω on S

= C ∪ {∞} such that Ω ≤ f

′

(s) ds ∧ dt. To see this,

let us change coor dina tes so that ∞ becomes 0: setting Ψ : C

∗

→ C

∗

: z 7→ 1/z, a

slightly tedious but straightforward computation gives

∗

′

(s) ds ∧ dt) = f

′

(s/|z|

)

|z|



1 +

(2st)

|z|



ds ∧ dt

≥ f

′

(s/|z|

)

|z|

ds ∧dt for z = s + it ∈ C \ {0 }.

(9.7)

We need to show t hat this 2- form can be bounded away from 0 as z → 0. Let us

choose f ∈ T such that

(9.8) f( r) = ±



ǫ −



for ±r ≥ 1

and extend f arbitrarily to [−1, 1] such that f

′

> 0. We can then ﬁnd a constant

c > 0 such that f

′

satisﬁes

′

(r) > min

for all r ∈ R.

160 Chris Wendl

Plugging this into (9.7) gives

∗

′

(s) ds ∧ dt) ≥ min



|z|



ds ∧dt,

which clearly blows up as |z| → 0. With this established, we observe that for any

number C > 0, the fact that f

′

(s) ds ∧dt has inﬁnite area implies we can choose an

area form Ω on S

with

Ω ≤ f

′

(s) ds ∧ dt on S

\ {∞} and

Ω > C.

We now have two possibilities:

(1) If

∗

Ω < ∞, then Theorem

9.2 implies that the singularities of Φ :

Σ →

C at Γ are all removable, i.e. Φ extends to a holomor phic map (Σ, j) →

, i), which has a well-deﬁned mapping degree k ≥ 0. Then

∗

′

(s) ds ∧ dt) ≥

∗

Ω =

∗

Ω = k

Ω > kC.

Since C > 0 can be chosen a r bitrarily large, this implies

∗

= ∞

unless k = 0, meaning Φ is constant.

(2) If

∗

Ω = ∞ (meaning there is an essential singularity, cf. Exercise

9.5),

then since Φ

∗

′

(s) ds ∧ dt) ≥ Φ

∗

Ω, (

9.6) implies

∗

= ∞.

Since u is constant whenever Φ is, this completes the proo f for

Σ = C.

Σ = R × S

, then it remains to deal with the case where t he facto r izat ion

u = u

◦Φ does not exist because γ is periodic. If the minimal period is T > 0, then

let us in this case redeﬁne u

as a n embedded J-holomorphic trivial cylinder

: R × S

: (s, t) 7→ (T s, γ(T t)).

Since the new u

is embedded, we can now write u = u

◦Φ for a unique holomorphic

map Φ : R ×S

→ R × S

. Identif ying R × S

biholomorphically with S

\ {0, ∞},

we claim that Φ extends to a holomorphic map S

→ S

. Indeed, by the removable

singularity theorem, this is true if and only if

R×S

∗

Ω < ∞ for some area form

Ω on S

. Notice that u

∗

= T

· f

′

(T s) ds ∧ dt, deﬁnes an area form o n R × S

with ﬁnite area for any f ∈ T since

∞

−∞

′

(s) ds < ∞; this is equivalent to the

observation t hat trivial cylinders always have ﬁnite energy. Using the biholomorphic

map (s, t) 7→ e

2π(s+it)

to identif y R ×S

with C

∗

= S

\{0, ∞} and using coordinat es

z = x + iy on the latter, another tedious but straightforward computation gives

∗

4π

′



2π

log |z|



|z|

dx ∧ dy fo r z = x + iy ∈ C

∗

Now suppose f ∈ T is chosen as in (

9.8). Then one can check that the positive

function in front of dx ∧dy in t he above f ormula goes to +∞ as |z| → 0; this means

that one can ﬁnd an area form Ω o n C with Ω ≤ u

∗

on C

∗

. The singularity at

+∞ ∈ S

can be handled in a similar way, thus we can ﬁnd an area form Ω on S

Lectures on Symplectic Field Theory 161

such that Ω ≤ u

∗

on R × S

. Now since E(u) < ∞, we have

R×S

∗

Ω ≤

R×S

∗

R×S

∗

< ∞,

so by Theorem

9.2, Φ ha s a holomorphic extension S

→ S

, which is then a map

of degree k ≥ 0 with Φ

−1

({0, ∞}) ⊂ {0, ∞}. If k = 0 then Φ is constant, a nd so

is u. Otherwise, Φ is surjective and thus hits both 0 and ∞, but it can only do this

at either 0 o r ∞, thus it either ﬁxes both or interchanges them. After composing

with a biho lomorphic map of S

preserving R ×S

, we may assume without loss of

generality that Φ(0) = 0 and Φ(∞) = ∞. This makes Φ a polynomial with only one

zero, hence as a map on C ∪{∞}, Φ(z) = cz

for some c ∈ C

∗

. Up to biholomorphic

equivalence, Φ(z) is then z

, which appears in cylindrical coordinat es as the map

(s, t) 7→ (ks, kt), so u is now the trivial cylinder

u(s, t) = u

(ks, kt) = (kT s, γ(kT t))

over the k-fo ld cover of γ. 

Remark 9.10. It may be useful for some a pplicatio ns to observe that L emma

9.9

does not require M to be compact. In contrast, the compactness arguments in this

lecture almost always depend on the assumption that W and M

are compact—

without t his, one wo uld need add some explicit assumption to g uarantee local C

bounds on sequences of holomorphic curves, e.g. the assumption in Theorem 9.2

that u( D \ {0}) is contained in a compact subset.

Before continuing, it is worth noting that neither of the two deﬁnitions of energy

stated above (one for curves in

W and the o ther for symplectizations) is unique,

i.e. each can be tweaked in various ways such that the results o f this section still

hold. Indeed, the original deﬁnitions app ear ing in [

Hof93, BEH

03] are slightly

diﬀerent, but equivalent to these. The next lemma illustrates one further example

of this freedom, which will be useful in some of the a rguments below.

Lemma 9.11. Given a stable Hamiltonian structure H = (ω, λ) on M, a suﬃ-

ciently small constant ǫ > 0 as in (

9.3), and J ∈ J(H), co nsider the alterna tive

notion of energy for J-holomo rphic curves u : (

Σ, j) → (R × M, J) deﬁned by

(u) = sup

f∈T

∗

where ω

= d (f(r) λ) + ω and



f ∈ C

∞

(R, (a, b))



′

> 0



for some constants −ǫ ≤ a < b ≤ ǫ. Then if E(u) d enotes the energy as written in

Remark

9.7, there exists a constant c > 0, depending on the data a, b, ǫ and H but

not on u, such that

cE(u) ≤ E

(u) ≤ E( u).

162 Chris Wendl

Proof. The second of the two inequalities is immediate since T

⊂ T . For the

ﬁrst inequality, note that since ǫ > 0 is small, we can assume there exists a constant

c > 1 such that for every X ∈ T (R × M) and every κ ∈ [−ǫ, ǫ],

(9.9)

(ω + κ dλ)(X, JX) ≤ ω(X, JX) ≤ c(ω + κ dλ)(X, JX).

This uses (

9.3) and the fa ct that dλ a nnihilat es ker ω. Now suppose f ∈ T , choose

a constant δ ∈ (0, b − a] and deﬁne

f ∈ T

f(r) =

2ǫ

f(r) +

a + b

Then

′

(r) =

2ǫ

′

(r), and given a J-holomorphic curve u :

Σ → R × M, we can

write ω

= ω + f(r) dλ + f

′

(r) dr ∧ λ and use (

9.9) to estimate

∗

(ω + f( r) dλ) +

∗

′

(r) dr ∧ λ)

≤ c

∗

ω +

2ǫ

∗



′

(r) dr ∧ λ



≤ c

∗



ω +

f(r) dλ



2ǫ

∗



′

(r) dr ∧ λ



If c

≥

2ǫ

b−a

, then we can choose δ := 2ǫ/c

≤ b −a a nd rewrite the last expression a s

∗



ω +

f(r) dλ



2ǫ

∗



′

(r) dr ∧ λ



= c

∗



ω +

f(r) dλ +

′

(r) dr ∧ λ



= c

∗

≤ c

(u).

On the other hand if c

2ǫ

b−a

, we can set δ := b − a and write

∗



ω +

f(r) dλ



2ǫ

∗



′

(r) dr ∧ λ



≤

2ǫ

b − a

∗



ω +

f(r) dλ +

′

(r) dr ∧ λ



2ǫ

b −a

∗

≤

2ǫ

b − a

(u).



With this preparation out of the way, we now begin in earnest with the proof

of Theorem

9.8. Assume u :

D →

W is a J-holomor phic punctured disk satisfying

E(u) < ∞. Using t he maps ϕ

: Z

→

D deﬁned in (

9.4), we shall write

:= u ◦ ϕ

: Z

→

and observe tha t these reparametrizations have no impact on the energy, i.e.

E(u

) = sup

f∈T (h,r

)

(u ◦ ϕ

)

∗

= sup

f∈T (h,r

)

∗

= E(u).

Lectures on Symplectic Field Theory 163

Fix a Riemannian metric on

W that is translation-invariant on the cylindrical ends,

and ﬁx the standard metric on the half-cylinders Z

. We will use these metrics

implicitly whenever referring to quantities such as |du

(z)|.

Lemma 9.12. The re exists a constant C > 0 such that |du

(s, t)| ≤ C for all

(s, t) ∈ Z

Proof. We use a bubbling argument as in the proof of Lemma

9.3. Supp ose the

contrary, so t here exists a sequence z

= (s

, t

) ∈ Z

with R

:= |du

)| → ∞.

Choose a sequence ǫ

> 0 with ǫ

→ 0 but ǫ

→ ∞, a nd using Lemma

9.4,

assume without loss of generality that

|du

(z)| ≤ 2R

for all z ∈ D

Deﬁne a rescaled sequence of J-holomorphic disks by

: D

→

W : z 7→ u ◦ ϕ

+ z/R

These satisfy |dv

| ≤ 2 on their domains, but they are not necessarily C

-bounded

since t heir imag es may escape to inﬁnity. We distinguish three possibilities, at least

one of which must hold:

Case 1: v

(0) has a bounded subsequence.

Then the corresponding subsequence of v

: D

→

W is uniformly C

-bounded

on every compact subset and thus (by elliptic regularity) has a further subsequence

convergent in C

∞

lo c

∞

: C →

with |dv

∞

(0)| = lim

k→∞

|dv

(0)| = 1. But by the same argument we used in the

proof of Lemma

9.3, the fact that

∗

< ∞ for any choice of f ∈ T (h, r

)

implies

∗

∞

= 0,

hence v

∞

is constant, and this is a cont r adiction.

Case 2: v

(0) has a subsequence diverging to {+∞} × M

Restricting to this subsequence, suppose

(0) ∈ {r

} × M

so r

→ ∞, and assume without loss of generality that r

> r

for all k. Let

∈ (0, ǫ

] for each k denote the largest radius such that v

) ⊂ (r

, ∞)×M

Then

→ ∞ since |dv

| is bounded. Now using the R-translation maps τ

deﬁned

in (

9.5), deﬁne

˜v

:= τ

−r

◦ v

: D

→ R ×M

Since we’re using a translation-invariant metric on [r

, ∞) × M

, ˜v

is now a uni-

formly C

lo c

-bounded sequence of maps into R×M

. Elliptic regularity thus provides

a subsequence convergent in C

∞

lo c

∞

: C → R × M

164 Chris Wendl

which is J

-holomorphic, where J

∈ J(H

) denotes the restriction of J to [r

, ∞)×

, extended over R × M

by R-invariance. We claim,

(9.10) E(v

∞

) < ∞ and

∗

∞

= 0,

where E(v

∞

) is now deﬁned as in Remark

9.7. By Lemma 9.11, the ﬁrst part of the

claim will follow if we can ﬁx a constant a ∈ (−ǫ, ǫ) and establish a uniform bound

∗

∞

Ω

≤ C,

with Ω

:= ω

+ d



f(r) λ



, for all smooth and strictly increasing f unctions f :

R → (a, ǫ). For convenience in the following, we shall assume a > h(r

). Now if f

is such a function, then for any R > 0,

∗

∞

Ω

= lim

k→∞

∗

−r

Ω

= lim

k→∞

∗

Ω

where f

(r) := f(r − r

). Notice that the dependence of the last integral on f

limited to the interval (r

, ∞) since v

) ⊂ (r

, ∞) × M

. Then since f > a >

h(r

) by assumption, there exists for each k a function h

∈ T (h, r

) that matches

outside some neighborhood of (−∞, r

] and thus satisﬁes

∗

Ω

∗

≤

∗

)

∗

≤

∗

≤ E(u).

This is true for every R > 0 and thus proves the ﬁrst part of (

9.10). To establish the

second part, ﬁx R > 0 again and pick any f ∈ T (h, r

). Observe that since we can

assume (after perhaps passing to a subsequence) the disks D

) are all disjoint,

0 = lim

k→∞

)

∗

= lim

k→∞

∗

= lim

k→∞

˜v

∗

≥ lim

k→∞

˜v

∗

= lim

k→∞

˜v

∗

Ω

where now f

(r) := f(r + r

). Wr iting Ω

= ω

+ d



(r) λ



= ω

+ f

(r) dλ

′

(r) dr ∧ λ

, we can choose f such that f

′

(r) = f

′

(r + r

) → 0 as k → ∞,

so the third term contributes nothing to the integral. For the second term, let

:= lim

k→∞

(r) = lim

r→∞

f(r), so the calculation above becomes

0 ≥

∗

∞

(ω

+ f

dλ

) .

Now observe that since f

∈ [−ǫ, ǫ], condition (

9.3) implies that the 2-form ω

dλ

is nondegenerate on ξ

, and it also annihilates ∂

and R

, so the vanishing

of this integr al implies that v

∞

is everywhere tangent to ∂

and R

over D

. But

R > 0 was arbitrary, so this is true on the whole plane, which is equivalent to

∗

∞

= 0. With the claim established, we apply Lemma

9.9 and conclude that

∞

is constant, contr adicting the fact that |dv

∞

(0)| = 1 .

Case 3: v

(0) has a subsequence diverging to {−∞} × M

−

This is simply the mirror image of case 2: writing the restriction of J to (−∞, −r

]×

Lectures on Symplectic Field Theory 165

−

as J

−

, one can follow t he same bubbling argument but tr anslate up and instead

of down, giving rise to a limiting nonconstant J

−

-holomorphic plane v

∞

: C →

R ×M

−

that has ﬁnite energy but

∗

∞

−

= 0, in contradiction to Lemma

9.9. 

Consider now a sequence s

→ ∞ and construct the J-holomorphic half-cylinders

: [−s

, ∞) × S

→

W : (s, t) 7→ u

(s + s

, t).

The derivatives |du

| are uniformly bounded due to Lemma

9.12, though again, u

might fail to be uniformly bounded in C

. We distinguish three cases.

Case 1: u

(0, 0) has a bounded subsequence.

Then the corresponding subsequence of u

is uniformly C

-bounded on compact

subsets and thus has a further subsequence converging in C

∞

lo c

(R × S

) to a J-

holomorphic cylinder

∞

: R ×S

→

W .

For any f ∈ T (h, r

) and any c > 0, we have

[−c,c]×S

∗

∞

= lim

k→∞

[−c,c]×S

∗

≤ lim

k→∞

[−s

/2,∞)×S

∗

= lim

k→∞

/2,∞)×S

∗

= 0

(9.11)

since

∗

< ∞. It follows that

R×S

∗

∞

= 0, so u

∞

is a constant map to

some point p ∈

W , implying that after passing to a subsequence of s

, ·) → p in C

∞

W ) as k → ∞.

Case 2: u

(0, 0) has a subsequence diverging to {+∞} ×M

Passing to the corresponding subsequence of u

, suppose

(0, 0) ∈ {r

} × M

so r

→ ∞. Since the derivatives |du

| are uniformly bounded, we can then ﬁnd a

sequence of intervals [−R

−

, R

] ⊂ [−s

, ∞) such that

([−R

−

, R

] × S

) ⊂ [r

, ∞) × M

and R

→ ∞.

Now the t r anslated sequence

−r

◦ u

[−R

−

]×S

: [−R

−

, R

] × S

→ R × M

is uniformly C

-bounded on compact subsets and thus has a subsequence coverging

in C

∞

lo c

to a J

-holomorphic cylinder

∞

: R × S

→ R × M

where J

again denotes the restriction of J t o [r

, ∞) ×M

, extended over R ×M

by R-tra nslation. We claim that this cylinder satisﬁes

E(u

∞

) < ∞ and

R×S

∗

∞

= 0.

166 Chris Wendl

The proof of this should be an easy exercise if you understood the proofs of (9.10)

and (

9.11) above, so I will leave it as such. Lemma 9.9 now implies that u

∞

is either

constant or is a reparametrization of a trivial cylinder

: R × S

→ R × M

: (s, t) 7→ (T s, γ(T t))

for some Reeb orbit γ : R/T Z → M

with period T > 0. More precisely, all the

biholomorphic reparametrizations of R × S

are of the form (s, t) 7→ (±s + a, ±t +

b), thus after shifting the parametrization of γ, we can write u

∞

without loss of

generality in the f orm

(9.12) u

∞

(s, t) = ( ±T s + a, γ(±T t))

for some constant a ∈ R and a choice of signs to be determined below (see Lemma

9.16).

Case 3: u

(0, 0) has a subsequence diverging to {−∞} × M

−

Writing J

−

:= J|

(−∞,−r

]×M

−

∈ J( H

−

) a nd imitating the argument for case 2, we

suppose u

(0, 0) ∈ {−r

} × M

−

with r

→ ∞ and obtain a subsequence fo r which

◦ u

converges in C

∞

lo c

(R × S

) to a J

−

-holomorphic cylinder u

∞

: R × S

→

R ×M

−

, where u

∞

is either a constant or takes the form (

9.12) for some orbit Reeb

γ : R/T Z → M

−

of period T > 0.

Here is one easy consequence of the discussion so far. Use the Riemannian metric

W to deﬁne a metric dist

(·, ·) on the space of continuous loops S

→

W .

Lemma 9.13. Given δ > 0, there exists s

≥ 0 such that f or every s ≥ s

, the

loop u

(s, ·) : S

→

W satisﬁes

dist

(s, ·), ℓ) < δ,

where ℓ : S

→

W either is cons tant or is a loop of the form ℓ(t) = (r, γ(±T t)) in

, ∞)×M

or (−∞, r

]×M

−

for som e constant r ∈ R and Reeb orbit γ : R/T Z →

of period T > 0.

Proof. If not, then there exists a sequence s

→ ∞ such that each of the loops

, ·) lies at C

-distance at least δ away from any loop of the above form. How-

ever, the preceding discussion then gives a subsequence for which u(s

, ·) becomes

arbitrarily C

∞

-close to such a loop, so this is a contradiction. 

Lemma 9.14. If u :

D →

W is not bounded, then it is proper.

Proof. We use the monotonicity lemma. Suppose there exists a sequence

, t

) ∈ Z

such that u

, t

) diverges to {+∞} × M

. This implies s

→ ∞,

and we claim then that for every R ≥ r

, there exists s

≥ 0 such that

((s

, ∞) × S

) ⊂ (R, ∞) × M

If not, then we ﬁnd R ≥ r

and a sequence (s

′

, t

′

) ∈ Z

with s

′

→ ∞ such

that u

′

, t

′

) 6∈ (R, ∞) × M

for every k. By continuity, we are free to suppose

′

, t

′

) ∈ {R}×M

for all k since Lemma

9.13 implies u

({s

}×S

) ⊂ (2R, ∞)×

for k suﬃciently large. Using Lemma

9.13 again, we also have

({s

′

} × S

) ⊂ (R − 1, R + 1) × M

Lectures on Symplectic Field Theory 167

for all k large. Assuming 2R > R + 2 without loss o f generality, we can therefore

ﬁnd inﬁnitely many pairwise disjoint annuli of the form [s

′

, s

]×S

⊂ Z

containing

open sets that u maps properly to small balls centered at points in {R + 2} × M

Choosing any f ∈ T (h, r

), the monotonicity lemma implies that each of these

contributes at least some ﬁxed amount to

∗

, contradicting the assumption

that E(u) < ∞.

A similar argument works if u

, t

) diverges to {−∞}×M

−

, proving that for

every R ≥ r

, there exists s

≥ 0 with

((s

, ∞) ×S

) ⊂ (−∞, −R) ×M

−



If u is bounded, t hen the singularity at 0 is removable by Theorem

9.2. If not,

then Lemma

9.14 implies that it maps neighborhoods of the puncture to neighbor-

hoods of either {+∞}× M

or {−∞} × M

−

, and we shall refer to the puncture as

po sitive or neg ative accordingly.

Lemma 9.15. If the p unc ture is positive/negative, then the li mit

Q := lim

s→∞

(s, ·)

∗

∈ R

exists.

Proof. If the puncture is positive, ﬁx s

≥ 0 such that u

([s

, ∞) × S

) ⊂

, ∞) × M

. Then by Stokes’ theorem, it suﬃces to show t hat the integral

,∞)×S

∗

dλ

exists, which is true if

(9.13)

,∞)×S



∗

dλ



< ∞.

We claim ﬁrst that

,∞)×S

∗

< ∞. Indeed, for any s > s

and f ∈ T (h, r

we have

E(u) ≥

,s]×S

∗

,s]×S

∗

,s]×S

∗

d (f(r) λ

) .

Applying Stokes’ theorem, the second term becomes the sum of some number not

dependent on s and the integral

(s, ·)

∗

(f(r) λ

) =

[f ◦ u

(s, ·)] u

(s, ·)

∗

which is bounded as s → ∞ since f and |du

| are both b ounded. This proves that

,s]×S

∗

is also bounded as s → ∞, and since u

∗

≥ 0, the claim follows.

Now observe that since dλ

annihilates the kernel of ω

and the latter tames J

on ξ

, there exists a constant c > 0 such that |u

∗

dλ

| ≤ c|u

∗

|, implying (

9.13).

An analogo us argument works if the puncture is negative. 

The fact that

W is noncompact is not a pro ble m for this application of the monotonicity

lemma, as we are only using it in the compact subset W

⊂

W .

168 Chris Wendl

The number Q ∈ R deﬁned in the above lemma matches what we referred to in

the statement of Theorem

9.8 as the charge of the puncture.

Lemma 9.16. If the puncture is nonremovable and Q 6= 0, then the puncture is

po sitive/negative if an d only if Q > 0 or Q < 0 respectively. In either case, give n any

sequence s

→ ∞ with u

, 0) ∈ {±r

}×M

, one can ﬁnd a sequen ce R

∈ [0, s

]

with R

→ ∞ such that u

maps [s

− R

, ∞) × S

into the positive/negative

cylindrical end for every k, and the sequence of half-cylinders

: [−R

, ∞) ×S

→ R ×M

or u

: (−∞, R

] × S

→ R × M

−

deﬁned by u

(s, t) = τ

∓r

◦u

(s±s

, t) has a subsequence con v erge nt in C

∞

lo c

(R ×S

)

to a J

-holomorphic cylinder of the form

∞

: R ×S

→ R ×M

: (s, t) 7→ (T s + a, γ(T t))

for some constant a ∈ R and Reeb orb it γ : R/T Z → M

with period T := ±Q.

Proof. Assume the puncture is either positive or negative and Q 6= 0. In the

discussion preceding Lemma

9.13, we showed that the sequence u

′

(s, t) := τ

∓r

◦

(s + s

, t) deﬁned on [−R

, ∞) × S

has a subsequence convergent in C

∞

lo c

to a

-holomorphic cylinder u

′

∞

: R × S

→ R × M

which is either constant or of the

form

(9.14) u

′

∞

(s, t) = ( σT s + a, γ(σT t))

for some a ∈ R, σ = ±1 and a Reeb orbit γ : R/T Z → M

of period T > 0. We

then have

0 6= Q = lim

s→∞

(s, ·)

∗

= lim

k→∞

′

(0, ·)

∗

′

∞

(0, ·)

∗

so u

′

∞

cannot be constant , and from (

9.14) we deduce Q = σT , hence u

′

∞

(s, t) =

(Qs + a, γ(Qt)). Writing u

(s, t) = (u

(s, t), u

(s, t)) ∈ R × M

for s suﬃciently

large, it follows that every sequence s

→ ∞ admits a subsequence for which

∂

, ·) → Q in C

∞

, R),

and consequently ∂

(s, ·) → Q in C

∞

, R) as s → ∞. This proves that the sign

of Q matches the sign of the puncture whenever Q 6= 0. The stated formula for u

∞

now follows by adjusting all the appropriate signs in the case Q < 0. 

Lemma 9.17. If the p unc ture is nonremovab l e, then Q 6= 0.

Proof. Assume on the contrary that u is a proper map, say with a positive

puncture, but Q = 0. In this case, the arg ument of the previous lemma shows that

the limiting map u

∞

: R × S

→ R × M

will always be constant, thus for every

sequence s

→ ∞, t here exists a point p ∈ M

such that u

, 0) ∈ {r

} × M

with r

→ ∞ and

−r

◦ u

, ·) → (0, p) ∈ R × M

in C

∞

, R × M

) as k → ∞.

In particular, this implies that all derivatives of u

decay t o 0 as s → ∞. Intuitively,

this should suggest to you that portions of u

near inﬁnity will have improbably

small symplectic area, perhaps violating the monotonicity lemma—t his will t urn out

Lectures on Symplectic Field Theory 169

to be true, but we have to be a bit clever with our argument since u

is unbounded.

We will make this argument precise by translating pieces of u

downward so that

we only compute it s symplectic area in [0, 2] × M

. Fix a function f : R → (−ǫ, ǫ)

with f

′

> 0 and set Ω

= ω

+ d (f(r) λ

Given a small number δ > 0, we can ﬁnd s

≥ 0 such that |du

(s, t)| < δ for all

s ≥ s

and each of the loops u

(s, ·) for s ≥ s

is δ-close to a constant in C

Assume u

, 0) ∈ {R}×M

and choose s

> s

such that u

, 0) ∈ {R+2}×M

which is possible since u

(s, t) → {+∞} × M

as s → ∞. Now consider the J

holomorphic annulus

:= τ

−R

◦ u

]×S

: [s

, s

] × S

→ R × M

We claim that

]×S

∗

Ω

can be made arbitrarily small by choosing δ suitably

small. Indeed, we can use Stokes’ theorem to write this integral as

]×S

∗

Ω

]×S

∗

]×S

∗

d (f(r) λ

)

]×S

∗

, ·)

∗

(f(r) λ

) −v

, ·)

∗

(f(r) λ

)] .

The second term is small because f (r) is bounded and |v

(s, ·)

∗

| is small in pro-

portion to |dv

(s, t)| = |du

(s, t)| for s ≥ s

. For the ﬁr st term, observe that since

both of the loops v

, ·) fo r i = 0, 1 are nearly constant, they are contractible and

can be ﬁlled in with disks ¯v

: D → R × M

for which



¯v

∗



may be assumed

arbitrarily small. Moreover, since all of the loops v

(s, ·) are similarly contractible,

the union of these two disks with the annulus v

deﬁnes a closed cycle in M

that is

trivial in H

), hence the integral of the closed 2-fo rm ω

over this cycle vanishes,

implying

]×S

∗

¯v

∗

−

¯v

∗

which is therefore arbitrarily small, and this proves the claim.

To ﬁnish, notice that since v

maps its boundary component s to small neighbor -

hoods o f {0} × M

and {2} × M

, one can ﬁx a suitable choice of radius r

> 0

such that v

must pass through a point in p ∈ {1}×M

for which the boundar y of

is outside the ball B

(p). The monotonicity lemma then bounds the symplectic

area of v

from below by a constant times r

, but since we can also make this area

arbitrarily small by choosing δ smaller, this is a contr adiction.

As usual, the case of a negative puncture can be handled similarly. 

We’ve now proved every statement in Theorem

9.8 up to the ﬁnal detail about

the case where the asymptotic orbit is nondegenerate or Morse-Bott . The com-

plete proof of this part requires delicate analytical results from [

HWZ96,HWZ01,

HWZ96, Bou02], but we can explain the ﬁrst step for the nondegenerate case.

In the following, we say that a closed Reeb orbit γ : R/T Z → M

is isolated if,

after rescaling the domain to write it as an element of C

∞

, M

), there exists

a neighborhood γ ∈ U ⊂ C

∞

, M

) such that all closed Reeb orbits in U are

reparametrizations of γ.

170 Chris Wendl

Lemma 9.18. Suppose the puncture is nonremovable, write

(s, t) = ( u

(s, t), u

(s, t)) ∈ R × M

for s ≥ 0 suﬃcie ntly large, and suppose s

→ ∞ is a sequence an d γ : R/T Z → M

is a Reeb o rbit such that

, ·) → γ(T ·) in C

∞

, M

If γ is isolated, then for every neighborhood U ⊂ C

∞

, M

) of the set of parametriza-

tions {γ(· + θ) | θ ∈ S

}, we ha ve u

(s, ·) ∈ U for all suﬃciently large s.

Proof. Note ﬁrst that if γ is isolated, then its image admits a neighborhood

im γ ⊂ V ⊂ M

such that no point in V \ im γ is cont ained in another Reeb

orbit of period T . Indeed, we could otherwise ﬁnd a sequence of T -periodic Reeb

orbits passing throug h a sequence of points in V \ im γ that converge to a point

in im γ. Since their derivatives are determined by the Reeb vector ﬁeld and are

therefore bounded, the Arzel`a-Ascoli theorem then gives a subsequence of these

orbits converging to a reparametrization of γ, contradicting the assumption that γ

is isolated.

Arguing by contradiction, suppose now that there exists a sequence s

′

→ ∞

with u

, ·) 6∈ U for all k. We can nonetheless restrict to a subsequence for which

′

, ·) converges to some Reeb orbit ˜γ : R/T Z → M

. Then ˜γ is disjoint from

γ, and by continuity, one can ﬁnd a sequence s

′′

→ ∞ for which each u

′′

, 0)

lies in the region V some ﬁxed distance away fr om im γ. There must then be a

subsequence for which u

′′

, ·) converg es to another T -periodic or bit , but this is

impossible since no such orbits exist in V \ im γ. 

9.3. Degenerations of holomorphic curves

To motivate the SFT compactness theorem, we shall now discuss three examples

of phenomena that can prevent a sequence of ho lomorphic curves from having a

compact subsequence. The theorem will then tell us that these three things are, in

essence, the only things that can g o wrong.

Throughout this section and the next, assume J

→ J ∈ J

(ω

, r

, H

−

)

is a C

∞

-convergent sequence of ta me almost complex structures on the completed

cobordism

W . More generally, one can also allow the data ω, h and H

to vary

in C

∞

-convergent sequences, but let’s not clutter the notation too much. We shall

denote the restrictions of J to the cylindrical ends by

:= J|

,∞)×M

∈ J(H

), J

−

:= J|

(−∞,−r

]×M

−

∈ J(H

−

Suppose

:= [(Σ

, j

, Γ

−

, Θ

, u

)] ∈ M

g,m

, A

, γ

−

)

is a sequence of J

-holomorphic curves in

W with ﬁxed genus g ≥ 0 and m ≥ 0

marked points, va rying relative ho mo logy classes A

∈ H

(W,

∪

−

) a nd ﬁxed

collections of asymptotic orbits γ

= (γ

, . . . , γ

). Observe that the energies

E(u

) depend only on the orbits γ

and relative homology classes A

, so in partic-

ular, E(u

) is uniformly bounded whenever the relative homology class is also ﬁxed.

The fundamental question of this section is:

Lectures on Symplectic Field Theory 171

Question. If E(u

) is uniformly bounded and no s ubsequence o f u

co nverges to

an elem ent of M

g,m

(J, A, γ

, γ

−

) for any A ∈ H

(W,

∪

−

), what can happen?

9.3.1. Bubbling. Suppose (Σ

, j

, Γ

−

, Θ

) = (Σ, j, Γ

, Γ

−

, θ) is a ﬁxed se-

quence of domains, and choose Riemannian metrics on

Σ = Σ \ Γ and

W that are

translation-invar iant on the cylindrical ends of both. Suppose there exists a point

∈

Σ such that u

(ζ

) is contained in a compact subset for all k. Suppose also

that the maps u

Σ →

W are locally C

-bounded outside some ﬁnite subset

′

= {ζ

, . . . , ζ

} ⊂

Σ,

i.e. for every compact set K ⊂

Σ \ Γ

′

, there exists a constant C

> 0 independent

of k such that

|du

| ≤ C

on K.

Then elliptic regularity gives a subsequence that converges in C

∞

lo c

(

Σ \ Γ

′

) to a J-

holomorphic curve

∞

Σ \ Γ

′

→

with E(u

∞

) ≤ lim sup E(u

) < ∞, thus all the punctures Γ

∪ Γ

−

∪ Γ

′

of u

∞

are

either removable or positively or negatively asymptotic to Reeb orbits. We cannot

be sure that the asymptotic behavior of u

∞

at Γ

is the same as for u

, but let’s

assume this for now (§

9.3.2 below discusses some things that can happen if this does

not hold). Then to complete the picture, we need to understand not only what u

∞

is doing at the additional punctures Γ

′

, but also what is happening to u

near these

points as its ﬁrst derivative blows up. For t his we can apply the familiar r escaling

trick: choose for each ζ

a sequence z

→ ζ

such that |du

)| =: R

→ ∞, a long

with a sequence ǫ

→ 0 with ǫ

→ ∞, and using Lemma

9.4, assume without loss

of generality that |du

(z)| ≤ 2R

for all z in the ǫ

-ball about z

. For convenience,

we can choose a holo mo r phic coordinate system identifying a neighbor hood of ζ

with D ⊂ C and placing ζ

at the origin, so z

→ 0 in these coordinates, and

assume without loss of generality that they identify our chosen metric near ζ

with

the Euclidean metric. Now setting

(z) = u(z

+ z/R

) for z ∈ D

gives a sequence of J

-holomorphic maps v

: D

→

W whose energies and ﬁrst

derivatives are both unifor mly bounded. As in the a r guments of §2, we now have

three possibilities:

• If u

) has a bounded subsequence, t hen the corresponding subsequence

of v

converges in C

∞

lo c

∞

: C →

W with ﬁnite

energy.

• If u

) has a subsequence diverging to {±∞} × M

, then translating

by the R-action produces a limiting ﬁnite-energy plane v

∞

in t he posi-

tive/negative symplectization R × M

Viewing C a s the punctured sphere S

\ {∞}, the singularity of v

∞

at ∞ may be

removable, in which case v

∞

extends to a J-holomorphic sphere and we say that u

has “bubbled oﬀ a sphere” at ζ

. Alternatively, v

∞

may be positively or negatively

asymptotic to a Reeb orbit at ∞.

172 Chris Wendl

Figure 9.2 shows two scenarios that could occur for a sequence in which |du

blows up at three points Γ

′

= {ζ

, ζ

}. Both scenarios show u

∞

with ζ

and ζ

as removable singularities and ζ

as a negative puncture, but the behavior of the

vario us v

∞

reveals a wide spectrum of possibilities. In the lower-left picture, the

points u

) are bounded and bubble oﬀ a sphere v

∞

: S

→

W . The picture shows

that v

∞

passes through u

∞

(ζ

) at some point; this does not follow from our argument

so far, but in this situat ion one can use a more careful analysis of u

near ζ

to show

that it must be true, i.e. “bubbles connect”. At ζ

, we have u

) → {−∞} ×M

−

and v

∞

is a plane in R ×M

−

with a positive puncture asymptotic to the same orbit

as ζ

; the coincidence of these orbits is another detail that does not follow from the

analysis above but turns out to be true in the general picture. The situation at

allows two diﬀerent interpretations: v

∞

could be the plane with negative end in

R ×M

, meaning u

) → {+∞}×M

, and the picture then shows an additional

plane in

W with a positive end approaching the same a symptotic orbit as v

∞

as well

as a point passing thr ough u

∞

(ζ

). One would need to choose a diﬀerent rescaled

sequence near ζ

to ﬁnd this extra plane, but as we will see, the SFT compactness

theorem dictates that some such object must be there. Alternatively, u

) could

also be bounded at ζ

, in which case v

∞

must be the plane in

W with positive

end, and the extra plane above this is something that one could ﬁnd via a diﬀerent

choice of rescaled sequence. In general, the range of actual possibilities can involve

arbitrarily many additional curves that could be discovered via diﬀerent choices of

rescaled sequences: e.g. there could be entire “bubble trees” as shown in the lower-

right picture, where each v

∞

is only one of several curves that arise a s limits of

diﬀerent parametrizations of u

near ζ

. O ne good place to read about the analysis

of bubble trees is [

HWZ03, §4].

9.3.2. Breaking. Figure

9.2 already shows some phenomena that could be in-

terpreted as “breaking” in the Floer-theoretic sense, but breaking can also ha ppen

when no derivatives are blowing up, simply due to the f act that our domains are non-

compact. Figures 9.3 and 9.4 show three such scenarios, where we assume again that

(Σ

, j

, Γ

−

, Θ

) = ( Σ, j, Γ

, Γ

−

, Θ) is a ﬁxed sequence of domains, and

Σ = Σ\Γ

and

W carry Riemannian metrics t hat are translation-invariant on the cylindrical

ends such that

|du

| ≤ C everywhere on

for some constant C > 0 independent o f k. This is a stronger conditio n than we had

in §

9.3.1, and if there exists a point ζ

∈

Σ such that u

(ζ

) is bounded, it implies

that u

∞

converges in C

∞

lo c

(

Σ) to a J-ho lomorphic map

∞

Σ →

with E(u

∞

) ≤ lim sup E(u

) < ∞. Convergence in C

∞

lo c

is, however, not very strong:

there may in general b e no r elat ion between the asymptotic behavior of u

∞

and u

at corresponding punctures, e.g. the top scenario in Figure

9.3 shows a case in

which a negat ive puncture of u

becomes a removable singularity of u

∞

. Whenever

this happens, there must be more to the story: in this example, one can choose

Lectures on Symplectic Field Theory 173

R × M

−

R × M

−

R × M

−

1 ζ

∞

Figure 9.2. Two possible pictures of spheres and/or planes that

can bubble oﬀ when the ﬁrst derivative blows up near three points.

holomorphic cylindrical coordinates (s, t) ∈ (−∞, 0] × S

⊂

Σ near the negative

puncture of u

and ﬁnd a sequence s

→ ∞ such that the sequence of half-cylinders

(−∞, s

] × S

→

W : (s, t) 7→ u

(s − s

, t)

174 Chris Wendl

is uniformly C

-bounded and thus converges in C

∞

lo c

(R × S

) to a ﬁnite-energy J-

holomorphic cylinder v

−

: R × S

→

W . In the picture, v

−

turns out to have a

removable singularity at +∞ mapping to the same point as the removable singularity

of u

∞

, and it s negative puncture appro aches the same orbit as the negative puncture

of u

More complicated things can happen in general: the bot t om scenario in this same

ﬁgure shows a case where all three singularities of u

∞

are removable, thus it extends

to a closed curve, while at one of the positive cylindrical ends [0, ∞) × S

⊂

Σ of

, we can ﬁnd a sequence s

→ ∞ such that the half-cylinders

[−s

, ∞) × S

→

W : (s, t) 7→ u

(s + s

, t)

are uniformly C

-bounded and converge in C

∞

lo c

(R ×S

) to a J-holomorphic cylinder

: R × S

→

W with one removable singularity and one positive puncture. At

the other positive end, we can perform the same trick in two distinct ways for two

sequences s

→ ∞, one diverging faster than t he ot her: the result is a pair of J-

holomorphic cylinders v

, v

: R × S

→

W , the former with both singularities

removable (thus forming a holomorphic sphere in the picture), and the latter with

one removable singularity and one positive puncture.

It can get weirder. Remember that

W is also noncompact!

In each of the above scenarios, we tacitly assumed that all of the various se-

quences obtained by reparametrizing portions of u

were locally C

-bounded, thus

all of the limits were curves in

W . But it may also happen that some of these se-

quences are C

lo c

-bounded while others locally diverge toward {±∞} × M

; in fact,

two such sequences that both diverge toward, say, {+∞}× M

, might even locally

diverge inﬁnitely far f rom each other, meaning one of them approaches {+∞}×M

quantitatively faster than the ot her. This phenomenon leads to the notion of limiting

curves with multiple levels.

In Figure

9.4, we see a scenario in which u

satisﬁes the same conditions as above,

except that instead of u

(ζ

) b eing bounded, it diverges to {+∞} × M

. It follows

that after applying suitable R-translations, a subsequence converges in C

∞

lo c

(

Σ) to a

-holomorphic curve

∞

Σ → R × M

with ﬁnite energy. In the example, all three of its punctures are nonremovable, but

two of them approach orbits that have nothing to do with the asymptotic orbits

of u

. Now observe that since u

has a negative cylindrical end (−∞, 0] × S

⊂

Σ,

one can necessarily ﬁnd a sequence s

→ ∞ such that u

(−s

, 0) is bounded, and

the sequence of half-cylinders

(−∞, s

] × S

→

W : (s, t) 7→ u

(s − s

, t)

is then unifo rmly C

-bounded and thus has a subsequence convergent in C

∞

lo c

(R×S

)

to a ﬁnite-energy J-holomorphic cylinder v

: R × S

→

W . In the picture, v

has

both a positive and a negative puncture, but its negative end again approaches a

diﬀerent Reeb orbit from the negative ends of u

, so one can deduce that there

must be still more happening near −∞: there exists another sequence s

′

→ ∞ with

Lectures on Symplectic Field Theory 175

∞

−

Figure 9.3. Even with ﬁxed conformal structures on t he domains

and without bubbling, a sequence o f punctured holomorphic curves in

W can break to produce multiple curves in

W with extra removable

punctures. The picture shows two such scenarios.

′

−s

→ ∞ such that suitable R-translations of the half-cylinders

(−∞, s

] × S

→ (−∞, −r

] × M

−

: (s, t) 7→ u

(s −s

′

, t)

deﬁne uniformly C

-bounded maps into R×M

−

, giving a subsequence that converges

in C

∞

lo c

(R × S

) to a ﬁnite- energy J

−

-holomorphic cylinder

−

: R × S

→ R × M

−

Finally, the fact that u

∞

has a positive asymptotic or bit diﬀerent from those of u

indicates that something more must also be happening near + ∞: in the example,

one of the positive ends [0, ∞) × S

⊂

Σ admits a sequence s

→ ∞ such that

, 0) ∈ {r

} × M

for some r

→ ∞, and suitable R-translations of

[−s

, ∞) ×S

→ [r

, ∞) × M

: (s, t) 7→ u

(s + s

, t)

become a uniformly C

-bounded sequence of half-cylinders in R × M

, with a sub-

sequence converging in C

∞

lo c

(R × S

) to a ﬁnite- energy J

-holomorphic cylinder

: R ×S

→ R ×M

that connects the errant asymptotic orbit of u

∞

to the corresponding orbit of u

. One

can now perform the same tr ick at the other positive end of

Σ, as there necessarily

also exists a sequence s

′

→ ∞ in this end such that u

′

, 0) ∈ {r

} × M

for

the same sequence r

→ ∞ as in the above discussion. The resulting limit curve

176 Chris Wendl

R × M

−

∞

−

Figure 9.4. Diﬀerent portions o f a breaking sequence of curves may

also become inﬁnitely far apar t in the limit, so that some live in

while others live in the symplectization of M

or M

−

: R ×S

→ R × M

however is not guaranteed to be interesting: in the picture,

it turns out to be a trivial cylinder.

The type of degeneration shown in Figure

9.4 happens whenever the sequence u

does interesting things in multiple regions of its domain that are sent increasingly

far away from each other in the image. The usual picture of

W that collapses

the cylindrical ends to a ﬁnite size therefore becomes increasingly inadequate for

visualizing u

as k → ∞: the middle picture in Figure

9.4 deals with this by

expanding the scale of the cylindrical ends so that the convergence to upper and

lower levels becomes visible.

9.3.3. The Deligne-Mumford space of Riemann surfaces. We next need

to relax the assumption that the Riemann surfaces (Σ

, j

, Γ

⊔Γ

−

⊔Θ

) are ﬁxed.

Recall that for integers g ≥ 0 a nd ℓ ≥ 0, the moduli space of pointed Riemann

surfaces is the space of equivalence classes

g,ℓ

= {(Σ, j, Θ)}



∼,

where (Σ, j) is a closed connected Riemann surface of g enus g, Θ ⊂ Σ is an or-

dered set of ℓ distinct points, and (Σ, j, Θ) ∼ (Σ

′

, j

′

, Θ

′

) whenever there exists a

biholomorphic map ϕ : (Σ, j) → (Σ

′

, j

′

) taking Θ t o Θ

′

with the ordering preserved.

This space is fairly easy to understand in the ﬁnitely many cases with 2g + ℓ < 3,

e.g. M

0,ℓ

is a one-point space for each ℓ ≤ 3. We say that (Σ, j, Θ) is stable when-

ever χ(Σ \ Θ ) < 0, which means 2g + ℓ ≥ 3. In the stable case, one can show

that every pointed Riemann surface has a ﬁnite automorphism gro up, and M

g,ℓ

is a

Lectures on Symplectic Field Theory 177

smooth orbifold o f dimension 6g −6 + 2ℓ. It is generally not compact, but it admits

a natural compactiﬁcation

g,ℓ

⊃ M

g,ℓ

known as the Deligne-Mumford c ompactiﬁcation. We shall now give a sketch

of this construction from t he perspective of hyperbolic geometry; for more details,

see [Hum97, SS92].

We recall ﬁrst the f ollowing standard result.

Theorem (Uniformization theorem). Every simp l y connected Riemann surface

is biholomorphically equivalent to either the Riemann sphere S

= C ∪ {∞}, the

co mplex plane C or the upper half plane H = {Im z > 0} ⊂ C.

The uniformization theorem implies that every Riemann surface can be presented

as a quotient of either ( S

, i), (C, i) or (H, i) by some f reely acting discrete group

of biholomorphic transformations. The only punctured surface

Σ = Σ \ Θ that has

as its universal cover is S

itself. It is almost as easy to see which surfaces are

covered by C, as the only biholomorphic tr ansformations on (C, i) with no ﬁxed

points are the translations, so every freely acting discrete subgroup of Aut(C, i) is

either trivial, a cyclic group of translations or a latt ice. The resulting quotients are,

respectively, (C, i), (R × S

, i)

∼

(C \ {0}, i) and the unpunctured tori (T

, j). All

stable pointed Riemann surfaces are thus quotients o f (H, i).

Proposition 9.19. There exists on (H, i) a compl e te Riemannian metric g

co nstant c urvature −1 that deﬁnes the same conformal structure as i and has the

pro perty that all conformal transformations on (H, i) are also i sometries of (H, g

Proof. We deﬁne g

at z = x + iy ∈ H by

where g

is the Euclidean metric. The conformal transformat ions on (H, i) are given

by fractional linear t ransformations

Aut(H, i) =



ϕ(z) =

az + b

cz + d



a, b, c, d ∈ R, ad − bc = 1



{±1}

= SL(2, R)/{±1} =: PSL(2, R),

and one can check that each of these deﬁnes an isometry with respect to g

. One

can also compute that g

has curvature −1, and the geodesics of g

are precisely

the lines and semicircles tha t meet R orthogona lly, parametrized so that they exist

for all forward and backward time, thus g

is complete. For more details on all o f

this, the book by Hummel [

Hum97] is highly recommended. 

By lifting to universal covers, this implies the following.

Corollary 9.20. For every pointed Riemann surface (Σ, j, Θ) such that χ(Σ \

Θ) < 0, the punctured Riemann surface (Σ \ Θ, j) admits a complete Riemannian

metric g

of constant curvature −1 that deﬁnes the same conformal structure as j,

and has the property that all biholom o rphic transform ations on (Σ \ Θ, j) are also

isometries of (Σ \ Θ, g

178 Chris Wendl

Figure 9.5. Two distinct pair-of-pants decomp ositions for t he same

genus 1 R iemann surface with three marked points. The decomposi-

tions are shown from two perspectives: the pictures at the right are

meant to give a more accurate impression of the Poincar´e metric,

which becomes singular and forms a cusp at each marked point .

The metric g

in this corollary is often called the Poincar´e metric . It is uniquely

determined by j.

Every class in π

(

Σ) contains a unique geodesic for g

. Now suppose C ⊂

Σ is a

union of disjoint embedded geodesics such tha t each connected component of

Σ \C

has the homotopy type of a disk with two holes. The components are then called

singular pairs of pants, and the result is called a pair-of-pants decomposition

of (

Σ, j). Two examples for the case g = 1 and ℓ = 3 are shown in Figure

9.5.

A pa ir -of-pants decomposition for (Σ, j, Θ ) gives rise to a local par ametrization

of M

g,ℓ

near [(Σ, j, Θ)], known as the Fenchel-Nielsen coordinates. These consist

of two parameters that can be associated to each of t he geodesics γ ⊂ Σ in the

decomposition, namely the length ℓ(γ) > 0 of the geodesic and a twist parameter

θ(γ) ∈ S

, which describes how the two neighboring pairs of pants are glued together

along γ. Note t hat by computing Euler characteristics, there a re always exactly

−χ(Σ \Θ) = 2g −2 + ℓ pairs of pants in a decomposition, so that the total number

of geodesics involved is [3(2g − 2 + ℓ) − ℓ] /2 = 3g −3 + ℓ, thus one can read oﬀ the

formula dim M

g,ℓ

= 6g − 6 + 2ℓ from this geometric picture.

One can also see the noncompactness of M

g,ℓ

in this picture quite concretely:

the twist parameters belong to a compact space, but each length parameter can

Lectures on Symplectic Field Theory 179

potentially shrink to 0 or blow up to ∞ as j (and hence g

) is deformed. It turns

out that the latter possibility is an illusion, but one may need to switch to a diﬀerent

pair-of-pants decomposition to see why:

Theorem. For every pair of integers g ≥ 0 and ℓ ≥ 0 with 2g + ℓ ≥ 3, there

exists a constant C = C(g, ℓ) > 0 such that every [(Σ, j, Θ)] ∈ M

g,ℓ

admits a pair-

of-pants decomposition in which all geodesics bounding the pairs of pants have length

at most C.

This theorem implies that from a hyperbolic perspective, the only meaningful

way for stable pointed Riemann surfaces to degenerate is when some of the bounding

geodesics in a pair-of-pants decomposition shrink to length zero. Figure

9.6 shows

several examples of degenerate Riemann surfaces that can arise in this way for g = 1

and ℓ = 3, giving elements of the space that we will now deﬁne as

1,3

Definition 9.21. A nodal Riemann surface with ℓ ≥ 0 marked points and

N ≥ 0 nodes is a tuple (S, j, Θ , ∆) consisting of:

• A closed but not necessarily connected Riemann surface (S, j);

• An ordered set of ℓ points Θ ⊂ S;

• An unordered set of 2N points ∆ ⊂ S \ Θ equipped with an involution

σ : ∆ → ∆. Each pair {z, σ(z)} for z ∈ ∆ is referred to as a node.

Let

S denote the closed surface obtained by performing connected sums o n S at

each node {z

, z

−

} ⊂ ∆. We then say that (S, j, Θ, ∆) is connected if and only if

S is connected, and the genus of

S is called the arithmetic genus of (S, j, Θ, ∆).

We say that (S, j, Θ, ∆) is stable if every connected component of S \ (Θ ∪ ∆)

has negative Euler characteristic. Finally, two nodal Riemann surfaces (S, j, Θ, ∆)

and (S

′

, j

′

, Θ

′

, ∆

′

) are considered equivalent if there exists a biholomorphic map

ϕ : (S, j) → (S

′

, j

′

) taking Θ to Θ

′

with the ordering preserved and taking ∆ to ∆

′

such that nodes are mapped to nodes.

The nodes {z

, z

−

} ⊂ ∆ are typically represented in pictures as self-intersections

of S, cf. Figure

9.6. We can think of the stable nodal surfaces as precisely those

which admit (possibly singular) pair-of-pants decompositions. All noda l Riemann

surfaces we consider will be assumed connected in the sense deﬁned above unless

otherwise noted; note that S itself can nonetheless be disconnected, as is the case

in f our out of the six nodal surfaces shown in Figure

9.6.

We now intr oduce some further terminology and notation that will be useful

in the next section as well. Whenever

Σ = Σ \ Γ is obtained by puncturing a

Riemann surface (Σ, j) at ﬁnitely many points Γ ⊂ Σ, we shall deﬁne the circle

compactiﬁcation

Σ :=

Σ ∪

[

z∈Γ

where f or each z ∈ Γ, the circle δ

is deﬁned as a “ha lf-projectivization” of the

tangent space at z:

:= (T

Σ \ {0})

∗

180 Chris Wendl

Figure 9.6. Starting from each of the pair-of-pants decompositions

for the g = 1 and ℓ = 3 case from Figure

9.5, shrinking geodesic

lengths to zero produces various examples of stable nodal Riemann

surfaces belonging to

1,3

with the positive real numbers R

∗

acting by scalar multiplication. To understand

the topology of

Σ, one can equivalently deﬁne it by choosing holomorphic cylindrical

coordinates [0, ∞) × S

⊂

Σ near each z, and replacing the open half-cylinder with

[0, ∞] × S

, where δ

is now the circle at inﬁnity {∞} × S

. There is no natural

choice of global smooth structure on

Σ, but it is homeomorphic to an oriented surface

with boundary and carries both smooth and conformal structures on its interior, due

to the obvious identiﬁcation

Σ =

Σ \

[

z∈Γ

⊂ Σ.

Lectures on Symplectic Field Theory 181

The conformal structure of Σ at each z ∈ Γ does induce on each of the circles δ

orthogonal structure, meaning a preferred class of homeomorphisms to S

that

are all related to each other by rotations. One can therefore speak of ort hogonal

maps δ

→ δ

′

for z, z

′

∈ Γ , which ar e always homeomorphisms and can either

preserve or reverse orientation.

Now if (S, j, Θ, ∆) is a nodal R iemann surface, we let

S = S \ ∆ and f orm the

circle compactiﬁcation

S, which has the topology of a compact oriented surface with

boundary. Given a node {z

, z

−

} ⊂ ∆, a decoration for {z

, z

−

} is a choice of

orientation reversing orthogonal map

Φ : δ

→ δ

−

We say that (S, j, Θ, ∆) is a decorated nodal surface if it is equipped with a

choice of decoration Φ for every node, or partially decorated if Φ is deﬁned for

some subset of the nodes. A partial decoration Φ gives rise to another compact

oriented surface



∼,

where the equivalence relation identiﬁes δ

with δ

−

via Φ for each decorated node

, z

−

} ⊂ ∆. Note that if every node is decorated, then

has the topology of a

closed connected and oriented surface whose genus deﬁnes the arithmetic genus of

(S, j, Θ, ∆) according to Deﬁnition

9.21. We shall denote the collection of special

circles in

where boundray components δ

, δ

−

⊂ ∂

S have been ident iﬁed by

⊂

Since

\ (∂

∪ C

) has a natural identiﬁcation with

S, it inherits smooth and

conformal structures which degenerate along C

and ∂

. We will say that two

partially decorated nodal Riemann surfaces (S, j, Θ, ∆, Φ) and (S

′

, j

′

, Θ

′

, ∆

′

, Φ

′

) are

equivalent if (S, j, Θ, ∆) and (S

′

, j

′

, Θ

′

, ∆

′

) are equivalent via a biholomorphic map

ϕ : (S, j) → (S

′

, j

′

) that extends continuously from

S →

′

to a homeomorphism

→

′

Now if 2g + ℓ ≥ 3, deﬁne

g,ℓ

as the set of equivalence classes of stable nodal

Riemann surfaces with ℓ marked points and arithmetic genus g. There is a natural

inclusion

g,ℓ

⊂

g,ℓ

by r egarding each pointed Riemann surface (Σ, j, Θ) as a nodal Riemann surface

(Σ, j, Θ, ∆) with ∆ = ∅. The most important property of

g,ℓ

is that it admits

the structure of a compact metrizable topological space for which the inclusion

g,ℓ

֒→

g,ℓ

is cont inuous onto an open subset. Rather than formulating all of

this in precise terms, let us state the main corollary that is important to know in

practice.

Theorem 9.22. Fix g ≥ 0 and ℓ ≥ 0 with 2g + ℓ ≥ 3. Then for any sequence

[(Σ

, j

, Θ

)] ∈ M

g,ℓ

, there exists a stable nodal Riemann surface [(S, j, Θ, ∆)] ∈

g,ℓ

such that a f ter restricting to a subsequence,

[(Σ

, j

, Θ

)] → [(S, j, Θ, ∆)]

182 Chris Wendl

in the follo wing sense: (S, j, Θ, ∆) admits a decoration Φ such that for suﬃcie ntly

large k, there are ho meomorphisms

ϕ :

→ Σ

smooth outside of C

, which map Θ to Θ

pre s erving the ordering and satisfy

∗

→ j in C

∞

lo c

(

\ C

∆

As one mig ht gather from the above statement, one could just a s well deﬁne a

compact metrizable topology on the space of equivalence classes of decorated nodal

Riemann surfaces and then characterize the topology of

g,ℓ

via the natural pro-

jection that forgets the decorations.

Exercise 9.23. The space M

0,4

has a natural identiﬁcation with S

\{0, 1, ∞},

deﬁned by choosing the unique identiﬁcation of any 4-pointed Riemann sphere

, j, (z

, z

)) with C ∪ {∞} such that z

, z

are identiﬁed with 0, 1, ∞

respectively, while z

is sent to some point in S

\{0, 1, ∞}. Show that this extends

continuously to an identiﬁcation of

0,4

with S

. What do the t hree nodal curves

0,4

\ M

0,4

look like in terms of pair-of-pants decompositions?

9.4. The SFT compactness theorem

We now introduce the natural compactiﬁcation of M

g,m

(J, A, γ

, γ

−

9.4.1. Nodal curves. A punctured J-holomorphic nodal curve in (

W , J)

with m ≥ 0 marked points consists of the data (S, j, Γ

, Γ

−

, Θ, ∆, u), where

• (S, j, Γ

⊔ Γ

−

⊔ Θ, ∆) is a nodal Riemann surface, with |Θ| = m;

• u : (

S, j) → (

W , J) for

S := S \ (Γ

∪ Γ

−

) is an asymptotically cylindrical

J-holomorphic map with p ositive punctures Γ

and negative punctures Γ

−

such that for each node {z

, z

−

} ⊂ ∆, u(z

) = u(z

−

Equivalence of two nodal curves

, j

, Γ

−

, Θ

, ∆

, u

) ∼ (S

, j

, Γ

−

, Θ

, ∆

, u

)

is deﬁned as the existence of an equivalence of nodal Riemann surfaces ϕ : (S

, j

, Γ

⊔

−

⊔ Θ

, ∆

) → (S

, j

, Γ

⊔ Γ

−

⊔ Θ

, ∆

) such that u

= u

◦ ϕ. We say that

(S, j, Γ

, Γ

−

, Θ, ∆, u) is connected if and only if the nodal Riemann surface (S, j, Γ

⊔

−

⊔Θ, ∆) is connected, and its arithmetic genus is then deﬁned to be the arith-

metic genus of the latter. We say that (S, j, Γ

, Γ

−

, Θ, ∆, u) is stable if every con-

nected compo nent of S \(Γ

∪Γ

−

∪Θ ∪∆) on which u is constant has negative Euler

characteristic. Note that the underlying nodal Riemann surface (S, j, Γ

⊔Γ

−

⊔Θ, ∆)

need not be stable in general.

Nodal curves are sometimes also referred to as holomorphic buildings of hei ght 1.

These are the objects that form the Grom ov compactiﬁcation of M

g,m

(J, A) when W

is a closed symplectic ma nif old. One can now roug hly imagine how the compactness

theorem in that setting is proved: given a converging sequence of almost complex

structures J

→ J and a sequence [( Σ

, j

, Θ

, u

)] ∈ M

g,m

, A

) with unifo rmly

bounded energy, we can ﬁrst add some auxiliary marked points if necessary to assume

that 2g +m ≥ 3. Now a subsequence of the domains [(Σ

, j

, Θ

)] ∈ M

g,m

converges

Lectures on Symplectic Field Theory 183

to an element of the Deligne-Mumford space [(S, j, Θ, ∆)] ∈ M

g,m

. Concretely,

this means that for large k, our sequence in M

g,m

, A

) admits representatives

(Σ, j

′

, Θ, u

′

), with Σ a ﬁxed surface with ﬁxed marked points Θ ⊂ Σ, and (S, j, Θ, ∆)

admits decoratio ns Φ so t hat one can identify

with Σ and ﬁnd

′

→ j in C

∞

lo c

(Σ \ C)

for some collection of disjoint circles C ⊂ Σ. The connected component s o f (Σ\C, j)

are then biholomorphically equivalent to t he connected components of (S \ ∆, j) ,

and if the newly reparametrized maps u

′

: Σ → W a r e uniformly C

lo c

-bounded o n

Σ \ C, then a subsequence converges in C

∞

lo c

(Σ \ C) to a limiting ﬁnite-energy J-

holomorphic map u

∞

: (S \∆, j) → (W, J), whose singularities a t ∆ are removable.

In particularly nice cases, this may be the end of the story, and our subsequence

of [(Σ

, j

, Θ

, u

)] ∈ M

g,m

, A

) converges to the nodal curve [(S, j, Θ, ∆, u

∞

)];

in particular the domain [(S, j, Θ, ∆)] in this case is stable and is thus an element

g,m

. But more complicated things can also happen, e.g. u

′

might not be

-bounded, in which case there is bubbling. The bubbles that arise will be either

planes or spheres, so they produce extra doma in compo nents with nonnegative Euler

characteristic, but since they are never constant, the limiting noda l curve is still

considered stable. Similarly, since Σ\C is not compact, there can also be breaking as

in Figure

9.3, producing more non-stable domain components which can b e cylinders

in addition to planes and spheres—but again, the limiting map on these components

will never be constant.

9.4.2. Holomorphic buildings. Only a small subset o f the phenomena ob-

served in §

9.3 can be described via nodal curves: we’ve seen that in general, we also

have to allow “broken” curves with mult iple “levels”. This notion can be formalized

as f ollows.

Given integers g, m, N

, N

−

≥ 0, a holomorphic building of height N

−

|1|N

with ar it hmetic genus g and m marked points is a tuple

u = (S, j, Γ

, Γ

−

, Θ, ∆

, ∆

, L, Φ, u),

with the va r ious data deﬁned as follows:

• The domain (S, j, Γ

⊔Γ

−

⊔Θ, ∆

⊔∆

) is a connected but not necessarily

stable nodal Riemann surface of arithmetic genus g, where |Θ| = m, and

the involution on ∆

⊔∆

is assumed to preserve the subsets ∆

and ∆

Matched pairs in these subsets are called the nodes and breaking pairs

respectively of u. The marked points of u are the point s in Θ, while Γ

and Γ

−

are its positive and negative punctures respectively.

• The level structure is a locally constant function

L : S → {−N

−

, . . . , −1, 0, 1 , . . . , N

}

that attains every value in {−N

−

, . . . , N

} except possibly 0, and satisﬁes:

(1) L(z

) = L(z

−

) for each node {z

, z

−

} ⊂ ∆

;

(2) Each breaking pair {z

, z

−

} ⊂ ∆

can be la belled such that L(z

) −

L(z

−

) = 1;

(3) L(Γ

) = {N

} and L(Γ

−

) = {−N

−

184 Chris Wendl

• The decoration is a choice of orientation- r eversing orthogonal map

−→ δ

−

for each breaking pair {z

, z

−

} ⊂ ∆

• The map is an asymptotically cylindrical pseudoholomorphic curve

u : (

S := S \ (Γ

∪ Γ

−

∪ ∆

), j) →

N∈{−N

−

,...,N

}

(

, J

where

(

, J

) :=











(R ×M

, J

) for N ∈ {1, . . . , N

(

W , J) for N = 0,

(R ×M

−

, J

−

) for N ∈ {−N

−

, . . . , −1},

and u sends

S ∩ L

−1

(N) into

for each N, with positive punctures at

and negative punctures at Γ

−

. Moreover,

u(z

) = u(z

−

) for every node {z

, z

−

} ⊂ ∆

and for each breaking pair {z

, z

−

} ⊂ ∆

labelled with L(z

)−L(z

−

) = 1,

u has a positive puncture at z

−

and a negative puncture at z

asymptotic

to the same orbit, such that if u

: δ

→ M

and u

−

: δ

−

→ M

denote

the induced asymptotic parametrizations of the orbit, then

= u

−

◦ Φ : δ

→ M

The following additional notation and terminology for the building u will be

useful to keep in mind. For each N ∈ {−N

−

, . . . , 0, . . . , N

}, denote



S \ (Γ

∪ Γ

−

∪ ∆

)



∩ L

−1

(N),

and denote the restriction of u to this subset by

→











R × M

if N > 0,

W if N = 0,

R × M

−

if N < 0.

Including Θ ∩L

−1

(N) and ∆

∩L

−1

(N) in the data deﬁnes u

as a (generally dis-

connected) nodal curve with mar ked points, whose positive punctures are in bijective

correspondence with the negative punctures of u

N+1

if N < N

. We call u

the

Nth level of u, and all it an upper or lower level if N > 0 or N < 0 respectively,

and the main level if N = 0. By convention, every holomorphic building in

W ha s

exactly one main level (which lives in

W itself) and arbitrary nonnegative numbers

of upper and lower levels (which live in the symplectizations R ×M

). One slightly

subtle detail is that it is possible for the main level to b e empty, meaning 0 is not in

the image of the level function L. The requirement that L should attain every o t her

value from −L

−

to L

is a convention to ensure tha t upper and lower levels are not

empty, so e.g. if a building has an empty main level and no lower levels, then the

lowest nonempty upper level is always labelled 1 instead of something arbitrary.

Lectures on Symplectic Field Theory 185

The p ositive punctures of the topmost level of u a r e Γ

, and the negative punc-

tures of the bottommost level are Γ

−

, so these give rise to lists of positive/negative

asymptotic orbits γ

= (γ

, . . . , γ

) in M

. There is also a relative homology class

[u] ∈ H

(W,

∪

−

To deﬁne this, recall from §

6.2 how it was deﬁned for smooth curves u :

Σ →

W : we

considered the retraction π :

W → W that collapses each cylindrical end to M

⊂

∂W , and noted that since u is asymptotically cylindrical, the map π ◦ u :

Σ → W

extends to a continuous map on the circle compactiﬁcation,

¯u :

Σ → W,

whose relative homology class gives the deﬁnition of [u]. The conditions on nodes

and breaking orbits allow us to perform a similar trick for the building u, using the

map

π :

N∈{−N

−

,...,N

}

→ W

which acts as the identity on W but collapses cylindrical ends of

W to ∂W and

similarly collapses each copy of R × M

to M

⊂ ∂W . Extending the decora-

tions Φ arbitrarily to decorations of the nodes ∆

, one can then take the circle

compactiﬁcation of

S := S \ (Γ

∪ Γ

−

∪ ∆

) a nd glue matching b oundary

components together along Φ to form a compact surface with boundary

such

that π ◦ u :

S → W extends to a continuous map

¯u :

→ W.

Its relative homology class deﬁnes [u] ∈ H

(W,

∪

−

We say that the building u is stable if two properties hold:

(1) Every connected component of S \(Γ

∪Γ

−

∪Θ ∪∆

∪∆

) on which the

map u is constant has negative Euler characteristic;

(2) There is no N ∈ {−N

−

, . . . , N

} for which the Nth level consists entirely

of a disjoint union of tr ivial cylinders without any marked points or nodes.

An equivalence between two holomorphic buildings

= (S

, j

, Γ

−

, Θ

, ∆

, L

, Φ

, u

), i = 0, 1

is deﬁned as an equivalence of partially decorated nodal Riemann surfaces

, j

, Γ

⊔ Γ

⊔ Θ

, ∆

⊔ ∆

, Φ

)

−→ (S

, j

, Γ

⊔ Γ

⊔ Θ

, ∆

⊔ ∆

, Φ

)

such that ϕ(Γ

) = Γ

, ϕ(Θ

) = Θ

, ϕ(∆

) = ∆

, ϕ(∆

) = ∆

, L

◦ϕ = L

, and

◦ ϕ = u

while

◦ ϕ = u

up to R-translation f or each N 6= 0.

Given lists of or bits γ

and a relative homology class A, the set of equivalence

classes of stable holomorphic buildings in (

W , J) with arithmetic genus g and m

186 Chris Wendl

marked point s, positively/negatively asymptotic to γ

and homologous to A will be

denoted by

g,m

(J, A, γ

, γ

−

Observe that for any A 6= 0, there is a natural inclusion M

g,m

(J, A, γ

, γ

−

) ⊂

g,m

(J, A, γ

, γ

−

) deﬁned by regarding J-holomorphic curves in M

g,m

(J, A, γ

, γ

−

)

as buildings with no upper or lower levels and no nodes. Such buildings are always

stable if A 6= 0 because they are not constant.

9.4.3. Convergence. For a general deﬁnition of the topology of

g,m

(J, A, γ

, γ

−

)

and the proof that it is bo th compact and metrizable, we refer to [

BEH

03] or the

more comprehensive treatment in [

Abb14]. The following statement contains all

the details about the topology that one usually needs to know in practice (see Fig-

ure

9.7).

Theorem 9.24. Fix integers g ≥ 0 and m ≥ 0, and a ssume all Reeb orbits in

(M, H

) and (M, H

−

) are nondegene rate. Then for any sequence

[(Σ

, j

, Γ

−

, Θ

, u

)] ∈ M

g,m

, A

, γ

−

)

of nonconstant J

-holomorphic curves in

W with uniformly bounded energy E(u

there exists a stable holomorphic building

∞

] = [(S, j, Γ

, Γ

−

, Θ, ∆

, ∆

, L, Φ, u

∞

)] ∈

g,m

(J, A, γ

, γ

−

)

such that after restricting to a subsequen ce, [(Σ

, j

, Γ

−

, Θ

, u

)] → [u

∞

] in the

following sense. The decoration s Φ at ∆

ca n be extended to decoration s at ∆

so that if

denotes the closed oriented topological 2-manifold o btained from S \

(∆

∪ ∆

) by gluing circle compactiﬁcations along Φ, then for k suﬃcien tly large,

there exist h omeomorphisms

→ Σ

that are smooth outside of C

, map Γ

⊔Γ

−

⊔Θ to Γ

⊔Γ

−

⊔Θ

with the ordering

pre s erved, and satisfy

∗

→ j in C

∞

lo c

(

\ C

Moreover for N = {−N

−

, . . . , 0, . . . , N}, let

:= u

◦ ϕ

→

W ,

with



S \ (Γ

∪ Γ

−

∪ ∆

)



∩ L

−1

(N) regarded as a subset of

\ C

Then:

(1) v

→ u

∞

in C

∞

lo c

(

W );

(2) For each ±N > 0, v

has image in the po sitive/negative cylin drical end for

all k suﬃciently large, and there exists a seq uence r

→ ±∞ such that the

resulting R-translations converge:

−r

◦ v

→ u

∞

in C

∞

lo c

(

, R × M

Lectures on Symplectic Field Theory 187

The rates of divergence of the seq uences r

→ ±∞ are related by

N+1

− r

→ +∞ for all N < N

Finally, let

denote the co mpact topological surface with boundary deﬁned as the

circle compactiﬁcation of

\(Γ

∪Γ

−

), and let Σ

denote the circle compactiﬁcation

:= Σ

\ (Γ

∪ Γ

−

). Then for all k large, ϕ

extends to a continuous map

¯ϕ

→ Σ

such that

¯u

◦ ¯ϕ

→ ¯u

∞

in C

(

, W ).

Remark 9.25. The t heorem is also true under the more general hypothesis

that the Reeb vector ﬁelds are Morse-Bott. In this case, one can also allow the

asymptotic Reeb orbits of the sequence to vary, as long as the sum of their periods is

uniformly bounded—such a bound plays the role of an energy bound and g uarantees

a convergent subsequence of orbits via the Arzel`a-Ascoli theorem.

Remark 9.26. Stability of the limit in Theorem

9.24 is guaranteed for the same

reasons as in our discussion of Gromov compactness in §

9.4.1: stable domains de-

generate to stable nodal domains as geodesics in pair-of-pants decompositions shrink

to zero length, while bubbling and breaking produce additional domain components

that ar e not stable but on which the maps are never trivial. Moreover, stability guar-

antees the uniqueness of the limiting building for any convergent sequence, i.e. it is

the reason why

g,m

(J, A, γ

, γ

−

) is a Hausdorﬀ space. Indeed, if u

converges to a

stable building u

∞

, then under the notion of convergence described in the theorem,

it will also converge to a building u

′

∞

constructed out of u

∞

by adding to S an extra

spherical component, att aching it to the rest by a single node and extending t he map

∞

to be constant on the extra component. One can also insert extra levels into

∞

that consist only of trivial cylinders, and u

will still converge to the resulting

building. But these modiﬁcations produce buildings that are not stable and thus

are not elements of M

g,m

(J, A, γ

, γ

−

9.4.4. Symplectizations, stretching and so forth. A few minor modiﬁca-

tions to the above discussion are necessary t o compactify the moduli space of curves

in a symplectization (R×M, J) for J ∈ J(H). It is possible to view this as a special

case of a completed symplectic cobordism, but this perspective produces a certain

amount of extraneous data that is not meaningful. The key observation is that in

the presence of an R-action, one should really compactify M

g,m

(J, A, γ

, γ

−

)



instead of M

g,m

(J, A, γ

, γ

−

). The compactiﬁcation

g,m

(J, A, γ

, γ

−

) then con-

sists of holomorphic buildings as deﬁned in §

9.4.2, but since all levels live in the

same symplectization R × M, there is no longer a distinguished main level or any

meaningful notion of upper vs. lower levels; the level structure is simply a function

L : S → {1, . . . , N} for some N ∈ N, and equivalence of buildings must permit

R-translations within each level. Fo r these reasons, the SFT compactness theorem

in symplectizations has a few qualitative diﬀerences, but is still very much analogous

to Theorem

9.24.

188 Chris Wendl

R × M

−

R × M

−

R × M

−

Figure 9.7. Convergence to a building with arithmetic genus 2, one

upper level and t hree lower levels.

To complete the picture, we should mention one more type of compactness theo-

rem that appears in [

BEH

03], which is colloquially described as stretching the

neck. The geometric idea is as follows: suppose (W, ω) is a closed symplectic

manifold and M ⊂ W is a stable hypersurface that separates W into two pieces

W = W

−

∪

, with an induced stable Hamiltonian structure H = (ω, λ) that

orients M as the boundary of W

−

A neighborhood of M in (W, ω) can then be

identiﬁed symplectically with

, ω

) := ( ( −ǫ, ǫ) × M, d(rλ) + ω)

for suﬃciently small ǫ > 0. The idea now is to replace N

with lar ger collars of the

form

((−T, T ) × M, d (f(r)λ) + ω) ,

with C

-small functions f chosen with f

′

> 0 so that the collar can be glued in

smoothly to replace (N

, ω

). This collar looks like a piece of the symplectization of

(M, H), thus we are free to choose tame almost complex structures whose restrictions

to the inserted collar belong t o J(H). Symplectic manifolds constructed in this

The assumption that M ⊂ W s eparates W is ine ssential, but makes certain details in this

discussion more conve nie nt.

Lectures on Symplectic Field Theory 189

way are all symplectomorphic, but their almost complex structures degenerate as

one takes T → ∞. Given a sequence T

→ ∞ and a corresponding degenerating

sequence J

, a sequence u

of J

-holomorphic curves with bounded energy converges

to yet another form of holo mo rphic building, this time involving a bottom level in

−

:= W

−

∪

([0, ∞) × M) with positive punctures approaching orbits in M, some

ﬁnite number of middle levels that live in the symplectization of M, and a top level

that lives in

:= ((−∞, 0] × M) ∪

with negative punctures approaching M.

A very po pula r example for applications arises from Lagrangian submanifolds

L ⊂ W . By the Weinstein neighborhood theorem, L always has a neighborhood

−

symplectomorphic to a neighborhood of the zero-section in T

∗

L, so M := ∂W

−

is a contact-type hypersurface conta cto morphic to the unit cotangent bundle of L.

Stretching the neck then yields T

∗

L as the completion of W

−

, and W \L as the com-

pletion of W

:= W \

−

. This construction has often been used in order to study

Lagrangian submanifolds via SFT-type methods, see e.g. [

EGH00, Theorem 1.7.5]

and [

Eva10, CM].

LECTURE 10

Cylindrical contact homology and the tight 3-tori

Contents

10.1. Contact structures on T

and Giroux torsion 191

10.2. Deﬁnition of cylindrical contact homology 194

10.2.1. Preliminary remarks 194

10.2.2. A compactness result for cylinders 197

10.2.3. The chain complex 202

10.2.4. The homology 202

10.2.5. Chain maps 204

10.2.6. Chain homotopies 205

10.2.7. Proof of invariance 206

10.3. Computing HC

∗

, ξ

) 208

10.3.1. The Morse-Bott setup 208

10.3.2. A digression on the Floer equation 210

10.3.3. Admissible data for (T

, ξ

) 219

We’ve now developed enough of the technical machinery of holomorphic curves

to be able to give a rigor ous construction of t he most basic version of SFT and apply

it to a problem in contact t opology.

10.1. Contact str uctures on T

and Giroux torsion

As a motivating goal in this lecture, we will prove a result about the classiﬁcation

of contact structures on T

= S

× S

. Denote the three global coordinates

on T

valued in S

= R/Z by (ρ, φ, θ), and for any k ∈ N, consider t he contact

structure

:= ker α

, where α

:= cos ( 2πkρ) dθ + sin(2πkρ) dφ.

It is an easy exercise to verify tha t these all satisfy the contact condition α

∧dα

> 0;

see Figure

10.1 f or a visual representation. The following result is originally due to

Giroux [

Gir94] and Kanda [Kan97].

Theorem 10.1. For each pair of positive integers k 6= ℓ, the contact manifolds

, ξ

) and (T

, ξ

ℓ

) are not con tactomorphic.

One of the reasons this result is interesting is that it cannot be proved using

any so-called “classical” invariants, i.e. invariants coming from algebraic topology.

An example of a classical invar iant would be the Euler class of the oriented vector

bundle ξ

→ T

, or anything else that depends only on the isomorphism class of this

191

192 Chris Wendl

Figure 10.1. The contact structures ξ

on T

can be constructed

by gluing k copies of t he same model [0, 1]×T

to each other cyclically.

bundle. The following observation shows that such invariants will never distinguish

from ξ

ℓ

Proposition 10.2. For e very k, ℓ ∈ N, ξ

and ξ

ℓ

are homotopic through a

smooth family of oriented 2-plan e ﬁelds on T

Proof. In f act, all the ξ

can be deformed smoothly t o ker dρ, via the homotopy

ker [(1 − s) α

+ s dρ] , s ∈ [0, 1].



Remark 10.3. One can check in fact that the 1-form in the homotopy given

above is cont act for every s ∈ [0, 1 ) , so Gray’s stability theorem implies that every

is isotopic to an arbitrarily small perturbat ion of the foliation ker dρ. In [

Gir94],

Giroux used this observation to show that all of them are what we now call weakly

symplectica lly ﬁllab l e. If ker dρ were also contact, then Gray’s theorem would imply

that ξ

and ξ

ℓ

are always isotopic. Thus Theorem

10.1 indicates the impo ssibility

of modifying a homotopy from ξ

to ξ

ℓ

into o ne that passes only through contact

structures.

Let us place this discussion in a larger context. Using the coordinates (ρ, φ, θ)

on R × T

, a pair of smooth functions f, g : R → R gives rise to a contact form

α = f(ρ) dθ + g(ρ) dφ

whenever the function D(ρ) := f (ρ)g

′

(ρ) −f

′

(ρ)g(ρ) is everywhere positive. Indeed,

we have α ∧dα = D(ρ) dρ ∧dφ ∧dθ, and one easily derives a similar formula for the

Reeb vector ﬁeld,

D(ρ)

′

(ρ) ∂

− f

′

(ρ) ∂

] .

The condition D > 0 means geometrically that the path (f, g) : R → R

winds

counterclockwise around the origin with its angular coordinate strictly increasing.

The simplest special case is the contact form

:= cos ( 2πρ) dθ + sin(2πρ) dφ,

Lectures on Symplectic Field Theory 193

which matches the formula for α

on T

given above. Let ξ

:= ker α

on R ×T

Definition 10.4. The Giroux torsion GT(M, ξ) ∈ N ∪ {0, ∞} of a contact

3-manifold (M, ξ) is the supremum of the set of positive int egers k such that there

exists a contact embedding



[0, k] × T

, ξ



֒→ (M, ξ).

We write GT(M, ξ) = 0 if no such embedding exists for any k, and GT(M, ξ) = ∞

if it exists for all k.

Example 10.5. The tori (T

, ξ

) for k ≥ Z a r e contactomorphic to (R×T

, ξ

)/kZ,

with kZ acting by translation o f the ρ-coordinate. Thus GT(T

, ξ

) ≥ k −1.

A 2-torus T ⊂ (M, ξ) embedded in a contact 3-manifold is called pre-Lagrangian

if a neighborhood of T in (M, ξ) admits a contactomo r phism to a neighborhood of

{0}×T

in (R ×T

, ξ

), identifying T with {0}×T

. The neighborhood in R ×T

can be arbitrarily small, thus the existence of a pre-Lagrangian torus does not imply

GT(M, ξ) > 0; in fact, pre-Lagrangian tori always exist in abundance, e.g. as bound-

aries of neighborhoods of transverse knots (using the contact model provided by the

transverse neighborhood theorem). But given any pre-Lagrangian torus T ⊂ ( M, ξ),

one can make a local modiﬁcation of ξ near T to produce a new contact structure (up

to isotopy) with positive Giroux torsion. Deﬁne (M

′

, ξ

′

) f r om (M, ξ) by replacing

the small neighborhood ((−ǫ, ǫ) ×T

, ξ

) with ((−ǫ, 1 + ǫ) ×T

, ξ

), then identify

′

with M by a choice of compactly supported diﬀeomorphism (−ǫ, 1+ǫ) → (−ǫ, ǫ).

There is now an obvious contact embedding of ([0, 1] × T

, ξ

) into (M, ξ

′

), hence

GT(M, ξ

′

) ≥ 1. Moreover, one can adapt the proof o f Prop.

10.2 above to show

that ξ

′

is homotopic to ξ through a smooth family of or iented 2-plane ﬁelds. The

operation changing ξ to ξ

′

is known as a Lutz twist along T . In this language, we

see that for each k ∈ N, (T

, ξ

k+1

) is obtained from (T

, ξ

) by performing a Lutz

twist along {0}× T

The invariant GT(M, ξ) is easy to deﬁne, but hard to compute in general. The

natural guess,

GT(T

, ξ

) = k − 1,

turns out to be correct, as was shown in [

Gir00], so this is one way to prove

Theorem

10.1, but not the approach we will take. The following example shows

that one must in any case be careful with such guesses.

Example 10.6. For each k ∈ N, deﬁne a model o f S

× S

×S

∼



[0, k + 1/2] ×T



∼

where the equivalence relation identiﬁes (ρ, φ, θ) ∼ (ρ, φ

′

, θ) for ρ ∈ {0, k + 1/2} and

every θ, φ, φ

′

∈ S

. Near ρ = 0 and ρ = k+1/2, this means thinking of (ρ, φ) as polar

coordinates, so the two subsets {ρ = 0} and {ρ = k + 1/2} become circles of the

form S

×{const} embedded in S

×S

. Since the φ-coordina t e is singular at these

two circles, the contact fo r m α

needs to be modiﬁed slightly in this region before

it will descend to a smooth contact form on S

×S

: this can be done by a C

-small

194 Chris Wendl

modiﬁcation of the form f(ρ) dθ + g(ρ) dφ, and the resulting contact structure is

then uniquely determined up t o isotopy. We shall call this contact manifold

× S

, ξ

Now observe that for each k ∈ N, (S

× S

, ξ

k+1

) is obtained from (S

× S

, ξ

) by

a Lutz twist. However, both contact manifolds are also over twisted: recall that a

contact 3-manifold (M, ξ) is overtwisted whenever it contains an embedded closed

2-disk D ⊂ M such that T ( ∂D) ⊂ ξ but T D|

∂D

⋔ ξ. (Exercise: ﬁnd a disk with this

property in (S

× S

, ξ

)!) Eliashberg’s ﬂexibility theorem for overtwisted contact

structures [

Eli89] implies that whenever ξ and ξ

′

are two contact structures on a

closed 3-manifold that a re both overtwisted and are homotopic as oriented 2 -plane

ﬁelds, t hey are actually isotopic. As a consequence, the contact structures ξ

×S

deﬁned above for every k ∈ N are all isotopic to each other. As tends to be

the case with most interesting h-principles, the isotopy is very hard to see concretely,

but it must exist.

Exercise 10.7. Show that if (M, ξ) is a closed overtwisted contact 3-manifold,

then G T(M, ξ) = ∞.

In contrast to the S

×S

example ab ove, the contact manifolds (T

, ξ

) are not

overtwisted, they are tight—in fact, the classiﬁcation of contact structures on T

by Giroux [

Gir94, Gir99, Gir00] and K anda [Kan97] states that these are all of

the tight contact structures on T

up to contactomorphism. We will use cylindrical

contact homology to show that they are not contactomorphic to each other. The

reader should keep Example 10.6 in mind and try to spot the reason why the same

argument cannot work for (S

× S

, ξ

Remark 10.8. It has b een conjectured that the converse of Exercise

10.7 mig ht

also hold, so every closed tight contact 3-manifold would have ﬁnite Giroux torsion.

This conj ecture is wide open.

10.2. Deﬁnition of cylindrical contact homology

10.2.1. Preliminary remarks. Cylindrical contact homology is the natural

“ﬁrst attempt” at using holomorphic curves in symplectizations to deﬁne a Floer-

type invariant of contact manifolds (M, ξ). The idea is to deﬁne a cha in complex

generated by Reeb orbits in M and a diﬀerential ∂ that counts holomor phic cylinders

in R × M. We already know some pretty good reasons why this idea cannot work

in general: in order to prove ∂

= 0, we need to be a ble to identify the space of

rigid “broken” holomorphic cylinders (these a r e what is counted by ∂

) with the

boundary of the compactiﬁed 1-dimensional space of index 2 cylinders (up to R-

translation). But this compactiﬁed boundary has more than just broken cylinders

in it, see Figure

10.2. In order to deﬁne cylindrical contact homology, one must

therefore restrict to situations in which complicated pictures like Fig ur e

10.2 cannot

occur. The ﬁrst useful remark in this direction is that since we are working with

a stable Hamiltonian structure o f t he form (dα, α) for a contact form α, a certain

subset of the scenarios allowed by the SFT compactness theorem can be excluded

immediately. Indeed:

Lectures on Symplectic Field Theory 195

Figure 10.2. A family of holomorphic cylinders can converge in

the SFT topology to buildings that include more complicated curves

than cylinders—this is why cylindrical contact homolo gy is not well

deﬁned for all contact manifolds.

Proposition 10 .9. If J ∈ J(α) and u : (

Σ, j) → (R×M, J) is an asymptotically

cylindrical J-holomorphic curve, then u h as at least one positive puncture.

Let us give two proofs of this result, since both contain useful ideas. As prepara-

tion for the ﬁrst proof, recall the deﬁnition of energy for curves in symplectizations

of contact manifolds that we wrote down in Lecture

E(u) := sup

f∈T

∗

d(e

f(r)

α),

where

T :=



f ∈ C

∞

(R, (−1, 1))



′

> 0



This formula is not identical to the deﬁnition of energy used in Lecture

9, but it

is equivalent in the sense that any uniform bounds on one imply similar unifo r m

bounds on the other.

First proof of Proposition

10.9. Denote the positive and negative punc-

tures of u :

Σ → R × M by Γ

and Γ

−

respectively, and suppose u is asymptotic

at z ∈ Γ

to the orbit γ

with period T

> 0. Choose any f ∈ T and denote

:= lim

r→±∞

f(r) ∈ [−1, 1]. Since d(e

f(r)

α) tames J ∈ J(α), Stokes’ theorem

gives

(10.1) 0 ≤ E(u) = e

z∈Γ

− e

−

z∈Γ

−

hence Γ

cannot be empty. 

Remark 10.10. The proof via Stokes’ theorem works just as well if instead of

R × M, u lives in the completion of an exact symplectic cobordism (W, ω) with

concave boundary (M

−

, ξ

−

= ker α

−

) and convex boundary (M

, ξ

= ker α

196 Chris Wendl

Recall that this means ∂W = −M

−

⊔ M

, and ω = dλ for a 1-form λ that restricts

to positive conta ct forms λ|

T M

= α

. As in Lecture

1, we will write

J(W, ω, α

, α

−

) ⊂ J(

W )

for the space of almost complex structures J on

W := ((−∞, 0] × M

−

) ∪

−

W ∪

([0, ∞) × M

) that a re compatible with ω on W and belong to J(α

) on the cylin-

drical ends. The energy o f a J-ho lomorphic curve u : (

Σ, j) → (

W , J) is then

E(u) := sup

f∈T

∗

dλ

where T := {f ∈ C

∞

(R, (−1, 1)) | f

′

> 0 and f(r) = r near r = 0} and











f(r)

on [0, ∞) × M

λ on W ,

f(r)

−

on (−∞, 0] ×M

−

The above proof now generalizes verbatim to show that u must always have a positive

puncture. Notice that in both settings, the argument also gives a unifor m bound

for the energy in terms of the periods of the positive asymptotic orbits.

Remark 10.11. We can also prove Pro p.

10.9 using the fact that u

∗

dα ≥ 0 for

any u : (

Σ, j) → (R × M, J) with J ∈ J(α). Indeed, Stokes’ theorem then gives

(10.2) 0 ≤

∗

dα =

z∈Γ

−

z∈Γ

−

The quantity

∗

dα is sometimes called the contact area of u. This version of

the argument however does not easily generalize to arbit r ary exact cobordisms.

The second proof is based on the maximum principle for subharmonic functions.

Proposition 10.12. Suppose J ∈ J(α) and u = (u

, u

) : (

Σ, j) → (R ×M, J)

is J-holomorphic , where

Σ has no bo und ary. Then u

Σ → R has no local maxima.

Proof. In any local holomorphic coordinates (s, t) on a region in

Σ, the non-

linear Cauchy-Riemann equation for u is equivalent to the system of equations

∂

− α(∂

) = 0,

∂

+ α(∂

) = 0,

∂

+ Jπ

∂

= 0,

where π

: T M → ξ denotes the projection along the Reeb vector ﬁeld. This gives

−∆u

= −∂

−∂

= −∂

[α(∂

)] + ∂

[α(∂

)]

= −dα(∂

, ∂

) = −dα(π

∂

, Jπ

∂

) ≤ 0

since J|

is tamed by dα|

, hence u

is subharmonic. The result thus follows from

the maximum principle, see e.g. [

Eva98]. 

Lectures on Symplectic Field Theory 197

Second proof of Proposition

10.9. If u = (u

, u

) :

Σ → R × M ha s

no p ositive puncture then u

Σ → R is a proper function bounded above, and

therefore has a local maximum, contradicting Proposition

10.12. 

Remark 10.13. The proof via the maximum principle does not generalize to

arbitrary exact cobordisms (W, dλ), but it does work in Stein cobordisms, i.e. if λ

and J are related by λ

= −dF ◦J for some plurisubharmonic function F :

W → R,

then F ◦ u :

Σ → R is subharmonic (cf. [

CE12]).

With these preliminaries understo od, the next two exercises reveal one natural

setting in which breaking of cylinders can be kept under control. Both exercises are

essentially combina torial.

Exercise 10.14. Suppose u is a stable J-holomorphic building in a completed

symplectic cobordism

W with the following properties:

(1) u has arithmetic genus 0 and exactly one positive puncture;

(2) every connected component of u has at least one positive puncture.

Show that u has no nodes, and all of its connected component s have exa ctly one

positive puncture.

Exercise 10.15. Suppose tha t in addition t o the conditions of Exercise

10.14,

u has exactly one negative puncture a nd no connected component of u is a plane.

Show that every level of u then consists of a single cylinder with one positive and

one negat ive end.

Exercise

10.15 makes it reasonable to deﬁne a Floer-type theory counting only

cylinders in any setting where planes can be excluded, for instance because the Reeb

vector ﬁeld has no contractible orbits. This is not always possible, e.g. Hof er [

Hof93]

proved that on overtwisted contact manifolds, there is always a plane (which is why

the Weinstein conjecture holds). So the invariant we construct will not be deﬁned

in such settings, but it happens to be ideally suited to the study of (T

, ξ

10.2.2. A compactness result for cylinders. Fix a closed contact manifold

(M, ξ) of dimension 2n −1 and a primitive homot opy class of loops h ∈ [S

, M]. By

primitive, we mean that h is not equal to Nh

′

for any h

′

∈ [S

, M] and an integer

N > 1, and this assumption will be crucial for technical reasons in the following.

Given a contact form α fo r ξ, let

(α)

denote the set of closed Reeb orbits homotopic to h, where two Reeb o rbits are

identiﬁed if they diﬀer only by parametrization.

Definition 10.16. Given a cont act manifold (M, ξ) and a primitive homotopy

class h ∈ [S

, M], we will say that a contact form α for ξ is h-admissible if:

(1) All orbits in P

(α) are nondegenerate;

It is to be expec ted that cylindrical contact homology can be deﬁned also for non-primitive

homotopy clas ses, but this would require more sophisticated metho ds to address transversality

problems. The assumption that h is primitive allows us to assume that all holomorphic curves in

the discussion are somewhere injective, hence they ar e always regular if J is generic.

198 Chris Wendl

(2) There are no contractible closed Reeb orbits.

Similarly, we will say that (M, ξ) is h-admissible if a conta ct form with the above

properties exists.

Definition 10.17. Given h ∈ [S

, M] and an h-admissible contact form α on

(M, ξ), we will say that an almost complex structure J ∈ J(α) is h-regular if every

J-holomorphic cylinder in R×M with a positive and a nega t ive end both asymptotic

to orbits in P

(α) is Fredholm regula r .

Proposition 10.18. If h ∈ [S

, M] is a primitive homotopy clas s of loops and

α is h-admissible on (M, ξ), then the space of h-regular almo s t complex structures

is comeager in J(α).

Proof. Since h is primitive, the asymptotic orbits for t he relevant holomorphic

cylinders cannot be multiply covered, hence all of these cylinders are somewhere

injective. The result therefore follows from the standard transversality results proved

in L ecture

8 for somewhere injective curves in symplectizations. 

Proposition 10.19. Given an h-admissibl e contact form α, an h-regular almos t

co mplex structure J ∈ J( α) and an orbit γ ∈ P

(α), suppose u

is a sequence of

J-holomorphic cylinders in R ×M w i th one positive puncture at γ and o ne negative

puncture. Then u

has a subseq uence convergent in the SFT topology to a broken

J-holomorphic cylinder, i.e. a stable building u

∞

whose levels u

∞

, . . . , u

∞

are each

cylinders with one positive and one negative puncture. Moreover, each level satisﬁes

ind(u

∞

) ≥ 1, thus for large k in the convergent subsequence,

ind(u

) =

N=1

ind(u

∞

) ≥ N

Proof. Let’s start with some ba d news: the standard SFT compactness the-

orem is not applicable in this situation, because we have not assumed that α is

nondegenerate, nor even Morse Bott—there is no a ssumption at all about Reeb or-

bits in homotopy classes o t her than h and 0. This fa ir ly loose set of hypotheses is

very convenient in applications, as nondegeneracy of a contact form is generally a

quite diﬃcult conditio n t o check. The price we pay is that we will have to prove

compactness manually instead of applying the big theorem (see Remark

10.20). For -

tunately, it is not that hard: the crucial point is tha t in the situation at hand, there

can be no bubbling at all.

Indeed, we claim that the given sequence u

: (R × S

, i) → (R × M, J) must

satisfy a uniform bound

|du

| ≤ C

with respect to any translation-invariant Riemannian metrics on R ×S

and R ×M.

To see this, note ﬁrst that since a ll the u

have the same positive asymptotic orbit γ,

their energies are uniformly bounded via (

10.1). Thus if |du

)| → ∞ for some

sequence z

∈ R × S

, we can perform the usual rescaling trick from Lecture

9 and

deduce the existence of a nonconstant ﬁnite-energy plane v

∞

: C → R × M. Its

singularity at ∞ cannot be removable since this would produce a nonconstant J-

holomorphic sphere, violating Proposition

10.9. It follows t hat v

∞

is asymptotic to a

Lectures on Symplectic Field Theory 199

Reeb or bit at ∞, but this is a lso impossible since α does not admit any contractible

orbits, and the claim is thus proved.

Suppose now that γ has period T

> 0, and observe that by nondegeneracy, the

set

(α, T

) :=



γ ∈ P

(α)



γ has period at most T



is ﬁnite. Let

(α), A

(α, T

) ⊂ (0, ∞)

denote the set of all periods of orbits in P

(α) and P

(α, T

) respectively. By

(

10.2), the negative asymptotic orbit of each u

is in P

(α, T

), so we can take a

subsequence and assume that these are all the same orbit; call it γ

−

∈ P

(α, T

)

and its period T

−

∈ A

(α, T

). If T

−

= T

then u

∗

dα ≡ 0 for all k, implying that

all u

are the trivial cylinder over γ and thus trivially converg e. Assume therefore

−

< T

. Then since u

∗

dα ≥ 0, Stokes’ theorem implies that for each k, the function

R → R : s 7→

(s, ·)

∗

is increasing and is a surjective map onto (T

−

, T

). The uniform bound on the

derivatives implies that for a ny sequences s

, r

∈ R with u

, 0) ∈ {r

}×M, the

sequence

: R × S

→ R × M : (s, t) 7→ τ

−r

◦ u

(s + s

, t)

has a subsequence convergent in C

∞

lo c

(R × S

) to some ﬁnite- energy J-holomorphic

cylinder

∞

: R × S

→ R × M,

which necessarily satisﬁes

∞

(s, ·)

∗

α = lim

k→∞

(s + s

, ·)

∗

α ∈ [T

−

, T

]

for every s ∈ R. This proves that v

∞

is nonconstant, with a positive puncture at

s = ∞ and negative puncture at s = −∞, and both of its asymptotic orbits are

in P

(α, T

If v

∞

is not a trivial cylinder, then it therefore satisﬁes

R×S

∗

∞

dα ≥ δ,

where δ is any positive number less than t he smallest distance between neighboring

elements of A

(α, T

Let us call a sequence s

∈ R nontrivial whenever the limiting cylinder v

∞

obtained by the a bove procedure is no t a trivial cylinder, and call two such sequences

and s

′

co mpatible if s

−s

′

is not bounded. We claim now t hat if s

, . . . , s

is a

collection of nontrivial sequences that are all compatible with each other, then

m <

2(T

− T

−

)

Recall from Lecture

9 that we deno te the R-translation action on R×M by τ

(r, x) := (r+c, x).

For an alternative argument that v

∞

must have a p ositive puncture at s = ∞ and negative

at s = −∞, see Figure

10.3.

200 Chris Wendl

Indeed, we can assume after ordering our collection appropriately and restricting

to a subsequence that s

N+1

− s

→ ∞ for each N = 1, . . . , m − 1, and let v

∞

R × S

→ R × M denote the limits of the corresponding convergent subsequences.

Then we can ﬁnd R > 0 such that

[−R,R]×S

∞

)

∗

dα >

and thus

−R,s

+R]×S

∗

dα >

for each N = 1, . . . , m for suﬃciently large k. But these domains are also all disjoint

for suﬃciently large k, implying

−T

−

R×S

∗

dα ≥

N=1

−R,s

+R]×S

∗

dα >

δm

We’ve shown that there exists a maximal collection of nontrivial sequences

, . . . , s

∈ R satisfying s

N+1

− s

→ ∞ for each N, such that if u

, 0) ∈

} ×M, then after restricting to a subsequence, the cylinders

(s, t) := τ

−r

◦ u

(s + s

, t)

each converge in C

∞

lo c

(R × S

) as k → ∞ to a nontrivial J-holomorphic cylinder

∞

: R × S

→ R × M. Let γ

denote the asymptotic orbit of u

∞

at s = ±∞. We

claim,

= γ

−

N+1

for each N = 1 , . . . , N

− 1.

If γ

6= γ

−

N+1

for some N, choose a neighborhood U ⊂ M of the image of γ

that

does not intersect any other o r bit in P

(α, T

). Then since each u

is continuous,

there must exist a sequence s

′

∈ R with

′

−s

→ ∞ and s

N+1

−s

′

→ ∞

such that u

′

, 0) lies in U for all k but stays a positive distance away from the

image of γ

. A subsequence of (s, t) 7→ u

(s + s

′

, t) then converges after suitable

R-translations to a cylinder u

′

∞

: R × S

→ R × M that cannot be trivial since

′

∞

(0, 0) is not contained in any orbit in P

(α, T

). This contradicts the assumption

that our collection s

, . . . , s

is maximal. A similar arg ument shows

−

= γ

−

and γ

= γ,

so the curves u

∞

, . . . , u

∞

form the levels of a stable holomorphic building u

∞

. A

similar argument by contradiction also shows that the sequence u

must converge

in t he SFT topology to u

∞

Finally, note that since all the breaking orbits in u

∞

are homotopic to h a nd J

is h-r egular, the levels u

∞

are Fredholm regular. Since all of them also come in 1-

parameter families of distinct curves related by the R-action, this implies ind(u

∞

) ≥

1 for each N = 1, . . . , N

. 

Lectures on Symplectic Field Theory 201

Figure 10.3. A degenerating sequence of holomorphic cylinders

: R ×S

→ R ×M cannot have a limiting level with a puncture of

the “wrong” sign unless u

violates the maximum principle for large k.

Remark 10.20. Nondegeneracy or Morse-Bott conditions are r equired fo r sev-

eral reasons in the proof of SFT compactness, and indeed, the theorem is not true

in general without some such assumption. One can see this by considering what

happens to a sequence u

of J

-holomorphic curves where J

→ J

∞

is compati-

ble with a sequence of nondegenerate contact forms α

converging to one that is

only Morse-Bott. A compactness theorem for this scenario is proved in [

Bou02],

but it requires more general limiting objects t han holomorphic buildings. On the

other hand, it is useful for certain kinds of applicatio ns to know when one can do

without nondegeneracy assumptions and prove compactness anyway. There ar e two

main advantages to knowing that all Reeb orbits are nondegenerate or belong to

Morse-Bott families:

(1) It implies t ha t the set o f all periods of closed orbits, the so-called action

spectrum of α, is a discrete subset of (0, ∞) ; in fact, for any T > 0, the

set of all periods less than T is ﬁnite. Using the relations (10 .1) and (10.2),

this implies lower bounds o n the possible energies of limiting components

and thus helps show t hat only ﬁnitely many such components can arise.

(2) Curves asymptotic to nondegenerate or Morse-Bott orbits also satisfy ex-

ponent ial convergence estimates proved in [

HWZ96, HWZ01, HW Z96,

Bou02], and similar asymptotic estimates yield a result about “long cylin-

ders with small area” (see [

HWZ02] and [BEH

03, Prop. 5.7]) which

helps in proving that neighboring levels connect to each other along break-

ing orbits.

Our situation in Proposition

10.19 was simple enough to avoid using the “long

cylinder” lemma, and we did use the discreteness of the actio n spectrum, but only

needed it f or orbits in P

(α) since we were able to rule out bubbling in the ﬁrst

step. An alternative would have been to assume that all orbits (in all homotopy

classes) with period up to the period of γ a r e nondegenerate: then (

10.2) implies

202 Chris Wendl

that degenerate orbits never play any role in the main arguments of [BEH

03], so

the big theorem becomes saf e to use.

10.2.3. The chain complex. We now deﬁne a Z

-graded chain complex with

coeﬃcients in Z

and generators hγi fo r γ ∈ P

(α), i.e.

∗

(M, α) :=

γ∈P

(α)

The degree of each generator hγi ∈ CC

∗

(M, α) is deﬁned by

|hγi| = n − 3 + µ

(γ) ∈ Z

where µ

(γ) ∈ Z

denotes the pa r ity o f the Conley-Zehnder index with respect to

any choice of trivialization. The choice t o write n −3 in front of this is a convention

that will make no diﬀ erence at all in this lecture, but it is consistent with a Z-

grading that we will be able to deﬁne under suitable assumptions in Lecture

12. To

deﬁne the diﬀerentia l on CC

∗

(M, α), choose an h-regular almost complex structure

J ∈ J(α). Given Reeb orbits γ

, γ

−

∈ P

(α) and a number I ∈ Z, let

(J, γ

, γ

−

)

denote the space of all R-equivalence classes of index I holomorphic cylinders in (R×

M, J) asymptotic to γ

at ±∞, i.e. the union of all components M

0,0

(J, A, γ

, γ

−

)/R

for which vir-dim M

0,0

(J, A, γ

, γ

−

) = I. Since J is h-regular, all the curves in

(J, γ

, γ

−

) are Fredholm regula r, so if I ≥ 1, M

(J, γ

, γ

−

) is a smooth mani-

fold with

dim M

(J, γ

, γ

−

) = I − 1.

Similarly, M

(J, γ

, γ

−

) only contains trivial cylinders and is thus empty unless

= γ

−

, and M

(J, γ

, γ

−

) is always empty for I < 0. In particular, M

(J, γ

, γ

−

)

is a discrete set whenever γ

6= γ

−

, and by Proposition

10.19, it is also compact,

hence ﬁnite. We can therefore deﬁne

∂hγi =

′

∈P

(α)

(J, γ, γ

′

)hγ

′

where for any set X, we denote by #

X the cardinality of X modulo 2. The operator

∂ has odd degree with respect to the grading since every index 1 holo mo r phic cylinder

u with asymptotic orbits γ

and γ

−

satisﬁes

ind(u) = 1 = µ

(γ

) − µ

(γ

−

)

for suitable choices of the trivialization τ .

10.2.4. The homology. Following the standard Floer theoretic prescription,

the relation ∂

= 0 should arise by viewing the compactiﬁcation

(J, γ

, γ

−

) for

each γ

, γ

−

∈ P

(α) as a compact 1 -manifold whose boundary is identiﬁed with the

set of rig id broken cylinders, as these are what is counted by ∂

. Here

(J, γ

, γ

−

)

Lectures on Symplectic Field Theory 203

is deﬁned as the closure of M

(J, γ

, γ

−

) in the space of all J-holomorphic buildings

in R × M modulo R-t r anslation. Proposition

10.19 gives a natural inclusion

(J, γ

, γ

−

) \ M

(J, γ

, γ

−

) ⊂

∈P

(α)

(J, γ

, γ

) × M

(J, γ

, γ

−

We therefore need an inclusion in the other direction, and for this we need to say

a word about gluing. We have not had time to discuss gluing in earnest in these

notes, and we will not do so now either, but the basic idea should be familiar from

Floer homology: given u

∈ M

(J, γ

, γ

) and u

−

∈ M

(J, γ

, γ

−

), one would

like to show that there exists a unique (up to R-translation) one-parameter family

∈ M

(J, γ

, γ

−

)}

R∈[R

,∞)

such that u

converges as R → ∞ to the building

∞

with bottom level u

−

and top level u

. One starts by constructing a family of

preglued maps

˜u

: R ×S

→ R ×M,

meaning a smooth family of maps which converge in the SFT topology as R → ∞ to

∞

but are only approximately J-holomorphic. More precisely, ﬁx parametrizations

of u

−

and u

and a parametrization of the orbit γ

: R/T Z → M such that

(s, t) = exp

(T s,γ

(T t))

(s, t) for s ≪ 0,

−

(s, t) = exp

(T s,γ

(T t))

−

(s, t) for s ≫ 0,

where h

are vector ﬁelds along the trivial cylinder satisfying lim

s→∓∞

(s, t) = 0.

By interpolat ing between suitable reparametrizations of h

and h

−

, one can now

deﬁne ˜u

such that

˜u

(s, t) = τ

2RT

◦ u

(s −2R, t) for s ≥ R,

˜u

(s, t) ≈ (T s, γ

(T t)) for s ∈ [−R, R],

˜u

(s, t) = τ

−2RT

◦ u

−

(s + 2R, t) for s ≤ −R,

∂

˜u

→ 0 as R → ∞.

Given regularity of u

and u

−

, one can now use a quantita t ive version of the implicit

function t heorem (cf. [

MS04, §3.5]) to show that a distinguished J-holomorphic

cylinder u

close to ˜u

exists for all R suﬃcient ly large. For a more detailed synopsis

of the analysis involved, see [

Nel13, Chapter 7], and [AD14, Chapters 9 and 13]

for the analogous story in Floer homology. The result is:

Proposition 10.21. For an h-admissible α, an h-reg ular J ∈ J(α) and a ny two

orbits γ

, γ

−

∈ P

(α), the space

(J, γ

, γ

−

) admits the structure of a compact

1-dimensional manifol d with boundary, where its boundary points can be identiﬁed

naturall y with

∈P

(α)

(J, γ

, γ

) × M

(J, γ

, γ

−

). 

Corollary 10.22. The homomorphi sm ∂ : CC

∗

(M, α) → CC

∗−1

(M, α) satis-

ﬁes ∂

= 0. 

We shall denote the homology of this chain complex by

∗

(M, α, J) := H

∗



∗

(M, α), ∂



204 Chris Wendl

The goal of the rest of this section is to prove that up to natural isomorphisms,

∗

(M, α, J) depends on (M, ξ) and h but no t on the auxiliary data α a nd J.

10.2.5. Chain maps. For any constant c > 0, there is an obvious bijection

between the generators of CC

∗

(M, α) and CC

∗

(M, cα), as t he rescaling changes

periods of orbits but not the set of closed orbits itself. Moreover, if J ∈ J(α) and

∈ J(cα) are deﬁned to match on ξ, then there is a biholomorphic diﬀeomorphism

(R × M, J) → (R × M, J

) : (r, x) 7→ (cr, x),

thus giving a bijective correspondence between the moduli spaces of J-holomorphic

and J

-holomorphic curves. It follows that our bijection of chain complexes is also

a chain map and therefore deﬁnes a canonical isomorphism

(10.3) HC

∗

(M, α, J) = HC

∗

(M, cα, J

Next suppose α

−

and α

are two distinct contact f orms for ξ, hence

= e

for some ﬁxed contact form α and a pair of smooth functions f

: M → R. After

rescaling α

by a constant, we are free to assume f

> f

−

everywhere. Fix h-regular

almost complex structures J

∈ J(α

) and let

∂

: CC

∗

(M, α

) → CC

∗−1

(M, α

)

denote the resulting diﬀerentials on the two chain complexes. The region

W :=



(r, x) ∈ R × M



−

(x) ≤ r ≤ f

(x)



now deﬁnes an exact symplectic cobordism from (M, ξ) to itself: more precisely,

setting



(x), x) ∈ W



x ∈ M



gives ∂W = −M

−

⊔ M

, and the Lio uville form λ := e

α satisﬁes λ|

T M

= α

Choose a generic dλ-compatible almost complex structure J on the completion

that restricts to J

on the cylindrical ends. Now given γ

∈ P

(α

) and γ

−

∈

(α

−

) and a number I ∈ Z, we shall denote by

(J, γ

, γ

−

)

the union of all components M

0,0

(J, A, γ

, γ

−

) that have virtual dimension I. Note

that we a r e not dividing by any R-action here since J need not be R-invariant.

Since γ

are still guaranteed to be simply covered, curves in M

(J, γ

, γ

−

) are

again a lways somewhere injective and therefore regular, hence M

(J, γ

, γ

−

) is a

smooth manifold with

dim M

(J, γ

, γ

−

) = I

if I ≥ 0, and M

(J, γ

, γ

−

) = ∅ for I < 0. The compactiﬁcation

(J, γ

, γ

−

) is

described via the following straightforward generalization of Proposition 10.19:

Lectures on Symplectic Field Theory 205

Proposition 10.23. For J as described above, suppose u

is a sequence of J-

holomorphic cylinders in

W wi th one positive puncture at an orbit γ ∈ P

(α

) and

one negative puncture. Then u

has a subsequence con vergent in the SFT topology

to a broken J-holomorphic cylinder, i.e. a stable building u

∞

whose levels u

∞

for

N = −N

−

, . . . , −1, 0, 1 , . . . , N

are each c ylin d ers with one posi tive and one negative

puncture, livi ng in R × M

for ±N > 0 and

W for N = 0. Moreover, the levels

satisfy ind(u

∞

) ≥ 0 and ind(u

∞

) ≥ 1 for N 6= 0, thus for large k in the convergent

subsequence ,

ind(u

) =

N=−N

−

ind(u

∞

) ≥ N

−

+ N



It follows that the set M

(J, γ

, γ

−

) is always ﬁnite, and we use this t o deﬁne a

map

: CC

∗

(M, α

) → CC

∗

(M, α

−

) : hγi 7→

′

∈P

(α

−

)

(J, γ, γ

′

)hγ

′

This map preserves degrees since it counts index 0 curves, and we claim that it is a

chain map:

◦ ∂

= ∂

−

◦ Φ

This follows from t he fact that by Proposition

10.23 (in conjunction with a cor r e-

sponding gluing theorem),

(J, γ

, γ

−

) is a compact 1-manifold whose boundary

consists of two types of broken cylinders, depending whether the index 1 curve

appears in an upper or lower level:

∂

(J, γ

, γ

−

) =

∈P

(α

)



, γ

) × M

(J, γ

, γ

−

)



∪

∈P

(α

−

)



(J, γ

, γ

) × M

−

, γ

−

)



Counting broken cylinders of the ﬁrst type gives the coeﬃcient in front of hγ

−

i in

◦ ∂

(hγ

i), and the second type gives ∂

−

◦ Φ

(hγ

i).

It follows that Φ

descends to a homomor phism

(10.4) Φ

: HC

∗

(M, α

, J

) → HC

∗

(M, α

−

, J

−

10.2.6. Chain homotopies. We claim that the map Φ

in (

10.4) does no t

depend on J. To see this, suppose J

and J

are two generic choices of compatible

almost complex structures on

W that both match J

on the cylindrical ends. The

space of almost complex structures with these properties is contractible, so we can

ﬁnd a smooth path

}

s∈[0,1]

connecting them. For I ∈ Z, consider the parametric moduli space

({J

}, γ

, γ

−

) :=



(s, u)



s ∈ [0, 1], u ∈ M

, γ

−

)



206 Chris Wendl

As we observed in Remark 7.4, a generic choice of the homotopy {J

} makes

({J

}) a smooth manifold with

dim M

({J

}, γ

, γ

−

) = I + 1

whenever I ≥ −1, and M

({J

}, γ

, γ

−

) = ∅ when I < −1. Adapting Proposi-

tion

10.23 to allow for a converging sequence of almost complex structures, it im-

plies that M

−1

({J

}, γ

, γ

−

) is compact and thus ﬁnite, so we can use it to deﬁne

a homomorphism of odd degree by

H : CC

∗

(M, α

) → CC

∗+1

(M, α

−

) : hγi 7→

′

∈P

(α

−

)

−1

({J

}, γ, γ

′

)hγ

′

We claim that this is a chain homotopy between Φ

and Φ

, i.e.

−Φ

= ∂

−

◦ H + H ◦ ∂

This follows by looking at the boundary of the compactiﬁed 1-dimensional space

({J

}, γ

, γ

−

), which consists of four types of objects:

(1) Pairs (0, u) with u ∈ M

, γ

−

), which ar e count ed by Φ

(2) Pairs (1, u) with u ∈ M

, γ

−

), which ar e count ed by Φ

(3) Pairs (s, u) with u a broken cylinder with upper level u

∈ M

, γ

)

and main level u

∈ M

−1

, γ

−

) for some s ∈ (0, 1); these are counted

by H ◦ ∂

(4) Pairs (s, u) with u a broken cylinder with lower level u

−

∈ M

−

, γ

−

)

and main level u

∈ M

−1

, γ

) for some s ∈ (0, 1); these are counted

by ∂

−

◦ H.

The sum Φ

+ Φ

+ ∂

−

◦ H + H ◦ ∂

therefore counts (modulo 2) the boundary

points of a compact 1-manifold, so it vanishes.

Since the action of Φ

on homology no longer depends on J, we will denote it

from now on by

Φ : HC

∗

(M, α

, J

) → HC

∗

(M, α

−

, J

−

It is well deﬁned for any pair of h-admissible contact forms α

and h-regular J

∈

J(α

) since one can ﬁrst rescale α

to assume α

= e

α with f

> f

−

, using the

canonical isomor phism (

10.3).

10.2.7. Proof of invariance. We claim that for any h-admissible α and h-

regular J ∈ J(α), t he cobordism map

Φ : HC

∗

(M, α, J) → HC

∗

(M, α, J)

is the ident ity. Indeed, the literal meaning of this statement is that for any c > 1,

the composition of the canonical isomorphism (10.3) with the ma p

Φ : HC

∗

(M, cα, J

) → HC

∗

(M, α, J)

deﬁned by counting index 0 cylinders in a trivial cobordism from (M, α, J) to

(M, cα, J

) is the identity. Writing c = e

for a > 0, the Liouville cobordism in

question is simply

(W, dλ) = ([0, a] × M, d(e

α)),

Lectures on Symplectic Field Theory 207

and one can choose a compatible almost complex structure on this which matches

J and J

on ξ while taking ∂

to g(r)R

for a suitable function g with g(r) = 1

near r = 0 and g(r) = 1/c near r = a. The resulting almost complex manifold

is biholomorphically diﬀeomorphic to the usual symplectization (R × M, J), so our

count of index 0 cylinders is equivalent to the count of such cylinders in (R ×M, J).

The latter are simply the trivial cylinders, all of which are Fredholm regular, so

counting these deﬁnes the identity map on the chain complex.

Finally, we need to show that f or any three h-admissible pair s (α

, J

) with i =

0, 1, 2, the cobordism maps Φ

: HC

∗

(M, α

, J

) → HC

∗

(M, α

, J

) satisfy

(10.5) Φ

◦ Φ

= Φ

We will only sketch this part: the idea is to use a stretching construction. After

rescaling, suppo se without loss of generality that α

= e

α with f

> f

. Then

the cobordism



(r, x)



(x) ≤ r ≤ f

(x)



contains a contact-type hypersurface



(x), x)



x ∈ M



⊂ W

As described at the end of Lecture

9, one can now choose a sequence of compatible

almost complex structures {J

}

N∈N

that are ﬁxed outside a neighborhood

of M

but degenerate in this neighborhood as N → ∞, equivalent to replacing a

small tubular neighborhood of M

with increasingly large colla r s [−N, N] × M in

which J

belongs to J(α

). The resulting chain maps

: CC

∗

(M, α

, J

) → CC

∗

(M, α

, J

)

are chain homotopic for all N, but as N → ∞, the index 0 cylinders counted by

these maps converge to buildings with two levels, t he top one an index 0 cylinder

in the completion of a cobordism from ( M, α

, J

) to (M, α

, J

), while the bottom

one a lso has index 0 and lives in a cobordism from (M, α

, J

) to (M, α

, J

). The

composition Φ

◦ Φ

counts these broken cylinders, so this proves (

10.5).

In particular, we conclude now that each of the cobordism maps

Φ : HC

∗

(M, α

, J

) → HC

∗

(M, α

−

, J

−

)

is an isomorphism, since composing it with a cobordism map in the opposite di-

rection must give the identity. The isomorphism class of HC

∗

(M, α, J) is therefore

independent of the auxiliary da t a (α, J), and will be denoted by

∗

(M, ξ).

This is the cylindrical contact homology o f (M, ξ) in the homoto py class h. It

is deﬁned for any primitive homoto py class h ∈ [S

, M] and closed contact manifold

that is h-admissible in the sense of Deﬁnition

10.16. It is also invariant under

contactomorphisms in the following sense:

Proposition 10 .24. Suppose ϕ : (M

, ξ

) → (M

, ξ

) is a contactomorphism

with ϕ

∗

= h

, where h

∈ [S

, M] is a primitive homotopy class of loops, and

, ξ

) is h

-admissible. Then (M

, ξ

) is h

-admissible, and HC

∗

, ξ

)

∼

∗

, ξ

208 Chris Wendl

Proof. Given an h

-admissible contact form α

on (M

, ξ

) and an h

-regular

∈ J(α

), the contact fo r m α

:= ϕ

∗

on M

is h

-admissible since ϕ deﬁnes a

bijection from P

(α

) to P

(α

) and also a bijection between the sets of contractible

Reeb orbits for α

and α

. Since ϕ

∗

= ξ

, α

is a contact form for (M

, ξ

), hence

the latter is h

-admissible. The diﬀeomorphism ˜ϕ := Id × ϕ : R × M

→ R × M

then maps ∂

to ∂

, R

to R

and ξ

to ξ

, thus J

:= ˜ϕ

∗

∈ J(α

), so ˜ϕ deﬁnes

a biholomorphic map (R × M

, J

) → (R × M

, J

) and thus a bijection between

the sets of holomorphic cylinders in each. It f ollows tha t J

is h

-regular, and the

bijection P

(α

) → P

(α

) deﬁnes an isomorphism between the chain complexes

deﬁning HC

∗

, α

, J

) and HC

∗

, α

, J

). 

10.3. Computing HC

∗

, ξ

)

10.3.1. The Morse-Bot t setup. The contact form α

on T

deﬁned at the

beginning of this lecture has Reeb vector ﬁeld

(ρ, φ, θ) = cos(2πkρ) ∂

+ sin(2πkρ) ∂

Its Reeb orbits therefore preserve and deﬁne linear foliations on each of the tori

{ρ} × T

. In particular, none of the closed orbits are contractible, though all of

them a re also degenerate, as they all come in S

-parametrized families foliating

{const} × T

. For certain homotopy classes h ∈ [S

, T

], this yields a very easy

computation of HC

∗

, ξ

), namely whenever h contains no periodic o r bit s:

Theorem 1 0.25. Suppose h ∈ [S

, T

] is any primitive homotopy class of loops

such that the projection p : T

→ S

: (ρ, φ, θ) 7→ ρ satisﬁes p

∗

h 6= 0 ∈ [S

, S

]. Then

is h-admissible and the resulting contact homology HC

∗

, ξ

) is trivial. 

Now for the interesting part. Every primitive class h ∈ [S

, T

] not covered

by Theorem

10.25 contains closed orbits of R

, all of t hem degenerate since they

come in S

-parametrized families foliating the tori {const}×T

. This ma kes it not

immediately clear whether (T

, ξ

) is h-admissible, though the following observation

in conjunction with Proposition

10.24 shows that if HC

∗

, ξ

) can be deﬁned, it

will be the same for all the homoto py classes under consideration.

Lemma 10.26. Suppose h

, h

∈ [S

, T

] are primitive homotopy class es that are

bo th ma pped to the trivial class under the pro j ection T

→ S

: (ρ, φ, θ) 7→ ρ. Then

there exists a co ntactomorphism ϕ : (T

, ξ

) → (T

, ξ

) satisfying ϕ

∗

= h

Proof. We can represent h

for i = 0, 1 by loo ps of the form γ

(t) = (0, β

(t)) ∈

×T

, where the loops β

: S

→ T

are embedded and thus represent generators

of π

) = Z

. One can thus ﬁnd a matrix



m n

p q



∈ SL(2, Z) such that the

diﬀeomorphism

ϕ : T

→ T

: (ρ, φ, θ) 7→ ( ρ, mφ + nθ, pφ + qθ)

satisﬁes ϕ

∗

= h

. We have

∗

= [q cos(2πkρ) + n sin(2πkρ)] dθ + [p cos(2πkρ) + m sin(2πkρ)] dφ

=: F (ρ) dθ + G(ρ) dφ.

Lectures on Symplectic Field Theory 209

The loop (F, G) : S

→ R

satisﬁes



F (ρ)

G(ρ)





q n

p m



cos(2πkρ)

sin(2πkρ)



where



q n

p m



∈ SL(2, Z), thus (F, G) winds k times about the origin. Any choice

of homotopy from (F, G) to (cos(2πkρ), sin(2πkρ)) through loops (F

, G

) : S

→ R

winding k t imes about the origin with positive rot ational velocity then gives rise to

a homotopy from ϕ

∗

to α

through contact forms F

(ρ) dθ + G

(ρ) dφ. Gray’s

stability theorem therefore yields a conta cto mo r phism ψ : (T

, ξ

) → ( T

, ker ϕ

∗

)

with ψ smoothly isotopic to the identity. The map ϕ ◦ψ is thus a cont actomorphism

of (T

, ξ

) with (ϕ ◦ ψ)

∗

= ϕ

∗

= ϕ

∗

= h

. 

In light of the lemma, we are free from now on to restrict our attention to the

particular homotopy class

h := [t 7→ (0, 0, t)],

which is the homotopy class of the 1-periodic orbits foliating the k t ori

:= {m/k} × T

, m = 0, . . . , k − 1

since R

(m/k, φ, θ) = ∂

. Though the orbits on these t ori are degenerate, it is not

hard to show that they all satisfy the Morse-Bott condition; in fact, α

is a Morse-

Bott contact form. We will explain a self-contained computation of HC

∗

, ξ

) in

the next two sections without using the Morse-Bott condition—but ﬁrst, it seems

worthwhile to sketch how one can guess the answer using Morse-Bott data.

Bourgeois’s t hesis [

Bou02] gives a prescription for calculating contact homology

in Morse-Bot t settings, i.e. for deducing what orbits and what holomorphic curves

will appear under certain standard ways of perturbing the Morse-Bott contact form

to make it nondegenerate. Notice ﬁrst that the only orbits in P

(α

) are t he ones

that foliate the k tori T

, . . . , T

k−1

, and they all have period 1. By (

10.2), it follows

that for any J ∈ J(α

), there can be no nontrivial J-holomorphic cylinders connect-

ing two orbits in P

(α

). This makes the calculation of HC

∗

, ξ

) sound trivial,

but of course there is more to the story since α

is not admissible; indeed, the chain

complex CC

∗

, α

) is not even well deﬁned. The prescription in [

Bou02] now

gives the following. Each of the families of orbits in T

, . . . , T

k−1

is parametrized

by S

, and by a standard perturbation technique, any choice o f a Morse function

: S

→ R for m = 0, . . . , k − 1 yields a contact form α

′

that is C

∞

-close to α

matches it outside a neighborhood of T

, but has a nondegenerate R eeb orbit on T

for each critical point of f

, while every other closed orbit in the perturbed region

can be assumed to have arbitrarily large period. Mor eover, there is a corresponding

perturbation from J ∈ J(α

) to J

′

∈ J(α

′

) such that every gradient ﬂow line of the

function f

: S

→ R gives rise to a J

′

-holomorphic cylinder in R × T

connecting

the corresponding nondegenerate Reeb orbits along T

. In the present situation,

since no J-holomorphic cylinders of the relevant type exist before the perturbation,

the only ones after the perturbation are those t hat come from gradient ﬂow lines.

Now imagine performing a similar perturbation near every T

, . . . , T

k−1

, using

Morse functions f

, . . . , f

k−1

: S

→ R that each have exactly two critical points.

210 Chris Wendl

For the perturbed contact form α

′

, P

(α

′

) now consists of exactly 2k o r bit s

, . . . , γ

k−1

∈ P

(α

′

where we denote by γ

and γ

−

the o r bits on T

corresponding to the maximum

and minimum of f

respectively. For the obvious choice of trivialization τ for the

contact bundle along γ

, one can relate the Conley-Zehnder indices to the Morse

indices of the cor r esponding critical points, giving

(γ

) = 0, µ

(γ

−

) = 1, m = 0, . . . , k − 1.

Moreover, the two gradient ﬂow lines connecting ma ximum and minimum f or each

give rise two exactly two holomorphic cylinders in M

′

, γ

−

, γ

) for each m =

0, . . . , k − 1, and these are a ll t he curves that a re counted for the diﬀerential on

∗

, α

′

, J

′

). Counting modulo 2, we thus have

∂hγ

i = 0 f or all m = 0, . . . , k − 1,

implying

∗

, α

′

, J

′

) =

(

∗ = odd,

∗ = even .

Let us state this as a theorem.

Theorem 10.27. Suppose h ∈ [S

, T

] i s a pri mitive homotopy cla ss that maps

to the trivial class under the projection T

→ S

: (ρ, φ, θ) 7→ ρ. Then (T

, ξ

) is

h-admissible and

∗

, ξ

)

∼

(

∗ = odd,

∗ = even.

Theorem

10.1 is an immediate corollary of this: indeed, if ϕ : (T

, ξ

) → (T

, ξ

ℓ

)

is a contactomorphism, choose any h ∈ [S

, T

] for which Theorem

10.27 applies,

and let h

:= ϕ

∗

h ∈ [S

, T

]. Then HC

∗

, ξ

ℓ

)

∼

2ℓ

implies via Proposition

10.24

that HC

∗

, ξ

)

∼

2ℓ

. But Theorems

10.25 and 10.27 imply that the latter is

also either 0 or Z

, hence k = ℓ.

10.3.2. A digression on the Floer equation. In preparat ion for giving a self-

contained proof of Theorem

10.27, we now explain a general procedure for relating

holomorphic cylinders in a symplectization to solutions of the Floer equation. This

idea is loosely inspired by arg uments in [

EKP06].

To motivate what follows, not ice that on a neighborhood of T

= {0} × T

⊂

, ξ

), we can write

= cos(2πkρ) (dθ + β) ,

where β := tan(2πkρ) dφ deﬁnes a Liouville form on the annulus A := [−1/8, 1/8] ×

with coordinates (ρ, φ). This makes the neighborhood A ×S

⊂ ( T

, ξ

) a special

case of the following general construction.

Lectures on Symplectic Field Theory 211

Definition 10.28. Suppose V is a 2n-dimensional manifold with an exact sym-

plectic form dβ. The contact manifold (V × S

, ker(dθ + β)) is then called the

contact iz ation of (V, β).

Here θ denotes the coordinate on the S

factor.

It’s easy to check that dθ + β is indeed a contact form on V × S

whenever dβ

is symplectic on V : the latter means (dβ)

> 0 on V , so

(dθ + β) ∧ [d(dθ + β)]

= (dθ + β) ∧ (dβ)

= dθ ∧ (dβ)

> 0.

Now here’s a cute trick one can play with contactizations. For t he rest of this

subsection, assume

(V, dβ)

is an arbitra r y compact 2n-dimensional exact symplectic ma nif old with boundary.

Fix a smooth function

H : V × S

→ R,

which we shall think of in the following as a time-dependent Hamiltonian H

H(·, θ) : V → R on (V, dβ). The 2- form on V × S

deﬁned by

Ω = dβ + dθ ∧ dH = d(β − H dθ)

is then ﬁberwis e symplectic, meaning its restriction to each of the ﬁbers of the

projection map V ×S

→ S

is symplectic. We claim that for every ǫ > 0 suﬃciently

small,

:= dθ + ǫ(β − H dθ)

deﬁnes a contact for m on V × S

. This is a var iation on the construction that was

used by Thurston and Winkelnkemper [

TW75] to deﬁne contact forms out of open

boo k decompositions, and the proof is simple enough: since dλ

= ǫΩ, we just need

to check tha t λ

∧ Ω

> 0 for ǫ > 0 suﬃciently small, and indeed,

∧ Ω

= dθ ∧ (dβ)

+ ǫ(β − H dθ) ∧ Ω

> 0

since the ﬁrst term is a volume form and ǫ is small. To see the relation between λ

and the contactization, we can write

= (1 − ǫH) dθ + ǫβ = (1 − ǫH)



dθ +

1 − ǫH



and observe tha t

1−ǫH

β is also a Lio uville form on V whenever H is θ-independent

and ǫ > 0 is suﬃciently small.

The Reeb vector ﬁelds R

for λ

vary with ǫ, but their directions do not, since

dλ

= ǫΩ has the same kernel for every ǫ. Moreover, while λ

ceases to be a cont act

form when ǫ → 0, the Reeb vector ﬁelds still have a well-deﬁned limit: they converge

as ǫ → 0 to the unique vector ﬁeld R

satisfying

dθ(R

) ≡ 1 and Ω(R

, ·) ≡ 0.

The latter can be written more explicitly as

= ∂

+ X

Elsewhere in the literature, the contactization is also often deﬁned as V ×R instead of V ×S

The usage here is consistent with [

MNW13].

212 Chris Wendl

where X

is the time-dependent Hamiltonian vector ﬁeld determined by H

, i.e. via

the condition

dβ(X

, ·) = −dH

As one can easily compute, the reason for this nice behavior as ǫ → 0 is that the R

are also the Reeb vector ﬁelds for a smooth family of stable Hamiltonian structures:

Proposition 10.29. The pairs H

:= (Ω, λ

) for ǫ ≥ 0 suﬃciently small deﬁne a

smooth family of stable Hamiltonian structures whose Reeb vec tor ﬁ e l ds are R

. 

We shall write the hyperplane distributions induced by H

:= ker λ

⊂ T (V × S

These are contact structures fo r ǫ > 0 small, and the space J(H

) o f R-invariant

almost complex structures on R × (V × S

) compatible with H

is then identical

to J(λ

). On the other hand for ǫ = 0, Ξ

= ker dθ is a fo liation, namely it is the

vertical subbundle of the trivial ﬁbration V ×S

→ S

. To interpret H

, notice that

its closed Reeb orbits in the homotopy class of γ : S

→ V × S

: t 7→ (const, t) are

all of the form γ(t) = (x(t), t) where x : S

→ V is a contractible 1- periodic orbit

of X

. Moreover, suppose J ∈ J(H

), which is equivalent to a choice o f compatible

complex structure on the symplectic bundle (Ξ

, Ω|

), or in other words, an S

parametrized f amily of dβ-compatible almost complex structures {J

}

θ∈S

on V .

Then if

u = (f, v, g) : R × S

→ R ×(V × S

)

is a J-holomorphic cylinder asymptotic at {±∞} × S

to two orbits of the form

described above, the nonlinear Cauchy-Riemann equation for u turns out to imply

that (f, g) : R×S

→ R×S

is a holomorphic map with degree 1 sending {±∞}×S

to {±∞}×S

, and we can therefore choo se a unique biholomorphic reparametriza-

tion of u so that (f, g) becomes the identity map. Having done this, the equation

satisﬁed by v : R × S

→ V is now

∂

v + J

(v)( ∂

v − X

(v)) = 0,

in other words, the Floer equation for the dat a {J

}

θ∈S

and {H

}

θ∈S

To complete the analogy, notice that since Ω is exact, we can write down a

natural symplectic action functional with respect to each H

: C

∞

, V × S

) → R : γ 7→

∗

(β − H dθ).

For loops o f the form γ(t) = (x(t), t) with x : S

→ V contractible, this reduces

(give or take a sign—see Remark

10.32) t o the usual formula for the Floer actio n

functional

(10.6) A

(γ) =

∗

β −

H(x(t)) dt =

¯x

∗

dβ −

H(x(t)) dt,

where ¯x : D → V is any map satisfying ¯x|

∂D

= x. Stokes’ theorem gives an easy

relation between the action and the so-called Ω-ene rgy if u : R ×S

→ R ×(V ×S

)

Lectures on Symplectic Field Theory 213

is a J-holomorphic curve for J ∈ J(H

) and is positively/negatively asymptotic to

orbits γ

: S

→ V × S

at s = ±∞: we have

0 ≤

R×S

∗

Ω = A

(γ

) −A

(γ

−

If u(s, t) = (s, v(s, t), t), then the left hand side is identical to the deﬁnition of energy

in Floer homology, namely

(v) :=

R×S

dβ(∂

v, ∂

v − X

(v)) ds ∧ dt =

R×S

dβ(∂

v, J

(v)∂

v) ds ∧ dt,

thus giving the f amiliar relation

(10.7) E

(v) = A

(γ

) − A

(γ

−

To relate this to the usual notion of energy with respect to a stable Hamiltonian

structure, we write the usual formula

(u) := sup

ϕ∈T

∗





ϕ(r)λ



+ Ω



with T :=



ϕ ∈ C

∞

(R, (−ǫ

, ǫ

))



′

> 0



for some constant ǫ

> 0 suﬃciently

small. Notice ﬁrst tha t for any ﬁxed ǫ, Stokes’ theorem gives a bound for E

(u) in

terms of the asymptotic orbits of u since Ω is exact. Finally, in the case ǫ = 0 with

u(s, t) = (s, v(s, t), t), we ﬁnd

(u) = sup

ϕ∈T

R×S

′

(s) ds ∧ dt +

R×S

∗

Ω = 2ǫ

+ E

(v),

so bo unds on E

(u) are equiva lent to bounds on the Floer homological energy E

(v).

The basic fa ct that Floer trajectories v : R × S

→ V with E

(v) < ∞ are asymp-

totic to contra ctible 1-periodic Hamiltonian orbits can now be regarded as a corollary

of our Theorem

9.6.

The above discussion gives a one-to-one correspo ndence between a certain mod-

uli space of unparametrized J-holomor phic cylinders in R ×(V ×S

) and the mod-

uli space of Floer trajectories between contractible 1-periodic orbits in (V, dβ) with

Hamiltonian f unction H. If we can adequately understand the moduli space of Floer

trajectories—in particular if we can classify them and prove that they are regular—

then the idea will be to extend this classiﬁcation via the implicit function theorem to

any J

∈ J(λ

) suﬃciently close to J for ǫ > 0 small. As the reader may be aware,

classifying Floer trajectories is also not easy in general, but it does become easy

under certain conditions. Simple examples of cont ractible 1 -periodic Hamilto nian

orbits are furnished by the constant loops γ(t) = x at critical points x ∈ Crit(H),

and for each such orbit, γ

∗

has a canonical homo t opy class of unitary trivializa-

tions, the so-called constant trivialization. The following fundamental r esult is

commonly used in proving the isomorphism from Hamiltonian Floer homology to

singular homology.

Theorem 10.30. Suppose H : V → R is a smooth Morse function with no

critica l poi nts o n the bo und ary, J is a ﬁxed dβ-compatible almost complex s tructure

on V , and the gradient ﬂow of H with respect to the metric dβ(·, J·) is Morse-Smale

214 Chris Wendl

and transverse to ∂V . Given δ > 0, let H

:= δH : V → R, with Hamiltonian vector

ﬁeld X

δ = δX

, and consider the stable Hamil toni an structure

:= (dβ + dθ ∧ dH

, dθ)

on V × S

with induced Reeb vector ﬁeld R

= ∂

+ X

δ . Then for all δ > 0

suﬃciently small, the following statements hold.

(1) The 1-periodic R

-orbit γ

: S

→ V × S

: t 7→ (x, t) arising from any

critica l point x ∈ Crit(H) is nondegenerate, and its Conley-Zehnd e r index

relative to the constant trivializa tion τ i s related to the Morse index ind(x) ∈

{0, . . . , 2n} by

(10.8) µ

(γ

) = n − ind(x).

(2) Any trajectory γ : R → V satisfying the negative gradient ﬂow question

˙γ = −∇H

(γ) gives rise to a Fredholm regular solution v : R × S

→ V :

(s, t) 7→ γ(s) of the time-ind ependent Floer equation

(10.9) ∂

v + J(v)(∂

v − X

δ (v)) = 0,

and the virtual dimensions of the spa ces of Floer trajectories near v and

gra dient ﬂow trajectories near γ are the same.

(3) Every 1-periodic orbit of X

δ in

V is a constant loop at a critical point

of H.

(4) Every ﬁnite-energy solution v : R×S

→

V of (

10.9) is of the f orm v(s, t) =

γ(s) for some negative gradient ﬂow trajectory γ : R → V .

Proof. The following proo f is based on arguments in [

SZ92], see in particular

Theorem 7.3.

For the ﬁrst statement, let γ(t) = (x, t) for x ∈ Crit(H) and recall from Lecture

the formula for the asymptotic operato r of a 1-periodic orbit,

: Γ(γ

∗

) → Γ(γ

∗

) : η 7→ −J



∇

η − ∇



where ∇ is any symmetric connection on V ×S

. Identifying Γ(γ

∗

) in the natural

way with C

∞

, T

V ), using the trivial connection a nd writing R

(z, θ) = ∂

(z) = ∂

+ δJ(z)∇H(z), A

becomes the operator

= −J∂

− δ∇

H(x)

on C

∞

, T

V ), where ∇

H(x) : T

V → T

V denotes the Hessian of H at x.

Choosing a unitary basis for T

V ident iﬁes this with −J

∂

−δS for some symmetric

2n-by- 2n ma t r ix S and the standard complex structure J



0 −1

1 0



, so ker A

corresponds to the space of 1-periodic solutions to ˙η = δJ

Sη. The Morse condition

implies that S is nonsingular, so the eigenvalues of δJ

S are all nonzero, but they

are also small since δ is small. It follows t ha t nontrivial solutions of ˙η = δJ

Sη

cannot be 1-periodic if S is nonsingular and δ is suﬃciently small, thus proving that

ker A

is trivial, hence γ is nondegenerate.

Lectures on Symplectic Field Theory 215

To calculate µ

(γ), note that λ ∈ σ(A

) if and only if there exists a nontrivial

1-periodic solution η to the equation

˙η = J

(δS + λ)η.

If δ and λ are both close to 0, then the same argument again implies that no such

solution exists unless δS + λ is singular, meaning λ ∈ σ(−δS). On the other hand,

any constant loo p η(t) ∈ ker(λ+δS) furnishes an element of the λ-eigenspace of A

so we obtain a bijection between the spectra of A

and −δS in some neighborhood

of 0. It follows that if S

denotes a pair of nonsingular symmetric matrices deﬁning

asymptotic operators A

= −J

∂

− δS

, then the spectral ﬂows are related by

spec

−

, A

) = −µ

spec

−

, S

)

when δ > 0 is suﬃciently small. Denoting the maximal negative-deﬁnite subspace

of S

by E

−

), this relation implies

dim E

−

) − dim E

−

) = µ

−

) − µ

Now suppose S

is a coo rdinate expression for the Hessian ∇

H(x), hence dim E

−

) =

ind(x) and µ

) = µ

(γ). Choosing S

−



1 0

0 −1



then gives dim E

−

) =

n a nd µ

−

) = 0 by deﬁnition, so µ

(γ) = n − ind(x) follows.

The second statement follows in a similar manner by writing down and compar-

ing the linearized operators for the Floer equation and the negative gradient ﬂow

equation. Let’s leave this as a n exercise.

For the third statement, suppose we have a sequence δ

→ 0 and a sequence o f

loops x

: S

→

V satisfying ˙x

= X

) = δ

). Pick a number c > 0

small enough for part (1) of the t heorem to hold with δ = c, choose a sequence of

integers N

∈ N such that

→ c,

and consider the loops y

: S

→

V : t 7→ x

t). These satisfy

˙y

= N

and since X

is C

∞

-bounded o n V and N

is also bounded, the Arzel`a-Ascoli

theorem provides a subsequence with

→ y

∞

in C

∞

, V ),

where y

∞

: S

→ V satisﬁes ˙y

∞

= X

∞

) for H

:= cH : V → R. But y

∞

also constant: indeed, since y

(t + 1/N

) = y

(t) and N

→ ∞, we can ﬁnd for any

t ∈ S

a sequence q

∈ Z satisfying q

→ t, so

(10.10) y

∞

(t) = lim

k→∞

) = lim

k→∞

(0) = y

∞

(0).

Since the constant orbit y

∞

is nondegenerate by part (1) of the theorem, there can

only be one sequence of solutions to ˙y

= X

) converging t o y

∞

, and we

conclude t hat y

is also constant for all k suﬃciently large.

We will now use a similar trick to prove the fourth statement in the theorem.

We shall work under the additional assumption that

(10.11) |ind(x) − ind(y)| ≤ 1 for all pairs x, y ∈ Crit ( H),

216 Chris Wendl

which suﬃces for the application in §10.3 .3 below.

Suppose to the contrary that there exists a sequence of positive numbers δ

→ 0

with ﬁnite-energy solutions v

: R × S

→

V of the equation ∂

+ J(v

)(∂

−

)) = 0, where each v

(s, t) is not t- independent. By part (3) of the theorem,

we can restrict to a subsequence and assume each v

for large k is asymptotic to

a ﬁxed pair of critical points x

= lim

s→±∞

(s, ·) ∈ Crit(H), and x

6= x

−

since

would otherwise by constant and therefore t-independent. Choose a sequence

∈ N with

→ ∞ and N

→ c,

where c > 0 is chosen suﬃciently small for the ﬁrst three statements in the theorem

to hold with δ = c. Deﬁne w

: R × S

→ V by

(s, t) = v

s, N

t).

Then w

satisﬁes ano ther time-independent Floer equation,

(10.12) ∂

+ J(w

) (∂

− X

)) = 0,

where the Hamiltonian functions H

converge to H

. The standard compactness

theorem for Floer t rajectories should now imply that a subsequence o f w

converges

to a broken Floer trajectory whose levels will be t-independent. Since the setting

may seem a bit nonstandard, here are some details.

The sequence w

is uniformly C

-bounded since V is compact. We claim that

it is also C

-bounded. If not, then there is a sequence z

= (s

, t

) ∈ R × S

with

|dw

)| =: R

→ ∞, and we can use the usual rescaling trick from Lecture

9 to

deﬁne a sequence

: D

→ V : z 7→ w

+ z/R

)

for a suitable sequence ǫ

→ 0 with ǫ

→ ∞ and |dw

(z)| ≤ 2R

for all z ∈

). The latter implies that f

satisﬁes a local C

-bound independent of k, and

since

∂

+ J(f

)



∂

−

J(f

)



elliptic regularity (see Remark

10.31 below) provides a subsequence fo r which f

converges in C

∞

lo c

(C, V ) to a J-holomorphic plane f

∞

: C → V , which is nonconstant

since

|df

∞

(0)| = lim

k→∞

|df

(0)| = 1 .

Since v

and therefore w

are all asymptotic to ﬁxed constant orbits x

, we have a

uniform bound on the Floer energies of w

) = A

) − A

−

) = N

[H(x

−

) − H(x

)] ,

(10.13)

Lifting this assumption requires gluing, whereas we shall only ne e d the usual implicit function

theorem for Fredholm regular solutions of the Floer equa tion.

Lectures on Symplectic Field Theory 217

where the right hand side is bounded since N

→ c. Using change o f variables and

the fact t hat dβ(∂

, J(f

) ∂

) ≥ 0, this implies a uniform bound

dβ(∂

,J(f

) ∂

) ds ∧ dt =

)

dβ(∂

, J(v

) ∂

) ds ∧ dt

≤

R×S

dβ(∂

, J(v

) ∂

) ds ∧ dt = E

) ≤ C,

thus

∗

∞

dβ =

dβ(∂

∞

, ∂

∞

) ds ∧dt =

dβ(∂

∞

, J(f

∞

) ∂

∞

) ds ∧ dt < ∞.

The removable singularity theorem now extends f

∞

to a nonconstant J-holomo r phic

sphere f

∞

: S

→ V , but this violates St okes’ theorem since J is tamed by an exact

symplectic form.

We’ve now shown that the sequence w

: R ×S

→ V is unifor mly C

-bounded,

and it has bounded energy due to (

10.13). Pick any sequence s

∈ R a nd consider

the sequence of translated Floer trajectories

˜w

(s, t) := w

(s + s

, t).

These are also uniformly C

-bounded, so by elliptic regularity (see Remark 10 .31

again), a subsequence converges in C

∞

lo c

(R×S

) to a map w

∞

: R×S

→ V satisfying

∂

∞

+ J(w

∞

) (∂

∞

− X

∞

)) = 0,

and it has ﬁnite energy E

∞

) < ∞ due to (

10.13), implying that w

∞

is asymp-

totic to a pair of 1-periodic orbit s of X

as s → ±∞. By the same arg ument used

in (

10.10) above, w

∞

is also t-independent. It follows that w

∞

(s, t) = γ

∞

(s) for

some nonconstant g radient ﬂow trajectory γ

∞

: R →

V . Depending on t he choice of

sequence s

, this trajectory may or may not be constant, but we can always choose

to guarantee that γ

∞

is not constant: indeed, since each w

is asymptotic to two

separate critical point s at ±∞, s

∈ R can be chosen such t hat w

, 0) stays a

ﬁxed distance away from every critical point of H, and then

∞

(0, 0) = lim

k→∞

, 0) 6∈ Crit(H

One can now adapt the argument of Proposition

10.19 to ﬁnd various sequences s

∈

R that yield potentia lly separate limiting trajectories forming the levels of a broken

trajectory, which is the limit of w

in the Floer topology. But since all the levels

are t-independent and the gradient ﬂow of H

is Morse-Smale, condition (

10.11)

implies that the most complicated (and therefore the only) limit possible involves

a single level w

∞

(s, t) = γ(s), which is a gradient ﬂow trajectory between critical

points whose Morse indices diﬀer by 1. This trajectory is Fredholm regular and has

index 1 due to par t (2) of the theorem, thus by the implicit function theorem, the

only solutions to (

10.12) that can converge to w

∞

are the obvious reparametrizations

of γ, i.e. they are also t-independent. This is a contradiction. 

Remark 10.31. In previous lectures we’ve used the theorem that “C

-bounds

imply C

∞

-bounds” to prove compactness for J-holomorphic curves, but not for

218 Chris Wendl

solutions of inhomogeneous Cauchy-Riemann type equations such as the Floer tra-

jectories w

and rescalings f

in t he above proof. There is an easy trick to reduce

these to our standard setup: as we’ve already seen, solutions of the Floer equation

are equivalent to honest pseudoholomorphic curves in the symplectization of a cer-

tain stable Hamiltonian structure, which is a manifold of two dimensions higher.

A similar trick can be used for any inhomogeneous Cauchy-Riemann type equa-

tion

∂

f = ν, reducing it to an honest Cauchy-Riemann type equation at the cost of

adding two dimensions. This trick was used already by Gromov, see [

Gro85, 1.4.C].

Remark 10.3 2. You may notice with some horror that (10.8) diﬀers by a sign

from what is stated in [

SZ92]. As far as I can tell, the discrepancy arises from

the fact that while Floer homology is traditionally deﬁned in terms of a negative

gradient ﬂow for the a ctio n functional, SFT is based on a posi tive gradient ﬂow—

this is also why the a ctio n functional in ( 10.6) diﬀers by a sign from what we saw

in Lecture

1. If one takes as an axiom that the Conley-Zehnder index should serve

as a “relative Morse index” for the action functional, then changing the sign of the

functional also reverses the signs of Conley-Zehnder indices, so as a result there

appear to be two parallel sign conventions for Conley-Zehnder indices in diﬀerent

sectors of the literature. I’m sorry. It’s not my fault.

Returning now to the family H

, choose δ > 0 suﬃciently small for Theo-

rem

10.30 to hold and deﬁne a modiﬁed family of stable Hamiltonian structures

on V × S

= (Ω

, λ

where

Ω

:= dβ + dθ ∧ dH

and λ

:= dθ + ǫ(β − H

dθ).

Denote the induced hyperplane distributions and Reeb vector ﬁelds by Ξ

and R

respectively. We have only changed the Hamiltonian H by rescaling, so all previous

statements about H

also apply to H

, in particular λ

is contact and J(H

) = J(λ

)

for all ǫ > 0 suﬃciently small, though the upper bound for the allowed range of ǫ

may now depend on δ. Once δ > 0 is ﬁxed by the requirements of Theorem

10.30,

we are still free to take ǫ > 0 is small as we like.

Theorem 10.33. Assume the same hypotheses as in Theorem

10.30, including

(

10.11), and denote the unique extension of J to a n R-invariant almost complex

structure in J(H

) by J

. Given δ suﬃciently small and a ny smooth family of

co mpatible R-invariant almost com plex structures J

∈ J(H

) matching J

at ǫ = 0,

there exists ǫ

> 0 such that every critical point x ∈ Crit(H) gives rise to a smooth

family of nondegenerate closed R

-orbits

: S

→ V × S

ǫ ∈ [0, ǫ

]

with x

(t) = ( x, t), an d every gradient ﬂow trajectory γ : R → V for H gives rise to

a smooth famil y of Fredholm regular J

-holomorphic cylinders

: R × S

→ R × (V × S

) ǫ ∈ [0, ǫ

]

with u

(s, t) = ( s, γ(δs), t). Moreover, for all ǫ ∈ [0, ǫ

], every closed R

-orbit homo-

topic to t 7→ (const, t) belongs to one of the families x

up to parametrization, and

Lectures on Symplectic Field Theory 219

every J

-holomorphic cylinder with a positive and a negative end asymptotic to orbits

of this type belongs to one of the families u

, up to biholomorphic parametrization.

Proof. The ﬁrst part is immediate from the implicit function theorem since the

orbits x

(t) = ( x, t) are nondegenerate and t he curves u

(s, t) = ( s, γ(δs), t) are Fred-

holm r egula r by Theorem

10.30. For the uniqueness statement, o bserve that if ǫ

→ 0

and γ

is a sequence of R

-orbits in the relevant homotopy class, then their p eriods

are uniformly bounded, so Arzel`a -Ascoli gives a subsequence convergent to a closed

-orbit, which is a nondegenerate orbit of the form x

(t) = (x, t) for x ∈ Crit(H)

by Theorem

10.30, so sequences converging to this orbit are unique by the implicit

function theorem. A similar ar gument proves uniqueness of J

-holomorphic cylin-

ders: if ǫ

→ 0 and u

is a J

-holomorphic sequence, then ﬁrst by the uniqueness

of the orbits, we can extract a subsequence for which all u

are asymptotic at both

ends to orbits in ﬁxed families x

converging to x

(t) = (x

, t) as k → ∞. Since

Ω is exact, Stokes’ theorem then gives a uniform bound on the energies E

Since all R

-orbits in the relevant homotopy class are nondegenerate and none are

contractible, one can now prove a s in Proposition

10.19 that u

has a subsequence

convergent to a ﬁnite-energy stable J

-holomorphic building u

∞

consisting only of

cylinders. Its levels ar e asymptotic to orbits of the form x(t) = (x, t) for x ∈ Crit(H),

thus they can be parametrized as (s, t) 7→ (s, v(s, t), t) fo r v : R ×S

→ V satisfying

the H

-Floer equation, hence v( s, t) = γ( δs) by Theorem

10.30. Now since ∇H is

Morse-Smale and indices of critical points can only diﬀer by at most 1, the building

∞

can have at most one nontrivial level u

∞

(s, t) = (s, γ(δs), t), implying u

→ u

∞

Since u

∞

is Fredholm regular, the implicit function theorem does the rest. 

10.3.3. Admissible data for (T

, ξ

). We now complete the computation of

the cylindrical contact homology HC

∗

, ξ

). We can assume via Lemma

10.26

that h is the homotopy class of the orbits in the special set of tori

= {m/k} × T

⊂ T

, m = 0, . . . , k − 1.

Let’s focus for now on the case k = 1, as the general case will simply be a k-fold

cover of this. Thanks to the Morse-Bott discussion in §

10.3.1, we know what we’re

looking for: we want an h-admissible contact form α for (T

, ξ

) such that P

(α)

contains exactly two orbits, both in T

⊂ T

, along with an h-regular J ∈ J(α) such

that the diﬀerential on CC

∗

, α) counts exactly two J-holomorphic cylinders that

connect the two orbits in T

. L et A denote the annulus

A = [−1, 1] × S

with coo r dina tes (ρ, φ). This will play the role of the Liouville manifold (V, dβ) from

the previous section, and we set

β := ρ dφ.

For the Hamiltonian H : A → R, choose a Morse function with the following

properties:

(1) H has a minimum at x

= (0, 0), an index 1 critical point at x

= (0, 1/2),

and no other critical points;

(2) H(ρ, φ) = |ρ| for 1/2 ≤ |ρ| ≤ 1;

220 Chris Wendl

(3) The gradient ﬂow of H with respect to the standard Euclidean metric on

[−1, 1] × S

is Morse-Smale.

Fix a number δ > 0 suﬃciently small so that Theorem

10.30 applies for Floer

trajectories of H

:= δH in A, and since it will turn o ut to be useful in Lemma

10.34

below, assume without loss of generality

δ ∈ Q.

Then following the prescription described above, we consider the family of stable

Hamiltonian structures H

= (Ω

, λ

) on A × S

for ǫ ≥ 0 small, where

= (1 − ǫδH) dθ + ǫρ dφ, Ω

= dρ ∧dφ + δ dθ ∧ dH,

with induced Reeb vector ﬁelds R

and hyperplane distributions Ξ

:= ker λ

Choose J

∈ J(H

) to be any smoot h family such that J

matches the stan-

dard complex structure on A deﬁned by J

∂

= ∂

. Then for all ǫ > 0 suﬃciently

small, Theorems

10.30 and 10.33 give a complete classiﬁcation of all closed R

-orbits

in A × S

homotopic to t 7→ (0, 0, t), as well as a classiﬁcation of all J

-holomorphic

cylinders asymptotic t o them. Up to parametrization, there are exactly two such

orbits,

: S

→ A × S

, i = 0, 1 ,

which correspond to the Morse critical points x

and x

and thus by (

10.8) have

Conley-Zehnder indices

(γ

) = 1 − ind(x

) = 1 − i ∈ {0, 1}

relative t o the constant trivialization τ. There are also exactly two J

-holomorphic

cylinders

: R × S

→ R ×(A × S

corresponding to the two negative gradient ﬂow lines that descend from x

to x

thus the u

are index 1 curves with a negative end approaching γ

and a positive

end approaching γ

. If we can suitably embed this model into (T

, ξ

) and show that

all the orbits and curves needing to be counted a r e contained in t he model, then we

will have a complete description of HC

∗

, ξ

), with two generators hγ

i and hγ

of even and odd degree respectively, satisfying

∂hγ

i = 2hγ

i = 0 and ∂hγ

i = 0

since the former counts two curves and the latter counts none.

Lemma 1 0.34. For any ǫ > 0 suﬃciently small, there exists a contact embedding

(A × S

, ker λ

) ֒→ (T

, ξ

)

identifying the homotopy class of the loops t 7→ (0, 0, t) in A ×S

with h. Moreover,

the contact fo rm λ

and almost co mplex structure J

∈ J(H

) can then be extended to

an h-admissible contact form α on (T

, ξ

) and an h-regular almost complex structure

J ∈ J(α) such that γ

and γ

are the o nly orbits in P

(α), and all J-holomorphic

cylinders with a positive and a n egative end asymptotic to either of these orbits are

co ntained in the interior of A × S

Lectures on Symplectic Field Theory 221

Proof. We’ve chosen β and H so that in the region 1/2 ≤ |ρ| ≤ 1,

α := λ

= (1 − ǫδ|ρ|) dθ + ǫρ dφ =: f(ρ) dθ + g(ρ) dφ,

so the Reeb vector ﬁeld on this region has the form

D(ρ)

′

(ρ) ∂

−f

′

(ρ) ∂

). Notice

that

′

(ρ)

′

(ρ)

= ∓

ǫδ

= ∓δ,

and we assumed δ ∈ Q, so the Reeb orbit s in this region are all periodic. Next, pick

a large number N ≫ 1 and extend α to a contact fo r m on [−N, N] × S

× S

via

the same formula. Now extend the pa th (f, g) : [−N, N] → R

to R such that it has

period 2N + 2 and winds once around the origin over the interval [−N − 1, N + 1],

with positive angular velocity. This produces a contact form α on



(2N + 2)Z



× S

which takes the form f (ρ) dθ + g(ρ) dφ outside of |ρ| ≤ 1/2. We claim in fact that

α is homotopic through contact forms to one that takes this form glo bally, where

(f, g) may be assumed to be a smooth loop winding once around the orig in. To see

this, one need only ho mo t op H in the region |ρ| ≤ 1/2 to a Morse-Bott function that

depends only on the ρ-coordinate; the contact condition holds for all Hamiltonians

in this ho mo topy as long as ǫ > 0 is suﬃciently small. With this understood, the

obvious diﬀeomorphism

→ T

: (ρ, φ, θ) 7→



2N + 2

, φ, θ



pushes ker α forward to a conta ct structure isotopic to one of the fo rm F (ρ) dθ +

G(ρ) dφ for a loop (F, G) : S

→ R

winding once around the origin, so ta king a

homotopy of this loop to (cos(2πρ), sin(2πρ)) and applying Gray’s stability theorem

produces a contactomorphism

, ker α) → (T

, ξ

)

that is isotopic to the above diﬀeomorphism.

The construction clearly guarantees that no closed Reeb orbit of α outside A×S

is homotopic to the preferred class h, and there ar e also no contractible orbits, so

α is an h- admissible contact form on T

. Cho ose any extension of J

to some

J ∈ J(α) on T

. We claim now that if N is cho sen suﬃciently large, then no

J-holomorphic cylinder in R × T

with one positive end at either of the orbits γ

can ever venture outside the region R × (−1/2 , 1/2) × T

. Suppose in pa r ticular

that u is such a curve and its image intersects R × {1/2} × T

. Since the entire

region [1/2 , N] ×T

is foliated by closed Reeb orbits, we can deﬁne Υ to be the set

of Reeb orbits γ in that region for which the image of u intersects R × γ. This is a

closed subset of the connected topological space of all Reeb orbits in [1/2, N] × T

indeed, if γ

∈ Υ is a sequence converging to some orbit γ

∞

, then u(z

) ∈ R × γ

for some sequence z

∈ R ×S

, which must be contained in a compact subset since

the asymptotic orbits of u lie outside of [1/2, N] × T

, hence z

has a convergent

subsequence z

→ z

∞

∈ R × S

with u(z

∞

) ∈ R × γ

∞

, proving γ

∞

∈ Υ. We claim

that Υ is also an open subset of the space of orbits in [1/2, N] × T

. This follows

222 Chris Wendl

from positivity of intersections, as every R × γ is also a J-holomorphic curve: if

u(z) ∈ R ×γ, then for every other closed orbit γ

′

close enough to γ, there is a p oint

′

∈ R × S

near z with u(z

′

) ∈ R × γ

′

. This proves that, in fact, u passes through

R × γ for every or bit γ in t he region [1/2, N] × T

. We will now use this to show

that if N is suﬃciently large, the contact area of u will be larger than is allowed by

Stokes’ theorem.

Let us write

u(s, t) = (r(s, t), ρ(s, t), φ(s, t), θ(s, t)) ∈ R ×





(2N + 2)Z



×S

× S

and choose two points ρ

∈ [1/2, 1] and ρ

∈ [N − 1 , N] which are both regular

values of the function ρ : R × S

→ R/(2N + 2)Z. The intersections of u with the

orbits in [1/2, N] ×T

imply that the function ρ(s, t) attains every value in [1/2, N],

and since the asymptotic limits of u lie outside t his region,

U := ρ

−1

([ρ

, ρ

]) ⊂ R × S

is then a nonempty and compact smoot h submanifold with boundary

∂U = −C

⊔ C

where C

:= ρ

−1

(ρ

) f or i = 1, 2. Restricting u t o the multicurves C

then gives a

pair of smooth maps

: C

→ T

: (s, t) 7→ (φ(s, t), θ(s, t)), i = 1, 2,

which are homologous to each other. Denote the generators o f H

) corresponding

to the φ- and θ-coordinates by ℓ

and ℓ

respectively, and suppose [w

] = mℓ

+ nℓ

for m, n ∈ Z. The key observation now is that the restriction of α to each of the

tori {ρ

} × T

is a closed 1-f orm, thus for each i = 1, 2,

∗

α depends only on

the homology class mℓ

+ nℓ

∈ H

) and not a ny further on the maps w

. In

particular,

∗

α = f(ρ

)n + g(ρ

for i = 1, 2. We now compute,

∗

dα =

∗

α −

∗

α = n[f(ρ

) −f(ρ

)] + m[g(ρ

) − g(ρ

)]

= n[(1 − ǫδρ

) −(1 − ǫδρ

)] + m[ǫρ

−ǫρ

]

= ǫ(ρ

− ρ

)(m − nδ)

This integral ha s to be positive since u

∗

dα ≥ 0 and u is no t a trivial cylinder, thus

m − nδ > 0. Moreover, δ was assumed rational, so if δ = p/q for some p, q ∈ N, we

have

m − nδ =

(mq − np) ≥

implying

R×S

∗

dα ≥

∗

dα ≥

(ρ

− ρ

) ≥

ǫ(N − 2)

Having chosen δ (which determines q) and ǫ in advance, we are free to make N a s

large as we like. But by (

10.2),

R×S

∗

dα cannot be any larger than the period

Lectures on Symplectic Field Theory 223

of its positive asymptotic orbit, which does not depend on N. So this gives a

contradiction, proving that u cannot touch the region {ρ ≥ 1/2}. The mirror image

of this argument shows that u also cannot touch the region {ρ ≤ −1/2}. 

With Lemma

10.34 in hand, the calculatio n of HC

∗

, α, J) for suﬃciently

large N is straightf orward: there is one odd generator and one even generator, with

a trivial diﬀerential, g iving

∗

, ξ

)

∼

(

∗ = odd,

∗ = even.

This calculation can now be extended to (T

, ξ

) by a cheap trick: using the contac-

tomorphism (T

, ker α) → (T

, ξ

), let us identify T

with T

and write α = F α

for some function F : T

→ (0, ∞). Then the k-fold covering map

: T

→ T

: (ρ, φ, θ) 7→ (kρ, φ, θ)

maps t he homotopy class h to itself and pulls back ξ

to ξ

, so Φ

∗

α is a contact form

for ξ

. It is also h-admissible: indeed, Φ

∗

α admits no contractible orbits since they

would project down to contractible orbits on (T

, α), and every orbit in P

(Φ

∗

α)

projects to one in P

(α), hence they are all no ndegenerate. The almost complex

structure Φ

∗

J ∈ J(Φ

∗

α) then makes the map Id ×Φ

: (R ×T

, Φ

∗

J) → (R ×T

, J)

holomorphic, so every Φ

∗

J-holomorphic cylinder counted by HC

∗

, Φ

∗

α, Φ

∗

projects to a J-holomorphic cylinder counted by HC

∗

, α, J), and conversely,

each orbit in P

(α) and each J-holomorphic cylinder has exactly k lifts to the cover.

The generators of CC

∗

, Φ

∗

α) thus consist of 2k orbits, k odd and k even, with

2k connecting Φ

∗

J-holomorphic cylinders that cancel each other in pairs, giving a

trivial diﬀerential. In summary:

∗

, ξ

)

∼

(

∗ = odd,

∗ = even.

LECTURE 11

Coherent orientations

Contents

11.1. Gluing maps and coherence 225

11.2. Permutations of punctures a nd bad orbit s 230

11.3. Orienting moduli spaces in general 232

11.4. The determinant line bundle 234

11.5. Determinant bundles of moduli spaces 237

11.6. An algorithm for coherent orientations 238

11.7. Permutations and bad orbits revisited 240

11.1. Gluing maps and coherence

This lecture will be concerned with orienting the moduli spaces

M(J) := M

g,m

(J, A, γ

, γ

−

)

of J-holomorphic curves in a completed symplectic cobordism

W , in cases where

they are smooth. We assume as usual that all Reeb orbits are nondegenerate so

that the usual linearized Cauchy-Riemann operators are Fredholm.

For SFT and other F loer-type theories, it is no t enough to know that each

component of M(J) is orientable—relations like ∂

= 0 rely on having certain

compatibility conditio ns between the orientations on diﬀerent components. The

point is that whenever a space of broken curves is meant to be interpreted as the

boundary of some other compactiﬁed moduli space, we need to make sure that it

carries the boundary orientation. This compatibility is what is known as coherence,

and in order to deﬁne it properly, we need to return to the subject of gluing.

Our discussion of gluing in L ecture 10 was fairly simple because it was limited

to somewhere injective holomorphic cylinders that could only break along simply

covered Reeb orbits. Recall however that more g eneral holomorphic buildings carry

a certain amount of extra structure that was not relevant in tha t simple case. Even

in a building u that has only two nontrivial levels u

−

and u

, the breaking punctures

carry decorations: i.e. if {z

, z

−

} is a breaking pair in u, then the decoration deﬁnes

an orientation-r eversing orthogonal map

−→ δ

−

between the two “circles at inﬁnity” δ

associated to the punctures z

(see §

9.3.3).

This extra informat ion is uniquely determined if the breaking orbit is simply covered,

but at a multiply covered breaking orbit there is ambiguity, and the decoration

225

226 Chris Wendl

cannot be deduced fro m knowledge of u

−

and u

alone. We therefore need to

consider moduli spaces of curves with a bit of extra structure.

For each Reeb orbit γ in M

or M

−

, choose a point on it s image

∈ im γ ⊂ M

For a J-holomorphic curve u : (

Σ = Σ \ (Γ

∪ Γ

−

), j) → (

W , J) with a puncture

z ∈ Γ

asymptotic to γ, an asymptotic marker is a choice of a ray ℓ ⊂ T

Σ such

that

lim

t→0

u(c(t)) = (±∞, p

)

for any smooth path c(t) ∈ Σ with c(0) = z and 0 6= ˙c(0) ∈ ℓ. If γ has covering

multiplicity m ∈ N, then there are exactly m choices of asymptotic markers at z,

related to each other by the action on T

Σ by the mth roots of unity. We shall

denote

(J) := M

g,m

(J, A, γ

, γ

−

) :=



(Σ, j, Γ

, Γ

−

, Θ, u, ℓ)



∼,

where (Σ, j, Γ

, Γ

−

, Θ, u) represents an element of M

g,m

(J, A, γ

, γ

−

), ℓ denotes an

assignment of asymptotic markers to every puncture z ∈ Γ

, and

(Σ

, j

, Γ

−

, Θ

, u

, ℓ

) ∼ (Σ

, j

, Γ

−

, Θ

, u

, ℓ

)

means t he existence of a biholomorphic map ψ : (Σ

, j

) → ( Σ

, j

) which deﬁnes

an equivalence of (Σ

, j

, Γ

−

, Θ

, u

) with (Σ

, j

, Γ

−

, Θ

, u

) and satisﬁes

∗

ℓ

= ℓ

. There is a natural surjection

(J) → M(J)

deﬁned by for getting the markers. We will say tha t an element u ∈ M

(J) is

Fredholm regular whenever its ima ge under the map to M(J) is regular. Let

$,reg

(J) = M

$,reg

g,m

(J, A, γ

, γ

−

) ⊂ M

(J)

denote the open subset consisting of Fredholm regular curves with asymptotic mark-

ers. Note that comp onents of M(J) and M

(J) consisting of closed curves are

identical spaces; components with punctures have the following simple relationship

to each other.

Proposition 11.1. Each component of M

$,reg

(J) consisting of curves with at

least one puncture admits the structure of a smooth manifold, whose dimension on

ea ch connected component matches that of M

reg

(J). Moreover, the natural ma p

$,reg

(J) → M

reg

(J)

is smooth, and the preimage of a curve u ∈ M

reg

(J) with asymptotic orbits {γ

}

z∈Γ

of covering multiplicities {κ

}

z∈Γ

co ntains exactly

z∈Γ

|Aut(u)|

distinct eleme nts.

Lectures on Symplectic Field Theory 227

Proof. The smooth structure of M

$,reg

(J) arises from the same argument we

used in Lecture

7 for M

reg

(J), supplemented by the following remarks: ﬁr st, ev-

ery nont r ivial automor phism ψ ∈ Aut(u) fo r u ∈ M(J) acts nontrivially on the

asymptotic markers. Indeed, ψ is required to ﬁx each of the punctures and is a bi-

holomorphic map with ψ

≡ Id for some k ∈ N, thus it takes the form z 7→ e

2πim/k

suitable holomorphic coordinat es near each puncture for suitable integers m, k ∈ Z.

If m = 0, then unique continuation implies ψ ≡ Id, and otherwise ψ changes the

asymptotic marker at every puncture. With this understood, one can deﬁne as

in §

7.4 a local identiﬁcation of M

(J) with

∂

−1

(0)/ Aut(Σ, j

, Γ ∪Θ), where

∂

−1

(0)

includes inf ormation about asymptotic markers and is a smooth manifold by t he im-

plicit function t heorem, but Aut(Σ, j

, Γ ∪Θ) acts on it freely, producing a quotient

with no isotropy.

Finally, if (Σ, j, Γ ∪Θ, u) represents an element of M(J) with asymptotic orbits

{γ

}

z∈Γ

, then the number of possible choices of asymptotic markers is precisely

z∈Γ

. However, not a ll of these produce inequivalent elements of M

(J): indeed,

the previous paragraph shows that Aut(u) acts freely on the set of all choices of

markers, so that the total number of inequivalent choices is as stated. 

Suppose u

and u

−

are two (possibly disconnected and/or nodal) ho lomorphic

curves, with asymptotic markers, such that the number of negative punctures of u

equals the number of positive punctures of u

−

, and the asymptotic orbit of u

its ith negative puncture matches that of u

−

at its ith p ositive puncture for every i.

Then the pair (u

−

, u

) naturally determines a holomorphic building: indeed, t he

breaking punctures admit unique decorations determined by identifying the markers

on u

with the markers at corresponding punctures o f u

−

Let us now consider a concrete example of a gluing scenario. Figure

11.1 shows

the degeneration of a sequence of curves in M

3,4

(J, A

, (γ

, γ

), γ

−

) to a building

u ∈

3,4

(J, A + B + C, (γ

, γ

), γ

−

) with one main level and one upper level. The

main level is a connected curve u

∈ M

1,2

(J, A, (γ

, γ

), γ

−

), and the upp er level

consists of two connected curves

∈ M

1,1

, B, γ

, (γ

, γ

)), u

∈ M

0,1

, C, γ

, γ

One can endow each of these curves with asymptotic markers compatible with the

decoration of u; this is a non-unique choice, but e.g. if one chooses markers for u

arbitrarily, then the markers a t the negative punctures of u

and u

are uniquely

determined. Now if a ll three curves are Fredholm regular, then a substantial general-

ization of the gluing procedure outlined in Lecture 1 0 provides open neighbo rhoods

and U

∈ U

⊂ M

1,2

(J, A, (γ

, γ

)),

[(u

, u

)] ∈ U

⊂



1,1

, B, γ

, (γ

, γ

)) × M

0,1

, C, γ

, γ

)



which are smooth manifolds of dimensions

dim U

= vir-dim M

1,2

(J, A, (γ

, γ

)),

dim U

= vir-dim M

1,1

, B, γ

, (γ

, γ

)) + vir-dim M

0,1

, C, γ

, γ

) −1,

228 Chris Wendl

−

R × M

Figure 11.1. The degeneration scenario behind the gluing map (

11.1)

along with a smooth embedding

(11.1) Ψ : [R

, ∞) × U

×U

֒→ M

3,4

(J, A + B + C, (γ

, γ

), γ

−

deﬁned for R

≫ 1 . This is an example of a gluing map: it has the property that

for any u ∈ U

and v ∈ U

, Ψ(R, u, v) converges in the SFT topolo gy a s R → ∞

to the unique building (with asymptotic markers) having main level u and upper

level v, and moreover, every sequence of smooth curves degenerating in this way is

eventually in the image of Ψ.

In analogous ways one can deﬁne gluing maps for buildings with a main level and

a lower level, or more than two levels, or multiple levels in a symplectization (always

dividing symplectization levels by the R-action). It’s important to notice that in all

such scenarios, the domain and target of t he gluing ma p have the same dimension,

e.g. the dimension of both sides of (11.1) is the sum of the virtual dimensions of t he

three moduli spaces concerned.

Definition 11.2. A set of orientations for the connected components of M

(J)

and M

) is called coherent if all gluing ma ps are orientation preserving.

Stated in this way, this deﬁnition is based on the pretense that we never have to

worry about non-regular curves in any components of M

(J), and that is of course

false—sometimes regularity cannot be achieved, in particular for multiply covered

curves. As we’ll see though in §

11.4, the question of orientations can be reframed in

a way that completely disjoins it from the question of regularity, thus we will later

be able to state a more general version of the above deﬁnition that is independent

Lectures on Symplectic Field Theory 229

of regularity (see Deﬁnition 11.14). The main result whose proof we will outline in

the next few sections is then:

Theorem 11.3. Coherent orientations exist.

But there is also some ba d news. The space M

(J) with asymptotic markers

is not actually the space we want to orient. In fact, even the usual moduli space

M(J) has a certain amount of extra information in it that we’d rather not keep

track of when we don’t have to, for instance the ordering of the punctures. Can we

forget this information without forgetting the orientation of the moduli space? Not

always:

Proposition 11.4. Suppose

= (γ

, . . . , γ

), and

is a similar ord ered l ist

of Reeb orbits obtained from

by exchan ging γ

with γ

for som e 1 ≤ j < k ≤ k

Then for any choice of coherent orientations , the natural map

g,m

(J, A,

, γ

−

) → M

g,m

(J, A,

, γ

−

)

deﬁned by permuting the corresponding punctures z

, z

∈ Γ

along with their as-

ymptotic markers is orientation reversing i f and only if the numbers

n − 3 + µ

(γ

)

for i = j, k are both odd. A similar statement holds for permutations of negative

punctures.

This result is the reason for the super-commutative algebra that we will see in

the next lecture. What about forgetting the markers? It turns out that we can

sometimes do that as well, but again not always.

Proposition 11.5. Suppo se M

g,m

(J, A, γ

, γ

−

) → M

g,m

(J, A, γ

, γ

−

) is the

map deﬁned by multiplying the asymptotic marker by e

2πi/m

at one of the punctures

for which the asymptotic orbit is an m-fo l d cover γ

of a simpl e orbit γ. For any

choice of coherent ori e ntations, this ma p reverse orientation if and only if m is even

and µ

(γ

) −µ

(γ) is odd.

Note that in both of the above propositions, only the odd/even parity of the

Conley-Zehnder indices matters, so there is no need to choose trivializations. Propo-

sition

11.5 mot ivates one of the more mysterious technical deﬁnitions in SFT.

Definition 1 1.6. A closed nondegenerate Reeb orbit γ is called a bad orbit if

it is an m-fold cover of some simple or bit γ

′

where m is even and µ

(γ) − µ

(γ

′

)

is odd. Orbits tha t are not bad are called good.

The upshot is that coherent orientations can be deﬁned on the union of all

components M

g,m

(J, A, γ

, γ

−

) for which all of the orbits in the lists γ

and γ

−

are

good. This does not mean that moduli spaces involving bad orbits cannot be dealt

with—in fact, such moduli spaces have the convenient property that the number of

distinct choices of asymptotic markers is always even, and every such choice can be

cancelled by an alt ernat ive choice that induces the opposite orientation. Fo r this

reason, while bad orbits certainly can appear in breaking of holomorphic curves, we

will see that they do not need to serve as generators of SFT.

230 Chris Wendl

11.2. Permutations of punctures and bad orbits

Before addressing the actual construction of coherent orienta tions, we can al-

ready give heuristic proofs of Propositions 11.4 a nd 1 1.5. They are not fully rigor-

ous because they ar e based on the same pretense as Deﬁnition

11.2, namely that

all curves we ever have to worry about (including multiple covers) are regula r . But

we will be able to turn these into precise arguments in §

11.7, after discussing the

determinant line bundle.

Heuristic proof of Proposition

11.4. To simplify the notation, suppose

consists of only two orbits, so

= (γ

, γ

) and

= (γ

, γ

). Consider the

gluing scenario shown in Figure

11.2, where u ∈ M

g,m

(J, A, (γ

, γ

), γ

−

) needs to

be glued to a disjoint union of two planes

∈ M

0,0

, B, ∅, γ

), u

∈ M

0,0

, C, ∅, γ

You might object that there’s no guarantee that such planes must exist in R ×M

e.g. the orbits γ

and γ

might not even be contractible. This concern is valid so

far as it goes, but it misses the point: since we’re talking about gluing rather than

compactness, we do not need any seriously global infor mation about

W and M

as the gluing process doesn’t depend on anything o ut side a small neighborhood of

the curves we’re considering. Thus we are free to change the global structure of

elsewhere so that the planes u

and u

will exist.

If you still can’t imagine

how one might do this, try not to worry about it and just think of Figure

11.2 as a

thought-experiment: it’s a situation that certainly does sometimes happen, so when

it does, let’s see what it implies about orientations.

Assuming all three curves in the picture are r egula r , there will be smooth open

neighborhoods

u ∈ U

⊂ M

g,m

(J, A, (γ

, γ

), γ

−

)

[(u

, u

)] ∈ U

⊂



0,0

, B, ∅, γ

) ×M

0,0

, C, ∅, γ

)



and a gluing map

: [R

, ∞) ×U

×U

֒→ M

g,m

(J, A + B + C, ∅, γ

−

which must be orientation preserving by assumption. But reversing the order of the

product M

0,0

, B, ∅, γ

)×M

0,0

, C, ∅, γ

) and letting u

′

∈ M

g,m

(J, A, (γ

, γ

), γ

−

)

denote the image of u under the map that switches the order of its positive punc-

tures, there are also smooth open neighborhoods

′

∈ U

⊂ M

g,m

(J, A, (γ

, γ

), γ

−

)

[(u

, u

)] ∈ U

⊂



0,0

, C, ∅, γ

) × M

0,0

, B, ∅, γ

)



and a gluing map

: [R

, ∞) ×U

×U

֒→ M

g,m

(J, A + B + C, ∅, γ

−

Of course by the maximum principle, planes with only negative ends will not exist in R ×M

if this is the symplectization of a contact manifold. But we could also change the contact data to

a stable Hamiltonian structure for which such planes are allowed.

Lectures on Symplectic Field Theory 231

−

R × M

Figure 11.2. The gluing thought-experiment used for proving

Propositions 11.4 and 11.5.

If both of these gluing maps preserve orientation, then the eﬀect on orientations of

the map fro m M

g,m

(J, A, (γ

, γ

), γ

−

) to M

g,m

(J, A, (γ

, γ

), γ

−

) deﬁned by inter-

changing the positive punctures must be the same as that of the map

0,0

, B, ∅, γ

) × M

0,0

, C, ∅, γ

) → M

0,0

, C, ∅, γ

) ×M

0,0

, B, ∅, γ

)

, u

) 7→ (u

, u

The latter is orientation reversing if and only if both moduli spaces of planes are

odd dimensional, which means n − 3 + µ

(γ

) is odd for i = 1, 2. 

Heuristic proof of Proposition

11.5. Let us reuse the t hought-experiment

of Figure

11.2, but with diﬀerent details in focus. Suppose γ

in the picture is an

m-fold covered orbit γ

, where γ is simply covered, and suppose that u

is also an

m-fold cover, taking t he form

(z) = v(z

)

for a somewhere injective plane v ∈ M

0,0

, B

, ∅, γ). We’re going to assume again

that all curves in the discussion are regular, including the multiple cover u

; while

this doesn’t sound very plausible, we will see once the determinant line bundle enters

the picture in §

11.4 that it is an irrelevant detail. Now, u

has a cyclic automorphism

group

Aut(u

) = Z

⊂ U(1)

which acts freely on the set o f m choices of asymptotic ma r ker for u

. Then if we

act with the same element of Z

on u

and on the corresponding asymptotic marker

232 Chris Wendl

for u, the building is unchanged, as it has the same decoration. Coherence therefore

implies that the eﬀect on orientations of the map fro m M

g,m

(J, A, (γ

, γ

), γ

−

) to

itself deﬁned by acting with the canonical generator of Z

⊂ U(1) on the marker at

is the same as the eﬀect of the map M

0,0

, mB

, ∅, γ

) → M

0,0

, mB

, ∅, γ

)

deﬁned by composing u

: C → R × M

with ψ(z) := e

2πi/m

The derivative of this map from M

0,0

, mB

, ∅, γ

) to itself at u

deﬁnes a

linear self-map

Ψ : T

0,0

, mB

, ∅, γ

) → T

0,0

, mB

, ∅, γ

)

with Ψ

= 1. The latter implies that Ψ cannot reverse orientation if m is odd. If

m is even, observe t hat the representation theory of Z

gives a decomp osition

0,0

, mB

, ∅, γ

) = V

⊕ V

−1

⊕ V

rot

where Ψ acts on V

±1

as ±1, and V

rot

is a direct sum of real 2-dimensional subspaces

on which Ψ acts by rotations (and therefore preserves orientations). Thus Ψ reverses

the orientation of T

0,0

, mB

, ∅, γ

) if and only if dim V

−1

is odd. As we will

review in the next section, T

0,0

, mB

, ∅, γ

) is a space of holomor phic sec-

tions of u

∗

T (R ×M

) modulo a subspace deﬁned via the linearized auto mo rphisms

of C, so V

consists o f precisely those sections η that satisfy η = η ◦ψ, meaning they

are m-fo ld covers of sections of v

∗

T (R×M

). This deﬁnes a bijective correspondence

between V

and T

0,0

, B

, ∅, γ), so

dim V

−1

= dim M

0,0

, mB

, ∅, γ

) −dim M

0,0

, B

, ∅, γ) (mod 2).

The result then comes from plugging in the dimension formulas for these two moduli

spaces. 

11.3. Or ienting moduli spaces in general

We now discuss concretely what is involved in orienting a moduli space of J-

holomorphic curves.

Recall from Lecture

7 that whenever a curve u

: (

Σ = Σ \Γ, j

) → (

W , J) with

marked points Θ ⊂

Σ is Fredholm regular, a neighborhood of u

in M(J) can be

identiﬁed with

∂

−1

(0)



where G

= Aut(Σ, j

, Γ ∪ Θ) and

∂

is the smooth Fredholm section

T × B

k,p,δ

→ E

k−1,p,δ

: (j, u) 7→ T u + J ◦T u ◦ j,

deﬁned on the product of a G

-invariant Teichm¨uller slice T through j

with a

Banach manifold B

k,p,δ

of W

k,p

-smooth maps

Σ →

W satisfying an exponential

decay condition at the cylindrical ends. Here G

acts on

∂

−1

(0) by

(11.2) G

∂

−1

(0) →

∂

−1

(0) : (ϕ, (j, u)) 7→ (ϕ

∗

j, u ◦ ϕ).

Regularity means tha t the linearizatio n D

∂

, u

) : T

T ⊕ T

k,p,δ

→ E

k−1,p,δ

)

surjective, and the implicit function theorem then gives a natural identiﬁcation

M(J) = ker D

∂

, u

)



aut(Σ, j

, Γ ∪ Θ),

Lectures on Symplectic Field Theory 233

where aut(Σ, j

, Γ ∪Θ) denotes the Lie alg ebra of G

, which acts on ker D

∂

, u

) by

diﬀerentiating (

11.2).

This action actually deﬁnes an inclusion of aut(Σ, j

, Γ ∪Θ)

into ker D

∂

, u

) whenever u

is not constant, thus we can regard aut(Σ, j

, Γ∪Θ)

as a subspace of ker D

∂

, u

As outlined in Proposition

11.1, the space M

(J) with asymptotic markers a d-

mits a similar local description: here one only needs to enhance the structure of the

Banach manifold B

k,p,δ

with information about asymptotic markers at each punc-

ture, so the Banach manifold needed to describe M

(J) is a ﬁnite covering space

of B

k,p,δ

. The rest of the discussion is identical, except for the fact that when markers

are included, G

always acts freely on

∂

−1

(0).

We now make a useful observation about the spaces aut(Σ, j

, Γ ∪ Θ) and T

T :

namely, they both carry natural complex structures and are thus canonically or i-

ented. This follows from the fact that both the automorphism group G

and the

Teichm¨uller space T (Σ, Γ∪Θ) = J(Σ)



Diﬀ

(Σ, Γ∪Θ) are naturally complex mani-

folds. On the linearized level, one way to see it is via the fact—mentioned previously

in §

6.3—that aut(Σ, j

, Γ ∪Θ) and T

]

T (Σ, Γ ∪Θ) can be naturally identiﬁed with

the kernel and cokernel respectively of the natural linear Cauchy-Riemann type

operator on ( Σ, j

(11.3) D

: W

k,p

Γ∪Θ

(T Σ) → W

k−1,p

(End

(T Σ)),

which is the linearization at Id of the nonlinear operator that detects holomorphic

maps (Σ, j

) → (Σ, j

). This operator is equivalent to the operator that deﬁnes the

holomorphic structure of T Σ, thus it is complex linear. To handle the punctures

and marked points, one needs to restrict the nonlinear operator to the space of

k,p

-smooth maps Σ → Σ that ﬁx every point in Γ ∪ Θ, thus the domain of the

linearization becomes the ﬁnite-codimensional subspace

k,p

Γ∪Θ

(T Σ) :=



X ∈ W

k,p

(T Σ)



Γ∪Θ

= 0



This subspace is still complex, thus so is (

11.3), and its kernel and cokernel inherit

natural complex structures.

The complex structure on aut(Σ, j

, Γ∪Θ) means that deﬁning an orientation on

the tangent space T

(J) is equivalent to deﬁning one on ker D

∂

, u

). The

latter operator takes the form

∂

, u

) : T

T ⊕ T

k,p,δ

→ E

k−1,p,δ

)

: (y, η) 7→ J ◦ T u

◦ y + D

η,

where D

: W

k,p,δ

∗

W ) ⊕ V

→ W

k−1,p,δ

(Hom

Σ, u

∗

W )) is the usual lin-

earized Cauchy-Riemann operat or at u

, with V

denoting a complex (#Γ)-dimensional

space of smoo th sections that are constant near inﬁnity.. The remarks above and

the fact t hat u

is J-ho lomorphic imply that the ﬁrst term in this operator,

T → E

k−1,p,δ

)

: y 7→ J ◦ T u

◦ y

The presence o f aut(Σ, j

, Γ ∪ Θ) in this discussion is only r e levant in the ﬁnite set of “non-

stable” cases where χ(

Σ \ Θ) ≥ 0, since otherwise G

is ﬁnite and thus aut(Σ, j

, Γ ∪ Θ) is trivial.

234 Chris Wendl

is a complex-linear map. Now if D

happens also to be a complex-linear map, t hen

we are done, because ker D

∂

, u

) will then be a complex vector space and inherit

a natural orientation.

In general, D

is not complex linear, though it does have a comple x-linear part,

η :=

η − JD

(Jη)) ,

which is also a Cauchy-Riemann type operator . The space of all Cauchy-Riemann

type operators on a ﬁxed vector bundle is aﬃne, so one can interpo late from D

to D

through a path of Cauchy-Riemann type operator s, though they may not all

be Fr edholm—t his depends on the asymptotic operato r s at the punctures. In the

special case however where there are no punctures, one can easily imagine making

use of this idea: if

Σ = Σ is a closed surface, then the obvious homotopy from D

to its complex-linear part yields a homotopy from D

∂

, u

) to its complex-linear

part, and if every operator along this homotopy happens to be surjective, then the

canonical orientation deﬁned on the kernel of t he complex-linear operator determines

an orientation on ker D

∂

, u

There are two obvious problems with the above discussion:

(1) We have no way to ensure tha t every operator in the homotopy from

∂

, u

) to its complex-linear part is surjective;

(2) If there are punctures, then we cannot even expect every operator in this

homotopy to be Fredholm.

The ﬁrst pr oblem motivates the desire to deﬁne a notion of orientations for a

Fredholm operator T that does not require T to be surjective but reduces to the

usual notion of orienting ker T whenever it is. The solution to this problem is

the determinant line bundle, which we will discuss in the next section. With this

object in hand, the above discussion for the case of closed curves can be made

rigorous, so that all smooth moduli spaces of closed J-holomorphic curves inherit

canonical orientations. One of the advantages of using the determinant line bundle

is that the question of orientations becomes entirely disjoined from the question of

transversality: if one can orient the determinant line bundle then moduli spaces of

regular curves inherit orientations, but orienting the determinant bundle does not

require knowing in advance whether the curves are regular.

The second problem is obviously signiﬁcant because in the punctured case, mod-

uli spaces of J-holomorphic curves sometimes have odd real dimension, making it

clearly impossible to homotop D

∂

, u

) through Fredholm opera t ors to one that

is complex linear. The solution in this case will be to deﬁne orientations algorithmi-

cally via the coherence condition, and we will describe a suitable algorithm for this

in §

11.6.

11.4. The determinant line bundle

Fix real Banach spaces X and Y and let Fred

(X, Y ) denote the space of real-

linear Fredholm operators, viewed as an o pen subset of the Banach space L

(X, Y )

of all bounded linear operators. We’ll use the following notat ion throughout: if V

Lectures on Symplectic Field Theory 235

is an n-dimensional real vector space, then the top-dimensional exterior power of V

is denoted by

max

V := Λ

This 1-dimensional real vector space is spanned by any wedge product of the form

∧. . . ∧v

where (v

, . . . , v

) is a basis of V . Denoting the dual space of V by V

∗

note that there is a canonical isomorphism (Λ

max

V )

∗

= Λ

max

∗

. If dim V = 0, then

we adopt the convention Λ

max

V = R.

Definition 11.7. Given T ∈ Fred

(X, Y ), the determinant line of T is the

real 1-dimensional vector space

det(T) = (Λ

max

ker T) ⊗ (Λ

max

coker T)

∗

Our main goal in this section is to prove:

Theorem 11.8. There exists a topological vector bundle det(X, Y )

−→ Fred

(X, Y )

of real rank 1 such that π

−1

(T) = det(T) for each T ∈ Fr ed

(X, Y ).

Observe that whenever T ∈ Fred

(X, Y ) is surjective, det(T) = Λ

max

ker T, so

an orientation of det(T) is equivalent to an orientation of ker T. More generally,

an orientation of det(T) is equivalent to an orientation for ker T ⊕ coker T. If T is

an isomorphism, then det(T) is simply R, so an orientation of det(T) amounts to a

choice of sign ±1.

To construct local trivializations of det(X, Y ) → Fred

(X, Y ), we start with

the case where X and Y are both ﬁnite dimensional. Note that in this case, every

linear map is Fredholm, including the zero map, and its determinant is simply

max

X ⊗ (Λ

max

Y )

∗

Lemma 11.9. Suppose X and Y are real vec tor space s of ﬁnite dimensions n and

m respectively. Then for e very T ∈ L

(X, Y ), there exists a canonical isomorphism

(Λ

max

ker T) ⊗ (Λ

max

coker T)

∗

= (Λ

max

V ) ⊗ (Λ

max

W )

∗

Proof. Suppose dim ker T = k and dim coker T = ℓ, so ind(T) = k−ℓ = n−m,

thus n − k = m − ℓ. We deﬁne a linear map Φ : (Λ

X) ⊗ (Λ

Y )

∗

→



ker T



⊗



ℓ

coker T



∗

via the following procedure. Fix x ∈ Λ

X a nd y

∗

∈ ( Λ

Y )

∗

and

suppose both a re nontrivial. Then for any nontrivial element k ∈ Λ

ker T, there

exists a unique element v ∈ Λ

n−k

(X/ ker T) such that for any subspace V ⊂ X

complementary to ker T, the element

v ∈ Λ

n−k

V ⊂ Λ

n−k

X o btained from v by

inverting the na t ur al isomorphism V → X/ ker T induced by the projection X →

X/ ker T satisﬁes

k ∧

v = x.

The map T descends to an isomorphism X/ ker T → im T and thus induces an iso-

morphism Λ

n−k

(X/ ker T) → Λ

m−ℓ

(im T) ⊂ Λ

m−ℓ

Y , which takes v to a nontrivial

element Tv. There is then a unique element c ∈ Λ

ℓ

coker T = Λ

ℓ

(Y/ im T) such

that for any subspace W ⊂ Y complementary to im T, the element

c ∈ Λ

ℓ

W ⊂ Λ

ℓ

obtained from c by inverting the isomorphism W → Y/ im T induced by the pro-

jection Y → Y / im T satisﬁes

∗

(

c ∧ Tv) = 1.

236 Chris Wendl

Now deﬁne Φ as the unique linear map such that

Φ(x ⊗y

∗

) = k ⊗c

∗

where c

∗

∈ (Λ

ℓ

coker T)

∗

is deﬁned by c

∗

that this deﬁnition does not depend on a ny choices: indeed, if we replace k by λk

for some λ ∈ R \ {0} in the above procedure, then v is replaced by

v, hence Tv

becomes

Tv, c becomes λc and c

∗

therefore becomes

∗

, so that k⊗c

∗

is replaced

(λk) ⊗



∗



= k ⊗ c

∗



To construct local trivializatio ns of det(X, Y ) in the inﬁnite-dimensional case,

recall the following construction from Lecture

3. Given T

∈ Fred

(X, Y ), we can

write X = V ⊕ K and Y = W ⊕ C where K = ker T

, C

∼

coker T

, W = im T

and T

: V → W is a n isomorphism. We shall use these splittings to write any

other operator T ∈ Fred

(X, Y ) as

T =



A B

C D



and let U ⊂ Fred

(X, Y ) denote the open neighborhood of T

for which the block

A : V → W is invertible. This gives rise to a pair of smooth maps

Φ : U → L

(K, C) : T 7→ D − CA

−1

and

F : U → L

(V ⊕ K) = L

(X) : T 7→



1 −A

−1

0 1



such that F (T) is always invertible and maps {0}⊕ker Φ(T) isomorphically to ker T.

Similarly, there is a smooth map

G : U → L

(W ⊕ C) = L

(Y ) : T 7→



1 0

−CA

−1



such that G(T) is always invertible and maps im T isomorphically to W ⊕im Φ(T),

so it descends to an isomorphism of coker T to coker Φ(T). Given the canonical iso-

morphism det(Φ(T)) = Λ

max

K ⊗(Λ

max

∗

= det(T

) from Lemma

11.9, the result-

ing smooth families of isomorphisms ker T → ker Φ(T) and coker T → coker Φ(T)

determine a local trivialization

det(X, Y )|

→ U × det(T

I will leave it as an exercise for the reader to check that the r esulting transition

maps are cont inuous.

This detail should not be underestimated, e.g. [

MW, §7.4] observes that the local trivializa-

tions constructed in [

MS04, §A.2] are, unfortunately, no t continuously compatible. See [Zin] for

further disc ussion of this point. If you discover that my local trivializations are also not continu-

ously compatible, please let me know.

Lectures on Symplectic Field Theory 237

Exercise 11.10. Show that if X and Y are complex Banach spaces, then

the restriction of det(X, Y ) to the subspace of complex-linear Fredholm operators

Fred

(X, Y ) ⊂ Fred

(X, Y ) admits a canonical orientation compatible with the

complex structures of ker T and coker T for each T ∈ Fred

(X, Y ). Show also that

whenever T ∈ Fred

(X, Y ) is an isomorphism, the canonical orientation of det(T)

agrees with the standard orientation of R.

The orientation of det(T) for T ∈ Fred

(X, Y ) described in Exercise

11.10 is

called the complex orientation.

11.5. Determinant bundles of moduli spaces

Combining ideas from the previous two sections, let

det(J) → M

(J)

denote the topolo gical line bundle that associates to a ny u ∈ M

g,m

(J, A, γ

, γ

−

)

the determinant line of the Fredholm operator

: W

k,p,δ

∗

W ) ⊕ V

→ W

k−1,p,δ

(

Hom

Σ, u

∗

W )).

One can construct local trivializations for this bundle using Theorem

11.8 and any

choice of local trivializations for the Banach space bundles T B

k,p,δ

and E

k−1,p,δ

Proposition 11.11. Any orientation of det(J) → M

(J) canonically deter-

mines an orientation of M

reg

(J).

Proof. As explained in §

11.3, an orientation of M

reg

(J) near a part icular curve

: (

Σ, j

) → (

W , J) is equivalent t o a continuously varying choice of orientations

for the kernels

ker D

∂

(j, u) ⊂ T

T ⊕ T

k,p,δ

for all (j, u) ∈

∂

−1

(0), where T is a Teichm¨uller slice through j

. The operator

∂

(j, u) is of the form

L(y, η) := J ◦T u ◦ y + D

and thus is homotopic through Fredholm operators to

(y, η) := D

η,

namely via the homotopy L

(y, η) := sJ ◦T u ◦y +D

η for s ∈ [0, 1]. The kernel and

cokernel of L

are T

T ⊕ ker D

and coker D

respectively, and since T

T carries a

complex structure, the orientation of det(D

) naturally determines an orientation

of det(L

). Using the homotopy L

, this determines orientations of det(D

∂

(j, u))

and thus orientat ions of ker D

∂

(j, u) fo r all (j, u) near (j

, u

), and this orientation

does no t depend o n the choice of Teichm¨uller slice since the operators D

also do

not. 

From now on, when we speak of an orientation of M

(J), we will actually

mean a n orientation of t he bundle det(J) → M

(J). The above proposition implies

that this is equivalent to what we want in applications, but one advantage of talking

about det(J) is that there is no need to limit the discussion to curves that a r e

238 Chris Wendl

regular, i.e. the notion of an orientation of M

(J) now makes sense even though

(J) is not globally a smooth object.

Proposition 11.1 2. Suppose all Reeb orbits in γ

have the property that their

asymptotic operators are complex linear. Then M

g,m

(J, A, γ

, γ

−

) admits a natural

orientation, known as the complex orientation.

Proof. Having complex-linear asymptotic operators implies that the obvious

homotopy from each Cauchy- Riemann operator D

to its complex-linear part does

not change the a symptotic operators and is therefore a homotopy through Fred-

holm operators. We therefore have a continuo usly varying homotopy of each of the

relevant ﬁbers of det(J) to the determinant bundle over a family of complex-linear

operators, which inherit the complex orientation described in Exercise

11.10. 

Proposition

11.12 applies in particular to all moduli spaces of closed J- holomorphic

curves, and thus solves the orientation problem in that case.

11.6. An algorithm for coherent o rientations

We now brieﬂy describe the construction of coherent o r ientations due to Bour-

geois and Mohnke [BM04]. A slightly diﬀerent construction is described in [EGH00],

though it appears to have minor erro r s in some details.

Recall from Lecture

4 the notion of a n asymptotically Hermitian vector bundle

(E, J) over a punctured Riemann surface (

Σ, j). Here (

Σ, j) is endowed with the

extra structure of ﬁxed cylindrical ends (

, j)

∼

, i) f or each puncture z ∈ Γ

which determines a choice of asymptotic markers. Likewise, the bundle E comes

with an a symptotic bundle (E

, J

, ω

) → S

associated to each puncture, carrying

compatible complex and symplectic structures. We shall now endow E with a bit

more structure that is always naturally present in the case E = u

∗

W : namely,

assume each of the asymptotic bundles comes with a splitting

(11.4) (E

, J

, ω

) = ( C ⊕

, i ⊕

, ω

⊕ ˆω

where ω

is the standard symplectic structure on the trivial complex line bundle

(C, i) over S

, and (

, ˆω

) → S

is anot her Hermitian bundle. Fix a choice

}

z∈Γ

of nondegenerate asymptotic operator s on each of the bundles (

, ˆω

and deﬁne the topological space

CR(E, {A

}

z∈Γ

)

to consist of all Cauchy-Riemann type operators on E that are asymptotic at the

punctures z ∈ Γ to the asymptotic operators

(−i∂

) ⊕ A

: Γ(C ⊕

) → Γ(C ⊕

This is an aﬃne space, so it is contractible, and if δ > 0 is suﬃciently small

and V

⊂ Γ(E) denotes a complex (#Γ)-dimensional space of smooth sections

that take constant values in C ⊕ {0} ⊂ E

near each puncture z, then every

D ∈ CR(E, {A

}

z∈Γ

) determines a Fredholm operator

D : W

k,p,δ

(E) ⊕ V

→ W

k−1,p,δ

(Hom

(T Σ, E)).

Lectures on Symplectic Field Theory 239

It follows that a choice of orientation of the determinant line for a ny one o f these

operators determines an orientatio n for all of them. The point of this construction

is that every u ∈ M

(J) determines an operator D

belonging to a space of this

form.

We now construct a gluing operation for Cauchy-Riemann operators that lin-

earizes the gluing maps described in §11.1. Suppose (E

, J

) → (

= Σ

\Γ

, j

) for

i = 0, 1 is a pair of asymptotically Hermitian bundles of the same rank, endowed

with asymptotic splittings as in (

11.4) and asymptotic operators {A

}

z∈Γ

, and that

there exists a pair of punctures z

∈ Γ

and z

∈ Γ

−

such that some unitary bundle

isomorphism

∼

−→

identiﬁes A

with A

. Note that such an isomorphism is uniquely determined up

to homotopy whenever it exists. For R > 0, we can deﬁne a family of glued Riemann

surfaces

(

= Σ

\ Γ

, j

)

by cutting oﬀ the ends (R, ∞) × S

⊂

and (−∞, −R) × S

⊂

and g luing

{R} × S

⊂

to {−R} × S

⊂

. The glued Riemann surface contains an

annulus biholomorphic to ([−R, R] × S

, i) in place of the inﬁnite cylindrical ends

at the punctures z

and z

. The unita ry isomorphism

→

then determines

an isomorphism E

→ E

via the splitting (

11.4) and hence an asymptotically

Hermitian bundle

, J

) → (

, J

Using cutoﬀ functions in the neck [−R, R] × S

, any Cauchy-Riemann operators

∈ CR(E

, {A

}

z∈Γ

) for i = 0, 1 now determine a f amily of operators

∈ CR(E

, {A

}

z∈Γ

)

uniquely up to homotopy. Analogously to the gluing maps in §11.1, one can a r-

range this construction so that the operators D

converge in some sense to the pair

, D

) as R → ∞, which has the following consequence:

Lemma 11.13 ([BM04, Corollary 7]). For R > 0 suﬃciently large, there is a

natural isomorphism

det(D

) ⊗ det(D

) → det(D

)

that is d eﬁned up to homotopy. 

Up to some additional direct sums and quotients by ﬁnite-dimensional complex

vector spaces, this isomorphism should be understood as the linearization of a glu-

ing map between moduli spaces, generalized to a setting in which the holomorphic

curves involved need not be regular. To orient M

(J) coherently, it now suﬃces to

choose orientations for the operators in CR(E, {A

}

z∈Γ

) that vary continuously un-

der deformations of j and E and are preserved by t he isomorphisms of Lemma

11.13.

This mo tivates the f ollowing generalization of Deﬁnition

11.2.

Definition 11.14. A system of coherent orientations is an assignment to

each asymptotically Hermitian bundle (E, J) → (

Σ, j) with asymptotic splittings as

in (

11.4) and asymptotic o perators {A

}

z∈Γ

of an orientation for the determinant

240 Chris Wendl

line of each D ∈ CR(E, {A

}), such that these orienta t ions vary continuously with

D as well as the data j and J, and such that the isomorphisms in Lemma

11.13 are

always orientation preserving.

The prescription of [

BM04] to construct such systems is now as follows.

(1) For any trivial bundle E over

Σ = C with ∞ as a negative puncture and any

asymptotic operator A

∞

, choose an arbitrary cont inuous family of orien-

tations for the operators in CR(E, {A

∞

}), subject only to the requirement

that these should match the complex orientation whenever A

∞

is complex

linear.

(2) For any trivial bundle E

−

over

Σ = C with ∞ as a positive puncture, any

asymptotic operator A

∞

and any D

−

∈ CR(E

−

, {A

∞

}), let E

denote the

trivial bundle over C with a negative puncture as in step (1), choose any

∈ CR(E

, {A

∞

}) and construct the resulting family of glued operators

∈ CR(E

where t he E

are trivial bundles over S

. Since S

has no punctures, D

has a natural complex orientation, so deﬁne the orientation of D

−

to be

the one that is compatible via Lemma

11.13 with this and the orientation

chosen for D

in step (1).

(3) For an arbitrary (E, J) → (

Σ, j), glue positive and negative planes to

Σ to

produce a bundle over a closed surface

Σ, and deﬁne t he or ientation of any

D ∈ CR(E, {A

}

z∈Γ

) to be compatible via Lemma

11.13 with the choices

in steps (1) and (2) and the complex orientation for operators over

Σ.

It should be easy to convince yourself that if we now vary the bundle (E, J) →

(

Σ, j) or the operators on this bundle (but not the asymptotic operators!) contin-

uously, the capping procedure described in step (3) above produces a continuous

family of Cauchy-Riemann type operators on bundles over closed Riemann surfaces.

Since these all carry the complex orientation, t he resulting orientations of the orig-

inal operators vary continuously. It is similarly clear from the construction that

any Cauchy-Riemann operator whose asymptotic operators are all complex linear

will end up with the complex orientation. Bourgeois and Mohnke use t his fact to

prove that any system of orientations constructed in this way is compatible with

all possible linear gluing maps arising from Lemma

11.13. The idea is to reduce

it to the complex-linear case by gluing cylinders to the ends of any asymptotically

Hermitian bundle so that the asymptotic operators can be changed at will; see

[BM04, Prop osition 8].

11.7. Permutations and bad orbits revisited

The heuristic proofs in §

11.2 can now be ma de precise in the following way.

Suppose D ∈ CR(E, {A

}

z∈Γ

), and D

′

is the same operator after interchanging

two of the punctures in Γ. Imagine gluing (E, J) → (

Σ, j) to trivial bundles E

and

over planes in order to cap oﬀ the two punctures that are being interchanged,

and choose Cauchy-Riemann operators D

and D

on these planes to form a glued

operator on the capped surfa ce. This capping procedure is done one plane at a time,

Lectures on Symplectic Field Theory 241

and the order of the two punctures determines which pla ne is glued ﬁrst. Compati-

bility with the isomorphisms of Lemma

11.13 then dictates that the orientations of

det(D) and det(D

′

) match if and only if the orientations of det(D

) ⊗ det(D

) and

det(D

)⊗det(D

) match. Since o r ientations of det(D

) for i = 1, 2 are equivalent to

orientations of ker D

⊕coker D

, reversing the order of the tensor product changes

orientations if and only if both of these direct sums are odd dimensional, which

means ind(D

) and ind (D

) are both odd. If the bundles have complex rank n and

the asymptotic operators are A

for k = 1, 2, we have

ind(D

) = nχ(C) ±µ

((−i∂

⊕ A

) ±δ) = n − 1 ± µ

which matches n − 3 + µ

) modulo 2. This proves Proposition

11.4.

Similarly for Proposition

11.5, we consider the action of the generator ψ ∈ Z

on det(D) where ψ rotates the cylindrical end by 1/m at some puncture where the

trivialized asymptotic operator A is of the form −i∂

−S(mt) for a loop of symmetric

matrices S(t). Capping oﬀ this puncture with a plane carrying a Cauchy-R iemann

operator D

∞

, coherence dictates that the same tra nsfor mation must act the same

way on the orientation of det(D

∞

). Since ψ

= 1, ψ cannot r everse this orientation

if m is odd. To understand the case of m even, note ﬁrst that we are free to

choose D

∞

so that it is an m-fold cover, meaning it is related to the branched cover

ϕ : C → C : z 7→ z

∞

(η ◦ ϕ) = ϕ

∗

∞

for some other Cauchy-Riemann operator

∞

, which is asymptotic to

A := −i∂

−

S(t). Now the group Z

generated by ψ acts on ker D

∞

and coker D

∞

, so represen-

tation theory tells us

ker D

∞

= V

⊕ V

−1

⊕ V

rot

coker D

∞

= W

⊕ W

−1

⊕ W

rot

where ψ acts on V

±1

and W

±1

as ±1 and acts as orientation-preserving rotations on

rot

and W

rot

. It follows that ψ reverses the orientat ion of ker D

∞

⊕coker D

∞

if and

only if dim V

−1

−dim W

−1

is odd. Now observe that there are natural isomorphisms

= ker

∞

, W

= coker

∞

hence

dim V

−1

−dim W

−1

= ind(D

∞

) − ind(

∞

) (mod 2).

This diﬀ erence in Fredholm indices is precisely µ

(A) − µ

(

A) up to a sign, and

this completes the proof o f Proposition

11.5.

LECTURE 12

The generating function of S FT

Contents

12.1. Some important caveats on transversality 243

12.2. Auxiliary data, grading and supercommutativity 244

12.3. The deﬁnition of H and commutators 247

12.4. Interlude : How to count points in an orbifold 251

12.5. Cylindrical contact homology revisited 256

12.6. Combinatorics of gluing 259

12.7. Some remarks on torsion, coeﬃcients, and conventions 263

12.7.1. What if H

(M) has torsion? 263

12.7.2. Combinatorial conventions 263

12.7.3. Coeﬃcients: Q, Z or Z

? 264

It is time to begin deriving algebraic consequences fro m the analytical results of

the previous lectures. We saw the simplest possible example of this in Lecture

10,

where the behavior of holomorphic cylinders in symplectizations of contact mani-

folds without contractible R eeb orbits led to a rudimentary version of cylindrical

contact homology HC

∗

(M, ξ) with Z

coeﬃcients. Unfortunat ely, the condition on

contractible orbits means that this version of HC

∗

(M, ξ) cannot always be deﬁned,

and even when it can, it only counts cylinders—we wo uld only expect it to capture

a small fragment of the infor ma t ion contained in more general moduli spaces of

holomorphic curves. Extracting information from these general moduli spaces will

require enlarging our algebraic notion of what a Floer-type theory can look like.

12.1. Some important caveats on transversality

For this and the next lecture, we ﬁx the following fantastically optimistic as-

sumption:

Assumption 1 2.1 (science ﬁction). One can ch oose s uitably compatible almost

co mplex structures so that all pseudoholomorphic curves are Fredholm regular.

This assumption held in Lecture 10 for the curves we were interested in, because

they were all guaranteed for topolo gical reasons to be somewhere injective. It can

also be shown to hold under some very restrictive conditions on Conley-Zehnder

indices in dimension three, see [N el15, Nel13]. Both of those are very lucky situ-

ations, and as we’ve discussed befo re, the assumption cannot generally be achieved

merely by perturbing J generically—it must sometimes fail for curves that are mul-

tiply covered, and such curves always exist (see §

12.4 for more on this). The only

243

244 Chris Wendl

way in reality to ensure something like Assumption 12.1 is to perturb the nonlin-

ear Cauchy-Riemann equation more abstractly, e.g. by replacing

∂

u = 0 with an

inhomogeneous equation of the form

∂

u = ν

for a generic perturbation ν. This is the standard technique in certain versions of

Gromov-Witten theory, see e.g. [

RT95, RT97]. Alternatively, one can allow J to

depend g enerically on points in the domain rather than just points in the target,

as in [

MS04, §7.3]. Both approaches eliminate the initial problem with multiple

covers, but they both also run into serious and subtle diﬃculties concerning the r e-

lationship between M(J) and the strata of its compactiﬁcation

M(J). As observed

in [

Sal99, §5], the possibility of symmetry in strata of M(J) makes it necessary

for any suﬃciently general abstract perturbation scheme to involve multivalued per-

turbations, and it is important for these perturbations to be “coherent” in a sense

analogous to our discussion of orientations in the previous lecture. These notions

have no t yet a ll been developed in a suﬃciently consistent and general way to give

a rigorous deﬁnition o f SFT, though there has been much progress: this is the main

objective of the long -running polyfold project by Hofer-Wysocki-Zehnder [

Hof06].

Recently, a quite diﬀerent and much more topological approach has been proposed

by John Pardon [

Par].

For most of this lecture we will ignore these subtleties and simply adopt As-

sumption

12.1 as a convenient ﬁction, thus pretending that all components o f M(J)

are smooth orbifolds of the correct dimension and all gluing maps are smoot h. All

“theorems” stated under this assumption should be read with the caveat that they

are only true in a ﬁctional world in which the assumption holds. Even if it is a

ﬁction, o ne can get quite far with this point of view: it is still possible not only

to deduce t he essential structure o f what we assume will someday be a rigorously

deﬁned polyfold-based SFT, but also to infer the existence of certain contact in-

varia nts that have interesting rigorous applications requiring only well-established

techniques, e.g. the cobordism obstructions discovered in [

LW11].

12.2. Auxiliary data, grading and supercommutativity

The goal is to deﬁne an invariant o f closed (2n − 1)-dimensional conta ct mani-

folds ( M, ξ) with closed nondegenerate Reeb or bits as generators and a Floer-type

diﬀerential counting J-holomorphic curves in the symplectization (R × M, d(e

α)).

The auxiliary data we choose must obviously therefore include a nondegenerate con-

tact form α and a generic J ∈ J(α), f or which we shall assume Assumption 12.1

holds. For convenience, we will also assume throughout mo st of this lecture:

Assumption 12.2. H

(M) is torsion free.

This is needed mainly in o r der to be able to deﬁne an integer gr ading, though

without this assumption, it is still always possible to deﬁne a Z

-grading—see §

12.7.1

for more on what to do when Assumption 12 .2 does not hold. We now supplement

the auxiliary data (α, J) with the following additional choices:

Lectures on Symplectic Field Theory 245

(1) Coherent orientatio ns as in Lecture

11 for t he moduli spaces M

(J) with

asymptotic markers.

(2) A collection of reference curves

∼

, . . . , C

⊂ M

whose homology classes form a basis of H

(M).

(3) A unitary trivialization of ξ along each of the reference curves C

, . . . , C

denoted collectively by τ.

(4) A spanning surface C

for each periodic Reeb orbit γ: this is a smooth

map of a compact and or iented surface with boundary into M such that

∂C

] − [γ]

in t he sense of singular 2-chains, where m

∈ Z are the unique coeﬃcients

with [γ] =

] ∈ H

(M).

These choices determine the following. To any collections of R eeb orbits γ

(γ

, . . . , γ

) and any relative homology class A ∈ H

(M,

∪

−

) with ∂A =

[γ

] −

[γ

−

], we can now associate a cycle in absolute homology,

A +

−

∈ H

(M).

Indeed, the boundary of this real 2-chain is a sum of linear combinations of the

reference curves C

, which add up to zero because

[γ

] and

[γ

−

] are homolo-

gous. We shall abuse notation and use this correspondence to associate the absolute

homology class

[u] ∈ H

(M)

to any asymptotically cylindrical holomor phic curve u in R × M. Adapting the

previous notation,

g,m

(J, A, γ

, γ

−

)

for A ∈ H

(M) will now denote a moduli space of curves whose r elative homology

classes glue to the chosen capping surfaces to form A.

Secondly, the cho sen trivializations τ along the reference curves can be pulled

back and extended over every capping surface C

, giving trivializations of ξ along

every orbit γ uniquely up to homotopy. We shall deﬁne

(γ) ∈ Z

from now on to mean the Conley-Zehnder index of γ relative to this trivialization.

Exercise 12.3. Show that if H

(M) has no torsion and u :

Σ → R × M is

asymptotically cylindrical, then its relative ﬁrst Chern number with respect to the

trivializations τ described above satisﬁes

∗

T (R × M)) = c

([u]),

where c

([u]) denotes the evaluation of c

(ξ) ∈ H

(M) on [u] ∈ H

(M).

246 Chris Wendl

By Exercise

12.3, the index o f a curve u : (

Σ = Σ \ Γ, j) → (R × M, J) with

[u] = A ∈ H

(M) and a symptotic orbits {γ

}

z∈Γ

can now be written as

(12.1) ind ( u) = −χ(

Σ) + 2c

(A) +

z∈Γ

(γ

) −

z∈Γ

−

(γ

In order to keep track of ho mo logy classes of holomorphic curves algebraically,

we can deﬁne our theory to have coeﬃcients in the group ring Q[H

(M)], or more

generally,

R := Q[H

(M)/G]

for a given subgroup G ⊂ H

(M). Elements of R will be written as ﬁnite sums

∈ R, c

∈ Q, A

∈ H

(M)/G,

where the multiplicative structure of the group ring is derived from the additive

structure of H

(M)/G by e

:= e

A+B

. The most common examples of G are

(M) and the trivial subgroup, giving R = Q or R = Q[H

(M)] respectively. We

will see a geometrically meaningful example in between these two extremes in the

next lecture.

Finally, we deﬁne certain formal variables which have degrees in Z or Z

for

some N ∈ N, and will serve as generators in our graded algebra. To each closed

Reeb orbit γ we associate two variables, q

, p

, whose integer-valued degrees are

| = n − 3 + µ

(γ), |p

| = n − 3 − µ

(γ).

To remember these numbers, think of the index of a J-holomorphic plane u positively

or negatively asymptotic to γ, with [u] = 0.

We also assign a n integer grading to the group ring Q[H

(M)] such that r ational

numbers have degree 0 and

| = −2c

(A), for A ∈ H

(M).

If c

(A) = 0 for every A ∈ G, in particular if c

(ξ) = 0, then this descends to an

integer grading on the ring R = Q[H

(M)/G]. Otherwise, R inherits a Z

-grading,

where

N := min



(A) > 0



A ∈ G



A Z

-grading is well deﬁned in every case.

The algebra will include one additional formal variable ~, which is deﬁned to

have degree

|~| = 2(n − 3) .

The degrees of ~ and the p

and q

varia bles should all be interpreted modulo 2N

if c

(ξ)|

6= 0.

The algebra of SFT uses monomials in the variables p

and q

respectively to

encode sets of positive and negative asymptotic orbits of holomorphic curves, while

the gro up ring R = Q[H

(M)/G] is used to keep track of the homology classes of

such curves, and powers of ~ are used to keep track of their genus. More precisely,

Lectures on Symplectic Field Theory 247

given g ≥ 0, A ∈ H

(M) and ordered lists of Reeb orbits γ

= (γ

, . . . , γ

), we

encode the moduli space M

g,0

(J, A, γ

, γ

−

) formally via the product

(12.2) e

g−1

−

:= e

g−1

−

. . . q

−

. . . p

where we are abusing notation by identifying A with its equivalence class in H

(M)/G

if G is nontrivial. Notice that according t o the above deﬁnitions, this expression has

degree

g−1

−

| = |e

| + (g − 1) |~| +

−

i=1



(n − 3 ) + µ

(γ

−

)



i=1



(n − 3 ) − µ

(γ

)



= −2c

(A) + (2g − 2 + k

+ k

−

)(n − 3) −

i=1

(γ

) +

−

i=1

(γ

−

)

= −vir-dim M

g,0

(J, A, γ

, γ

−

(12.3)

interpreted modulo 2N if c

(ξ)|

6= 0. The orientation results in Lecture

11 suggest

introducing a supercommutativity relation for the variables q

and p

: deﬁning the

graded commutator bracket by

(12.4) [F, G] := F G − (−1)

|F ||G|

GF,

we deﬁne a relation on the set of all monomials of the form q

−

by setting

(12.5) [q

, q

] = [p

, p

] = 0

for all pairs of orbits γ

and γ

. As a consequence, permuting the orbits in the lists

changes the sign of t he monomial (

12.2) if and only if it changes the orientat ion

of the corresponding moduli space. In particular, any product t hat includes multiple

copies of an odd generator q

or p

is identiﬁed with 0. This accounts fo r the fact that

any rigid moduli space M

g,0

(J, A, γ

, γ

−

) with two copies of γ among its positive or

negative asymptotic orbits contains zero curves when counted with the correct signs:

every curve is cancelled by a curve that looks identical except for a permutation of

two of its punctures.

12.3. The deﬁnition of H and commutators

To write down the SFT generating function, let

(J) := M(J)



∼

denote the space of equivalence classes where two curves are considered equivalent

if they have parametrizations that diﬀer only in the ordering of the punctures. This

space is in some sense mo re geometrically natural than M(J) or M

(J), but due to

the orientation results in the previous lecture, less convenient for technical reasons.

248 Chris Wendl

Given u : (

Σ, j) → (R × M, J) representing a nonconstant element of M

(J) with

no marked points, it is natural to deﬁne

Aut

(u) ⊂ Aut(Σ, j)

as the (necessarily ﬁnite) group of biholomorphic transformations ϕ : (Σ, j) → (Σ, j)

satisfying u = u ◦ ϕ; in particular, elements of Aut

(u) are allowed to permute the

punctures, so Aut

(u) is generally a larger group than the usual Aut(u). For k ∈ Z,

let

(J) ⊂ M

(J)

denote the subset consisting of index k curves that have no marked points and whose

asymptotic orbits are all g ood (see Deﬁnition

11.6 in Lecture 11).

We now deﬁne the SFT generating function as a formal power series

(12.6) H =

u∈M

(J)/R

ǫ(u)

|Aut

(u)|

g−1

−

where t he terms of each monomial a r e determined by u ∈ M

(J) as follows:

• g is the genus of u;

• A is the equivalence class of [u] ∈ H

(M) in H

(M)/G;

• γ

= (γ

, . . . , γ

) are the asymptotic orbits of u after arbitrarily ﬁxing

orderings of its positive and negative punctures;

• ǫ(u) ∈ {1, −1} is determined by the chosen coherent orientations on M

(J).

Speciﬁcally, g iven the chosen ordering of the punctures and an arbitrary

choice of asymptotic markers at each puncture, u determines a 1-dimensional

connected component of M

(J), and we deﬁne ǫ(u) = +1 if and only if the

coherent orientation of M

(J) matches its tautological orientation deter-

mined by the R-action.

Note that while both ǫ(u) and the corresponding mo nomial q

−

depend on a

choice of orderings of the punctures, their product does not depend on this choice.

Moreover, ǫ(u) does not depend on the choice of asymptotic markers since curves

with bad asymptotic orbits are excluded from M

(J). Since every monomial in H

corresponds to a holomorphic curve of index 1, (

12.3) implies

|H| = −1.

There are various combinatorially more elaborate ways t o rewrite H. For any

Reeb orbit γ, let

:= cov(γ) ∈ N

denote its covering multiplicity, and for a ﬁnite list of orbits γ = (γ

, . . . , γ

), let

i=1

Given u ∈ M

(J) with k

≥ 0 positive/negative punctures asymptotic to the set

of orbits γ

= (γ

, . . . , γ

), there are k

−

!κ

−

ways to order the punctures

and choose asymptotic markers, but some of them are equivalent since (by an easy

Lectures on Symplectic Field Theory 249

varia t ion on Proposition 11 .1) the ﬁnite group Aut

(u) acts freely on this set of

choices. As a result, (

12.6) is the same as

(12.7) H =

u∈M

(J)/R

ǫ(u)

−

!κ

−

g−1

−

where M

(J) denotes the space of all index 1 curves without marked points in

(J), and the rest of the mononomial is determined by the condition that u

belongs to M

g,0

(J, A, γ

, γ

−

), with no need for any arbitrary choices. Another way

of writing this is

(12.8) H =

g,A,γ

,γ

−



g,0

(J, A, γ

, γ

−

)





−

!κ

−

g−1

−

where t he sum ranges over all integers g ≥ 0, homolo gy classes A ∈ H

(M) and

ordered tuples of Reeb orbits γ

= (γ

, . . . , γ

), and #



g,0

(J, A, γ

, γ

−

)





∈

Z is the signed count of index 1 connected components in M

g,0

(J, A, γ

, γ

−

). Fo r

ﬁxed g and γ

, the union of these spaces for all A ∈ H

(M) is ﬁnite due t o SFT

compactness, as the energy of curves in (R ×M, d(e

α)) is computed by integrating

exact symplectic forms and thus (by Stokes) admits a uniform upper bound in terms

of γ

. For this reason, (

12.8) deﬁnes a forma l power series in the p variables and

in ~, with coeﬃcients that are polynomia l s in the q va r iables and the group ring R.

We played a slight ly sneaky trick in writing down (

12.7) and (12.8): these sum-

mations to not exclude bad orbits, whereas (

12.6) was a sum over curves u that are

not asymptotic to any bad orbits— a necessary exclusion in that case because ǫ(u)

would otherwise depend o n choices of asymptotic markers. The reason bad orbits

are a llowed in (12.8) is that their total contribution adds up to zero: indeed, bad

orbits are always multiple covers with even multiplicity, so whenever u ∈ M

(J)

has a puncture approaching a bad orbit with multiplicity 2m, there ar e exactly

2m − 1 other elements of M

(J) that diﬀer only by a dj ustment o f the marker at

that one puncture, and by Proposition

11.5, half of these cancel out the other half

in the signed count. We’ve already seen that a similar remark explains the harmless

absence from (

12.8) of terms with multiple facto r s of any odd generator q

or p

Remark 12.4. Readers famliar with Floer homology may see a resemblance

between the group ring R = Q[H

(M)/G] and the Novikov r ings that often appear

in Floer homology, tho ugh R is not a Novikov ring since it only allows ﬁnite sums.

In Floer homology, the Novikov ring sometimes must be included because counts of

curves may fail to be ﬁnite, though they only do so if the energies of those curves

blow up. The situation above is somewhat diﬀerent: since the symplectization is an

exact symplectic manifold, Stokes’ theorem implies t hat energy cannot blow up if

the po sitive asymptotic orbits ar e ﬁxed, and o ne therefore obtains well-deﬁned curve

counts no matter the choice of the coeﬃcient ring R. The use of the gr oup ring is

convenient however for two reasons: ﬁrst, without it one cannot always deﬁne an

integer grading, and second, diﬀerent choices of coeﬃcients can sometimes be used

250 Chris Wendl

to detect diﬀerent geometric phenomena via SFT. We will see an example of t he

latter in Lecture

13.

The compactness and gluing theory of SFT is encoded algebraically by viewing H

as an element on a noncommutative operator algebra determined by the commutator

relations

, q

] = κ

, q

′

] = 0 if γ 6= γ

′

(12.9)

Here [ , ] again denotes the graded commutator (

12.4), so “commuting” generators

actually anticommute whenever they ar e both odd. The rest of the multiplicative

structure of this algebra is determined by requiring all elements of R and powers

of ~ (all of which are even generators) to commute with everything, meaning all

operators are R[[~]]-linear.

One concrete representation of this operator algebra is as follows: let A denote

the graded supercommutative unital algebra over R g enerated by the set





γ a good Reeb orbit



The ring of formal power series A[[~]] is then an R[[~]]-module. Deﬁne each of the

generators q

to be R[[~]]-linear operators on A[[~]] via multiplication fr om the left,

and deﬁne p

: A[[~]] → A[[~]] by

(12.10) p

= κ

∂

∂q

Here the R[[~]]-linear partial derivative operator is deﬁned via

∂

∂q

= 1,

∂

∂q

′

= 0 for γ 6= γ

′

and the graded Leibniz rule

∂

∂q

(F G) =

∂F

∂q

G + (−1)

||F |

∂G

∂q

for all homogeneous elements F, G ∈ A[[~]].

Exercise 12.5. Check that the operator p

: A[[~]] → A[[~]] deﬁned above has

the correct degree and satisﬁes the commutation relations (

12.5) and ( 12.9).

Notice that while H contains terms of order −1 in ~, every term also conta ins

at least one p

varia ble since all index 1 holomorphic curves in (R ×M, d(e

α)) have

at least one positive puncture. The substitution (

12.10) thus produces a diﬀerential

operator in which every term contains a nonnegative power of ~, giving a well-deﬁned

R[[~]]-linear operato r

SFT

: A[[~]]

−→ A[[~]].

The following may be regarded a s the fundamental theorem of SFT.

Theorem 12.6. H

= 0.

Lectures on Symplectic Field Theory 251

We will discuss in §12.6 how this relation follows from the compactness and

gluing theory of punctured holomorphic curves, and we will use it in Lecture

13 to

deﬁne various Floer-type contact inva r iants. The ﬁrst and most obvious of these is

the homology

SFT

∗

(M, ξ) := H

∗

(A[[~]], D

SFT

which will tur n out to be an invariant of (M, ξ) in the sense that any two choices

of α, J and the other auxiliary data described in §

12.2 gives rise to a functorial

isomorphism between the two graded homology groups. No t ice that while A[[~]] is

an algebra, its product structure does not descend to H

SFT

∗

(M, ξ) since D

SFT

is not a

derivation— indeed, it is a fo r ma l sum of diﬀerent ial operators of all orders, not just

order one. In the next lecture we will discuss various ways to produce homo logical

invariants out of H with nicer algebraic structures.

On the other ha nd, it is fairly easy to understand the geometric meaning of the

complex (A[[~]], D

SFT

) in Floer-theoretic terms. Each individual curve u ∈ M

(J)

with g enus g, homology class A ∈ H

(M) and asymptotic orbits γ

= (γ

, . . . , γ

)

contributes to D

SFT

the diﬀerential operator

ǫ(u)

|Aut

(u)|

g+k

−1

−

. . . q

−

∂

∂q

. . .

∂

∂q

Applying this operator to a monomial q

. . . q

∈ A[[~]] that does not contain all of

the generators q

, . . . , q

will produce zero, and its eﬀect on a product that does

contain all of these generators will be to eliminate them and multiply q

−

. . . q

−

whatever remains, plus some combinatorial factors and signs that may ar ise from

diﬀerentiating by the same q

more than once. Ignoring the combinatorics and

signs for the mo ment, this operation on q

. . . q

has a g eometric interpretation:

it counts all po tentially disconnected J-holomorphic curves of index 1 (i.e. disjoint

unions of u with trivial cylinders) that have γ

, . . . , γ

as their positive asymptotic

orbits; see Figure

12.1. In other words, the action of D

SFT

on each monomial q

for

γ = (γ

, . . . , γ

) is determined by a f ormula of the form

(12.11) D

SFT

∞

g=0

A∈H

(M)

′

k=1

g+k−1

(γ, γ

′

, k)q

′

where n

(γ, γ

′

, k) is a product of some combinatorial factors with a signed count of

generally disconnected index 1 holomorphic curves of genus g and homology class A

with positive ends at γ and negative ends at γ

′

, such that the nontrivial connected

component has exactly k positive ends. The presence of t he combinatorial factors

hidden in n

(γ, γ

′

, k) is a slightly subtle po int which we will try to clarify in the

following sections.

12.4. Interlude: How to count points in an orbifold

As in all versions of Floer theory, the proof that H

= 0 is based on the fact that

certain moduli spaces are compact oriented 1-dimensional manifolds with bound-

ary, and the signed count of their boundary points is therefore zero. We must be

252 Chris Wendl

Figure 12.1. Counting disjoint unio ns of index 1 curves u ∈

2,0

(J, A, (γ

, γ

), (γ

, γ

)) with some trivial cylinders contributes

a multiple of ~

to D

SFT

careful of course because, strictly speaking,

M(J) is not a manif old even when

Assumption

12.1 holds—it is an orbifold, with the possibility of singularities at mul-

tiply covered curves with no ntrivial automorphism groups. On the other hand, one

can show that (after excluding curves with bad asymptotic orbits) it is an oriented

orbifold, and oriented 1-dimensional o r bif olds happen to be very simple objects:

since smooth ﬁnite gro up actions on R cannot be nontrivial without reversing ori-

entation, all oriented 1-dimensional orbifolds are actually manifolds, suggesting the

simple f ormula

“#∂

(J) = 0.”

I have placed this formula in quotation marks for a reason. The reality of the

situation is somewhat more complicated.

This is in fact where it becomes important to remember t hat Assumption 12.1, in

the way that we stated it, really is not just science ﬁction but fa ntasy: transversality

is sometimes impossible to achieve for multiple covers, and we must therefore at least

have a sensible back-up plan for such cases. To see the problem, remember that our

local structure theorem for M(J) was proved by identifying it in a neighborhood o f

any curve u

: (

Σ, j

) → (R × M, J) with a set of the form

∂

−1

(0)



where

∂

: T ×B

k,p,δ

→ E

k−1,p,δ

is a smooth section of a Banach space bundle E

k−1,p,δ

over the product of a Teichm¨uller slice T through j

with a Banach manifold B

k,p,δ

of maps

Σ → R × M, and G is the group of automorphisms of j

, whose action on

the base

G × (T × B

k,p,δ

) → T × B

k,p,δ

: (ψ, (j, u)) 7→ (ψ

∗

j, u ◦ ψ)

preserves

∂

−1

(0). In fact, the action of G on T ×B

k,p,δ

is covered by a natural action

on the bundle E

k−1,p,δ

, and the reason fo r it preserving the zero-set is that

∂

is an

equivariant section,

∂

(ψ

∗

j, u ◦ ψ) = ψ

∗

∂

(j, u).

As you may k now if you’ve ever heard a talk abo ut polyfolds, there are some analytical

problems with this discussion if G is a Lie group of positive dimension: its action on the inﬁnite-

dimensional manifold B

k,p,δ

of non-smooth maps cannot then be considered smooth in any con-

ventional sense. This proble m leads to the introduction of sc-smooth structures, cf. [

HWZ07].

There is no problem however if G is ﬁnite, e.g. if the under lying Riemann surface is stable, which

we may as well assume for this discussion.

Lectures on Symplectic Field Theory 253

If G is ﬁnite, then another way to say this is that

∂

is a smooth Fredholm section

of the inﬁnite-dimensional orbibundle E

k−1,p,δ

/G over the orbifold (T × B

k,p,δ

)/G,

whose isotr opy g r oup at (j

, u

) is Aut(u

). This section is transverse to the zero-

section if and only if the usual regularity condition holds, making

∂

−1

(0)/G a sub orb-

ifold of (T ×B

k,p,δ

)/G whose isotropy group at (j

, u

) is some quotient of Aut(u

Remark 12 .7. Most sensible deﬁnitions of the term orbifold (cf. [

ALR07,Dav,

FO99]) require local models of the form U/G, where U is a G-invariant open subset

of a vector space on which the ﬁnite group G acts smoothly and eﬀectively—the

latter condition is necessary in order to have isotropy groups t hat are well-deﬁned

up to isomorphism at every point. In the above example, G acts eﬀectively on

T × B

k,p,δ

but might have a nontrivial subgroup H ⊂ G of transformations that

ﬁx every element of

∂

−1

(0), in which case the G-action on

∂

−1

(0) can be replaced

by an eﬀective actio n of G/H. The isotropy group of (j

, u

) ∈

∂

−1

(0)/G is then

Aut(u

)/(Aut(u

) ∩ H).

Now to see just how unreasonably optimistic Assumption

12.1 is, not ice that it’s

easy to think up examples of smooth orbibundles in which zeroes of sections can

never be regular if they have nontrivial isotropy.

Example 12.8. Let M = C/Z

with Z

acting as the antipodal map, and

consider the trivial complex line bundle E = M × C = (C × C)/Z

, where the Z

action on C × C identiﬁes (z, v) with (−z, v). A smooth function f : C → C then

represents a section of t he orbibundle E → M if and only if f(z) = f (−z) for all z.

This implies t hat if f(0) = 0, then df(0) = 0. It is possible to perturb f generically

to a section that is transverse to the zero-section, but such a perturbation can never

have zeroes at 0.

Of course, we do know how to assign Z-valued orders to degenerate zeroes of

sections, e.g. f(z) = z

deﬁnes a section of E → M with a zero of order 2 at 0.

Notice however that if we perturb this to f

(z) = z

+ ǫ for ǫ > 0 small, then f

has

two simple zeroes at points near the origin, but they are actually the same point in

C/Z

, giving a count of only 1 zero. This means that if we give the zero of f at the

origin its full weight, then we are counting wrongly—the resulting count will not be

homotopy invariant. The correct algebraic count of zero es is evidently

(12.12) #f

−1

(0) :=

z∈f

−1

(0)⊂M

ord(f; z)

∈ Q,

where ord(f ; z) ∈ Z is the order of t he zero (computed in t he usual way as a winding

number, or in higher dimensions as the degree of a map of spheres, cf. [

Mil97]), and

∈ N denotes the order of the isotropy g r oup at z.

Exercise 12.9. Convince yourself that for any smooth oriented orbibundle E →

M of real ra nk m over a compact, smooth and oriented m-dimensional orbifold M

without boundary, the count (

12.12) gives the same result for any section with

isolated zeroes.

Hint: The space of sections of an orbibundle is still a vector space, so any two

If you’re still not sure what an orbibundle is, a deﬁnition can be found in [

FO99, Chapter 1].

254 Chris Wendl

are homo t opic. Since M and [0, 1] are both compact, it suﬃces to focus on small

perturbations of a single section on a single orbifold chart.

For a slightly diﬀerent perspective on (

12.12), consider the special case of a closed

orbifold that is the quotient of a closed manifold

M by an eﬀective orientation-

preserving ﬁnite group action,

M =

M/G.

Suppose

E →

M is an oriented vector bundle with rank equal to dim M, and G also

acts on

E by orientation-preserving linear bundle maps that cover its action on

so the quotient

E =

E/G → M

is an orbibundle. A section f : M → E is then equivalent to a G-equivariant section

f :

M →

E, and the signed count of zeroes

−1

(0) =

z∈

−1

(0)⊂

ord(

f; z) ∈ Z

is of course the same for any section that has only isolated zeroes. It can also be

expressed in terms of f since any z ∈ f

−1

(0) ⊂ M has exactly |G|/κ

lifts to points

−1

(0) ⊂

M, implying

−1

(0) =

z∈f

−1

(0)⊂M

|G|

ord(f; z)

and thus #f

−1

(0) =

|G|

−1

(0). The invariance of ( 12.12) is now an immediate

consequence of the invaria nce of #

−1

(0), which follows from the standard argument

as in [

Mil97].

Now, if you enjoyed reading [

Mil97] as much as I did, then it may seem tempting

to try proving invaria nce of (

12.12) in general by choo sing a generic homotopy

H : [0, 1] × M → E between two generic sections f

and f

and showing that

−1

(0) ⊂ [0, 1] × M is a compact oriented 1-dimensional o rbifold with boundary.

As we observed at t he beginning of this section, H

−1

(0) is then actually a manifo ld,

so the signed count of its boundary points should be zero. But this would give

the wrong result: it would sugg est that

z∈f

−1

(0)⊂M

ord(f; z) should be homotopy

invariant, without the ratio nal weights, and we’ve a lr eady seen that this is not true.

What is going on here? The answer is that the ho mo gopy H cannot in general be

made transverse to the zero-section, now matter how generically we perturb it! It

is an illustration of the fundamental conﬂict between the notions of genericity and

eq uivariance.

Example 12.10 . Let M = C/Z

as in Example

12.8, but deﬁne the complex

orbibundle E → M by

E = (C × C)



(z, v) ∼ (−z, −v),

i.e. the Z

-action also acts antipodally on ﬁbers. Now a smooth function f : C → C

deﬁnes a section of E if and only if f(−z) = −f(z), hence all such sections have a

Lectures on Symplectic Field Theory 255

zero a t the origin. Compare the two sections

(x + iy) = x + iy, f

(x + iy) = (x

− x) + iy.

They have qualitatively the same behavior near inﬁnity, meaning in particular that

they are homotopic thro ugh a family of sections whose zeroes are conﬁned to some

compact subset, thus we expect the algebraic count of zeroes to be the same for both.

This is true if the count is deﬁned by (

12.12): we have #f

−1

(0) = #f

−1

(0) =

, in

particular the nega tive zero of f

at the origin counts fo r −1/2 while the positive

zero at (1, 0) ∼ (−1, 0 ) counts f or 1. We see that the inclusion of the rational weight s

is crucial for this result. Notice that if H : [0, 1] × M → E is a homotopy o f

sections fr om f

to f

, then H(τ, 0) = 0 for all τ, thus ∂

H(τ, 0) vanishes and

dH(τ, 0) = df

(0)

where f

= H(τ, ·). But df

(0) cannot be an isomorphism for all τ ∈ (0, 1) since

(0) preserves orientation while df

(0) reverses it. This is not a problem that can

be ﬁxed by making H more generic—the homotopy will never be transverse to the

zero-section, no matter what we do.

The need to address issues of the type raised by the above examples leads na t u-

rally to the notion of multisections as outlined in [

Sal99, §5] and [FO99], and this

is a major f eat ur e of the analysis under development by Hofer-Wysocki-Zehnder, see

for example [

HWZ10]. In Example 12.10 for instance, one can consider functions

f : C → Sym



, z

) ∼ (z

, z

which can be regarded as doubly-valued sections of E → M if f is Z

-equivariant

for the antipodal a ctio n of Z

on the symmetric product Sym

(C). Such a section

is considered single-valued at any point z where f(z) is of the form [(v, v)], so o ne

can now ima gine homotopies from f

to f

through doubly-valued sections. One

advantage of this generalization is that f can now take nonzero va lues of the form

[(v, −v)] a t the origin, e.g. if g : C → C is any odd function, then

f(z) := [(g(z) + c, g(z) − c)]

is a well-deﬁned multisection for every c ∈ C.

Exercise 12.11. Find a homotopy between the sections f

and f

of Exam-

ple

12.10 through doubly-valued sections, such that the homot opy is transverse to

the zero-section.

You may notice if you work out Exercise

12.11 that the zero set of the homotopy

in [0, 1] × M is still not submanifold or suborbifold. Instead, it naturally carries

the structure of a w eighted branched manifold with boundary. The rational weights

attached to every point in this object can be used to explain the weights appearing

in (

12.12) and thus give a Milnor-style proof that #f

−1

(0) ∈ Q is invariant.

We will no t discuss multisections or weighted branched manifolds any further,

but the main takeaway from this discussion should be that the “right” way to count

0-dimensional orbifolds algebraically is always some version of (

12.12), and the count

in general is a rational number, not an integer. We’ve discussed this above from

the perspective of obtaining a homotopy-invariant count, but the same logic a pplies

256 Chris Wendl

to any Floer-type theory since the relation ∂

= 0 is typically based on similar

arguments via 1-dimensional moduli spaces with boundary. While a more simplistic

notion of counting may produce well-deﬁned homology theories in isolated cases

where Assumption

12.1 holds (e.g. in [Nel15]), we cannot expect it to generalize

beyond these cases, due to the fundamental conﬂict between transversality and

equivariance. On the other hand, it will be possible in our situation to r emove

isotropy from the picture by lifting to moduli spaces with asymptotic markers; the

moduli space we’re interested in is always the quotient of this larger space by a

ﬁnite group action, so the situation is analogous t o replacing an orbibundle E =

E/G →

M/G by an ordinary vector bundle

E over a manifold

M. In the inﬁnite-

dimensional setting, transversality is still a hard problem, but having lifted to a

manifold and thus removed t he need for equivariance, there is no longer any a priori

reason why it cannot be solved by choosing suﬃciently generic perturbations. This

makes counting curves with rational weights seem a much more promising method

for deﬁning invaria nts, and we will adopt this perspective in the discussion to follow.

12.5. Cylindrical contact homology revisited

Under an extra assumption on the complex (A[[~]], D

SFT

), we can recover from

it a more general version of the cylindrical conta ct homology we saw in Lecture

10.

Suppose in part icular that there are no index 1 holo mo r phic planes in R × M, so

every term in ~H has at least one factor of either ~ or one o f the q

varia bles. Then

SFT

γ,γ

′





u∈M

0,0

(J,A,γ,γ

′

)/R

ǫ(u)

|Aut(u)|

′

∂

∂q





+ . . . ,

where the ﬁrst sum is over all pairs of good Reeb orbits γ and γ

′

, and the ellipsis

is a sum of terms that all include at least a positive power of ~ or two q

varia bles

or two partial derivatives. Let us abbreviate t he spaces M

0,0

(J, A, γ, γ

′

)/R of R-

equivalence classes of J-holomor phic cylinders by M

(γ, γ

′

), and notice that for

any u ∈ M

(γ, γ

′

), the automorphism group is a cyclic group of order equal to the

covering multiplicity

|Aut(u)| = κ

:= cov(u) ∈ N.

Thus for any single g enerato r q

, we have

SFT

= ∂

CCH

+ O(|q|

, ~),

where

(12.13) ∂

CCH

:= κ

′





u∈M

(γ,γ

′

)

ǫ(u)





′

The fact that D

SFT

= 0 thus implies

∂

CCH

= 0,

and the homology of the graded R-module generated by {q

| γ good} with diﬀer-

ential ∂

CCH

is an obvious generalization of the cylindrical contact homology from

Lecture

10. What we saw there was a special case of this where the combinatorial

Lectures on Symplectic Field Theory 257

factor κ

/κ

did not appear because we were restricting to a homotopy class in which

all orbits were simply covered, and all holomorphic cylinders were thus somewhere

injective.

The presence of the factor κ

/κ

deserves further comment. According to the

above formula, we have

∂

CCH

′

,γ

′′

,A,A

′

u∈M

(γ,γ

′

)

v∈M

′

(γ

′

,γ

′′

)

A+A

′

ǫ(u)ǫ(v)

′′

hence ∂

CCH

= 0 holds if and only if for all A ∈ H

(M) and all pairs of good orbits

, γ

−

(12.14)

B+C=A





(u,v)∈M

(γ

,γ

)×M

(γ

,γ

−

)

ǫ(u)ǫ(v)





= 0.

If γ

and γ

−

happen to be simply covered orbits, then u and v in this expression

always have trivial automorphism groups and it is clear what this sum means: every

such pair (u, v) ∈ M

(γ

, γ

)×M

(γ

, γ

−

) corresponds to exactly κ

distinct holo-

morphic buildings obtained by diﬀerent choices of decoration, so (

12.14) is the count

of boundary points of t he compactiﬁed 1-dimensional manif old of index 2 cylinders

(γ

, γ

−

)/R. This sum skips over all bad orbits γ

, but this is ﬁne because when-

ever the breaking orbit is bad, there are evenly many cho ices of decoration such that

half of these choices cancel the other half when counted with the correct signs.

To understand why this formula is still correct in the presence of automorphisms,

let us outline two equivalent approaches.

The easiest o ption is to instead consider moduli spaces with asymptotic markers,

which never have automorphisms: removing unnecessary factors of κ

and κ

−

then

transforms (

12.14) into

B+C=A

(γ

, γ

) · #M

(γ

, γ

−

) = 0.

Now since each pair (u, v) ∈ M

(γ

, γ

)×M

(γ

, γ

−

) carries a canonical decoration

and thus determines a holo mo rphic building, the division by κ

accounts for the

fact that #M

(γ

, γ

) · #M

(γ

, γ

−

) overcounts the set of broken cylinders from

to γ

−

with asymptotic markers at γ

by precisely this factor, as a simultaneous

adjustment of the marker at γ

in both u ∈ M

(γ

, γ

) and v ∈ M

(γ

, γ

−

)

produces the same decoration and therefore the same building.

The following alternative perspective will be more useful when we generalize

beyond cylinders in the next section. We can directly count po ints in ∂

(γ

, γ

−

though as we saw in §

12.4, rational weights should be included in the count whenever

there is isotropy. Let us write

(γ

, γ

−

) = M

(γ

, γ

−

)/G,

where G

∼

× Z

−

is a ﬁnite gr oup acting by adjustment of the asymptotic

markers. Since

(γ

, γ

−

) is a compact oriented 1-manifold with boundary under

Assumption

12.1, the signed count of its boundary points is 0. We can ignore

258 Chris Wendl

buildings broken along bad orbits in this count, since these always come in cancelling

pairs. Let us now transform this into a count of buildings (u|Φ|v) ∈ ∂

(γ

, γ

−

)

broken along good orbits γ

: here u ∈ M

(γ

, γ

) and v ∈ M

(γ

, γ

−

) for some

homology classes with B + C = A, and Φ is a decoration which describes how to

glue the ends of u and v at γ

. The automorphism group o f such a building is the

subgroup

Aut(u|Φ|v) ⊂ Aut(u) × Aut(v)

consisting of all pairs (ϕ, ψ) ∈ Aut(u) ×Aut(v) that deﬁne the same rotation at t he

two punctures asymptotic to γ

; note that this group does not actually depend on

the decoration Φ. Since we’re talking about cylinders, we can be much more speciﬁc:

we have Aut( u) = Z

and Aut(v) = Z

, and if both are regarded as subgroups of

U(1),

Aut(u|Φ|v) = Z

∩ Z

= Z

gcd(κ

,κ

)

which is injected into Aut(u)×Aut(v) by ψ 7→ (ψ, ψ) . The boundary of

(γ

, γ

−

)

can be understood likewise as a space of equivalence classes

[(u, v)] ∈



(γ

, γ

) × M

(γ

, γ

−

)



∼,

where two such pairs are equiva lent if their asymptotic markers at the ends as-

ymptotic to γ

determine the same decoration. Now observe t ha t the gr oup G

∼

×Z

−

also acts on buildings in ∂

(γ

, γ

−

), and the stabilizer of this action

at (u, v) is Aut(u|Φ|v), hence each (u|Φ|v) ∈ ∂

(γ

, γ

−

) gives rise to

|G|

gcd(κ

,κ

)

terms in the count of ∂

(γ

, γ

−

), implying

(12.15)

(u|Φ|v)∈∂

(γ

,γ

−

)

ǫ(u)ǫ(v)

gcd(κ

, κ

)

= 0.

Finally, notice that while each pair (u, v) ∈ M

(γ

, γ

) × M

(γ

, γ

−

) determines

buildings with κ

distinct choices of decoration, some of these buildings may be

equivalent: every pair of automorphisms ( ϕ, ψ) ∈ Aut(u) × Aut(v) transforms a

building (u|Φ|v) by potentially changing the decoration Φ, thus producing an equiv-

alent building. This action on buildings is trivial if and only if (ϕ, ψ) ∈ Aut(u|Φ|v),

hence every pair (u, v) ∈ M

(γ

, γ

) × M

(γ

, γ

−

) gives rise t o exactly



(Aut(u) × Aut(v))



Aut(u|Φ|v)



gcd(κ

, κ

)

elements of ∂

(γ

, γ

−

), so that (12.15) becomes

B+C=A





(u,v)∈M

(γ

,γ

)×M

(γ

,γ

−

)

ǫ(u)ǫ(v)

gcd(κ

, κ

)

gcd(κ

, κ

)





B+C=A





(u,v)∈M

(γ

,γ

)×M

(γ

,γ

−

)

ǫ(u)ǫ(v) κ





= 0,

reproducing (

12.14).

Lectures on Symplectic Field Theory 259

12.6. Combinatorics of gluing

Now let’s try to justify the formula H

= 0. The product of H with itself is the

formal sum over all pairs of index 1 curves u, v ∈ M

(J)/R of certain monomials:

in particular if these two curves respectively have genus g

and g

, homology classes

and A

, and asymptotic orbits γ

and γ

, then the corresponding term in H

ǫ(u)ǫ(v)

|Aut

(u)||Aut

(v)|

−2

−

Before we can add up a ll monomials of this form, we need to put all the q and p

varia bles in the same order: within each of the products q

−

, p

and so forth this is

simply a matter of permuting the variables and chang ing signs as appropriate, but

the interesting par t is the product p

−

, for which we can apply the commutation

relations (

12.9) to put all q variables before all p variables. Before discussing how

this works in general, let us consider a more speciﬁc example.

Assume γ

for i = 1, 2 are two speciﬁc orbits with n − 3 + µ

(γ

) even, so the

corresponding q and p var iables have even degree, and suppose

= (γ

, γ

), γ

−

= (γ

, γ

After applying t he relation p

= q

+ κ

~ a total of ﬁve times, one obtains

the expansion

= q

+ 4κ

+ 2κ

thus contributing a total o f three terms to H

, namely the products o f the factor

ǫ(u)ǫ(v)

|Aut(u)||Aut(v)|

with each of the expressions

−2

−

,(12.16)

4κ

−1

−

,(12.17)

2κ

−

.(12.18)

As shown in Figure

12.2, this sum of three terms can be interpreted as the count of

all possible ho lomrphic buildings obtained by gluing v on top of u together with a

collection of trivial cylinders. Indeed, since γ

and γ

−

include two matching orbits

(which also happen to be the same one), there are several choices to be made:

(1) The top-right picture shows what we might call the “stupid gluing,” in

which no ends of u a re matched with any ends of v, but all are instead

glued to trivial cylinders, thus producing a disconnected building. This

possibility is encoded by (

12.16), and we will see that in the total sum

forming H

, this term gets cancelled out by a similar term for the stupid

gluing of u on top of v.

(2) The lower-left picture shows the building obtained by gluing one end of

u to an end of v alo ng the matching orbit γ

. This option is encoded

by (

12.17), where t he factor 4κ

appears because there are precisely 4κ

distinct buildings of this type: indeed, there are four choices of which end

of u should be glued to which end of v, and for each of these, a further κ

260 Chris Wendl

−

Figure 12.2. Three possible ways of gluing the curves u and v

along with trivial cylinders to form index 2 curves.

choices of the decoration. The arithmetic genus of the resulting building is

+ g

, as represented by the factor ~

−1

(3) The lower-right picture is enco ded by (

12.18): here there are two choices of

bijections between the two pair s of punctures asymptotic to γ

, and taking

the choices of decoration at each breaking orbit into account, we obtain the

combinatoria l factor 2κ

. The presence of two nontrivial breaking orbits

increases the arithmetic genus to g

+1, as encoded in the factor ~

You may now be able to extrapolate from the above example why the commu-

tator algebra we’ve deﬁned enco des gluing of holomorphic curves in the symplec-

tization and thus leads to the relation H

= 0. Think of the algorithm by which

you change q

−

into a sum of products with all q’s appearing before p’s:

for the ﬁrst q you see appearing after a p, move it past each p for diﬀerent orbits

(changing signs as necessary) until it encounters a p for the same orbit. Now you

replace p

with (−1)

||q

+ κ

~, turning one pr oduct into a sum of two. This

represents a choice between two options: either you move q

past p

and apply the

Lectures on Symplectic Field Theory 261

usual sign change, or you eliminate them both but replace t hem with the combina-

torial factor κ

and an extra ~. Then you continue t his process until all q’s appear

before all p’s.

The key point is t ha t the process of gluing v on top of u in all possible ways is

governed by e xactly the same algorithm: ﬁrst consider the disjoint union of the two

curves as a single disconnected curve, with its punctures o r dered in the same way

in which their orbits appear in t he monomial. Now reorder negative punctures of v

and positive punctures of u, changing orientations a s appropriate, until yo u see two

such punctures next to each other approaching the same orbit γ. Here you have two

options: either glue them together, or don’t glue them but exchange t heir order. If

you exchange the order, then you may again have to change orientatio ns (depending

on the parity of n − 3 + µ

(γ)), but if you glue, then you have κ

distinct choices

of decoration and will also increase the arithmetic genus of the eventual building

by 1. In this way, every individual term in the ﬁnal expansion of q

−

represents a particular choice of which positive of ends of u should or should not be

glued to which negative ends of v. Additional factors of ~ appear to keep track of

the increase in arithmetic genus, and covering multiplicities o f t he breaking orbits

also appear due to distinct choices of decorations. At the end these must still be

divided by orders of automorphism groups in order to avoid counting equivalent

buildings separately. Fleshing out these details leads to t he following explanation

for the relation H

= 0:

Proposition 12 .12. Let ∂

(J) denote the space of two-level ho l omorphic

buildings in

M(J) that have total index 2 and no bad asymptotic or breaking orbits,

divided by the equivalence relation that forgets the order of the punctures. Then

u∈∂

(J)

ǫ(u)

|Aut

(u)|

g−1

−

where the terms in each monomial are determined by u ∈ ∂

(J) as follows:

(1) g is the arithmetic genus of u;

(2) A is the equivalence class of [u] ∈ H

(M) in H

(M)/G;

(3) γ

= (γ

, . . . , γ

) are the asymptotic orbits of u after arbitrarily ﬁxing

ord erings of i ts positive and negative punctures;

(4) ǫ(u) ∈ {1, −1} is the boundary orientation at u determined by the chosen

co herent o rientations on M

(J). Speci ﬁcally, given the c hosen orderin g of

the punctures and an arbitrary choice of asymptotic markers at each punc-

ture, u determines a boundary point of a 1-dimension al co nnected compo-

nent of

(J), and we deﬁne ǫ(u) = +1 if and only if the orientation of

(J) at this poi nt is outward.

Once again ǫ(u) and q

−

change signs in the same way under any reordering

of the punctures, so t heir product is well deﬁned, and there is no dependence on

choices of markers since bad orbits have been excluded.

262 Chris Wendl

Proof of Proposition 12.12. Our original f ormula for H gives rise to an

expansion

(u,v)∈M

(J)/R×M

(J)/R

ǫ(u)ǫ(v)

|Aut

(u)||Aut

(v)|

−2

−

As explained in the previous paragraph, the process of reordering p

−

to put all

q’s befo re p’s produces an expansion, each term of which can be identiﬁed with a

speciﬁc choice of which positive punctures of u should be g lued to which negative

punctures of v. If k punctures are glued, then the resulting power of ~ is g

−2+ k, corresponding to the fact that the resulting building has arithmetic genus

+k−1. We claim that t he term for k = 0 is cancelled out by the corresponding

term of H

that has the roles of u and v reversed. To see this, imagine ﬁrst the case

where u and v have no asymptotic orbits in common, hence no nontrivial gluings

are possible and all the q and p variables in the expression supercommut e with each

other. Then since both curves have index 1, the monomials q

−

and q

−

must

both have odd degree, implying

−

= −q

−

and thus the desired cancellation. If u and v do have orbits in common, then t he

result for the k = 0 terms is still not any diﬀerent from this: all signs still change

in t he same way when applying [p

, q

] = κ

~ to change p

into q

, we simply

ignore the extra term κ

~ since it is only relevant for gluings with k > 0. This

proves the claim, and consequently, that the expansion resulting from the curves u

and v has no term containing ~

−2

The combinatorial factors can be explained a s follows. The commutator expan-

sion for p

−

automatically produces combinator ial factors that count the diﬀerent

possible gluings, but if u and v have automorphisms, then not all of these give in-

equivalent buildings. This part of the discussion is a straightfor ward extension of

what we did for cylindrical contact ho mo logy at the end of §

12.5. Indeed, the a ctual

set of inequivalent buildings is the quotient of this larger set by an action of

(Aut

(u) × Aut

(v))



Aut

(u),

where for a building u formed by endowing the pa ir (u, v) with decorations, Aut

(u)

denotes the subgroup consisting of pairs (ϕ, ψ) ∈ Aut

(u) × Aut

(v) that preserve

pairs of breaking punctures along with their decorations. This is what changes the

factor

|Aut

(u)||Aut

(v)|

into

|Aut

(u)|

as in the statement of the proposition. 

The theorem that H

= 0 now follows once you believe the propag anda from

12.4, arguing that

u∈∂

(J)

ǫ(u)

|Aut

(u)|

is the correct way to count the bo undar y

points of M

(J). As we did with cylindrical contact homology, we can use the obvi-

ous projection

(J) → M

(J) to reduce t his to the fact that if the 1-dimensional

components of

(J) are manifolds (which is true if Assumption 12.1 holds), then

the integer-valued signed count of their boundary points va nishes.

Lectures on Symplectic Field Theory 263

12.7. Some remarks on torsion, coeﬃcients, and conventions

12.7.1. What if H

(M) has torsion? The main consequence for SFT if H

(M)

has torsion is that one cannot deﬁne an integer gr ading, though there is always a

canonical Z

-grading.

The setup in §12.2 must now be modiﬁed as follows. The

reference curves

, . . . , C

⊂ M

are required to fo rm a basis of H

(M)/torsion, so for every integral homology class

[γ], there is a unique collection of integers m

, . . . , m

such that [γ] =

] ∈

(M; Q). Instead of spanning surfaces for each orbit, one can deﬁne spanning

chains C

, which are singular 2-chains with rational coeﬃcients satisfying

∂C

] − [γ]

for the aforementioned set of integers m

∈ Z. Note that C

must in general

have nonintegral coeﬃcients since

] and [γ] mig ht not be homologous in

(M; Z), so C

cannot always be represented by a smooth map of a surface. One

consequence of this is that the absolute homology class associated to an asymptoti-

cally cylindrical holomorphic curve u :

Σ → R × M will now be rational,

[u] ∈ H

(M; Q),

and we must therefore take G to be a linear subspace

G ⊂ H

(M; Q).

Another consequence is that we cannot use capping chains to transfer trivializations

from the reference curves to the orbits, so there is no na tural way to deﬁne µ

(γ)

as a n integer. The easiest thing to do instead is to take the mod 2 Conley-Zehnder

index

(γ) ∈ Z

and deﬁne all degrees of generators as either even or odd with no further distinction.

In par t icular, we now have

| = n − 3 + µ

(γ) ∈ Z

, |p

| = n − 3 − µ

(γ) ∈ Z

while ~ and all elements of R = Q[H

(M; Q)/G] are even. With these modiﬁcations,

the rest of the discussion also becomes valid for the case where H

(M) has torsion,

and leads to Z

-graded contact invariants.

12.7.2. Combinatorial conventions. The combinatorial factors appearing in

our deﬁnition of H may at ﬁrst look slightly diﬀerent from what appears elsewhere in

the literature. Actually, most papers seem to ag ree on this detail, but va r ious subtle

diﬀerences and ambiguities in notation mean that it sometimes requires intense

concentration to recognize this fact.

The original propaganda paper [

EGH00] expresses everything in terms of moduli

spaces with asymptotic markers, and the f ormula for H in §2.2.3 of that paper (which

In fact ther e is a bit more than a Z

-grading, see [

EGH00, §2.9.1].

264 Chris Wendl

is expressed in a slightly more general fo r m involving ma rked points) agrees with

our (

12.8).

Cieliebak and Latschev [

CL09, §2] write down the same formula in terms of mod-

uli spaces that have no asymptotic markers but remember the order of the punctures,

thus it includes some factorials that do not appear in (

12.6) but is missing the κ

terms of (

12.8). The notation n

(Γ

−

, Γ

) used in [CL09] for curve counts must be

understood implicitly to include rational weights arising from automorphisms (or

multivalued perturbations, as the case may be).

My paper with Latschev [LW11] uses moduli spaces with asymptotic markers

and attempts to write down the same formula as in [

EGH00, CL09], but gets it

slightly wrong due to some missing κ

terms that should appear in front of each

∂

∂q

Mea culpa.

For cylindrical contact homology, the combinatorial factors in §

12.5 also a gree

with what appears in [

Bou03]. As observed by Nelson [Nel13, Remark 8.3], there

are other conventions for ∂

CCH

that appear in the literature and lead to equivalent

theories: in particular it is possible to replace (

12.13) with

∂

CCH

′





u∈M

(γ,γ

′

)

ǫ(u)





′

One can derive this from the same deﬁnition of H by applying a “change of co-

ordinates” to the algebra A[[~]], or equivalently, by choosing a slightly diﬀerent

representation o f the operator algebra deﬁned by the p

and q

varia bles. To avoid

confusion, let us write the generators of A as x

instead of q

, and then deﬁne the

operators q

and p

on A[[~]] by

= κ

, p

= ~

∂

∂x

These operators still satisfy [p

, q

] = κ

~ and thus deﬁne an equivalent theory, but

the resulting diﬀerential operat or D

SFT

on A[[~]] now includes extra factors of κ

for the negative punctures instead of the positive punctures.

12.7.3. Coeﬃcients: Q, Z or Z

? While we were able to use Z

coeﬃcients

for cylindrical cont act homology in a primitive homotopy class in Lecture

10, a quick

glance at any version of the formula for H should make the reader very skeptical

about doing this fo r more general versions of SFT. The existence of curves with

automorphisms means that H always contains terms with rational (but nonintegral)

coeﬃcients. And this is only what is true in the ﬁctional world of Assumption

12.1:

in the general version of the theory, we exp ect to have to replace expressions like

ǫ(u)

|Aut(u)|

with counts of 0-dimensional weighted branched orbifolds with rational

weights, arising as zero-sets of generic multisections. In this case we not only obtain

rational counts but may also lose all control over the sizes of the denominators.

A similar phenomenon occurs in general versions of Gromov-Witten t heory. For

instance, in the approach of Cieliebak-Mohnke [

CM07] for the rational Gromov-

Witten invariants of a closed symplectic manifold (W

, ω) with [ω] ∈ H

(W ; Q),

the invariants are deﬁned by replacing the usual moduli space M

0,m

(J, A) by a space

Lectures on Symplectic Field Theory 265

0,m+N

(J, A; Y ) consisting of J-holomorphic spheres u : S

→ W with some large

number of auxiliary marked points ζ

, . . . , ζ

required to satisfy the condition

u(ζ

) ∈ Y, i = 1, . . . , N.

Here Y

2n−2

⊂ W

is a J-holomorphic hypersurface with [Y ] = D · PD([ω]) ∈

2n−2

(W ) for some degree D ∈ N, and the number of extra marked points is deter-

mined by

N = A · [Y ] = Dh[ω], Ai,

so positivity of intersections implies that u only intersects Y at the auxiliary marked

points. These auxiliary points are convenient for technical reasons involving trans-

versality—their role is vaguely analogous to the way that asymptotic markers get

rid of isotr opy in SFT—but they are not geometrically meaningful, as we’d actually

prefer t o count curves in M

0,m

(J, A). Every such curve has N intersections with Y ,

so accounting for permutations, it lifts to N! distinct elements of M

0,m+N

(J, A; Y ),

and the correct count is therefore obtained as an integer count of curves in the latter

space divided by N!. Perturbing to achieve transversality breaks the symmetry,

however, so there is no guarantee that counting curves in M

0,m+N

(J, A; Y ) will

produce a multiple of N!, and moreover, N could be arbitrarily large since one

needs to take hypersurfaces of arbitrarily large degree in o r der to show that the

invariants don’t depend on this choice. For these reasons, the resulting Gromov-

Witten invariants are rational numbers rather than integers, and their denominators

cannot b e predicted or bounded.

The upshot of this discussion is that there is probably no hope of deﬁning SFT

with integer coeﬃcients in general, much less with Z

coeﬃcients—for this reason

the inclusion of orientations in the picture is unavoidable. That is the bad news.

The good news however is that whenever formulas like

ǫ(u)

|Aut(u)|

can be taken

literally as a count of curves, the chain complex (A[[~]], D

SFT

) can in fact be deﬁned

with Z coeﬃcients, and one can even reduce to a Z

version in order to ignore signs.

A special case of this was observed for cylindrical contact homology in [

Nel15, Re-

mark 1.5], and you may notice it already when you look at the formula (

12.13) for

∂

CCH

: the factor κ

/κ

is always an integer since the multiplicity of a holomorphic

cylinder always divides the covering multiplicity of both its asymptotic orbits. Sur-

prisingly, something similar turns out t o be tr ue for the much larger chain complex

of SFT. The following result is stated under Assumption

12.1 for safety’s sake, but in

light of the discussion in §

12.4, we should expect it to hold somewhat more generally.

Proposition 12.1 3. If Assumption

12.1 holds then the rational coeﬃcients

(γ, γ

′

, k) in the formula (

12.11) for D

SFT

are all integers.

Corollary 1 2.14. Under Assumption

12.1, there exist well-de ﬁned chain com-

plexes

[[~]], D

SFT

) and (A

[[~]], D

SFT

where for a general commutative ring R, A

denotes the graded supercommutative

unital algeb ra over R[H

(M)/G] generated by the q

variables for good Reeb orbits γ.

The diﬀerentials D

SFT

on A

[[~]] and A

[[~]] are deﬁned by the same formula as

on A[[~]], where in the Z

ca se we are free to se t all signs ǫ(u) equal to 1.

266 Chris Wendl

Proof of Proposition 12.13. We need to show that expressions of the f orm

|Aut

(u)|

∂

∂q

. . .

∂

∂q

produce integer coeﬃcients for every holomorphic curve u with asymptotic orbits

= (γ

, . . . , γ

) and every tuple γ = (γ

, . . . , γ

). It suﬃces to consider the

special case γ = γ

, as the derivative in question is only no nzero on monomials

that are divisible by q

. Up to a sign change, we can reorder the orbits and write

in t he form

= (γ

, . . . , γ

{z }

, . . . , γ

| {z }

)

for some ﬁnite set of distinct orbits γ

, . . . , γ

and numbers m

∈ N, i = 1, . . . , N.

We then have

|Aut

(u)|

∂

∂q

. . .

∂

∂q

. . . κ

|Aut

(u)|



∂

∂q



. . .



∂

∂q





. . . q



= ±

. . . κ

! . . . m

|Aut

(u)|

(12.19)

We claim that this number is always an integer. Indeed, if Aut

(u) is nontrivial, then

u :

Σ → R × M is a multiple cover u = v ◦ ϕ for some holo morphic branched cover

ϕ : (Σ, j) → (Σ

′

, j

′

) and somewhere injective curve v : (

′

= Σ

′

\Γ

′

, j

′

) → (R×M, J).

Automorphisms ψ ∈ Aut

(u) thus deﬁne biholomorphic maps o n (Σ, j) that permute

each of the sets of punctures asymptotic to the same orbit. Given any puncture

z ∈ Γ where u is asymptotic to γ

, the Aut

(u)-orbit of z consists of ℓ ≤ m

other

punctures also asymptotic to γ

, and its stabilizer is a cyclic subgroup of order

k = |Aut

(u)|/ℓ, acting on a neighborhood of z by biholomorphic rota tions. It

follows that κ

is divisible by k, hence

ℓ

|Aut

(u)|

∈ N,

and (

12.19) is a multiple of this. 

Remark 12.15. Since 1 = −1 in A

, anticommuting elements of A

[[~]] ac-

tually commute, so unless one imposes extra algebraic conditions in the case of Z

coeﬃcients, higher powers of odd generators p

and q

do not vanish. Nonetheless,

these powers still do not appear in H, so the complex (A

[[~]], D

SFT

) ignores curves

with multiple ends approaching an orbit of odd degree (and also bad orbits, for that

matter).

LECTURE 13

Contact invariants

Contents

13.1. The Eliashberg-Givental-Hofer package 268

13.1.1. Full S FT as a Weyl superalgebra 268

13.1.2. The semiclassical limit: rational SFT 272

13.1.3. The contact homology algebra 275

13.1.4. Algebraic overtwistedness 276

13.2. SFT generating functions for cobordisms 278

13.2.1. Weak, strong and stable cobordisms 278

13.2.2. Counting disconnected index 0 curves 283

13.3. Full SFT as a BV

∞

-algebra 288

13.3.1. Cobordism maps and invariance 289

13.3.2. Algebraic torsion 292

In the previous lecture, we introduced an operator algebra deﬁned via the su-

percommutators [p

, q

] = κ

~, then we deﬁned the SFT generating function

H =

u∈M

(J)/R

ǫ(u)

|Aut

(u)|

g−1

−

and proved (modulo transversality) that H

= 0. The generating function is a formal

power series whose coeﬃcients are r ational counts of holomorphic curves, and these

counts are strongly dependent on the cho ices of contact form α, almost complex

structure J ∈ J(α) and further auxiliary data such as coherent orient ations. Thus

in contrast to G r omov-Witten theory, the generating function does not deﬁne an

invariant, but one can follow the standard prescription of Floer-type theories and

deﬁne inva r iants via homology. We saw that fo r the natural represent ation A[[~]] of

the operator a lgebra deﬁned by setting p

= κ

∂

∂q

, H deﬁnes a diﬀerential operator

SFT

: A[[~]] → A[[~]] with D

SFT

= 0. One of our g oals in this lecture will be to

explain (again modulo transversality) why the resulting homology

SFT

∗

(M, ξ; R) = H

∗

(A[[~]], D

SFT

)

is an invariant of the contact structure. We will then use it to deﬁne simpler nu-

merical invariants that detect symplectic ﬁllability properties of contact manifolds.

But ﬁrst, A[[~]] is not the only possible r epresent ation of the operato r algebra

of SFT: other choices lead to diﬀerent inva riants with diﬀerent algebraic structures.

Let’s begin by describing the original hierarchy of contact invariants that were out-

lined in [

EGH00].

267

268 Chris Wendl

Remark 13.1. Throughout this lecture, we assume for simplicity that H

(M)

has no torsion, and the same assumption is made about cobordisms in §

13.2.2. Only

minor changes are necessary if this condition is lifted, e.g. one could then replace a ll

instances of H

(M) with H

(M; Q) and assume always that the grading is Z

; see

§12.7.1.

13.1. The Eliashberg-Givental-Hofer package

In the following, (M, ξ) is a (2n − 1)-dimensional closed contact manif old with

a contact form α and almost complex structure J ∈ J(α) for which the optimistic

transversality condition (Assumption 12.1) of Lecture 12 is assumed to hold. We

ﬁx also the auxiliary data described in §

12.2, plus a choice of subgroup G ⊂ H

(M)

which determines the coeﬃcient ring

R = Q[H

(M)/G].

Each of the diﬀerential graded algebras described below then carries the same grad-

ing that was described in that lecture, i.e. there is always a t least a Z

-grading, and

it lifts to Z if H

(M) is torsion free and c

(ξ)|

= 0, or possibly Z

if N ∈ N is the

smallest possible value for c

(A) with A ∈ G.

13.1.1. Full SFT as a Weyl superalgebra. We start with some seemingly

trivial algebraic observations. First, the relation H

= 0 is equivalent to

[H, H] = 0.

Remember that [ , ] is a super-commutator, so [F, F] = 0 holds automatically fo r

operators F with even degree, but H is odd, and for odd operators the commutator

is deﬁned by [F, G] = FG + GF, hence [H, H] = 2H

. Formally sp eaking [ , ] is a

super Lie bracket and thus satisﬁes the “super Jacobi identity”:

(13.1)



F, [G, K]



+ (−1)

|F||G|+|F||K|



G, [K, F]



+ (−1)

|F||K| +|G||K|



K, [F, G]



= 0.

A consequence of this is that in order to create a homolo gy theory out of H, we don’t

absolutely need to ﬁnd a representatio n of the entire operator algebra: it suﬃces

to ﬁnd a representation of the induced super Lie algebra. Indeed, suppose V is a

graded R[[~]]-module and L is a linear grading-preserving ma p that associat es to

operators F (expressed as power series functions of p’s, q’s and ~ with coeﬃcients

in R) an R[[~]]-linear map

: V → V

such that

[F,G]

= L

− (−1)

|F||G|

for every pair of operators F, G. Then the R[[~]]- linear map L

: V → V satisﬁes

, L

] =

[H,H]

= 0,

hence (V, L

) is a chain complex. The complex (A[[~]], D

SFT

) was a special case of

this, in which we represented the super Lie algebra via a faithful representation o f

the whole operator algebra.

Exercise 13.2. Verify (

13.1).

Lectures on Symplectic Field Theory 269

Remark 13.3 (supersymmetric sign rules). To see where the signs in (1 3.1)

come fr om, it suﬃces to know the following basic rule of superalgebra: for any

pair of Z

-graded vector spaces V and W , the natural “commutation” isomorphism

c : V ⊗ W → W ⊗ V is deﬁned on homogeneous elements by

c(v ⊗ w) = (−1)

|v||w|

w ⊗ v.

For any permutation of a ﬁnite tuple of Z

-graded vector spaces, one can derive the

appropriate isomorphism from this: in particular the cyclic permutation isomor-

phism σ : X ⊗ Y ⊗ Z → Y ⊗ Z ⊗ X takes the form

σ = (1 ⊗ c

) ◦ (c

⊗ 1) : x ⊗ y ⊗ z 7→ (−1)

|x||y|+|x||z|

y ⊗ z ⊗ x.

Writing the Jacobi identity as [·, [·, ·]] ◦ (1 + σ + σ

) = 0 then produces (

13.1). In

this sense, it only diﬀers from the usual Jacobi identity in being based on a diﬀerent

deﬁnition of the commutation isomorphism V ⊗ W → W ⊗ V . For more on this

perspective, see [Var04, §3.1].

Now here is a diﬀerent kind of example, where the representation does not respect

the product structure o f the operator algebra but does resp ect its Lie bracket. Let

W denote the graded unital algebra consisting of formal power series

γ,k

(q)~

where the sum ranges over all integers k ≥ 0 and all ordered sets γ = (γ

, . . . , γ

)

of good Reeb orbits for m ≥ 0, and the f

γ,k

are polynomial functions of the q

varia bles, with coeﬃcients in R. Note that t he case of the empty set of orbits is

included here, which means p

= 1. The multiplicative structure of W is deﬁned

via the usual (super)commutation relations, and its elements can be interpreted as

operators. If we now associate to each F ∈ W the R[[~]]-linear map

: W → W : G 7→ [F, G],

then the Jacobi identity (

13.1) implies

[F,G]

= D

−(−1)

|F||G|

This is just the graded version of the standard adjoint representation of a Lie algebra.

The only problem in applying this idea to deﬁne a diﬀerential

(13.2) D

: W → W : F 7→ [H , F]

is that H is not technically an element of W: indeed, H conta ins terms of order −1

in ~, thus

H ∈

On the other hand, the failure of supercommutativity in W is a “phenomenon o f

order ~,” i.e. since every nontrivial commutator contains a factor of ~, we have

[F, G] = O(~) for all F, G ∈ W.

Here and in the following we use the symbol

O(~

)

270 Chris Wendl

to denote any element of the form ~

F for F ∈ W. As a consequence, [H, F] ∈ W

whenever F ∈ W, hence (

13.2) is well deﬁned, and the Jacobi identity now implies

= 0.

The homology of the resulting chain complex gives another version of what is often

called full SFT,

∗

(M, ξ; R) := H

∗

(W, D

A proof (modulo tr ansversality) that this deﬁnes a contact invariant is outlined in

[

EGH00, §2], but it is algebraically somewhat more involved than for H

SFT

∗

(M, ξ; R),

so I will skip it since I don’t have any applications of H

∗

(M, ξ; R) in mind. As far

as I am aware, no contact topological applications of this invariant or computations

of it (outside the trivial case—see §

13.1.4 below) have yet appeared in the literature.

This is a pity, because H

∗

(M, ξ; R) actually has much mo re algebraic structure t han

SFT

∗

(M, ξ; R). Indeed, using the identities

[F, GK] = [F, G]K + (−1)

|F||G|

G[F, K],

[FG, K] = F[G, K] + (−1)

|G||K|

[F, K]G,

(13.3)

one sees that D

: W → W satisﬁes a graded Leibniz rule,

(FG) = (D

F)G + (−1)

|F|

F D

It follows that D

: W → W is also a derivation with respect to the bracket structure

on W, i.e.

[F, G] = [D

F, G] + ( −1)

|F|

[F, D

for all F, G ∈ W. As a consequence, the product and bracket structures on W

descend to H

∗

(M, ξ; R), giving it the structure of a Weyl superalgebra.

As a matt er of interest, we observe that (W, D

), as with (A[[~]], D

SFT

) in the

previous lecture, can be deﬁned with Z or Z

coeﬃcients whenever the transversality

results are good enough to take the usual expression

ǫ(u)

|Aut

(u)|

literally as a count

of holomorphic curves. This result is of limited interest since it cannot hold in gen-

eral cases where transversality for multiple covers is impossible without multivalued

perturbations—nonetheless I ﬁnd it amusing.

Proposition 13.4. If Assumption 12.1 in Lecture 1 2 hold s, then D

is also well

deﬁned if the ring R = Q[H

(M)/G] i s replaced by Z[H

(M)/G] or Z

(M)/G].

Proof. Since D

is a derivation, it suﬃces to check that for every good Reeb

orbit γ, D

and D

are each sums of monomials of the form ce

−

with

The same arguments used to deﬁne SFT chain complexes over the integers can also be applied

to the chain maps involved in the proof of invariance (see §

13.3.1), so the SFT invariants should

be deﬁned over the integers if transversality can be achieved for multiple covers. There are known

situations however in which this cannot hold: even if the chain complexes are well deﬁned over Z,

invariance may hold only over Q, due to the failure of transversality in cobordisms. See [

Hut].

Lectures on Symplectic Field Theory 271

coeﬃcients c ∈ Z. Suppose u ∈ M

(J) is an index 1 holomorphic curve with positive

and/or negat ive asymptotic orbits

= (γ

, . . . , γ

{z }

, . . . , γ

{z }

where γ

6= γ

for i 6= j. We can assume all the or bit s γ

are good and that m

= 1

whenever n − 3 + µ

(γ

) is odd. Up to a sign and factors of e

and ~ which are

not relevant to this discussion, u then contributes a monomial

|Aut

(u)|

−

. . . q

−

. . . p

to H. The commutator [H

, q

] vanishes unless γ is one of the orbits γ

, . . . , γ

so supp ose γ = γ

. If n − 3 + µ

(γ) is odd, then m := m

= 1, and (

13.3) with

, q

] = κ

~ implies

, q

] =

|Aut

(u)|



−

. . . q

−

. . . p

−1

, q



|Aut

(u)|

−

. . . q

−

. . . p

−1

The fraction in front of this expression is an integer since u can have only one

end asymptotic to γ, and κ

is thus divisible by the covering multiplicity of u. If

n − 3 + µ

(γ) is even, then we generalize this calculation by using (

13.3) to write

, q

] = mκ

m−1

so then,

, q

] =

|Aut

(u)|



−

. . . q

−

. . . p

−1

, q



|Aut

(u)|

−

. . . q

−

. . . p

−1

m−1

To see that

|Aut

(u)|

is always an integer, recall fr om our proof of Prop.

12.13 in

the previous lecture that transformations in Aut

(u) permute each of the sets of

punctures that are asymptotic to the same Reeb orbit. Suppose the set of positive

punctures of u asymptotic to γ is partitioned by the Aut

(u)-action into N subsets,

each consisting of ℓ

, . . . , ℓ

punctures, where ℓ

+ . . . + ℓ

= m. If z is a puncture

in t he ith of t hese subsets, then its stabilizer is a cyclic subgroup of order k

acting

on a neighborhood of z by biholomorphic rotations, where k

ℓ

= |Aut

(u)|. Each

of these orders k

necessarily divides the multiplicity κ

, so we can write k

= κ

for some a

∈ N. Putting all this together, we have

m =

i=1

ℓ

i=1

ℓ

= |Aut

(u)|

i=1

Following this same procedure, you should now be able to verify on your own

that the coeﬃcient appearing in [H

, p

] is also always an integer. The existence

272 Chris Wendl

of a chain complex with Z

coeﬃcients follows from this simply by projecting Z

to Z

. 

13.1.2. The semiclassical limit: r at ional SFT. The idea of rational sym-

plectic ﬁeld theory (RSFT) is to extract as much inf ormation as possible f rom genus

zero holomorphic curves but ig nore curves of higher genus. The algebra of SFT

provides a fairly obvious mechanism for this: RSFT should be what SFT becomes

in t he “limit as ~ → 0,” i.e. the classical approximation to a quantum theory. Let

P := W



~W,

so P is a graded unital algebra generated by the p

and q

varia bles and the co-

eﬃcient ring R, but it does not include ~ a s a generator. Since all commutators

in W are in ~W, the product structure of P is supercommutative. Let us use the

distinction between capital and lowercase letters to denote the quotient projection

W → P : F 7→ f.

We will make an exception for the letter “H”: recall that H is not an element of W

since its genus zero terms have order −1 in ~, but ~H ∈ W, so we will deﬁne

h =

ǫ(u)

|Aut

(u)|

−

∈ P

to be the image of ~H under the projection. The sum in this expression ranges

over all R-equivalence classes of index 1 curves with genus zero, so h will serve as

the generating function of RSFT. To encode gluing of genus zero terms, note ﬁrst

that the commutator operation would not be appropriate since it prodcues terms

for every possible gluing of two curves, including those which glue genus zero curves

along more than one breaking orbit to produce buildings with positive arithmetic

genus. We need instead to have an algebraic operation on P that encodes gluing

along o nly one breaking orbit at a time.

You already know what to expect if you’ve ever taken a quantum mechanics

course: in t he “classical limit,” commutators become Poisson brackets. To express

this properly, we need to make a distinction between diﬀerential operators operating

from the left or the right: let

−−→

∂

∂q

: W → W

denote the usual operator

∂

∂q

, which was previously deﬁned on A[[~]] but has an

obvious extension to W such that

−→

∂

∂q

′

= 0 for all γ

′

. This operator satisﬁes the

graded Leibniz rule

−−→

∂

∂q

(FG) =

−−→

∂

∂q

G + (−1)

||F|

−−→

∂

∂q

The related opera tor

←−−

∂

∂q

: W → W : F 7→ F

←−−

∂

∂q

Lectures on Symplectic Field Theory 273

is deﬁned exactly the same way on individual var iables p

and q

, but satisﬁes a

slightly diﬀerent L eibniz rule,

(FG)

←−−

∂

∂q

= F

←−−

∂

∂q

+ (−1)

||G|

←−−

∂

∂q

The point of writing

←−

∂

∂q

so that it acts from the right is to obey the usual conventions

of superalgebra: signs change whenever the order of two odd elements (or operators)

is interchanged. Partial deriva tives with respect to p

can be deﬁned analogously

on W. With this notation in hand, the graded Poisson bracket on W is deﬁned

(13.4) {F, G} =

←−−

∂

∂p

−−→

∂

∂q

G − (−1)

|F||G|

←−−

∂

∂p

−−→

∂

∂q

where the sum r anges over all good Reeb orbits. In the same manner, the diﬀerential

operators and the bracket { , } can also be deﬁned on P.

It is easy to check that { , } on W almost satisﬁes a version of (

13.3): we have

{F, GK} = {F, G}K + (−1)

|F||G|

G{F, K}+ O(~),

{FG, K} = F{G, K} + (−1)

|G||K|

{F, K}G + O(~)

(13.5)

for all F, G, K ∈ W. The extra terms denoted by O(~) arise from the fact that

in proving (

13.5), we must sometimes reorder products FG by writing them as

(−1)

|F||G|

GF + [F, G], where [F, G] = O(~). Since the terms with ~ disappear in

P, the relations become exact in P:

{f, gk} = {f, g}k + (−1)

|f ||g|

g{f, k},

{fg, k} = f{g, k} + (−1)

|g||k|

{f, k}g

(13.6)

for all f, g, k ∈ P.

Proposition 13.5 . For all F, G ∈ W,

[F, G] = ~{f, g } + O(~

and { , } satisﬁes the conditions of a super Lie bracket on P.

Remark 13.6. In formulas like the one in the above proposition, we interpret

{f, g} ∈ P as an element of W via any choice of R-linear inclusion P ֒→ W that acts

as the identity on the generators p

, q

. There is ambiguity in this choice due to the

noncommutativity of W, but the ambiguity is in ~W and thus makes no diﬀerence

to the formula.

Proof of Proposition

13.5. The formula is easily checked when F and G

are individual variables of the form p

or q

; in fa ct the extra term O(~

) can be

omitted in these cases. The case where F and G are general monomials follows from

this via (

13.3) and (13.5) using induction on the number of va riables in the product.

This implies the general case via bilinearity.

274 Chris Wendl

Given the f ormula, the condition {f, g} + (−1)

|f ||g|

{g, f} = 0 and the Poisson

version of the super Jacobi identity (

13.1) follow from the corresponding properties

of [ , ]. 

The proposition implies that our genus zero generating function h ∈ P satisﬁes

0 = ~

[H, H] = [~H, ~H] = ~{h, h} + O(~

), thus

{h, h} = 0.

This relation can be interpreted as the count of boundary points of all 1-dimensional

moduli spaces of genus zero curves: indeed, any pair of genus two curves u, v ∈

(J)/R constributes to {h, h} a term of the form

|Aut

(u)||Aut

(v)|

−

←−−

∂

∂p

−−→

∂

∂q

−

plus a corresponding term with the roles of u and v reversed. This sums all the mono-

mials that one can construct by cancelling one p

varia ble from u with a matching

varia ble from v, in other words, constructing a building by gluing v on top of u

along o ne matching Reeb orbit.

The graded Jacobi identity will a gain imply that any r epresent ation of the super

Lie alg ebra (P, { , }) gives rise to a chain complex with h as its diﬀerential. For

example we can take the adjoint representation,

P → End

(P) : f 7→ d

, d

g := {f, g},

which satisﬁes d

{f ,g}

= d

−(−1)

|f ||g|

due to the Jacobi ident ity. Then d

= 0

since h has odd degree and {h, h} = 0, and the homology of rational SFT is

deﬁned as

RSFT

∗

(M, ξ; R) := H

∗

(P, d

We again refer to [

EGH00] for an argument that H

RSFT

∗

(M, ξ; R) is an invariant

of the contact structure. Notice that Proposition

13.5 yields a simple relationship

between the chain complexes (W, D

) and (P, d

), namely

(13.7) D

F = d

f + O(~),

where d

f is interpreted as an element of W via Remark

13.6. In other words, the

projection W → P : F → f is a chain map. Moreover, d

is a derivation on P with

respect to both the product a nd the Poisson bracket: this follows via Proposition

13.5

and (13.7) from the fact that D

satisﬁes the corresponding properties on W. We

conclude that H

RSFT

∗

(M, ξ; R) inherits the structure of a Poisson superalgebra, and

the map

∗

(M, ξ; R) → H

RSFT

∗

(M, ξ; R)

induced by the chain map (W, D

) → (P, d

) is both a n algebra homomorphism

and a homomorphism of gr aded super Lie algebras.

Lectures on Symplectic Field Theory 275

13.1.3. The contact homology algebra. Contact homology is the most pop-

ular tool in the SFT package and was probably the ﬁrst to be understood beyond t he

more straightforward cylindrical theory. In situations where cylindrical contact ho-

mology cannot be deﬁned due to bubbling of holomorphic planes, the next simplest

thing one can do is to deﬁne a theory t hat counts genus zero curves with one positive

end but arbitrary numbers of negat ive ends (cf. Exercise

10.14 in Lecture 10).

The proper algebraic setting for such a theory turns out t o be the algebra A

generated by the q

varia bles, and it can be derived from RSFT by setting all p

varia bles to zero. Using the obvious inclusion A ֒→ P, deﬁne ∂

: A → A by

∂

f = d

p=0

We can thus write d

f = ∂

f + O(p), where

O(p

)

will be used generally to denote any formal sum consisting exclusively of terms of

the form p

. . . p

f for f ∈ P. Now observe that for any good orbit γ,

= {h, p

} = −(−1)

′

←−−

∂

∂p

′

−−→

∂

∂q

′

= −(−1)

∂h

∂q

= O(p)

since every term in h has at least one p variable. It follows that d

(O(p)) = O(p),

so the fact t hat d

= 0 implies ∂

= 0, and contact homology is deﬁned as

∗

(M, ξ; R) := H

∗

(A, ∂

Since d

is a derivation on P, the formula d

f = ∂

f + O(p) implies that ∂

likewise a derivation on A, so HC

∗

(M, ξ; R) ha s the structure of a graded super-

commutative algebra with unit. Moreover, the proj ection P → A : f 7→ f|

p=0

is a

chain map, giving rise to an algebra homomorphism

RSFT

∗

(M, ξ; R) → HC

∗

(M, ξ; R).

The invariance of HC

∗

(M, ξ; R) will follow from the invar iance of H

SFT

∗

(M, ξ; R), to

be discussed in §

13.3.1 below.

To interpret ∂

, we can separate the part of h that is linear in p variables,

writing

h =

(q)p

+ O(p

where fo r each good Reeb orbit γ, h

(q) denotes a polynomial in q variables with

coeﬃcients in R. Since elements f ∈ A have no dependence on p variables, we then

have

f = {h, f} =

←−−

∂

∂p

−−→

∂

∂q

∂f

∂q

+ O(p),

hence

∂

f =

∂f

∂q

276 Chris Wendl

In par t icular, ∂

acts on each generator q

∈ A as

∂

= κ

ǫ(u)κ

Aut

(u)

−

where the sum is over all R-equivalence classes of index 1 J-holomorphic curves u

with genus zero, one positive end at γ, and negative ends γ

−

, and homology class

A ∈ H

(M)/G.

13.1.4. Algebraic overtwistedness. Even the simplest of the three diﬀeren-

tial graded algebras described above is too large to compute in most cases. The

major exception is the case of overtwisted contact manifolds.

Theorem 1 3.7. If (M, ξ) is overtwisted, then HC

∗

(M, ξ; R) = 0 for all choices

of the coeﬃc i ent ring R.

Remark 13.8. If X is a n algebra with unit, then saying X = 0 is equivalent to

saying that 1 = 0 in X.

The notion of overtwisted contact structures in dimension three was intro duced

by Eliashberg in [

Eli89], who proved that they are ﬂexible in the sense that their

classiﬁcation up to isotopy reduces to the purely obstruction-theoretic classiﬁcation

of almost contact structures up to homotopy. This means in eﬀect that an over-

twisted contact structure carries no distinctly contact geometric information, so it

should not be surprising when “interesting” contact invariants such as HC

∗

(M, ξ)

vanish. The three-dimensional case of Theorem

13.7 seems to have been among the

earliest insights about SFT: its ﬁrst a ppearance in the lit era ture was in [

Eli98], and

a proof later appeared in a paper by Mei-Lin Yau [

Yau06], which includes a brief

appendix sketching Eliashberg’s original proof. We will discuss Eliashberg’s proof

in detail in Lecture

16.

The deﬁnitive higher-dimensional no tion of overtwistedness was introduced a few

years ago by Borman-Eliashberg-Murphy [

BEM15], following earlier steps in this

direction by Niederkr¨uger [

Nie06] and others. There are now two known proofs of

Theorem

13.7 in higher dimensions: the ﬁrst uses the fact that since overtwisted

contact manifolds are ﬂexible, they always admit an embedding of a plastikstufe,

which implies vanishing of contact homology by an unpublished result of Bourgeois

and Niederkr¨uger (see [Bou09, Theorem 4.10] for a sketch of the argument) . The

second argument app eals to an even more recent result of Casals-Murphy-Presas

[

CMP] showing that (M, ξ) is overtwisted if and only if it is supported by a nega-

tively stabilized open book, in which case HC

∗

(M, ξ) = 0 was proven by Bourgeois

and van Koert [

BvK10].

It is not known whether the vanishing of contact homolog y characterizes over-

twistedness, i.e. there ar e not yet any examples of tight contact manifolds with

∗

(M, ξ) = 0. I will go out o n a limb and say that such examples seem unlikely

to exist in dimension three but are much more likely in higher dimensions; in fact

vario us candidates are known [

MNW13,CDvK], but we do not yet have adequate

methods to prove that any of them are tight. The ana logous question about Legen-

drian submanifolds and relative contact homology was recently answered by Ekholm

Lectures on Symplectic Field Theory 277

[Ekh], giving examples of Legendrians tha t are not loose in the sense of Murphy

[

Mur] but have vanishing L egendrian contact homology.

Nevertheless, the lack of known counterexamples has given rise to the following

deﬁnition.

Definition 13.9. A closed contact manifold (M, ξ) is algebraically over-

twisted if HC

∗

(M, ξ; R) = 0 for every choice of the coeﬃcient ring R.

Remark 13.10. The coeﬃcient ring is not always mentioned in statements of t he

above deﬁnition, but it should be. We will see in §13.3.2 below that this detail makes

a diﬀerence to issues like symplectic ﬁlling obstructions. Note that for any nested

pair of subgroups G ⊂ G

′

⊂ H

(M), the natural projection H

(M)/G

′

→ H

(M)/G

induces an algebra homomorphism

∗

(M, ξ; Q[H

(M)/G

′

]) → HC

∗

(M, ξ; Q[H

(M)/G]).

Since algebra homomorphisms necessarily map 1 7→ 1 and 0 7→ 0, the target of

this map must vanish whenever its domain does, so for checking Deﬁnition

13.9, it

suﬃces to check the case R = Q[H

(M)].

We’ve seen above that there exist algebra homomorphisms

(13.8) H

∗

(M, ξ; R) → H

RSFT

∗

(M, ξ; R) → HC

∗

(M, ξ; R),

thus the vanishing of either o f the algebras H

∗

(M, ξ; R) or H

RSFT

∗

(M, ξ; R) with

all coeﬃcient rings R is another suﬃcient condition for algebraic overtwistedness.

Bourgeois and Niederkr¨uger observed that, in fact, these conditions are also neces-

sary:

Theorem 13.11 ( [

BN10]). For any coeﬃcient ring R, the following conditions

are equivalent:

(1) HC

∗

(M, ξ; R) = 0,

(2) H

RSFT

∗

(M, ξ; R) = 0,

(3) H

SFT

∗

(M, ξ; R) = 0.

Proof. The implications (3) ⇒ (2) ⇒ (1) are immediate from t he algebra

homomorphisms (

13.8), thus it will suﬃce to prove (1) ⇒ (3). Suppose 1 = 0 ∈

∗

(M, ξ; R), which means ∂

f = 1 for some f ∈ A. Using the obvious inclusion

A ֒→ W, this means

f = 1 − G,

where G = O(p, ~), i.e. G is a sum of terms that a ll contain at least o ne p

varia ble

or a power of ~. It follows that G

= O(p

, ~

) for all k ∈ N, and the inﬁnite sum

∞

k=0

is therefore an element of W, as the coeﬃcient in front of any ﬁxed monomial ~

this sum is a polynomial function of the q var iables. This sum is then a multiplicative

inverse of 1 − G, and since

0 = D

f = 0 = −D

278 Chris Wendl

it also satisﬁes D

((1 − G)

−1

) = 0. Using the fact that D

is a derivation, we

therefore have



(1 −G)

−1



= (1 − G)

−1

(1 −G) = 1,

implying 1 = 0 ∈ H

SFT

∗

(M, ξ; R). 

13.2. SFT generating functions for cobordisms

All invariance proofs in SFT are based on a generating function analogous to

H that counts index 0 holomorphic curves in symplectic cobordisms. The basic

deﬁnition is a straightfor ward extension of what we saw in Lecture

12, but there

is an added wrinkle due to the fact that, in general, one must include disconnected

curves in the count.

13.2.1. Weak, strong and stable cobordisms. F ir st some remarks about

the category we are working in. Since t he stated purpose of SFT is to deﬁne invari-

ants of contact structures, we have been working since Lecture

12 with symplectiza-

tions of contact manifolds rather than more general stable Hamiltonian structures.

We’ve made use of this restriction on several occasions, namely so t hat we can

assume:

(1) All nontrivial holomo r phic curves in R ×M have at least one positive punc-

ture;

(2) The energy of a holomorphic curve in R × M can be bounded in terms of

its positive asymptotic orbits.

It will be useful however for certain applications to permit a slightly wider class of

stable Hamiltonian structure. Recall that a hypersurface V in an almo st complex

manifold (W, J) is called pseudo convex if the ma ximal complex subbundle

ξ := T V ∩ J(T V ) ⊂ T V

deﬁnes a contact structure on V whose cano nical conformal symplectic bundle struc-

ture tames J|

. For example, if α is a conta ct form on M and J ∈ J(α), then each of

the hypersurfaces {const}×M is pseudoconvex in (R×M, J). The contact structure

ξ induces an orientation on the hypersurface V ; if V comes with its own orient ation

(e.g. as a boundary component of W ), then we call it pseudoconvex if ξ is a positive

contact structure with respect to this or ientation, and pseudoconcave otherwise.

For example, if (W, ω) is a symplectic cobordism from (M

−

, ξ

−

) to (M

, ξ

) and

J ∈ J(W, ω, α

, α

−

), then M

is pseudoconvex and M

−

is pseudoconcave.

Definition 1 3.12. Given an odd-dimensional manifold M, we will say that an

almost complex structure J on R×M is pseudoconvex if {r}×M is a pseudoconvex

hypersurface in (R ×M, J) f or every r ∈ R, with the induced orientation such that

∂

and {r}× M are positively transverse.

If H = (ω, λ) is a stable Hamiltonian structure on M, t hen pseudoconvexity of

J ∈ J(H) imposes conditions on H, in particular λ must be a contact form. It also

requires J|

to be tamed by dλ|

, but unlike the case when J ∈ J(λ), J|

need not be

co mpatible with it , i.e. the positive bilinear form dλ(·, J·)|

need not be symmetric.

As always, J|

must be compatible with ω|

, but ω need not be an exact form for

Lectures on Symplectic Field Theory 279

this to hold—the freedom to chang e [ω] ∈ H

(M) will be the main beneﬁt of this

generalization, particularly when we discuss weak symplectic ﬁllings below.

Proposition 13.13. Suppose H = (ω, λ) is a stable Hamiltonian structure on

a closed manifold M and J ∈ J(H) is pseudoconvex. Then all nonconstant ﬁnite-

energy J-holomorphic curves in R × M have at least one positive puncture, and

their energies satisfy a unif orm upper bound i n terms of the periods of their positive

asymptotic orbits.

Proof. It is straightforwa rd t o check that either of the two proofs of Proposi-

tion

10.9 given in Lecture 10 generalizes to any J on R × M that is pseudoconvex.

In particular, pseudoconvexity implies tha t if u : (

Σ, j) → (R × M, J) is a J-

holomorphic curve, then u

∗

dλ ≥ 0, with equality only at points where u is tangent

to ∂

and the Reeb vector ﬁeld. Stokes’ theorem thus gives

(13.9) 0 ≤

∗

dλ =

z∈Γ

−

z∈Γ

−

where T

> 0 denotes the period of the asymptotic orbit at each positive/negat ive

puncture z ∈ Γ

. Since J|

is also ta med by ω|

and ω annihilates the Reeb vector

ﬁeld, we similarly have u

∗

ω ≥ 0 , with the same condition for equality, and the

compactness of M then implies an estimate of the form

0 ≤ u

∗

ω ≤ cu

∗

dλ

for every J- holomorphic curve u : (

Σ, j) → (R × M, J), with a constant c > 0 that

depends only on M, H and J. In light of (

13.9), this implies an upper bound on

∗

ω in terms of the periods T

for z ∈ Γ

. Writing ω

= ω +d(ϕ(r)λ) for suitable

-small increasing functions ϕ : R → R, we can then apply Stokes’ theorem to the

second t erm in

E(u) = sup

∗

ω + sup

∗

d(ϕ(r)λ) ,

implying a similar upper bound f or E(u). 

Corollary 13.14. For a ny stable Hamiltonian structure H = (ω, λ) with a

nondegenerate Reeb vector ﬁ eld R

and a pseudoconvex J ∈ J(H), one can use

closed R

-orbits and count J-holomorphic curves in R × M to deﬁn e the ch ain

co mplexes (A[[~]], D

SFT

), (W, D

), (P, d

) and (A, ∂

We shall denote the homologies of the above chain complexes with coeﬃcients

in R = Q[H

(M)/G] by

SFT

∗

(M, H, J; R), H

∗

(M, H, J; R), H

RSFT

∗

(M, H, J; R), HC

∗

(M, H, J; R).

We make no claim at this point about these homologies being invariant . For t he

examples that we actually care about, this will turn out to be an irrelevant question

due to Propo sition

13.16 and Exercise 13.32 below.

Example 13.15 . Suppose α is a contact form on (M, ξ) and H = (Ω, α) is a

stable Hamiltonian structure. Then for all constants c > 0 suﬃciently large, H

(Ω + c dα, α) is also a stable Hamiltonian structure and there exists a pseudoconvex

280 Chris Wendl

∈ J(H

). To see the latter, notice that H

′



Ω + dα, α



is another family of

stable Hamiltonian structures, with J(H

′

) = J(H

) for all c, and H

′

→ (dα, α) as

c → ∞. Thus one can select J

∈ J(H

) converging to some J

∞

∈ J(α) as c → ∞,

and these are pseudoconvex for c > 0 suﬃciently large since J

∞

is.

Proposition 13.16. In the setting of Example

13.15, assume α is nondegenerate

and J

∞

∈ J(α) is generic. If HC

∗

(M, ξ; R) = 0, then HC

∗

(M, H

, J

; R) also

vanishes for all c > 0 suﬃciently large.

Proof. We will assume in the following that the usual (unrealistic) transver-

sality assumptions hold, but the essential idea of the argument would not change in

the presence of abstract perturbations.

Let ( A, ∂

∞

) denote the contact homology chain complex generated by closed

-orbits, with ∂

∞

counting J

∞

-holomorphic curves in R × M. The assumption

∗

(M, ξ; R) = 0 means there exists an element f ∈ A with ∂

∞

f = 1. Here f is

a polynomial function of the q

varia bles, and ∂

∞

f counts a speciﬁc ﬁnite set of

Fredholm regular index 1 curves in (R ×M, J

∞

). Now let (A, ∂

) denote the chain

complex for HC

∗

(M, H

, J

; R), and notice that since the stable Hamiltonia n struc-

tures (dα, α) and H

deﬁne matching Reeb vector ﬁelds, the set of generators is un-

changed. There is also no change to this complex if we replace H

= (Ω+c dα, α) by

′



Ω + dα, α



: this changes the energies of individual J

-holomorphic curves,

but the sets of ﬁnite-energy curves are still the same in both cases. We can assume

→ J

∞

in C

∞

as c → ∞. The implicit function theorem then extends each of

the ﬁnitely many J

∞

-holomorphic curves counted by ∂

∞

f uniquely to a smooth 1-

parameter family of J

-holomorphic curves for c > 0 suﬃciently large.

We claim

that these are the only curves counted by ∂

f when c > 0 is large. Indeed, there

would otherwise exist a sequence c

→ ∞ for which additional J

-holomorphic

index 1 curves u

contribute to ∂

f, and since f has only ﬁnitely many terms rep-

resenting possible positive asymptotic orbits, we can ﬁnd a subsequence for which all

the u

have the same positive asymptotic orbits. A further subsequence then has all

the same negative asymptotic orbits as well since the Reeb ﬂow is nondegenerate and

the total period of the negative orbits is bounded by the total period of the positive

orbits. Finally, since the sequence of stable Hamiltonian structures H

′

converges to

(dα, α), the curves u

have uniformly bounded energy with respect to H

′

, so that

SFT compactness yields a subsequence converging to a J

∞

-holomorphic building of

index 1, which can only be one of the curves counted by ∂

∞

f. This contradicts the

uniqueness in the implicit function theorem and thus proves t he claim. We conclude

that for all c > 0 suﬃciently lar ge, ∂

f = 1. 

Definition 13.17. Assume (W, ω) is a symplectic cobordism with stable bound-

ary ∂W = −M

−

⊔ M

, with induced stable Hamiltonian structures H

= (ω

, λ

)

at M

, and suppose J is an almo st complex structure on the completion

W that is

ω-tame on W and belongs to J(H

) on the cylindrical ends. We will say that J

In cas e you are co nc erned about the parametric moduli space being an orbifold instead of a

manifold, just add asymptotic markers so that there is no isotropy, and divide by the appropriate

combinatorial factors to count.

Lectures on Symplectic Field Theory 281

is pseudoconvex near inﬁnity

if the R-invariant almost complex structures J

deﬁned by restricting J to [0 , ∞) × M

and (−∞, 0] × M

−

are bo th pseudoconvex.

Note that the condition on J in the above deﬁnition can only be satisﬁed if λ

are both positive contact forms on M

, but the 2 -forms ω

need not be exact.

Proving conta ct invaria nce of SFT r equires counting curves in trivial exact sym-

plectic cobordisms, but it is also natural to try to say t hings about non-exact

strong symplectic cobordisms using SFT.

These ﬁt naturally into our previously

established picture since every strong cobordism has collar neighborhoods near the

boundary in which it matches the symplectization of a cont act manifold. The fol-

lowing more general notion of cobordism is also natural from a conta ct topological

perspective, but ﬁts less easily int o the SFT picture.

Definition 13.18 ([

MNW13]). Given closed contact manifolds (M

, ξ

) and

−

, ξ

−

) of dimension 2n − 1, a weak symplectic cobordism from (M

−

, ξ

−

) to

, ξ

) is a compact symplectic manifo ld (W, ω) with ∂W = −M

−

⊔M

admitting

an ω-tame almost complex structure J for which the almost complex manifold (W, J)

is pseudoconvex at M

and pseudoconcave at M

−

, with

= T M

∩ J(T M

Weak cobo r disms are characterized by the existence of a tame almost complex

structure J whose restriction to ξ

is tamed by two symplectic bundle structures,

ω|

and dα

(for any choices of contact forms α

deﬁning ξ

). Notice that

in dimension 4, the second condition is mostly va cuous, and the weak cobordism

condition just reduces to

ω|

> 0.

In t his form, the low-dimensional case of Deﬁnition 13.18 has been ar ound since the

late 198 0’s, and there are many interesting results about it, e.g. examples of contact

3-manifolds that are weakly but not strongly ﬁllable [

Gir94, Eli96]. We will see in

13.3.2 that this distinction is detectable via SFT. Higher-dimensional examples of

this phenomenon were found in [

MNW13].

One major diﬀerence between weak and strong cobordisms is that the latter are

always exact near the boundary, as the Liouville vector ﬁeld is dual to a primitive

of ω. It turns out that up to deformat ion, weak ﬁllings that are exact at t he

boundary are the same thing as strong ﬁlling s—t his was ﬁrst observed by Eliashberg

in dimension three [

Eli91, Prop. 3.1], and was extended to higher dimensions in

[

MNW13]:

Proposition 13.19. Suppose (W, ω) is a weak ﬁlling of a (2n − 1)-dimensiona l

co ntact manifold (M, ξ) such that ω|

T M

is exact. Then after a homotopy of ω through

If I were being hypercorr ect about use of language, I might insist on saying that J is “pseu-

doconvex near +∞ and pseudoconcave ne ar −∞,” as the orientation reversal at the negative

boundary makes M

−

technically a pseudoconcave hypers urface in (

W , J), not pseudoconvex. But

this deﬁnition will only be useful to us in cases where M

−

= ∅, so my linguistic guilt is limited.

By strong cobordism, we mean the us ual notion of a compact symplectic manifold with convex

and/or concave bo undary components (see §

1.4). The word “strong” is included in order to contrast

this notion with its weaker cousin described in Deﬁnition

13.18.

282 Chris Wendl

a family of symplectic forms that va ry only in a collar neighborhood of ∂W and deﬁne

weak ﬁllings of (M, ξ), (W, ω) is a strong ﬁlling of (M, ξ).

Proof. Choose any contact form α for ξ, denote its Reeb vector ﬁeld by R

and let Ω = ω|

T M

. Identify a collar neighborhood of ∂W in W smoothly with

(−ǫ, 0] ×M, with the coordinate on ( −ǫ, 0] denoted by r, such that ∂

and R

span

the symplectic complement of ξ at ∂W and satisfy ω(∂

, R

) = 1. Then ω and

Ω + d(rα) are cohomologous symplectic forms on (−ǫ, 0] × M that match at r = 0,

hence a Moser deformation argument implies they are isotopic. We can therefore

assume without loss of generality that ω = Ω + d(rα) on the collar near ∂W .

By assumption, Ω = dη for some 1-for m η on M, and since (W, ω) is a weak

ﬁlling of (M, ξ = ker α), we can choose a complex structure J

on ξ that is tamed

by both dα|

and dη|

. Now choose a smooth cutoﬀ function β : [0, ∞) → [0, 1] that

has compact support and equals 1 near 0 . We claim tha t

ω := d(β(r)η) + d(rα)

is a symplectic form on [0, ∞) × M if |β

′

| is suﬃciently small. Indeed, writing

ω = dr ∧ (α + β

′

(r) η) + [β(r) dη + r dα], we have

= n dr ∧ α ∧ [β(r) dη + r dα]

n−1

+ nβ

′

(r) dr ∧ η ∧ [β(r) dη + r dα]

n−1

The ﬁrst term is positive and bounded away fr om zero since dη|

and dα|

both

tame J

, hence do does β dη + r dα|

. The second term is then harmless if |β

′

| is

suﬃciently small, proving ω

> 0.

This deﬁnes an extension of the original weak ﬁlling to a symplectic completion

W = W ∪

([0, ∞) × M), and for each r

≥ 0, the compact subdomains deﬁned

by r ≤ r

deﬁne weak ﬁllings o f ({r

} × M, ξ) since ω|

= (β(r

) dη + r

dα)|

also

tames J

. Notice that for r

suﬃciently large, the dη term disappears, so ω has a

primitive that restricts to {r

} × M as a contact form for ξ, meaning we have a

strong ﬁlling of this hypersurface. The desired deformatio n of ω can therefore be

deﬁned by pulling back via a smooth family of diﬀeomorphisms (−ǫ, 0] → (−ǫ, r

where r

varies from 0 to a suﬃciently larg e constant. 

Unlike strong cobordisms, being a weak cobordism is an open condition: if (W, ω)

is a weak cobordism, then so is (W, ω + ǫσ) for any ǫ > 0 suﬃciently small and a

closed 2-f orm σ, which need not be exact at ∂W . As a consequence, the cylindrical

ends of a completed weak cobordism cannot always be deformed to look like the

symplectization of a contact manifold. This is where Deﬁnition

13.17 comes in

useful. The proof of the next lemma is very much analogous to Proposition

13.19.

Lemma 13.20 ([

MNW13, Lemma 2.10]). Suppose (W, ω) is a weak ﬁlling of

a (2n − 1)-dimensional contact manifold (M, ξ), α is a contact form for ξ and Ω

is a closed 2-form on M with [Ω] = [ω|

T M

] ∈ H

(M). Then for any constant

c > 0 suﬃciently large, after a homotopy of ω through a family of symplectic forms

that vary only in a collar neighborhood of ∂W and deﬁne weak ﬁllings of (M, ξ),

ω|

T M

= Ω + c dα. 

The following result then provides a suitable model that can be used as Ω in

the above lemma when ω|

T M

is nonexact. The stat ement b elow is restricted to the

Lectures on Symplectic Field Theory 283

case where [ω|

T M

] is a rational cohomology class; the reason for this is that it relies

on a Donaldson-type existence result for contact submanifolds obtained as zero-sets

of approximately holomorphic sections, due to Ibort, Mar t i´ınez-Torres and Presas

[

IMTP00]. It seems likely that the rationality condition could be lifted with more

work, and in dimension three this is known to b e true; see [

NW11, Prop. 2.6].

Lemma 13.21 ([

CV15, Prop. 2.18]). For any rational cohomology cl ass η ∈

(M; Q) on a closed (2n − 1)-dimensional contact manifold (M, ξ), there exists

a closed 2-form Ω and a nondegenerate contact form α for ξ such that (Ω, α) is a

stable Hamiltonia n structure. 

Combining all of the above results (including Example

13.15) proves:

Proposition 13.22. Suppose (W, ω) is a weak ﬁlling of a (2n − 1)-dimensiona l

co ntact manifold (M, ξ) such that [ω|

T M

] ∈ H

(M) is rational or n = 2. Fix a

nondegenerate contact form α f or ξ. Then there exists a closed 2 - form Ω cohomol-

og ous to ω|

T M

such that H := (Ω, α) is a stable Hamiltonian structure, and for

all c > 0 suﬃciently l arge , ω can be deformed in a collar neighborhood of ∂W ,

through a family of symplectic forms deﬁ ning weak ﬁllings of (M, ξ), to a new weak

ﬁlling for which ∂W is also stable and inherits the stable Hamiltonian structure

:= (Ω + c dα, α). In particular, after this deformation, the completed stable ﬁll-

ing admits a tame almost complex s tructure that is pseudoconvex near inﬁnity and

may be assumed C

∞

-close to any given J ∈ J(α). 

We will use this in §

13.3.2 to deﬁne obstructions to weak ﬁllability via SFT.

Remark 13.23. There is apparently no analogue of Propositions

13.19 and 13.22

for negative boundary components of weak cobordisms, and this is one of a few

reasons why they are not often discussed. For example, if L is a Lagrangian torus

in the standard symplectic 4-ball D

, t hen the complement of a neighborhood of

L in B

deﬁnes a strong cobordism from the standard contact T

to S

. The

symplectic form on this cobordism is obviously exact, but if any result analogous to

Proposition

13.19 were to hold at the concave boundary, then we could deform it to

a Liouville cobo r dism. No such Liouville cobordism exists—it would imply that the

Lagrangian L ⊂ B

is exact, thus violating Gromov’s famous t heorem [

Gro85] on

exact Lagrangians.

13.2.2. Counting disconnected index 0 curves. Fix a symplectic cobor-

dism (W, ω) with stable boundary ∂W = −M

−

⊔ M

carrying stable Hamiltonian

structures H

= (ω

, λ

), along with a generic almost complex structure J that is

ω-tame on W , belongs to J( H

) on the cylindrical ends, and is pseudoconvex near

inﬁnity. This implies that the stabilizing 1-forms λ

are both contact forms. Let us

also assume t ha t the λ

are bo th nondegenerate, and that the induced R-invariant

almost complex structures J

∈ J(H

) are suﬃciently generic to achieve regularity

for all holomorphic curves under consideration. In particular, these assumptions

mean that all the usual SFT chain complexes are well deﬁned for (M

, H

, J

; R

)

with any choice of coeﬃcient ring R

= Q[H

)/G

]. Denote the corresponding

SFT g enerating functions by H

284 Chris Wendl

Recall from Lecture 12 that the auxiliary data on M

and M

−

includes a choice

of capping surface C

for each closed Reeb orbit γ (or a capping chain with rational

coeﬃcients if H

) has torsion). These surfaces satisfy

∂C

] − [γ],

where the m

are integers and C

⊂ M

are ﬁxed curves forming a basis of H

Assume H

(W ) is torsion free, in which case the same is true of H

) and

−

). (Only minor modiﬁcations a r e needed if this assumption f ails to hold,

see Remark

13.1.) We can then ﬁx the following additional auxiliary data:

(1) A collection of reference curves

∼

, . . . , C

⊂ W

whose homology classes from a basis of H

(W ).

(2) A unitary trivialization o f T W along each of the reference curves C

, . . . , C

denoted collectively by τ.

(3) A spanning surface S

for each of the positive/negative reference curves

⊂ M

, i.e. a smooth map of a compact and orient ed surface with

boundary into W such that

∂S

] − [C

]

in the sense of singular 2-chains, where m

∈ Z are the unique coeﬃcients

with [C

] =

] ∈ H

(W ).

Now to any collections of orbits γ

= (γ

, . . . , γ

) in M

and a relative homol-

ogy class A ∈ H

(W,

∪

−

) with ∂A =

[γ

] −

[γ

−

], we can associate an

absolute homology class in two steps: ﬁrst add A to suitable sums of the capping

surfaces C

producing a 2-chain whose boundary is a linear combination of positive

and negative reference curves, then a dd a suitable linear combination of the S

that the boundary becomes the trivial linear combination of C

, . . . , C

. With this

understood, we can now associate an a bsolute homology class

[u] ∈ H

(W )

to any asymptotically cylindrical J-holomorphic curve u : (

Σ, j) → (

W , J), and

this deﬁnes the notation M

g,m

(J, A, γ

, γ

−

) with A ∈ H

(W ). We now require the

trivializations of ξ

along each C

to be compatible with τ in the sense that they

extend to tr ivializations of T W along the capping surfaces S

. With this convention,

the Fredholm index formula takes the expected form

ind(u) = (n − 3)χ(

Σ) + 2c

([u]) +

i=1

(γ

) −

−

j=1

(γ

If H

(W ) has torsion, then this whole discussion can be adapted as in §

12.7.1 by re-

placing integral homology with rational homology and capping surfaces with capping

chains, and the Conley-Zehnder indices can be deﬁned modulo 2.

Lectures on Symplectic Field Theory 285

We will also need to impose a compatibility condition relating the coeﬃcient

rings R

= Q[H

)/G

] to a corresponding choice on the cobordism W . Choo se

a subgroup G ⊂ H

(W ) such that

(13.10) h[ω], Ai = 0 for all A ∈ G,

and such that the maps H

) → H

(W ) induced by the inclusions M

֒→ W

send G

into G. If [ω] 6= 0 ∈ H

(W ), then we will have to deal with noncompact

sequences of J-holomorphic curves that have unbounded energy, so it becomes nec-

essary to “complete” R to a Novikov ring

R, which contains R but also includes

inﬁnite formal sums

∞

i=1

such that h[ω], A

i → +∞ as i → ∞.

Note that the evaluatio n h[ω], Ai ∈ R is well deﬁned for A ∈ H

(W )/G due to

(

13.10).

Analogously to our deﬁnition of H in Lecture

12, the generating function for

index 0 curves in

W is deﬁned as a formal power series in the variables ~, q

(for

orbits in M

−

), and p

(for o r bit s in M

), with coeﬃcients in

(13.11) F =

u∈M

(J)

ǫ(u)

|Aut

(u)|

g−1

−

where M

(J) denotes the moduli space of connected J-holomorphic curves u in

W with ind(u) = 0 and only good asymptotic orbits, modulo permutations of the

punctures, and for each u:

• g is the genus of u;

• A is the equivalence class of [u] ∈ H

(W ) in H

(W )/G;

• γ

= (γ

, . . . , γ

) are the asymptotic orbits of u after arbitrarily ﬁxing

orderings of its positive and negative punctures;

• ǫ(u) ∈ {1 , −1} is the sign of u as a point in the 0-dimensional compo-

nent of M

(J) (after choosing an ordering of the punctures and asymptotic

markers), relative to a choice of coherent orientatio ns on M

(J).

As usual, the product ǫ(u)q

−

is independent of choices. We shall regard F as

an element in an enlarged operator algebra that includes q and p variables for good

orbits in both M

and M

−

, related to each other by the supercommutation relations

−

, q

] = [p

, q

−

] = [q

−

, q

] = [p

−

, p

] = 0

whenever γ

−

is an o r bit in M

−

and γ

is an orbit in M

. Since all curves counted

by F have index 0, F is homogeneous with degree

|F| = 0.

Notice that for any ﬁxed monomial q

−

, the corresponding set of curves in M

(J)

may be inﬁnite if ω is nonexact, but SFT compactness implies that the set of such

curves with any given bound on

∗

ω is bounded. As a consequence, the coeﬃcient

of q

−

in F belongs to the Novikov ring

286 Chris Wendl

Consider next the series

exp(F) :=

∞

k=0

We will be able to view this as a formal power series in q and p variables a nd a formal

Laurent series in ~ with coeﬃcients in

R, though it is not obvious at ﬁrst glance

whether its coeﬃcients a r e in any sense ﬁnite. We will deduce this after interpreting

it as a count of disconnected index 0 curves: ﬁrst, write

exp(F) =

∞

k=0

,...,u

)∈(M

(J))

ǫ(u

) . . . ǫ(u

)

|Aut

)|. . . |Aut

+...+g

−k

+...A

· q

−

. . . q

−

Observe that since each of the curves u

∈ M

(J) in this expansion has index 0,

the monomials q

−

all have even degree and thus the order in which they are

written does not matter. Now for a given collection of distinct curves v

, . . . , v

and

integers k

, . . . , k

∈ N with k

+ . . . + k

= k, the various permutations of

, . . . , u

) := ( v

, . . . , v

{z }

, . . . , v

{z }

) ∈ (M

(J))

occur

!...k

times in the above sum, so if we forg et the ordering, then the contri-

bution of this particular k-tuple of curves to exp(F) is

ǫ(u

) . . . ǫ(u

)

! . . . k

!|Aut

)|. . . |Aut

+...+g

−k

+...+A

−

. . . q

−

Notice next that the denominator k

! . . . k

!|Aut

)|. . . |Aut

)| is the order of

the automorphism group of the dis connected curve formed by the disjoint union of

, . . . , u

: t he extra factors k

! come from automorphisms that permute connected

components of the do ma in. Thus exp(F) can also be written as in (

13.11), but with

(J) replaced by the moduli space of potentially disconnected index 0 curves with

unordered punctures, and g −1 generalized to g

+. . .+g

−k fo r any curve that has

k connected components of genera g

, . . . , g

. One subtlety that was glossed over in

the above discussion: the sum also includes the unique curve with zero components,

i.e. the “empty” J- holomorphic curve, which app ear s as the initial 1 in the series

expansion of exp(F).

With this interpretation of exp(F) understood, we can now address the possibil-

ity that the inﬁnite sum deﬁning exp(F) might include inﬁnitely many terms for a

given monomial ~

−

, i.e. tha t there are inﬁnitely many disconnected index 0

curves with ﬁxed asymptotic orbits and a ﬁxed sum of the genera minus the number

of connected components. We claim that this can indeed, happ en, but only if the

curves belong to a sequence of homolo gy classes A

∈ H

(M)/G with h[ω], A

i → ∞,

hence the coeﬃcient of ~

−

in exp (F) belongs to the Novikov ring

R. The

danger here comes only from closed curves, since a disjoint union of two curves

with punctures always has strictly more punctures. Notice also that for any given

Lectures on Symplectic Field Theory 287

tuples of orbits γ

, there exists a number c ∈ R depending only on these o rbits a nd

the chosen capping surfaces such that every (possibly disconnected) J-holomorphic

curve u :

Σ →

W asymptotic to γ

satisﬁes

h[ω], [u]i ≥ c.

This follows from the fact that the integral of ω over the relative homology class of

u always has a nonnegative integrand.

Lemma 13.24. Given constants C ∈ R and k ∈ Z, there exists a number N ∈ N

such that if u : (Σ, j) → (

W , J) is a cl osed J-holomorphic curve satisfying

∗

ω ≤

C, with m connected components of genera g

, . . . , g

satisfying g

+. . .+g

−m = k,

then m ≤ N.

Proof. Note ﬁrst that for each integer g ≥ 0, there is an energy thresh-

old, i.e. a constant c

> 0 such that every nonconstant closed and connected J-

holomorphic curve u : Σ →

W of genus g has

∗

ω ≥ c

This is an easy consequence of SFT compactness: indeed, if there were no such

constant, then we would ﬁnd a sequence u

: Σ →

W of connected closed curves

with genus g such that

E(u

) =

∗

ω → 0;

here we have used the fact that Σ is closed and

∗

depends only on the homology

class o f u in order to simplify the usual deﬁnition o f energy for asymptotically

cylindrical curves. SFT compactness then gives a subsequence of u

that converges

to a stable holomorphic building in which every component has zero energy and is

therefore constant. Since there ar e no marked points in the picture, no such building

exists, so this is a contradiction.

Now if u is a disconnected curve satisfying t he stated conditions, the bound on

∗

ω combines with the energy threshold to give a bound for each g ≥ 0 on the

number o f connected components of u with genus g. In particular, there is a bound

on the number of components with genus 0 or 1. All other components contribute

positively to the left hand side of t he relation

i=1

− 1) = k, so this implies a

universal bound on m. 

Corollary 13.25. Fix constants C ∈ R an d k ∈ Z, a nd tuples of Reeb orbits

, and assume that the usual transversality conditions hold. Then there exist at

most ﬁnitely many potentially disconnected J-holomorphic curves u :

Σ →

W with

index 0 such that the number of connected com ponents m and the genera g

, . . . , g

of its co mponents satisfy g

+ . . . + g

− m = k.

Corollary 13.26. The expression exp(F) is a formal power series in q and p

variables and a formal Laurent series in ~, with coeﬃcients in the Novikov ring

288 Chris Wendl

The necessity of considering disconnected curves becomes clear when one tries to

translate the compactness and gluing theory of J-holomorphic curves in

W into alge-

braic relations. In particular, consider the 1-dimensional moduli space of connected

index 1 curves in

W with genus g. The boundary points of the compactiﬁcation of

this space consist of two types of buildings:

Type 1 : A main level of index 0 and an upper level of index 1;

Type 2 : A main level of index 0 and a lower level of index 1.

This is clear under the usual transversality assumptions since regular curves in

must have index at least 0, while regular curves in the symplectizations R × M

have index at least 1 unless they are trivial cylinders. The building must also be

connected and have arithmetic genus g, but there is nothing to guarantee that each

individual level is connected. In f act, we already saw this issue in Lecture

12 when

proving H

= 0, but it was simpler to deal with there, because disconnected regular

curves of index 1 in a symplectization always have a unique nontrivial compo nent,

while the rest are trivial cylinders. In the cobordism

W , on the other hand, a

disconnected index 0 curve can be formed by any disjoint union of index 0 curves,

all of which are nontrivial. Exponentiation provides a convenient way to encode all

data about disconnected curves in terms of connected curves.

Since the union of all buildings of types 1 and 2 described above forms the

boundary of a compact oriented 1-manifold, the count of these buildings is zero,

and this fact is encoded in the so-called master equation

(13.12) H

−

exp(F)|

−

− exp(F)H

= 0,

where the expressions “p

−

= 0” and “q

= 0” mean that we discard all terms in

−

exp(F) −exp ( F)H

containing any variables p

for o rbits in M

−

or q

for orbits

in M

. The resulting expression is therefore a formal power series in q variables

for or bits in M

−

and p variables for orbits in M

, representing a count of generally

disconnected index 1 holomorphic buildings in

W with the speciﬁed asymptotics.

The various ways to form such buildings by choices o f gluings is again encoded by

the commutator algebra. The master equation (

13.12) can be used to prove the

chain map property for counts of curves in cobordisms, thus it is an essential piece

of the invariance proo f for each of the homology theories introduced above.

Exercise 13.27. Fill in the details of the proof of (

13.12).

13.3. Full SFT as a BV

∞

-algebra

In this section we discuss the speciﬁc theory H

SFT

∗

(M, ξ; R), deﬁned as the ho-

mology of the chain complex (A[[~]], D

SFT

). The case G = H

(M) with trivial gr oup

ring coeﬃcients Q[H

(M)/G] = Q will be a bbreviated as

SFT

∗

(M, ξ) := H

SFT

∗

(M, ξ; Q).

As we deﬁned it, D

SFT

acts on A[[~]] by treating the g enerating function H as a

diﬀerential operator via the substitution

(13.13) p

= κ

∂

∂q

Lectures on Symplectic Field Theory 289

According to [CL09], this makes (A[[~]], D

SFT

) into a BV

∞

-algebra; we’ll have no

particular need to discuss here what that means, but one convenient featur e is the

expansion

(13.14) D

SFT

∞

k=1

(k)

SFT

in which each D

(k)

SFT

: A → A is a diﬀerential operato r of order ≤ k (see [

CL09, §5]).

For each k ∈ N, D

(k)

SFT

is a count o f all index 1 holomorphic curves that have genus

g ≥ 0 and m ≥ 1 positive punctures such that g + m = k. In particular, D

(1)

SFT

is simply the cont act ho mo logy diﬀerential ∂

, and the expansion (

13.14) implies

together with D

SFT

= 0 t hat (D

(1)

SFT

)

= 0, hence we again see the chain complex

for contact homology hidden inside a version of the “full” SFT complex.

13.3.1. Cobordism maps and invariance. One can use the master equation

(

13.12) to prove invariance of H

SFT

∗

(M, ξ; R) by a stra ightforward generalization o f

the usual Floer-theoretic argument. Suppose (W, dλ) is an exact symplectic cobor-

dism from (M

−

, ξ

−

) to (M

, ξ

) with λ|

T M

= α

, and choose a generic almost

complex structure J on

W that is dλ-compatible on W and restricts to the cylin-

drical ends as generic elements J

∈ J(α

). Let (A

[[~]], D

SFT

) denote the chain

complexes associated to the data (α

, J

), and for simplicity in this initial discus-

sion, choose the trivial coeﬃcient ring R = Q for both. We then deﬁne a map

Φ : A

[[~]] → A

−

[[~]] : f 7→ exp(F)f|

where the generating function exp(F) is regarded as a diﬀerential operator via the

substitution (

13.13), with e

:= 1 for all A ∈ H

(W ) since we are using t r ivial

coeﬃcients, and “q

= 0” means that after applying exp(F) to change f into a

function of q variables for orbits in both M

and M

−

, we discard all terms that

involve orbits in M

. The exactness of the cobordism implies that negative powers

of ~ do not appear in Φf, thus producing an element of A

−

[[~]]: indeed, since there

are no holomorphic curves in

W without positive punctures, every term in F contains

at least one p variable, so that negative powers of ~ do not appear in exp(F) after

applying (

13.13).

The master equation for F now translates into the fact that Φ is a chain map,

−

SFT

◦ Φ − Φ ◦ D

SFT

thus it descends to homology. The geometric meaning o f Φ is straightforward to

describe: a nalogous to (

12.11) in Lecture 12, we can write

(13.15) Φq

∞

g=0

′

g+k−1

(γ, γ

′

, k)q

′

where n

(γ, γ

′

, k) is a product of some combinatorial factors with a signed count

of disconnected index 0 holomorphic curves with connected components of genus

, . . . , g

satisfying g

+ . . . + g

− m = g − 1, and with positive ends at γ and

negative ends at γ

′

, where k is the number of positive ends.

290 Chris Wendl

Let’s discuss two applications of the cobordism map Φ. First, note that if W is a

trivial symplectic cobordism [0, 1] ×M, then the above discussion can easily be gen-

eralized with (A

, D

SFT

) both deﬁned over the same group ring R = Q[H

(M)/G]

for any choice of G ⊂ H

(M). There is no need to consider a Novikov ring in

deﬁning F here since the cobordism is exact. We therefore obtain a chain map

with arbitrary group ring coeﬃcients, and extending this discussion along standard

Floer-theoretic principles will imply that the chain map is an isomorphism: this can

be used in particular to prove that H

SFT

∗

(M, ξ; R) does not depend on the choices of

contact form and almost complex structure. There a r e two additional steps involved

in this ar gument: ﬁrst, one needs to use a chain homotopy to prove that Φ does

not depend on the choice o f almost complex structure J on

W . Given a g eneric

homotopy {J

}

s∈[0,1]

, the chain homotopy map

Ψ : A

[[~]] → A

−

[[~]]

is deﬁned as a diﬀerential operator in the same manner as Φ, but counting pairs

(s, u) where s ∈ [0, 1] is a parameter value for which J

is nongeneric and u is a

disconnected J

-holomorphic curve in

W with index −1. We saw how this works

for cylindrical contact homology in Lecture

10, but there is a new subtlety now

that should be mentioned: in principle, a d isconnected index −1 curve in

W could

have arbitrarily many components, including perhaps many with index −1 a nd oth-

ers with arbitrarily large index. Even worse, the compactiﬁed 1-dimensional space

of pairs (s, u) for J

-holomorphic curves u of index 0 may include buildings that

have symplectization levels of index greater than 1, balanced by disjoint unions of

many index −1 curves in the main level. This sounds hor r ible, but it can actu-

ally be ignored, for the following reason: ﬁrst, since there are only ﬁnitely many

pairs (s, u) where u is a connected J

-holomorphic curve with index −1, one can (if

transversality is achievable at all) use a genericity argument to assume without loss

of generality that for any given s ∈ [0, 1], at most one connected index −1 curve ex-

ists. This means that in any building that has multiple index −1 components, those

components are just multiple copies of the same curve. Now, since that curve has

odd index, it is represented by a monomial q

−

that contains an odd number of

odd generators, and any nontrivial product of such generators therefore disappears

in A since odd generators anticommute with themselves. This algebraic miracle

encodes a convenient fact about coherent orientations: whenever one of the horrible

buildings described ab ove appears, one can reorder two of the index −1 components

to produce from it a diﬀerent building t hat lives in a moduli space with the opposite

orientation. Gluing this building back together then produces a continuation of the

1-dimensional moduli space, so that the horrible building can actually be interpreted

as an “interior” point of the 1-dimensional space, rather than boundary. The actual

count of boundary points is then exactly what we want it to be: it is represented

algebraically by the chain homotopy relation!

Finally, compositions of cobordism maps can be understood via a stretching

argument that is not substantia lly diﬀerent f rom the case of cylindrical contact

homology. Since the trivial cobordism with R-invariant data gives a cobordism

map that just counts trivial cylinders and is therefore the identity, it follows that

Lectures on Symplectic Field Theory 291

cobordism maps relating diﬀerent pairs of dat a (α

, J

) are always invertible, and

this proves the invariance of H

SFT

∗

(M, ξ; R).

The second applicatio n concerns nontrivial exact cobordisms, and it is immediate

from the fact that Φ is a chain map:

Theorem 13.28. Any exact cobordism (W, dλ) from (M

−

, ξ

−

) to (M

, ξ

) gives

rise to a Q[[~]]-linear map

SFT

∗

, ξ

) → H

SFT

∗

−

, ξ

−



It is much more complicated to say what ha ppens in the event of a nonexact

cobordism, but slightly easier if we restrict o ur attention to ﬁllings, i.e. the case with

−

= ∅. Assume (W, ω) is a compact symplectic manif old with stable boundary M,

inheriting a stable Hamiltonian structure H = (Ω, α) for which α is a nondegenerate

contact form, and assume also that the completion

W admits an almost complex

structure J that is ω-ta me on W and has a pseudoconvex restriction J

∈ J(H) to

the cylindrical end. We saw in Proposition

13.22 that these conditions can always

be achieved for a weak ﬁlling after deforming the symplectic structure. Let

G := ker[ω] := {A ∈ H

(W ) | h[ω], Ai = 0},

and choose G

⊂ H

(M) to be any subgroup such t hat the map H

(M) → H

(W )

induced by t he inclusion M ֒→ W sends G

into G. In ot her words, G

can be any

subgroup of ker[Ω] ⊂ H

(M). Deﬁne the group rings

= Q[H

(M)/G

], R = Q[H

(W )/ ker[ω]],

with the Novikov completion of R denoted by

R. The map H

(M)/G

→ H

(W )/G

induced by M ֒→ W then gives a natural ring homomorphism

(13.16) R

→ R.

If ω is not exact, then it may no longer be true that every term in F has at least

one p varia ble. Let us write

F = F

+ F

where F

contains no p variables a nd F

= O(p), i.e. F

counts all closed curves

W , and F

counts everything else. Since F

and F

have even degree, they

commute, and thus

exp(F) = exp( F

) exp(F

where exp(F

) is an invertible element of R[[~, ~

−1

]] since exp ( −F

) exp(F

) = 1.

By the master equation,

exp(F

) exp(F

)H = O( q),

hence exp(F

)H = exp(−F

)O(q) = O(q) since exp( −F

) contains no p variables.

Using the substitution (

13.13), and using (13.16) to map coeﬃcients in R

to R, it

follows that exp(F

) gives rise to a diﬀerential operator

Φ : A[[~]] →

R[[~]] : f 7→ exp(F

)f|

q=0

292 Chris Wendl

which is a chain map to the SFT of the empty set with Novikov coeﬃcients, meaning

Φ ◦D

SFT

= 0.

This chain map counts the disconnected index 0 curves in

W whose connected

components all have at least one positive puncture.

Theorem 13.29. Suppose (W, ω) is a compact symplectic manifold with s ta-

ble boundary (M, H = (Ω, α)), whe re α i s a no ndegenerate co ntact form, and its

co mpletion

W admits an almost complex structure that is ω-tame on W and has

a generic and pseudoconv ex restriction J

∈ J(H) to the cylindrica l end. Let

denote the Novikov comp l e tion of Q[H

(W )/ ker[ω]], and let R

= Q[H

(M)/G

where G

⊂ H

(M) is any subgroup on which the evaluation of [Ω] ∈ H

(M)

vanishes. Then there exists an

R[[~]]-linear map H

SFT

∗

(M, H, J

; R

) → R[[~]]. 

13.3.2. Algebraic torsion. We can now generalize the notion of algebraic

overtwistedness. Notice that since every term in D

SFT

is a diﬀerential operator

of order at least 1,

SFT

f = 0 for all f ∈ R[[~]],

hence every element o f the extended coeﬃcient ring R[[~]] represents an element

of H

SFT

∗

(M, ξ; R) that may or may not be trivial. Since D

SFT

commutes with all

elements of R[[~]], t he subset consisting of elements that are t rivial in homology

forms an ideal. The following deﬁnition originates in [

LW11].

Definition 13.30. We say that a closed contact manifold (M, ξ) has algebraic

torsion of order k (or k-torsion for short) with coeﬃcients in R if

] = 0 ∈ H

SFT

∗

(M, ξ; R).

The numerical invariant

AT(M, ξ; R) ∈ N ∪{0, ∞}

is deﬁned to be the smallest integer k such that (M, ξ) has algebraic k-torsion but

no (k − 1)-torsion, or ∞ if there is no algebraic torsion of any order.

Several consequences of algebraic torsion can be read oﬀ quickly from the prop-

erties of SFT cobordism ma ps. Consider ﬁrst the case of trivial coeﬃcients R = Q,

which we shall refer to as untwisted algebraic torsion and abbreviate

AT(M, ξ) := AT(M, ξ; Q).

If (W, ω) is a strong ﬁlling of (M, ξ), then the hypotheses of Theorem

13.29 are

fulﬁlled even with G

= H

(M) since ω is exact at the boundary, thus we obtain

a Q[[~]]-linear map H

SFT

∗

(M, ξ) →

R[[~]], with R denoting the Novikov completion

of Q[H

(W )/ ker[ω]]. If [~

] = 0 ∈ H

SFT

∗

(M, ξ), then the cobordism map implies a

contradiction since ~

does not equal 0 in

R[[~]]. Similarly, if (W, dλ) is an exact

cobordism fro m (M

−

, ξ

−

) to (M

, ξ

), then the cobordism map H

SFT

∗

, ξ

) →

SFT

∗

−

, ξ

−

) of Theorem

13.28 is also Q[[~]]-linear, and thus any algebraic k-torsion

in ( M

, ξ

) is inherited by (M

−

, ξ

−

). This proves:

Lectures on Symplectic Field Theory 293

Theorem 13.31. Contact manifolds with AT(M, ξ) < ∞ are not strongly ﬁllabl e.

Moreover, if there exists an exac t symplectic cobordi sm from (M

−

, ξ

−

) to (M

, ξ

then AT(M

−

, ξ

−

) ≤ AT(M

, ξ

). 

It is known (see [Wen13]) that the second part of t he above theorem does not

hold for strong symplectic cobordisms in g eneral, so exactness of cobordisms is a

meaningful symplectic topological condition, not just a technical hypothesis. It is

also known thanks to a construction of Ghiggini [

Ghi05] that strong and exact

ﬁllability are not equivalent conditions, but Ghiggini’s proof of this uses Heegaard

Floer homology; thus far it is not known whether this phenomenon can be detected

via SF T or other holomorphic curve techniques.

There are also many known examples of contact manifolds that have untwisted

algebraic torsion but a r e weakly ﬁllable. The simplest are the tight tori (T

, ξ

)

for k ≥ 2, for which weak ﬁllings were ﬁrst constructed by Giroux [Gir94], but

Eliashberg [

Eli96] showed tha t strong ﬁllings do not exist, and we will see in Lec-

ture

16 that AT(T

, ξ

) = 1. The weak/strong distinction can often be detected via

the choice of coeﬃcients in SFT. We saw in §

13.2.1 that a weak ﬁlling of a contact

manifold (M, ξ) can always be deformed so as to have stable boundary with data

(H = (Ω, α), J

) for which α is a nondegenerate contact form a nd J

is C

∞

-close to

any given element of J(α). Pro position 13.16 showed that if (M, ξ) is algebraically

overtwisted, then the contact homology for the stable Hamiltonian data ( H, J

) can

also be made to vanish.

Exercise 13.32. Generalize the proof of Prop.

13.16 to show that if (M, ξ) ha s

algebraic k-torsion with coeﬃcients in R, then also [~

] = 0 ∈ H

SFT

∗

(M, H

, J

; R)

for suﬃciently large c > 0.

It then follows using Theorem

13.29 that a lgebraic torsion with suitably twisted

coeﬃcients a lso gives a n obstruction to weak ﬁlling . L et us say that (M, ξ) has fully

twisted algebraic k-torsion whenever [~

] = 0 ∈ H

SFT

∗

(M, ξ; Q[H

(M)]). Note that

in parallel with Remark 13.10, any nested pair of subgroups G ⊂ G

′

⊂ H

(M) gives

rise to a map

SFT

∗

(M, ξ; Q[H

(M)/G

′

]) → H

SFT

∗

(M, ξ; Q[H

(M)/G]),

which is a morphism in the sense that it ma ps the unit and all powers of ~ to

themselves. This implies that (M, ξ) has fully twisted k-torsion if and only if it has

k-to r sion for every choice of coeﬃcients.

Theorem 13.33. If (M, ξ) is a closed contact manifold with a ﬁnite o rder of

algebraic torsion w i th coeﬃcients in R = Q[H

(M)/G] for some subgroup G, then

(M, ξ) does not admit any weak symplectic ﬁlling (W, ω) for which [ω|

T M

] ∈ H

(M)

is rational and annihilates all elements of G. In particular, if (M, ξ) has fully twisted

algebraic torsion of some ﬁnite order, then it is not weakly ﬁllable.

Remark 13.34. The rationality condition in Theorem 13.33 can probably be

lifted, and is known to be unnecessary at least in dimension three. It is clear in any

case that if (M, ξ) admits a weak ﬁlling (W, ω), then one can always make a small

perturbation of ω to produce a weak ﬁlling for which [ω|

T M

] ∈ H

(M; Q).

294 Chris Wendl

We will see some concrete examples of algebraic torsion computations in Lec-

ture

16. Let us conclude this discussion for now with the observation that algebraic

torsion of order zero is a notion we’ve seen before:

Proposition 13.35. For any closed contact ma nifold (M, ξ) and group ring

R = Q[H

(M)/G], the follow i ng conditions are equivalent:

(1) (M, ξ) has algebraic 0-torsion (with coeﬃcients in R);

(2) (M, ξ) is al gebraically overtwisted (with coeﬃcients in R);

(3) H

SFT

∗

(M, ξ; R) = 0.

Proof. It is obvious that (

3) implies (1). Since D

SFT

f = ∂

f +O(~) for f ∈ A,

the R[[~]]-linear map

A[[~]] → A : F 7→ F|

~=0

deﬁnes a chain map (A[[~]], D) → (A, ∂

) and thus descends to an R[[~]]-linear

map H

SFT

∗

(M, ξ; R) → HC

∗

(M, ξ; R). The existence of this map proves that (

implies (

2).

To prove that (

2) implies (3), recall ﬁrst that if there exists f ∈ A with ∂

f =

1, then the fact that HC

∗

(M, ξ; R) = 0 follows easily since for any g ∈ A with

∂

g = 0, the graded Leibniz rule implies ∂

(fg) = (∂

f)g − f(∂

g) = g. This

works because ∂

is a derivation—but D

SFT

is not one, so the same trick will not

quite work for D

SFT

. The trick in proving H

SFT

∗

(M, ξ; R) = 0 will be to quantify

the failure o f D

SFT

to be a derivation. For our purposes, it suﬃces to know that

(13.17) D

SFT

(FG) = (D

SFT

F)G + (−1)

|F|

F(D

SFT

G) + O(~)

holds for all F, G ∈ A[[~]], which follows from the fact that ∂

is a derivation.

With this remark out of the way, suppose f ∈ A satisﬁes ∂

f = 1, in which case

(13.18) D

SFT

f = 1 + ~G

for some G ∈ A[[~]]. We claim then that for any Q ∈ A[[~]] with D

SFT

Q = 0, there

exists Q

∈ A[[~]] with

(13.19) D

SFT

(fQ) = Q + ~Q

and D

SFT

= 0. Indeed, (

13.19) follows from (13 .17) and (13.18) since D

SFT

Q = 0,

and D

SFT

= 0 then follows by applying D

SFT

to (

13.19) and using D

SFT

= 0.

Fixing Q

:= Q ∈ A[[~]], we can now deﬁne a sequence Q

∈ A[[~]] satisfying

SFT

= 0 for all integers k ≥ 0 via the inductive condition

SFT

(fQ

) = Q

+ ~Q

k+1

Then

∞

k=0

(−1)

∈ A[[~]], and

SFT

∞

k=0

(−1)

= Q.



LECTURE 14

Transversality and embedding controls in dimension four

The ﬁnal three lectures will be included in the published version of this book.

For updates on publication, see the author’s website

https://www.mathematik.hu-berlin.de/

wendl/publications.html#notes

295

LECTURE 15

Intersection theory for punctured holomorphic curves

The ﬁnal three lectures will be included in the published version of this book.

For updates on publication, see the author’s website

https://www.mathematik.hu-berlin.de/

wendl/publications.html#notes

297

LECTURE 16

Torsion computations and applications

The ﬁnal three lectures will be included in the published version of this book.

For updates on publication, see the author’s website

https://www.mathematik.hu-berlin.de/

wendl/publications.html#notes

299

APPENDIX A

Sobolev spaces

Contents

A.1. Approximation, extension and embedding theorems 301

A.2. Products, compositions, and rescaling 305

A.3. Spaces of sections of vector bundles 311

A.4. Some remarks on domains with cylindrical e nds 316

In this appendix, we review some of the standard properties of Sobolev spaces,

in particular using them to prove Propositions

2.7, 2.8 and 2.10 from §2 .2, and

elucidating the construction of Sobolev spaces of sections on vector bundles. A

good reference for the necessary background material is [

AF03].

A.1. Approximation, extension and embedding theorems

Unless otherwise noted, all functions in the following are a ssumed to be deﬁned

on a nonempty open subset

U ⊂ R

with its standard Lebesgue measure, and taking values in a ﬁnite-dimensional normed

vector space that will usually not need to be speciﬁed, though occasionally we will

assume it is R o r C so that one can deﬁne products of functions. The domain U

will also sometimes have additional conditions speciﬁed such as b oundedness or reg-

ularity at the boundary, tho ugh we will try not to add too many more restrictions

than are really needed. The most useful assumption t o impose on U is known as

the strong local Lipschitz condition: if U is bounded, then it means simply that

near every boundary point of U, one can ﬁnd smooth local coordinates in which

U looks like the region bounded by the graph of a Lipschitz-continuous function,

and in this case we call U a bounded Lipschitz domain. If U is unbounded,

then one needs to impose extra conditions guaranteeing e.g. uniformity of Lipschitz

constants, and the precise deﬁnition becomes a bit lengthy (see [

AF03, §4.9]). For

our purposes, all we really need to know about the strong local Lipschitz condition

is that that it is satisﬁed bo th by bo unded Lipschitz domains and by relatively

tame unbounded domains such as (0, 1) ×(0, ∞) ⊂ R

which have smooth boundary

with ﬁnitely many corners. We will repeatedly need to use the generalized version

of H¨older’s inequality, which states that for any ﬁnite collection of measurable

301

302 Chris Wendl

functions f

, . . . , f

(A.1)



i=1



≤

i=1

for 1 ≤ p ≤ p

, . . . , p

≤ ∞ with

i=1

This is an easy corollary of t he standard version,



|f|· |g|



≤ kfk

·kgk

whenever 1 ≤ p, q ≤ ∞ and 1 =

For an integer k ≥ 0 and real number p ∈ [1 , ∞] we deﬁne W

k,p

(U) as in §

2.2 to be

the Ba nach space of all f ∈ L

(U) which have weak partial derivatives ∂

f ∈ L

(U)

for all |α| ≤ k. For p = 2, these spaces are also often denoted by

(U) := W

k,2

(U),

and they admit Hilbert space structures with inner product

hf, gi

|β|≤k

h∂

f, ∂

We denote by

k,p

(U) ⊂ W

k,p

(U), H

(U) ⊂ H

(U)

the closed subspaces deﬁned as the closures of C

∞

(U) with respect to the relevant

norms. Since C

∞

(U) is dense in L

(U) for 1 ≤ p < ∞ (see e.g. [

LL01, §2.19]),

there is no diﬀerence between W

0,p

(U) and W

0,p

(U) for p < ∞, but in general

k,p

(U) 6= W

k,p

(U) for k ≥ 1, with a few not able exceptions such as the case

U = R

(cf. Corollary

A.2 below). Let

k,p

lo c

(U) :=



functions f on U



f ∈ W

k,p

(V) for all open subsets V ⊂ U

with compact closure

V ⊂ U



and we say that a sequence f

∈ W

k,p

lo c

(U) converges in W

k,p

lo c

to f ∈ W

k,p

lo c

(U) if

the restrictions to all precompact open subsets V ⊂

V ⊂ U converge in W

k,p

(V).

Recall that for k ∈ {0, 1, 2, . . . , ∞}, C

(U) denotes the space of functions on U with

continuous derivatives up to o rder k, while

(

U) ⊂ C

(U)

is the space of f ∈ C

(U) such that for all |α| ≤ k, ∂

f is bounded and uniformly

continuous.

Theorem A.1 ([

AF03, §3.17, 3.22]). For any open subset U ⊂ R

, and any

k ≥ 0, 1 ≤ p < ∞, the subspace

∞

(U) ∩ W

k,p

(U) ⊂ W

k,p

(U)

is dense. Moreover, if U ⊂ R

satisﬁes the strong local Lipschitz condition, then the

space

f ∈ C

∞

(U)



f =

for some

f ∈ C

∞

)

is also d ense in W

k,p

(U), so in particular,

∞

(

U) ∩ W

k,p

(U) ⊂ W

k,p

(U)

Lectures on Symplectic Field Theory 303

is dense. 

Corollary A.2. The space C

∞

) is dense in W

k,p

) for every k ≥ 0 and

p ∈ [1, ∞). 

Here is another useful char acterization of W

k,p

(U):

Theorem A.3 ([

AF03, §5.29]). Assume U ⊂ R

is an open subset satisfying the

strong local Lipschitz condition. Th en a function f ∈ W

k,p

(U) belongs to W

k,p

(U)

if and only if the function

f on R

deﬁned to match f on U and 0 everywhere else

be l ongs to W

k,p

). 

While it is obvious from the deﬁnitions that functions in W

k,p

(U) always admit

extensions of class W

k,p

over R

, this is much less obvious for functions in W

k,p

(U)

in general, and it is not true without suﬃcient assumptions about the regularity

of ∂U. For our purposes it suﬃces to consider the following case.

Theorem A.4 ( [

AF03, §5.22]). Assume U ⊂ R

is a bounded open subset such

that ∂

U is a submanifold of class C

for some m ∈ {1, 2, 3, . . . , ∞}. Then there

exists a linear o perator E that maps functions deﬁn ed almost everywh e re on U to

functions deﬁned a l most everywhere on R

and has the follow i ng properties:

• For every function f on U, Ef|

≡ f almost everywhe re;

• For every nonn egative integer k ≤ m and every p ∈ [1, ∞), E deﬁne s a

bo und ed linear operator W

k,p

(U) → W

k,p



Corollary A.5. Suppose U, U

′

⊂ R

are open subsets such that U has com pact

closure contained in U

′

. If U satisﬁes the hypo thes i s of Theorem

A.4, then the

resulting extension operator E can be chosen such that it maps each W

k,p

(U) for

k ≤ m and 1 ≤ p < ∞ into W

k,p

′

Proof. Choose a smooth function ρ : U

′

→ [0 , 1] that has compact support and

equals 1 on

U, then replace the o perator E given by Theorem A.4 with the operat or

f 7→ ρ · Ef . 

To state the Sobolev embedding theorem in its pro per generality, recall tha t for

0 < α ≤ 1, the H¨older seminorm of a f unction f o n U is deﬁned by

|f|

:= |f|

(U)

:= sup

x6=y∈U

|f(x) − f (y)|

|x − y|

and C

k,α

(U) is then deﬁned as the Banach space of functions f ∈ C

(

U) for which

the nor m

kfk

k,α := kfk

k + max

|β|=k

|∂

is ﬁnite. In reading the following statement, it is important to remember that

elements of W

k,p

(U) are technically not f unctions, but r ather equivalence classes

of functions deﬁned almost everywhere. Thus when we say e.g. that there is an

inclusion W

k,p

(U) ֒→ C

m,α

(U), the literal meaning is that for every function f

representing an element of W

k,p

(U), one can change the values of f in a unique way

304 Chris Wendl

on some set of measure zero in U so that after this change, f ∈ C

m,α

(U). Continuity

of the inclusion means tha t there is a bound of the form

kfk

m,α

≤ ckfk

k,p

for all f ∈ W

k,p

(U), where c > 0 is a constant which may in general depend on m,

α, k, p and U, but not on f .

Theorem A.6 ([

AF03, §4.12]). Assume U ⊂ R

is an open subset satisfying

the strong local Lipschitz co ndition, k ≥ 1 is an integer and 1 ≤ p < ∞.

(1) If kp > n and k − n/p < 1, then there exist continuous inclusions

k,p

(U) ֒→ C

0,α

(U) for each α ∈ (0, k − n/p],

k,p

(U) ֒→ L

(U) for each q ∈ [p, ∞].

(2) If kp < n and p

∗

> p is deﬁned by the condition

∗

−

then there ex ist continuous inclusions

k,p

(U) ֒→ L

(U), for each q ∈ [p, p

∗

(3) If kp = n, then there exist continuous inclusions

k,p

(U) ֒→ L

(U), for each q ∈ [p, ∞).

Moreover, the spaces W

k,p

(U) admit similar inclusion s under no assumption on the

open subset U ⊂ R

. 

Under the same assumption on the domain U, one can apply Theorem

A.6 to suc-

cessive derivatives of functions in W

k,p

(U) and thus o bt ain the f ollowing inclusions

for any integer d ≥ 0:

(A.2) W

k+d,p

(U) ֒→ C

d,α

(U) if kp > n and 0 < α ≤ k − n/p < 1,

(A.3) W

k+d,p

(U) ֒→ W

d,q

(U) if kp > n and p ≤ q ≤ ∞,

(A.4) W

k+d,p

(U) ֒→ W

d,q

(U) if kp < n and p ≤ q ≤ p

∗

, with

∗

−

(A.5) W

k+d,p

(U) ֒→ W

d,q

(U) if kp = n and p ≤ q < ∞.

This last inclusion can then be composed with (

A.2) for an arbitrarily large choice

of q, giving another inclusion

(A.6) W

k+d,p

(U) ֒→ C

d−1,α

(U) if kp = n and 0 < α < 1.

Remark A.7. The embedding theorem suggests that one should intuitively think

of W

k,p

(U) as consisting of functions with “k − n/p continuous derivatives,” where

the number k −n/p may in general be a non-integer and/or negative. This provides

a useful mnemonic for results about embeddings of one Sobolev space into another,

such as the following.

Lectures on Symplectic Field Theory 305

Corollary A.8. Assume U ⊂ R

is an open subset satisfying the strong local

Lipschitz condition, 1 ≤ p, q < ∞, and k, m ≥ 0 are integers satisfying

k ≥ m, p ≤ q, and k −

≥ m −

Then there exists a continuous incl usion W

k,p

(U) ֒→ W

m,q

(U). 

By the Arzel`a-Ascoli theorem, the natural inclusion

k,α

′

(U) ֒→ C

k,α

(U)

for α < α

′

is a compact operator whenever U ⊂ R

is bounded. It follows that if U ⊂

in (

A.2) is bounded and α is strictly less than the extremal value k − n/p, then

the inclusion (

A.2) is also compact. A similar statement holds for the inclusion (A.4)

when p ≤ q < p

∗

, and this is known as the Rellich-Kondrachov compactness

theorem. We summarize these as f ollows:

Theorem A.9 ([

AF03, §6.3]). Assume U ⊂ R

is a bounded Lipschitz d omain,

k ≥ 1 and d ≥ 0 are integers and 1 ≤ p < ∞.

(1) If kp > n and k − n/p < 1, then the inclusions

k+d,p

(U) ֒→ C

d,α

(U) for α ∈ ( 0, k − n/p),

k+d,p

(U) ֒→ W

d,q

(U) for q ∈ [p, ∞)

are compact.

(2) If kp ≤ n and p

∗

∈ (p, ∞] is deﬁned by the condition 1/p

∗

= 1/p − k/n,

then the inclusions

k+d,p

(U) ֒→ W

d,q

(U) for q ∈ [p, p

∗

)

are compact.

In particular, the continuous i nclusion W

k,p

(U) ֒→ W

m,q

(U) in Corollary

A.8 is

co mpact whenever the inequality k − n/p ≥ m − n/q is strict. 

A.2. Products, compositions, and rescaling

We now restate and prove Propositions

2.7, 2.8 and 2 .10 from §2.2. These are

all corollaries of the Sobolev embedding theorem, so in particular they hold for the

same class of domains U ⊂ R

, and the restrictions on U can be dropped at the cost

of replacing each space W

k,p

by W

k,p

We begin by generalizing Prop.

2.7, hence we consider Sobolev spaces of functions

valued in R or C so that po intwise products of functions are well deﬁned almost

everywhere. We say that there is a continuous product map,

(U) × . . . × W

(U) → W

k,p

(U),

or a continuous product pairing in the case m = 2, if for every set of functions

∈ W

(U) with i = 1, . . . , m, the pointwise product function f

· . . . · f

is in

k,p

(U) and there is an estimate of the form

· . . . · f

k,p ≤ ckf

· . . . · kf

306 Chris Wendl

for some constant c > 0 not dep ending on f

, . . . , f

. The case m = 2, k

= k

and p

= p

= p is especially interesting, as the space W

k,p

(U) is then a Banach

algebra. More generally, one can ask under what circumstances multiplication by

functions of class W

k,p

deﬁnes a bounded linear operator on functions o f class W

m,q

A hint about this comes from the world of classically diﬀerentiable functions: mul-

tiplication by C

-smooth functions deﬁnes a continuous ma p C

→ C

if and only

if k ≥ m. The corresponding answer in Sobolev spaces turns out to be that func-

tions of class W

k,p

need to have strictly more than zero derivatives in the sense of

Remark

A.7, and at least as many derivatives a s functions of class W

m,q

Theorem A.10. Assume U ⊂ R

is an ope n subset satisfying the strong local

Lipschitz condition, k, p, m and q sa tisfy the same numerical hypotheses as in

Corollary

A.8 (so in particular W

k,p

(U) embeds continuously into W

m,q

(U)), and

kp > n. Then there exists a continuous prod uct pairing

k,p

(U, C) × W

m,q

(U, C) → W

m,q

(U, C) : (f, g) 7→ f g.

The following preparatory lemma will be useful both for proving the product

estimate and for f urther results below. It is an easy consequence of Theorem A.6

and H¨older’s inequality.

Lemma A.11. Assume U ⊂ R

is an open subset satisfying the strong lo-

ca l Lipschitz condition, m ≥ 2 is an integer, and we are given positive numbers

, . . . , p

≥ 1 an d integers k

, . . . , k

≥ 0. Let I :=



i ∈ {1, . . . , m}



≤ n



Then for any q ≥ 1 s a tisfying

i∈I



−



≤

i=1

there is a continuous product map

(U) × . . . × W

(U) → L

(U).

Proof. By the generalized H¨older inequality (

A.1), it suﬃces to show that for

any q ≥ 1 in the stated ra nge, one can ﬁnd numbers q

, . . . , q

∈ [q, ∞] satisfying

1/q = 1/q

+ . . . + 1/q

for which Theorem

A.6 provides continuous inclusions

(U) ֒→ L

(U)

for each i = 1 , . . . , m. Whenever k

> n, this inclusion is valid with q

chosen freely

from the interval [p

, ∞], so 1/q

can then take any value subject to the constraint

0 ≤

≤

If on the other hand k

≤ n, then we can arrange 1/q

to take any value in the

range

−

≤

Adding these up, the range of values for

that we can achieve in this way covers

the stated interval. 

Lectures on Symplectic Field Theory 307

Proof of Theorem A.10. By density o f smooth functions, it suﬃces to prove

that an estimate of the form

kfgk

m,q

≤ ckfk

k,p

kgk

m,q

holds for all f ∈ C

∞

(U) ∩ W

k,p

(U) and g ∈ C

∞

(U) ∩ W

m,q

(U). Equivalently, we

need to show that for all f and g of this type and every multiindex α of degree

|α| ≤ m, there is a constant c > 0 independent of f and g such that

k∂

(fg)k

≤ ckfk

k,p

kgk

m,q

Since f and g are smooth, we are free to use the product rule in computing ∂

(fg),

which will then be a linear combination of terms of the form ∂

f · ∂

g where |α| =

|β| + |γ|, hence we have reduced the problem to proving a bound

k∂

f · ∂

≤ ckfk

k,p kgk

m,q

for every pair of multiindices β, γ with |β| + |γ| ≤ m. Since ∂

f ∈ W

k−|β|,p

(U) and

∂

f ∈ W

m−|γ| ,q

(U), the result follows if we can assume that for every pair of integers

a, b ≥ 0 satisfying a + b ≤ m, there exists a continuous product pairing

(A.7) W

k−a,p

(U) × W

m−b,q

(U) → L

(U).

If (k −a)p > n, then W

k−a,p

֒→ L

∞

and (

A.7) is immediate since W

m−b,q

֒→ L

(U).

For the remaining cases, we shall apply Lemma

A.11, noting that the condition

1/q ≤ 1/p + 1/q is trivially satisﬁed.

If (m −b)q > n but (k −a)p ≤ n, then the hypotheses of the lemma a re satisﬁed

if and only if

−

k − a

Since

−

≤

−

by assumption, we have

−

k − a

−

≤

−

≤

since a ≤ m, a nd equality holds only if a = m, b = 0 and k − n/p = m − n/q,

which implies mq > n. In this case W

m−b,q

= W

m,q

֒→ L

∞

, and the pairing (

A.7)

follows because W

k−a,p

= W

k−m,p

embeds continuously into L

: the latter follows

from Theorem

A.6 since

−

k−m

Finally, when (k − a)p ≤ n a nd (m − b)q ≤ n, the hypotheses of the lemma are

satisﬁed since



−

k − a





−

m −b



≤

−



−



where we’ve used the assumption kp > n and the fact that a + b ≤ m. 

The next result generalizes Proposition

2.8 and concerns the following question:

if f : U → R

is a f unction o f class W

k,p

whose graph lies in some open subset

V ⊂ U × R

, and Ψ : V → R

is another function, under what conditions can we

conclude t hat the function

U → R

: x 7→ Ψ(x, f(x))

308 Chris Wendl

is in W

k,p

(U, R

)? We will abbreviate this f unction in t he following by Ψ ◦(Id ×f),

and we would also like t o know whether it depends continuously (in the W

k,p

topology) on f and Ψ. The following t heorem is stated rather generally, but on ﬁrst

reading you may prefer to a ssume U ⊂ R

is bounded, in which case some of t he

hypot heses become vacuous. We will say that an open subset V ⊂ U × R

is a

star-shaped neighborhood of f : U → R

if it contains the graph of f

and

(x, v) ∈ V ⇒ (x, tv + (1 − t)f

(x)) ∈ V for all t ∈ [0, 1].

Theorem A.12. Assume U ⊂ R

is an ope n subset satisfying the strong local

Lipschitz condition, p ∈ [1, ∞) and k ∈ N satisfy kp > n, and V ⊂ U × R

is a star-sha ped neighborhood of some function f

∈ W

k,p

(U, R

). Assume also

k,p

(U; V) ⊂ W

k,p

(U, R

) is an open neighborhood of f

such that

(x, f(x)) ∈ V for all x ∈ U and f ∈ O

k,p

(U; V),

and O

(

V, R

) ⊂ C

(V, R

) is a subset such that a ll Ψ ∈ O

(V, R

) have the

following properties:

(1) There exists a bounded subset K ⊂ U such that Ψ(x, v) is independent of x

for all x ∈ U \ K;

(2) Ψ ◦ (Id × f

) ∈ L

(U, R

Then there is a well-de ﬁned and continuous map

(V, R

) × O

k,p

(U; V) → W

k,p

(U, R

) : (Ψ , f ) 7→ Ψ ◦ (Id × f ).

Proof. We will show ﬁrst that if f ∈ O

k,p

(U; V) is smooth, then Ψ ◦ (Id ×

f) belongs to W

k,p

(U, R

) for every Ψ ∈ O

(

V, R

). Since V is a star-shaped

neighborhood o f f

, we have

|Ψ(x, f( x))−Ψ(x, f

(x))| =





x, tf(x) + (1 − t)f

(x)





≤





x, tf(x) + (1 − t)f

(x)



|dt



· |f(x) − f

(x)|

≤ kΨk

(V)

· |f(x) − f

(x)|

for all x ∈ U, implying

kΨ ◦ (Id × f) − Ψ ◦ (Id × f

≤ kΨk

(V)

· kf − f

hence Ψ ◦ (Id × f) ∈ L

(U, R

For ℓ = 1, . . . , k, we can regard the ℓth derivative of Ψ with respect to variables

in R

as a bounded and uniformly continuous map from V into the vector space of

symmetric ℓ-multilinear maps from R

to R

, denoting this by

ℓ

Ψ : V → Hom((R

)

⊗ℓ

, R

Denote the partial derivatives with respect to variables in U ⊂ R

Ψ : V → R

Both of the conditions on Ψ ∈ O

(

V, R

) are vacuous if U ⊂ R

is bounded.

Lectures on Symplectic Field Theory 309

where β is a multiindex in n variables. Now for any multiindex α with |α| ≤ k, the

derivative ∂

(Ψ ◦(Id ×f)) is a linear combination of product functions of the form

(A.8) (D

ℓ

Ψ ◦ (Id × f))(∂

f, . . . , ∂

ℓ

f) : U → R

where ℓ + |γ| ∈ {1, . . . , |α|} a nd |β

| + . . . + |β

ℓ

| = |α| − |γ|. If ℓ = 0 but |γ| > 0,

then this expression is clearly in L

(U, R

) since it is continuous and D

Ψ(x, v) = 0

for x ∈ U \ K, where K is bounded. For ℓ ≥ 1, it satisﬁes



ℓ

Ψ ◦ (Id × f))(∂

f, . . . , ∂

ℓ



(U)

≤ kD

ℓ

Ψk

(V)



ℓ

j=1

|∂



(U)

if the product on the r ight hand side has ﬁnite L

-norm. The latter is trivially

true if ℓ = 1. To deal with the ℓ ≥ 2 case, note that ∂

f ∈ W

k−|β

|,p

(U) for each

j = 1, . . . , ℓ, so the necessary bound will follow from the existence of a continuous

product map

k−m

(U) × . . . × W

k−m

ℓ

(U) → L

(U)

for m

:= |β

|, and we claim that such a product map does exist whenever kp > n

and m

, . . . , m

ℓ

≥ 0 are integers satisfying m

+ . . . + m

ℓ

≤ k. To see this, note

ﬁrst tha t since W

k−m

֒→ L

∞

whenever (k − m

)p > n, it suﬃces to prove the

claim under the assumption that (k − m

)p ≤ n for every j = 1, . . . , ℓ. In this case,

Lemma

A.11 provides the desired product map if the condition

ℓ

j=1



−

k − m



≤

ℓ

j=1

is satisﬁed. And it is: using kp > n, ℓ ≥ 2 and m

+ . . . + m

ℓ

≤ k, we ﬁnd

ℓ

j=1



−

k − m



= ℓ



−



+ . . . + m

ℓ

≤

+ (ℓ − 1)



−



This pr oves that Ψ ◦ (Id × f) ∈ W

k,p

(U, R

Next, suppose f ∈ O

k,p

(U; V) is not necessarily smooth but f

∈ O

k,p

(U; V) is

a sequence of smooth functions converging to f in W

k,p

, while Ψ

∈ O

(

V, R

)

converges to Ψ ∈ O

(

V, R

) in C

. Then the same a rgument we used to estimate

kΨ ◦ (Id ×f) − Ψ ◦ (Id × f

shows that Ψ ◦ (Id × f

) → Ψ ◦(Id ×f) in L

, and

since f

is also C

-convergent, the compactly supported functions D

Ψ ◦ (Id × f

)

converge to D

Ψ ◦ (Id × f) in L

for each multiindex with 1 ≤ |γ| ≤ k. For ℓ ≥ 1

and |γ| + ℓ ≤ k, D

ℓ

◦ (Id × f

) converges to D

ℓ

Ψ ◦ (Id × f) in C

(

U, R

and each of the derivatives ∂

appearing in (

A.8) also converges in L

(U). In

light of the continuous product maps discussed above, it follows that each derivative

∂

(Ψ

◦ (Id × f

)) for |α| ≤ k is L

-convergent, and its limit is necessarily the

corresponding weak derivative ∂

(Ψ ◦ ( Id × f)), hence (see Exercise A.13 below)

Ψ ◦ (Id × f) ∈ W

k,p

(U, R

) and Ψ

◦ (Id ×f

)

k,p

−→ Ψ ◦ (Id × f). 

310 Chris Wendl

Exercise A.13. Show that if f

is a sequence of smooth functions on an open set

U ⊂ R

with f

→ f a nd ∂

→ g for some multiindex α and functions f, g ∈ L

(U),

then ∂

f = g in the sense of distributions.

The following result on coordinate transformations of the domain can b e proved

in an analogous way to Theorem

A.12, though it is considerably easier since there is

no need to worry about Sobolev product maps (and thus no need to assume kp > n

or impose regularity conditions on the domain).

Theorem A.14 ([

AF03, §3.41]) . Assume k ∈ N, 1 ≤ p ≤ ∞, and U, U

′

⊂

are open subsets with a C

-smooth diﬀeomorphism ϕ : U → U

′

such that all

derivatives of ϕ an d ϕ

−1

up to o rder k are bo und ed and uniformly continuous. Then

there is a well-deﬁned Banach s pace isomorphism

k,p

′

) → W

k,p

(U) : f 7→ f ◦ ϕ.



We now r estate and prove Proposition 2.10. We denote by

⊂ R

the open

balls of radius 1 and ǫ respectively about the origin.

Theorem A.15. Assume p ∈ [1, ∞) and k ∈ N satisfy kp > n, and for each

f ∈ W

k,p

(

) and ǫ ∈ (0, 1], deﬁne f

∈ W

k,p

(

) by

(x) := f(ǫx).

Then there exist constants C > 0 and r > 0 such that for eve ry f ∈ W

k,p

(

− f(0) k

k,p

(

)

≤ Cǫ

kf − f (0)k

k,p

(

)

for all ǫ ∈ (0, 1].

Proof. Let β denote a multiindex of order |β| = k. Then using a change of

varia bles, we have

k∂

− f(0) ) k

(

)

= ǫ

|∂

f(ǫx)|

= ǫ

kp−n

|∂

f(x)|

≤ ǫ

kp−n

k∂

(

)

≤ ǫ

kp−n

kf − f(0)k

k,p

(

)

and ǫ

kp−n

→ 0 as ǫ → 0 since kp − n > 0.

Next, suppose |β| = m ∈ {1, . . . , k −1}. Then ∂

f and ∂

are in W

k−m,p

(

and if (k − m)p < n, Theorem

A.6 gives a continuous inclusion

(A.9) W

k−m,p

(

) ֒→ L

(

)

with q > p satisfying 1/q + (k −m)/n = 1/p. Likewise, if (k −m)p ≥ n, then (

A.9)

is a continuous inclusion for arbitrarily large choices of q ≥ p. We will therefore

assume in general that (

A.9) holds with q ∈ (p, ∞) satisfying

Lectures on Symplectic Field Theory 311

where r =

k−m

if (k − m)p < n and otherwise r = p + δ for some δ > 0 which may

be chosen arbitrarily small. Given this, we use H¨older’s inequality and ﬁnd

k∂

− f(0) ) k

(

)

= ǫ

|∂

f(ǫx)|

= ǫ

mp−n

|∂

f(x)|

≤ ǫ

mp−n

k∂

(

)

k1k

(

)

≤ ǫ

mp−n

[Vol(D

)]

p/r

k∂

(

)

≤ cǫ

mp−n

[Vol(D

)]

p/r

k∂

k−m,p

(

)

≤ cǫ

mp−n

[Vol(D

)]

p/r

kf − f(0)k

k,p

(

)

for some constant c > 0. Writing Vol(D

) = Cǫ

for a suitable constant C > 0, the

exp onent on ǫ in this expression becomes

mp − n +

which is positive whenever r = p + δ with δ > 0 suﬃciently small since m ≥ 1, and

in t he case r = n/(k − m), it becomes simply kp − n > 0.

Finally, to bound the L

-norm of f

−f (0) itself, we can use the fact that f ∈ W

k,p

is H¨older continuous, i.e. it satisﬁes

|f(x) − f (0)| ≤ ckf − f (0)k

k,p

(

)

|x|

for all x ∈

for suitable constants c > 0 and α ∈ (0, 1). Thus

− f(0) k

(

)

|f(ǫx) − f(0)|

≤ c

kf − f(0)k

k,p

|ǫx|

αp

= c

kf − f(0)k

k,p

αp

|x|

αp

= ǫ

αp

Vol(S

n−1

)

αp + n

kf − f (0)k

k,p



A.3. Spaces of sections of vector bundles

In this section, ﬁx a ﬁeld

F := R or C,

assume M is a smooth n-dimensional manifold, possibly with boundary, and π :

E → M is a smooth vector bundle o f r ank m over F. This comes with a “bundle

atlas” A(π), a set whose elements α ∈ A(π) each consist of the following data:

(1) An open subset U

⊂ M;

(2) A smooth local coordinate chart ϕ

: U

∼

−→ Ω

, where Ω

is an open

subset of R

:= {(x

, . . . , x

) ∈ R

| x

≥ 0};

(3) A smooth local trivialization Φ

: E|

∼

−→ U

×F

312 Chris Wendl

Smoothness of ϕ

and Φ

means as usual that for every pair α, β ∈ A(π), the

coordinate transformations

βα

:= ϕ

−1

◦ ϕ

: Ω

αβ

∼

−→ Ω

βα

, Ω

αβ

:= ϕ

∩ U

)

and transition maps

βα

: U

∩ U

→ GL(m, F) such that Φ

◦ Φ

−1

(x, v) = (x, g

βα

(x)v)

for x ∈ U

∩ U

, v ∈ F

are smooth, and we shall assume the bundle atlas is maximal in the sense that

any triple (U, ϕ, Φ) that is smoothly compatible with every α ∈ A(π) also belongs

to A( π).

Any α ∈ A(π) now associates to sections η : M → E their local coordinate

representatives

:= pr

◦Φ

◦ η ◦ ϕ

−1

: Ω

→ F

where pr

: U

× F

→ F

is the projection, and the representa tives with respect

to two distinct α, β ∈ A(π) are related by

= (g

βα

◦ ϕ

−1

)(η

◦ ϕ

αβ

) on Ω

βα

⊂ Ω

For p ∈ [1, ∞] and each integer k ≥ 0, we t hen deﬁne the topological vector space

of sections of class W

k,p

lo c

k,p

lo c

(E) :=



η : M → E



sections such that η

∈ W

k,p

lo c

(

Ω

, F

)

for all α ∈ A(π)



where convergence η

→ η in W

k,p

lo c

(E) means tha t η

→ η

in W

k,p

lo c

(

Ω

, F

) for all

α ∈ A(π). Note that Ω

is no t necessarily an open subset of R

since it may cont ain

points in ∂R

= R

n−1

× {0}, but its interior

Ω

is open in R

, and W

k,p

lo c

(

Ω

) is

thus deﬁned as in §

A.1. Strictly speaking, element s of η ∈ W

k,p

lo c

(E) are not sections

but equivalence classes of sections deﬁned almost everywhere—the latter notion is

deﬁned with respect to any measure a r ising from a smooth volume element on M,

and it does not depend on this choice.

It t ur ns out that W

k,p

lo c

(E) can be given the structure of a Banach space if M is

compact. This follows from the fact that M can then be covered by a ﬁnite subset of

the atlas A(π), but we must be a little bit careful: not all charts in A(π) are equally

suitable for deﬁning W

k,p

-norms on sections, because e.g. even a nice smooth section

η ∈ Γ(E) may have kη

k,p

(

Ω

)

= ∞ if Ω

⊂ R

is unbounded. One way to deal

with this is as follows: we will say that α ∈ A(π) is a precompact chart if there

exists α

′

∈ A(π) and a compact subset K ⊂ M such that

⊂ K ⊂ U

′

When this is the case, Ω

⊂ R

is necessarily bounded, and the transition maps

between two precompact charts necessarily have bounded derivatives of all orders,

as they are restrictions to precompact subsets of maps that are smooth on larger

domains. If M is compact, then one can always ﬁnd a ﬁnite subset I ⊂ A(π)

consisting of precompact charts such that M =

α∈I

Lectures on Symplectic Field Theory 313

Definition A.16. Suppose E → M is a smooth vector bundle over a compact

manifold M, and I ⊂ A( π) is a ﬁnite set of precompact chart s such that {U

}

α∈I

is an open cover of M. We then deﬁne W

k,p

(E) as the vector space of all sections

η : M → E for which the norm

kηk

k,p := kηk

k,p

(E)

α∈I

kη

k,p

(

Ω

)

is ﬁnite.

The norm in the above deﬁnition depends on auxiliary choices, but it is easy to

see that the resulting deﬁnition of the space W

k,p

(E) and its topology do not. In

fact:

Proposition A.17. If M is compact, then W

k,p

(E) = W

k,p

lo c

(E), and a sequence

co nverges to η in W

k,p

lo c

(E) if and only if the norm given in Deﬁnition

A.16 satisﬁes

kη

−ηk

k,p

(E)

→ 0.

The proposition is an immediate consequence of the following.

Lemma A.18. Suppose M is a sm ooth manifold, π : E → M is a smooth vector

bundle, {β} ∪ J ⊂ A(π) is a ﬁnite collection of charts such that M =

α∈J

and

all coord i nate transformations and tran sition maps relating any two charts in the

co llection {β} ∪ J have bounded derivatives of all ord ers (e.g. it suﬃces to assume

all are precompact). Then there exists a constant c > 0 s uch that

kη

k,p

(

Ω

)

≤ c

α∈J

kη

k,p

(

Ω

)

for all sec tions η : M → E with η

∈ W

k,p

(

Ω

) for every α ∈ J.

Proof. Choose a partition of unity {ρ

: M → [0, 1]}

α∈J

subordinate to the

ﬁnite open cover {U

}

α∈J

. Now η =

α∈J

η, and each ρ

η is supported in U

, so

(ρ

η)

has suppo rt in Ω

βα

= ϕ

∩ U

). Thus using Theorem

A.14 with the f act

that g

βα

, ϕ

−1

, ϕ

αβ

and ϕ

βα

= ϕ

−1

αβ

are all smoot h functions with bounded derivatives

of all orders on the domains in question, we ﬁnd

kη

k,p

(

Ω

)



α∈J

(ρ

η)



k,p

(

Ω

)

≤

α∈J

k(ρ

η)

k,p

(

Ω

βα

)

α∈J

k(ρ

◦ ϕ

−1

)(g

βα

◦ ϕ

−1

)(η

◦ ϕ

αβ

k,p

(

Ω

βα

)

≤ c

α∈J

kη

k,p

(

Ω

αβ

)

≤ c

α∈J

kη

k,p

(

Ω

)



Corollary A.19. If M is compact, then the norm on W

k,p

(E) given by Deﬁ-

nition

A.16 is independent of all auxilia ry choices up to equivalence of norms. 

Theorem A.20. For any smooth v ector b undle π : E → M over a compact

manifold M, W

k,p

(E) is a Banach space .

314 Chris Wendl

Proof. If η

∈ W

k,p

(E) is a Cauchy sequence, then for some chosen ﬁnite

collection I ⊂ A(π) of precompact charts covering M, t he sequences η

for α ∈ I are

Cauchy in W

k,p

(

Ω

) and thus have limits ξ

(α)

∈ W

k,p

(

Ω

, F

). Choosing a part ition

of unity {ρ

: M → [0, 1]}

α∈I

subordinate to {U

}

α∈I

, we can now associate to each

α ∈ I a section η

∞,α

∈ W

k,p

(E) characterized uniquely by the condition that it

vanishes outside of U

and is represented in the trivialization on U

∞,α

= (ρ

◦ ϕ

−1

)ξ

(α)

We claim that ρ

→ η

∞,α

in W

k,p

(E) for each α ∈ I. Indeed, we have

(ρ

)

= (ρ

◦ ϕ

−1

)η

→ (ρ

◦ ϕ

−1

)ξ

(α)

= η

∞,α

in W

k,p

(

Ω

)

since η

→ ξ

(α)

. For all other β ∈ I not equal to α, (ρ

)

− η

∞,α

∈ W

k,p

(

Ω

, F

)

has suppo rt in Ω

βα

= ϕ

∩ U

), thus

k(ρ

)

−η

∞,α

k,p

(

Ω

)

= k(ρ

)

−η

∞,α

k,p

(

Ω

βα

)

≤ ck(ρ

)

− η

∞,α

k,p

(

Ω

)

where the inequality comes from Lemma

A.18 after replacing M with U

, and U

with U

∩ U

(note that the lemma does not require M to be compact). With the

claim established, we have

α∈I

→

α∈I

∞,α

in W

k,p

(E).



Exercise A.21 . For U ⊂ R

an open subset, the space W

k,p

lo c

(U) was deﬁned in

A.1, but one can give it an alternative deﬁnition in the present context by viewing

functions on U as sections of a trivial vector bundle over U, with the latter viewed

as a noncompact smooth n-manifold. Show that these two deﬁnitions of W

k,p

lo c

(U)

are equivalent.

Exercise A.22. Suppose U ⊂ R

is a bounded open subset with smooth bound-

ary, so its closure U ⊂ R

is a smooth compact submanifold with boundary, and let

E →

U be a trivial vector bundle. Show that there is a canonical Banach space iso-

morphism between W

k,p

(U) as deﬁned in §

A.1 and W

k,p

(E) as deﬁned in the present

section. Hint: Recall that sections in W

k,p

(E) are only required to be deﬁned almost

everywhere, so in particular if the domain M is a manifold with boundary, they need

not be well deﬁned on ∂M.

In light of Exercise

A.22, the natural generalization of W

k,p

(U) in the present

setting is

k,p

(E) := C

∞

(E|

M\∂M

i.e. it is the closure in the W

k,p

-norm of t he space of smooth sections that vanish

near t he boundary. Density of smooth sections will imply that this is the same as

k,p

(E) if M is closed, but in general W

k,p

(E) is a closed subspace of W

k,p

(E).

The partit ion of unity argument in Theorem

A.20 contains all the essential ideas

needed to generalize results about Sobolev spaces on domains in R

to compact

manifolds. We now state the essential results, leaving the proofs as exercises.

Lectures on Symplectic Field Theory 315

Theorem A.23. Assume M is a smooth compact n-manifold, possibly with

bo und ary, π : E → M is a smooth vector bundle of ﬁnite rank, k ≥ 0 is an in-

teger and 1 ≤ p < ∞. Then the Banach space W

k,p

(E) has the following properties.

(1) The space Γ(E) of smooth sec tion s is dense in W

k,p

(E).

(2) If N ⊂ M is a smooth compact n-dimension al submanifold with boundary,

then there ex ists a bounded linear operator

E : W

k,p

(E|

) → W

k,p

(E)

which is an extension o perator in the sense that Eη|

= η for all η ∈

k,p

(E|

). Moreover, a s ection η ∈ W

k,p

(E|

) belon g s to W

k,p

(E|

) if

and only if the section ˜η deﬁned to match η on N and to vanish on M \N

be l ongs to W

k,p

(E).

(3) If kp > n, then for each integer d ≥ 0, there e xists a continuous and compact

inclusion

k+d,p

(E) ֒→ C

(E).

(4) The natural inc lusion

k+1,p

(E) ֒→ W

k,p

(E)

is compact.

(5) Suppose F, G → M are smooth v ector bundles such that there exi s ts a

smooth bundle map

E ⊗ F → G : η ⊗ ξ 7→ η · ξ.

Then if kp > n and 0 ≤ m ≤ k, there exists a continuous product pairing

k,p

(E) × W

m,p

(F ) → W

m,p

(G) : ( η, ξ) 7→ η · ξ.

In pa rticular, products of W

k,p

sections give W

k,p

sections whenever kp > n.

(6) Suppose F → M is another smooth vec tor bundle, V ⊂ E is an open subset

that intersects every ﬁber of E, a nd we consider the spaces

k,p

(V) :=



η ∈ W

k,p

(E)



η(M) ⊂ V



and

(V, F ) :=



Φ : V → F | ﬁber-preserving maps of class C



where the latter is assigned the topology o f C

-convergence on compact sub-

sets. If kp > n, then W

k,p

(V) is an o pen subset of W

k,p

(E), and the map

(V, F ) × W

k,p

(V) → W

k,p

(F ) : (Φ, η) 7→ Φ ◦ η

is well deﬁned and continuous.

(7) If N is another smooth com pact manifol d and ϕ : N → M is a smooth

diﬀeomorphism, then there is a Banach space isomorphism

k,p

(E) → W

k,p

(ϕ

∗

E) : η 7→ η ◦ ϕ.



316 Chris Wendl

A.4. Some remarks on domains with c ylindrical ends

For bundles π : E → M with M noncompact, W

k,p

(E) is not generally well

deﬁned without making additional choices. When M =

Σ = Σ \ Γ is a punctured

Riemann surface and π : E →

Σ is equipped with an asymptotically Hermitian struc-

ture {(E

, J

, ω

)}

z∈Γ

, one nice way to deﬁne W

k,p

(E) was introduced in §

4.1: one

takes it to be the space of sections in W

k,p

lo c

(E) whose W

k,p

-norms on each cylindrical

end are ﬁnite with resp ect to a choice of asymptotic trivialization. This deﬁnition

requires the convenient fact that complex vector bundles over S

are always triv-

ial, though one can also do without t his by using the ideas in the previous section.

Indeed, any collection of local trivializations on the asymptotic bundle E

→ S

covering S

gives rise via the asymptotically Hermitian structure t o a collection of

trivializations on E covering the corresponding cylindrical end

. The key fact

is then that S

is compact, hence one can always choose such a covering to be ﬁ-

nite: combining this with a ﬁnite covering of

Σ in the complement of its cylindrical

ends by precompact chart s, we obtain a covering of

Σ by a ﬁnite collection of bun-

dle charts t hat are not all precompact, but nonetheless have the property that all

transition maps have bounded derivatives of all orders. This is enough to deﬁne a

k,p

-norm for sections of E →

Σ as in Deﬁnition

A.16 and to prove that it does

not depend on the choices of charts or local trivializations, though it does depend

on the asymptotically Hermitian structure.

With this deﬁnition understood, one can easily generalize the Sobolev embedding

theorem and other important statements in Theorem

A.23 to the setting of an

asymptotically Hermitian bundle over a punctured Riemann surface. We shall leave

the details of this generalization as an exercise, but take the opportunity to point

out a few important diﬀerences from t he compact case.

First, since

Σ is not compact, neither are t he inclusions

k+d,p

(E) ֒→ C

(E), W

k+1,p

(E) ֒→ W

k,p

(E).

The proof o f compactness fails due to the fact that cylindrical ends require local

trivializations over unbounded doma ins of the form ( 0, ∞) × (0, 1) ⊂ R

, for which

Theorem

A.9 does not hold. And indeed, considering unbounded shifts on the

inﬁnite cylinder

Σ = R ×S

, it is easy to ﬁnd a sequence of W

k,p

-bounded functions

with kp > 2 that do not have a C

-convergent subsequence. That is the bad news.

The good news is that if η ∈ W

k+d,p

(E) for kp > 2, then one can say considerably

more about η t han just that it is C

-smooth. Indeed, restricting t o one of the

cylindrical ends [0, ∞) × S

⊂

Σ, notice that ﬁniteness of the W

k+d,p

-norm over

implies

kηk

k+d,p

((R,∞)×S

)

→ 0 as R → ∞.

Since these domains are all naturally diﬀeomorphic for diﬀerent values of R, the

-norm of η over (R, ∞) ×S

is bounded by t he W

k+d,p

-norm via a constant that

does not depend on R, so this implies an asymptotic decay condition

kηk

([R,∞)×S

)

→ 0 as R → ∞

for every η ∈ W

k+d,p

(E).

Lectures on Symplectic Field Theory 317

Here is another useful piece of good news: since

Σ does not have boundary,

k,p

(E) = W

k,p

(E).

Theorem A.24. Given an asymptotically Hermitian bundle E over a punctured

Riemann surface

Σ, the space C

∞

(E) of smooth s ections with compact support is

dense in W

k,p

(E) for all k ≥ 0 and 1 ≤ p < ∞.

Proof. We can assume as in Deﬁnition

A.16 that the W

k,p

-norm for sections η

of E is given by

kηk

k,p

α∈I

kη

k,p

(Ω

)

where I ⊂ A(π) is a ﬁnite collection of bundle charts

α =



: U

∼

−→ Ω

, Φ

: E|

∼

−→ U

×C



such that each of the open sets Ω

⊂ C is either bounded or (for charts over the

cylindrical ends) of the form

Ω

= (0, ∞) ×ω

⊂ R

= C

for some bounded open subset ω

⊂ R. Now given η ∈ W

k,p

(E), Theorem

A.1

provides f or each α ∈ I a sequence η

∈ W

k,p

(Ω

) of smo oth functions with bounded

support such that η

→ η

in W

k,p

(Ω

). Choose a partition of unity {ρ

Σ →

[0, 1]}

α∈I

subordinate to the open cover {U

}

α∈I

and let

α∈I

(η

◦ ϕ

) ∈ W

k,p

(E).

These sections are smoo t h and have compact support since the η

have bounded

support in Ω

, and they converge in W

k,p

to η. 

APPENDIX B

The Floer C

space

The C

-topology for functions was introduced by Floer [

Flo88b] to provide a Ba-

nach manifo ld of perturbed geometric structures without departing from the smo oth

category: it is a way to circumvent the annoying fact that spaces of smooth functions

which ar ise naturally in geometric settings are not Banach spaces. The construction

of C

spaces generally depends on several arbitrary choices and is thus far from

canonical, but this detail is unimpor t ant since the C

space itself is never the main

object of interest. What is important is merely the properties that it has, namely

that it not only embeds continuously into C

∞

and contains an abundance of non-

trivial functions, but also is a separable Banach space and can therefore be used in

the Sa r d-Smale theorem for genericity arguments. We shall prove these facts in this

appendix.

Fix a smooth ﬁnite-rank vector bundle π : E → M over a ﬁnite-dimensional

compact manifold M, possibly with boundary. For each integer k ≥ 0, we denote by

(E) the Banach space of C

-smooth sections of E; note that the norm on C

(E)

depends on various auxiliary choices but is well deﬁned up to equivalence of norms

since M is compact. Now if (ε

)

∞

k=0

is a sequence of positive numbers with ε

→ 0,

set

(E) =



η ∈ Γ(E)



kηk

< ∞



where t he C

-norm is deﬁned by

(B.1) kηk

∞

k=0

kηk

k .

The norm for C

(E) is somewhat more delicate than for C

(E), e.g. its equivalence

class is not obviously independent of auxiliary choices. This remark is meant as

a sanity check, but it should not cause extra concern since, in practice, the space

(E) is typically rega rded as an auxiliary choice in it self. In many applications,

one ﬁxes an o pen subset U ⊂ M and considers the closed subspace

(E; U) =



η ∈ C

(E)



η|

M\U

≡ 0



Remark B.1 . The requirement for M to be compact can be r elaxed as long as

U ⊂ M has compact closure: e.g. in one situation of frequent interest in this book,

we take M to be the noncompact completion of a symplectic cobor dism. In this case

(E; U) can be deﬁned as a closed subspace of C

(E|

) where M

⊂ M is any

compact manifold with boundary that contains the closure of U. For this reason,

we lose no generality in cont inuing under the assumption that M is compact.

319

320 Chris Wendl

In order to prove things about C

(E), we will need to specify a more precise

deﬁnition of the C

-norms. To this end, deﬁne a sequence of vector bundles E

(k)

→

M for integers k ≥ 0 inductively by

(0)

:= E, E

(k+1)

:= Hom(T M, E

(k)

Choose connections and bundle metrics on both T M and E; these induce connec-

tions and bundle metrics on each of the E

(k)

, so that f or any section ξ ∈ Γ(E

(k)

the covariant derivative ∇ξ is now a section of E

(k+1)

. In particular for η ∈ Γ(E),

we can deﬁne the “kth covariant derivative” of η a s a section

∇

η ∈ Γ(E

(k)

Using the bundle metrics to deﬁne C

-norms for sections of E

(k)

, we can t hen deﬁne

kηk

(E)

m=0

k∇

ηk

(m)

)

where by convention ∇

η := η. We will assume throughout the following that the

-norms appearing in (

B.1) are deﬁned in this way.

Theorem B.2. C

(E) is a Banach space.

Proof. We need to show that C

-Cauchy sequences converge in the C

-norm.

It is clear from the deﬁnitions that if η

∈ C

(E) is Cauchy, then η

is also C

Cauchy for every k ≥ 0, hence its derivatives ∇

for every k are C

-convergent

to continuous sections ξ

of E

(k)

. This convergence implies that ξ

k+1

= ∇ξ

the sense of distributions, hence by the equivalence of classical and distributional

derivatives (see e.g. [

LL01, §6.10]), η

∞

:= ξ

is smooth with ∇

∞

= ξ

, so that

∇

→ ∇

∞

in C

(k)

) for all k.

We claim η

∞

∈ C

(E). Choose N > 0 such that kη

−η

< 1 for all i, j ≥ N.

Then for every m ∈ N and every i ≥ N,

k=0

kη

k ≤

k=0

kη

− η

k +

k=0

kη

≤ kη

− η

+ kη

< 1 + kη

Fixing m and letting i → ∞, we then have

k=0

kη

∞

≤ 1 + kη

for all m, so we can now let m → ∞ and conclude kη

∞

≤ 1 + kη

< ∞.

The argument that kη

→ η

∞

→ 0 as j → ∞ is similar: pick ǫ > 0 and N

such that kη

−η

< ǫ for all i, j ≥ N. Then for a ﬁxed m ∈ N, we can let i → ∞

in t he expression

k=0

kη

− η

k < ǫ, giving

k=0

kη

∞

− η

≤ ǫ.

This is true for every m, so we can take m → ∞ and conclude kη

∞

− η

≤ ǫ for

all j ≥ N. 

Lectures on Symplectic Field Theory 321

To show that C

(E) is also separable, we will follow a hint

from [HS95] and

embed it isometrically into another Banach space t hat can be more easily shown to

be separable. For each integer k ≥ 0, deﬁne t he vector bundle

(k)

= E

(0)

⊕ . . . ⊕ E

(k)

and let X

denote the vector space of all sequences

ξ := (ξ

, ξ

, . . .) ∈

∞

k=0

(k)

)

such that

kξk

∞

k=0

kξ

< ∞.

Exercise B.3. Adapt the proof of Theorem

B.2 to show that X

is also a Banach

space.

Lemma B.4. X

is separable.

Proof. Since C

(k)

) is separable for each k ≥ 0, we can ﬁx countable dense

subsets P

⊂ C

(k)

). The set

P :=



(ξ

, . . . , ξ

, 0, 0, . . .) ∈ X



N ≥ 0 and ξ

∈ P

for all k = 0, . . . , N



is then countable and dense in X

. 

Theorem B.5. C

(E) is separable.

Proof. Consider the injective linear map

(E) ֒→ X

: η 7→



η, (η, ∇η), (η, ∇η, ∇

η), . . .



This is a n isometric embedding and thus presents C

(E) as a closed linear subspace

of X

, hence the theorem follows from Lemma

B.4 and the fa ct that subspaces of

separable metric spaces are always separable. 

Note that given any open subset U ⊂ M, Theorems

B.2 and B.5 also hold for

(E; U), as a closed subspace of C

(E). So far in this discussion, however, there has

been no guarantee that C

(E) or C

(E; U) contains anything other than the zero-

section, though it is clear that in theory, one should always be able to enlarge the

space by choosing new sequences ε

that converg e to zero faster. The following result

says that C

(E; U) can a lways be ma de large enough t o be useful in applications.

Theorem B.6. Given an open subset U ⊂ M, the sequence ε

ca n be chosen to

have the following properties:

(1) C

(E; U) is dense in the s pace of continuous sections vanishin g outside U.

(2) Given any point p ∈ U, a neighborhood N

⊂ U of p, a number δ > 0 and

a continuous s ection η

of E, there exists a section η ∈ Γ(E) and a smooth

co mpactly supported function β : N

→ [0 , 1] such that

βη ∈ C

(E; U), β(p)η(p) = η

(p), and kη − η

< δ.

Thanks to Sam Lisi for e xplaining to me w hat the hint in [

HS95] was referring to.

322 Chris Wendl

Proof. Note ﬁrst that it suﬃces to ﬁnd two separate sequences ε

and ε

′

that

have the ﬁrst a nd second property resp ectively, as the sequence of minima min(ε

, ε

′

)

will then have both properties.

The following construction for the ﬁrst property is based on a suggestion by

Barney Bramham. Observe ﬁrst that the space C

(E; U) of continuous sections

vanishing outside U is a closed subspace of C

(E) and is thus separable, so we can

choose a countable C

-dense subset P ⊂ C

(E; U). Moreover, the space of smooth

sections vanishing outside U is dense in C

(E; U), hence we can assume without loss

of generality that the sections in P are smooth. Now write P = {η

, η

, . . .} and

deﬁne ε

> 0 for every integer k ≥ 0 to have the property

min



kη

, . . . ,

kη



Then every η

is in C

(E; U), as

kη

j−1

k=0

kη

∞

k=j

< ∞.

The second property is essentially local, so it can be deduced from Lemma

B.7

below. 

Lemma B.7. Suppose β :

→ [0 , 1] is a smooth function with compact support

on the open unit ball

⊂ R

and β(0) = 1. One can choo s e a sequence of posi tive

numbers ε

→ 0 such that for every η

∈ R

and r > 0 , the function η : R

→ R

deﬁned by

η(p) := β(p/r)η

satisﬁes

∞

k=0

kηk

k < ∞.

Proof. Deﬁne ε

> 0 so tha t for k ≥ 1,

kβk

Then

∞

k=1

kηk

k ≤

∞

k=1

kβk

∞

k=1



1/r



< ∞.



APPENDIX C

Genericity in the space of asymptotic operators

The purpose of this appendix is to prove Lemma

3.18, which was needed for

our deﬁnition of spectral ﬂow in §

3.2. The proof combines some ideas from that

section with the technique used in Lecture

7 to prove generic transversality of moduli

spaces via the Sard-Smale theorem. Some knowledge of that technique should thus

be considered a prerequisite for this appendix; if you have never seen it before and

were directed here aft er reading the statement of Lemma 3 .18, you might want to

skip this for now and come back after you’ve read as far as Lecture

Recalling the notation from Lecture

3, we ﬁx the real Hilb ert spaces

H = L

, R

), D = H

, R

the symmetric index 0 Fredholm operator

ref

= −J

∂

: D → H

and, given a smooth loop of symmetric matrices S : S

→ End

sym

), refer to any

operator of the form

A = −J

∂

− S : D → H

as a n asymptotic operator. Such operat ors belong to the space of symmetric

compact perturbations of T

ref

Fred

sym

(D, H, T

ref

) =



ref

+ K : D → H



K ∈ L

sym

(H)



which we regard as a smooth Banach manifold via its obvious identiﬁcation with

the space L

sym

(H) of symmetric b ounded linear operators on H. For k ∈ N, we

denote by

Fred

sym,k

(D, H, T

ref

) ⊂ Fred

sym

(D, H, T

ref

)

the ﬁnite-codimensional submanifold determined by the condition dim

ker A =

dim

coker A = k.

Here is the statement of Lemma

3.18 again.

Lemma. Fix a smooth map S : [−1, 1 ] × S

→ End

sym

) and consider the

1-parameter family of operators

:= −J

∂

− S(s, ·) ∈ Fred

sym

(D, H, T

ref

)

for s ∈ [−1, 1]. Then a f ter a C

∞

-small perturbation of S ﬁxed at s = ±1, one can

assume the following:

(1) For every s ∈ (−1, 1), all eigenv alues of A

(regarded as an unbounded

operator on H) are simple.

323

324 Chris Wendl

(2) All intersections of the path

(−1, 1) → Fred

sym

(D, H, T

ref

) : s 7→ A

with Fred

sym,1

(D, H, T

ref

) are transverse.

We shall now prove this by constructing a Floer- type space of C

-smooth (see

Appendix

B) perturbed families of asymptotic operator s, and using the Sard-Smale

theorem to ﬁnd a countable collection of comeager subsets whose intersection con-

tains perturbations achieving the desired conditions.

Choose a sequence of positive numbers (ε)

∞

k=0

with ε

→ 0 to deﬁne a separable

Banach space



B ∈ C

∞

([−1, 1] × S

, End

sym

))



kBk

< ∞ a nd B(±1, ·) ≡ 0



and assume via Theorem

B.6 that A

is dense in the Banach space of continuous

functions [−1, 1] × S

→ End

sym

) vanishing at {±1} × S

. We then consider

perturbed 1-parameter families of asymptotic operators of the form

:= A

+ B(s, ·) : D → H

for B ∈ A

, s ∈ [−1, 1]. For each k ∈ N and B ∈ A

, deﬁne the set

(B) =



(s, λ) ∈ (−1, 1) × R



dim

ker



− λ



= k



To show that eigenvalues are generically simple, we need to show that for a comeager

set of choices of B ∈ A

, V

(B) is empty f or all k ≥ 2. Given (s

, λ

) ∈ V

(B),

recall fr om §

3.2 that there exist decompositions

D = V ⊕ K, H = W ⊕ K

where K = ker



− λ



, W = im



− λ



is the L

-orthogonal complement

of K, and V = W ∩ D, so that any symmetric bounded linear operato r T in a

suﬃciently small neighborhood O ⊂ L

sym

(D, H) of A

− λ

can be written in

block form

T =



A B

C D



with A : V → W invertible, giving rise to a smooth map

Φ : O → End

sym

(K) : T 7→ D − CA

−1

whose zero-set is precisely the set of nearby symmetric operators with k-dimensional

kernel. A neighbor hood of (s

, λ

) in V

(B) can thus be identiﬁed with the zero-set

of the map

(s, λ) := Φ(A

− λ) ∈ End

sym

(K),

deﬁned for (s, λ) ∈ (−1, 1)×R suﬃciently close to (s

, λ

). Notice that the derivative

dΨ

(s, λ) : R ⊕ R → End

sym

(K) is Fredholm since its domain and targ et are both

ﬁnite dimensional, and it can only ever be surjective when k = dim

K = 1.

The following space will now play the role of a “universal moduli space” as in

Lecture

7: let



(s, λ, B) ∈ (−1, 1) × R × A



(s, λ) ∈ V

(B)



Lectures on Symplectic Field Theory 325

The proof that t his is a smooth Banach manifold depends on the following algebraic

lemma.

Lemma C.1. Fix an asymptotic operator A = −J

∂

−S an d a linear tran sfor-

mation

Υ : ker A → ker A

that is symmetric with respect to the L

-product. Then there e xists a smooth loo p

B : S

→ End

sym

) such that

hη, Bξi

= hη, Υξi

for all η, ξ ∈ ker A.

Proof. Note ﬁrst that every nontrivial loop η ∈ ker A ⊂ H

, R

) is smoo t h

and nowhere zero since it satisﬁes a linear ﬁrst-order ODE with smooth coeﬃ-

cients. It follows that if we ﬁx a basis (η

, . . . , η

) f or ker A, then the vectors

(t), . . . , η

(t) ∈ R

are also linearly independent for all t ∈ S

and thus span

a smooth S

-family of k-dimensional subspaces V

⊂ R

, each equipped with a

distinguished basis. It follows that there exists a unique smoo t h S

-family of linear

transformations

B(t) : V

→ V

such that for every η ∈ ker A,

B(t)η(t) = (Υη)(t)

for all t. Extend

B(t) arbitrarily to a smooth family of linear maps on R

The matrices

B(t) ∈ End

) need not be symmetric, but they do satisfy

hη,

Bξi

= hη, Υξi

for all η, ξ ∈ ker A.

Since Υ is symmetric, this implies mo r eover that for all η, ξ ∈ ker A,

hη, Υξi

= hξ, Υηi

= hξ,

Bηi

= hη,

ξi

The loop B :=

(

B +

) thus has the desired properties. 

Now using the previously described construction in the space of symmetric Fred-

holm operators, a neighborhood of any point (s

, λ

, B

) in V

can be identiﬁed with

the zero-set of a smooth map of the form

Ψ(s, λ, B) := Ψ

(s, λ) ∈ End

sym

(K),

deﬁned fo r all (s, λ, B) suﬃciently close to (s

, λ

, B

) in (−1, 1) × R × A

, where

K = ker



− λ



. The partial derivative of Ψ with respect to the third variable

at (s

, λ

, B

) is then a linear map

L := D

Ψ(s

, λ

, B

) : A

→ End

sym

(K)

of the form

(C.1) LB : K → K : η 7→ π

(B(s

, ·)η),

where π

: W ⊕K → K is the orthogonal proj ection. We claim tha t L is surjective.

Indeed, for any Υ ∈ End

sym

(K), Lemma

C.1 provides a smooth loop C

: S

→

End

sym

(K) such that

η) = Υη for all η ∈ K,

and this can be extended t o a smooth function C : [−1, 1] × S

→ End

sym

(K)

satisfying C(s

, ·) ≡ C

and C(±1, ·) ≡ 0 since s

6= ±1. The function C might fail

326 Chris Wendl

to be of class C

, but since it can be approximated arbitrarily well in the C

-norm

by functions in A

, we conclude that the image of L is dense in End

sym

(K). Since

the latter is ﬁnite dimensional, the claim follows.

The implicit function theorem now gives V

the structure of a smooth Banach

submanifold of (−1, 1)×R ×A

, and it is separable since the latter is also separable.

Consider the projection

(C.2) π : V

→ A

: (s, λ, B) 7→ B,

which is a smooth map of separable Banach manifolds whose ﬁbers π

−1

(B) are the

spaces V

(B). Using Lemma

7.18, the fact that each map Ψ

is Fredholm implies

that π is also a Fredholm map, so the Sard-Smale theorem implies that the regular

values of π form a comeager subset

reg,k

⊂ A

The intersection

reg

k∈N

reg,k

is then another comeager subset of A

, with the property that for each B ∈ A

reg

and every (s, λ) ∈ V

(B), dΨ

(s, λ) is (by L emma

7.18) surjective. As was observed

previously, this is impossible for dimensional reasons if k ≥ 2, implying that V

(B)

is then empty.

To ﬁnd perturbations that also achieve the transversality condition, we use a

similar a r gument: deﬁne for each B ∈ A

the subset

(B) =



s ∈ (−1, 1)



dim

ker A

= 1



along with the corresponding universal set



(s, B) ∈ (−1, 1) × A



s ∈ V

(B)



A neighborhood of any (s

, B

) in V

is then the zero-set of a smooth map of the

form

Ψ(s, B) = Φ(A

) ∈ End

sym

(ker A

deﬁned fo r all (s, B) ∈ (−1, 1) × A

close enough to (s

, B

). For a ﬁxed B ∈ A

near B

and s

∈ V

(B) near s

, a neighborhood of s

in V

(B) is then the zero-set

of Ψ

(s) := Ψ(s, B), and t he intersection of the path s 7→ A

∈ Fred

sym

(D, H, T

ref

)

with Fred

sym,1

(D, H, T

ref

) at s = s

is tra nsverse if and only if

dΨ

) : R → End

sym

(ker A

)

is surjective. At (s

, B

), the partial deriva t ive of Ψ with respect to B is again the

same operator

L = D

Ψ(s

, B

) : A

→ End

sym

(ker A

)

as in (

C.1), which we’ve already seen is surjective due to Lemma C.1. Thus one can

apply the Sard-Smale theorem to the projection

→ A

: (s, B) 7→ B,

obtaining a comeager subset A

reg,0

⊂ A

such that all paths A

+ B(s, ·) for B ∈

reg,0

satisfy the required transversality condition. The comeager subset A

reg,0

∩

Lectures on Symplectic Field Theory 327

reg

⊂ A

thus consists of perturbed families of operato rs for which all desired

conditions are satisﬁed, and it cont ains a sequence converging in the C

∞

-topology

to 0. This concludes the proof of Lemma

3.18.

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