FUNCTORIALITY IN CATEGORICAL SYMPLECTIC GEOMETRY MOHAMMED ABOUZAID AND NATHANIEL BOTTMAN Abstract. Categorical symplectic geometry is the study of a rich collection of invariants of sym-

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FUNCTORIALITY IN CATEGORICAL SYMPLECTIC GEOMETRY

MOHAMMED ABOUZAID AND NATHANIEL BOTTMAN

Abstract. Categorical symplectic geometry is the study of a rich collection of invariants of sym-

plectic manifolds, including the Fukaya -category, Floer cohomology, and symplectic cohomol-

ogy. Beginning with work of Wehrheim and Woodward in the late 2000s, several authors have

developed techniques for functorial manipulation of these invariants. We survey these functorial

structures, including Wehrheim–Woodward’s quilted Floer cohomology and functors associated to

Lagrangian correspondences, Fukaya’s alternate approach to deﬁning functors between Fukaya -

categories, and the second author’s ongoing construction of the symplectic -category. In the

last section, we describe a number of direct and indirect applications of this circle of ideas, and

propose a conjectural version of the Barr–Beck Monadicity Criterion in the context of the Fukaya

-category.

Contents

1. Introduction 1

2. Floer cohomology, the Fukaya -category, and the Operadic Principle 4

3. Quilted Floer theory and functors from Lagrangian correspondences 19

4. The symplectic -category 39

5. Applications 53

References 65

1. Introduction

Asymplectic manifold is a smooth even-dimensional manifold , together with a 2-form

that is closed ( ) and non-degenerate in the sense that its top exterior power is

a volume form ( pointwise). The original motivation for this deﬁnition came from celestial

mechanics, but much of modern symplectic geometry is independent of these physical origins.

Example 1.1. The fundamental example of a symplectic manifold is Euclidean space with the

Darboux symplectic form:

(1)

where is equipped with coordinates . This choice of notation goes back to

classical mechanics, where the coordinates record the position of a particle, and its momentum.

From the point of view of a mathematician, the might as well represent local coordinates on a

smooth manifold, in which case the coordinates can be understood as coordinates on the cotangent

ﬁbre. In this way, one obtains the canonical symplectic form

(2)

on the total space of the cotangent bundle of any smooth manifold.

arXiv:2210.11159v1 [math.SG] 20 Oct 2022

Starting with the Darboux symplectic form, one constructs a large class of examples as follows:

identify with complex aﬃne space , by setting and , and observe that

the symplectic form is given, in terms of the and operators of complex analysis (

and ) as

(3)

with (this amount to the statement that the Darboux form is the real part of the

standard Kähler form). Since the norm of a vector is invariant under rotation, one obtains an

induced symplectic form on the quotient of the unit sphere by the circle action. This

symplectic form on projective space is known as the Fubini-Study form FS, and may be expressed

directly in terms of coordinates on a standard aﬃne chart of projective space as

(4)

Via the complex geometry result that Kählerness is preserved by restriction to complex subman-

ifolds, one then obtains from complex submanifolds of projective space (i.e. projective algebraic

varieties) a large class of compact symplectic manifolds.

One of the fundamental questions in symplectic geometry is to understand the geometry of

the Lagrangian submanifolds (or simply Lagrangians), i.e. those embedded submanifolds

along which the symplectic form vanishes. (In this paper, we will assume that all Lagrangians are

oriented.)

Example 1.2. The fundamental examples of Lagrangians are the - and - planes in , equipped

with the standard symplectic form. This naturally generalises to the cotangent ﬁbre and the zero

section of the cotangent bundle . The zero section is an example of a more general class: the

graph of any closed 1-form on .

In the examples which arise from complex geometry, one may use real geometry to produce

examples by: ﬁx a smooth projective variety that is deﬁned by a set of equations with real

coeﬃcients. Considering as a symplectic manifold equipped with the restriction of the Fubini–

Study form, the real locus (whenever it is smooth) is a Lagrangian submanifold.

1.1. Symplectic invariants from pseudoholomorphic curves. Unlike Riemannian geometry,

symplectic geometry has no local symplectic invariants as a consequence of Darboux’s theorem

[Dar82]. Below, we state this theorem in combination with Weinstein’s Lagrangian neighborhood

theorem [Wei71], which is the analogous result for the local geometry of near a Lagrangian subman-

ifold. Weinstein’s theorem involves the notion of a symplectomorphism, which is a diﬀeomorphism

between two symplectic manifolds that satisﬁes .

Theorem 1.3. Any point in a symplectic manifold admits a neighbourhood which is symplectomor-

phic to a neighbourhood of the origin in . Similarly, any Lagrangian embedding in

of a closed manifold extends to a symplectomorphism between a neighborhood of the zero section

in and a neighborhood of in .

The reader new to this ﬁeld may get the sense from these theorems that symplectic geometry

is similar in ﬂavor to diﬀerential topology, but this is not the case. Indeed, a motif in symplectic

geometry is the interplay between ﬂexibility and rigidity. In the foundational paper [Gro85], Mikhail

Gromov opened the ﬂoodgates to a wide variety of rigidity results, by importing holomorphic tech-

niques from complex geometry. Consider, for instance, the following result.

Theorem 1.4 (Theorem 0.4.A , [Gro85]).For any closed embedded Lagrangian , there

exists a non-constant map , mapping the boundary to , and which is holomorphic with

respect to the standard complex structures on and .

If we write for the standard complex structure on , and for complex structure on the disc,

the holomorphicity condition on amounts to the requirement that the operator

(5)

vanish pointwise on the domain. We can easily deduce the following corollary, which establishes a

topological obstruction to Lagrangian embeddings into .

Corollary 1.5. Suppose that is a closed -manifold with . Then does not admit

a Lagrangian embedding into .

Proof. Deﬁne by , and note that is a primitive of .

Denote by the restriction of to . By Stokes’s theorem, we have:

(6)

where the inequality follows from the fact that the latter integral is equal to the area of the image

of , which is nonnegative by the holomorphicity condition, and strictly so by non-constancy.

Gromov’s key insight in [Gro85] is that one can use similar ideas even in the case of symplec-

tic manifolds that are not Kähler: if is any symplectic manifold, there is a contractible

(in particular, nonempty) space of -compatible almost complex structures, i.e. endomorphisms

of the tangent bundle satisfying the following properties:

(AC structure) .

( -compatible) The contraction deﬁnes a Riemannian metric on .

While one can study holomorphic maps from a Riemann surface to an arbitrary almost complex

manifold, the fundamental result proved by Gromov in [Gro85] is that the moduli space of such

maps admits a natural compactiﬁcation whenever the target is symplectic. This is the foundation

of all later developments extracting symplectic invariants from moduli spaces of holomorphic maps.

In this article, we will be primarily concerned with two symplectic invariants. The ﬁrst is the

Floer cohomology group1associated to a pair and of appropriate Lagrangians in

a symplectic manifold , which categoriﬁes their intersection number. Andreas Floer introduced

this invariant in the 1980s, and as an immediate consequence obtained a proof of one version of

the Arnold–Givental conjecture. Brieﬂy, is the homology of a chain complex

freely generated by the elements of . The diﬀerential is deﬁned by counting holomorphic

maps from to , with boundary conditions deﬁned by and . We will discuss Floer

cohomology in more detail in §2.1.

Lagrangian Floer cohomology groups form the morphism spaces of the second invariant which

we will consider, the Fukaya -category , whose objects are Lagrangians (appropriately

decorated). Its deﬁnition originated in work of Simon Donaldson and Kenji Fukaya in the early

1990s, and over the intervening three decades its structure and properties have been steadily devel-

oped. It plays a central role in Maxim Kontsevich’s Homological Mirror Symmetry (HMS) conjecture

[Kon95], which posits that in certain situations there are pairs of a symplectic manifold

and a complex algebraic variety for which a “derived” version of is equivalent to an in-

variant of called the derived category of coherent sheaves on . A great deal of work has gone

1This group is not deﬁned for arbitrary pairs , as its construction depends on a choice of additional data

which may not always exist. We suppress this point until §2.4 below.

into proving and reﬁning the HMS conjecture in various settings. In §2.2 we will give an overview

of the deﬁnition and of some of the properties of .

1.2. Functorial properties of pseudoholomorphic curve invariants, the Operadic Princi-

ple, and the plan for this paper. If one wants to develop a toolbox for computing pseudoholo-

morphic curve invariants, the following is an obvious question:

If one understands the Fukaya category of a symplectic manifold , which is geometrically

related to a possibly-diﬀerent manifold , is it possible to then compute ?

Until the late 2000s, the only answer with any of degree of generality was given by Seidel in [Sei08],

where he demonstrated an inductive method for computing the Fukaya -category of the total

space of a Lefschetz ﬁbration in terms of the Fukaya category of . While Seidel’s

work provides a powerful toolbox, it is limited to the setting of Lefschetz ﬁbrations, and we might

want a more ﬂexible framework. To that end, consider this variant on the above question:

What functorial properties are enjoyed by Floer cohomology, the Fukaya -category,

and other symplectic invariants deﬁned by counting pseudoholomorphic curves?

One approach to this question is given by Weinstein’s symplectic creed [Wei82], which states that

“Everything is a Lagrangian submanifold.” In particular, this suggests that when it comes to the

Fukaya category, we should attempt to associate functors to Lagrangian correspondences, i.e. La-

grangians . In the late 2000s, Wehrheim and Woodward

pursued this approach, which led them to develop their theory of pseudoholomorphic quilts. In §3,

we will describe this work in detail. In §4, we will describe the second author’s development of the

symplectic -category. Finally, in §5, we will survey a variety of applications of the theory of

pseudoholomorphic quilts.

Throughout this paper, we will emphasize the following principle:

The operadic principle in symplectic geometry: The algebraic nature of a sym-

plectic invariant deﬁned by counting rigid pseudoholomorphic maps is inherited from the

operadic structure of the underlying collection of domain moduli spaces.

Finally, we note that Kenji Fukaya made a major contribution to this ﬁeld in his 2017 preprint

[Fuk17]. Speciﬁcally, he associates functors to Lagrangians correspondences under very general

hypotheses. Fukaya used quilts to accomplish this, but he took a quite diﬀerent approach from that

taken by Wehrheim and Woodward. See §3.11 for an account of Fukaya’s work.

1.3. Acknowledgments. N.B. was supported by an NSF Standard Grant (DMS-1906220) during

the preparation of this article. He is grateful to the Max Planck Institute for Mathematics in Bonn

for its hospitality and ﬁnancial support. M.A. would like to thank Kobi Kremnizer for asking him,

many years ago, about whether there is a place for the Barr–Beck theorem in Floer theory. He was

supported by an NSF Standard Grant (DMS-2103805), the Simons Collaboration on Homological

Mirror Symmetry, a Simons Fellowship award, and the Poincaré visiting professorship at Stanford

University. The authors thank Kenji Fukaya, Yankı Lekili, and Paul Seidel for useful conversations.

2. Floer cohomology, the Fukaya -category, and the Operadic Principle

Default geometric hypotheses: In §§2.1–2.2, we assume our symplectic manifolds and

Lagrangians are closed and aspherical, i.e. satisfy , unless otherwise stated.

In §2.4, we relax this hypothesis and work with general closed symplectic manifolds.

In this section, we will introduce some fundamental objects in categorical symplectic geometry.

After introducing Floer cohomology in §2.1 and the Fukaya -category in §2.2, we will explain in

§2.3 that is the ﬁrst instance of the Operadic Principle mentioned in §1.2.

2.1. Floer cohomology. Given two Lagrangians and in a symplectic manifold, their La-

grangian Floer cohomology group , when deﬁned, categoriﬁes their intersection number.

Shortly, we will mention a major result that motivated Floer to deﬁne this invariant. Before this,

we need to introduce the notion of a Hamiltonian diﬀeomorphism.

Given symplectic manifolds and , a symplectomorphism is a diﬀeomorphism

satisfying . The inﬁnitesimal version of self-symplectomorphisms of is given by the

symplectic vector ﬁelds, i.e. those Xwith the property that is closed. Indeed, this

follows from Cartan’s magic formula:

(7)

An important class of symplectic vector ﬁelds is formed by the Hamiltonian vector ﬁelds, i.e. those

for which is not only closed, but exact. Note that since is nondegenerate, we can

associate to any smooth function a Hamiltonian vector ﬁeld deﬁned by solving the

equation . Given a path of functions , we can integrate the associated

vector ﬁelds to obtain a symplectomorphism . Such a map is called a Hamiltonian

diﬀeomorphism.

In 1988, Floer introduced Lagrangian Floer cohomology in order to prove the following

case of a conjecture due to Arnold ([Ad78, Appendix 9], [Ad65]) and typically referred to as the

Arnold–Givental conjecture.

Theorem 2.1 ( case of Theorem 1, [Flo88]).Suppose is a closed symplectic manifold and

that is a Lagrangian with . Fix a Hamiltonian diﬀeomorphism of such

that and intersect transversely. Then the following estimate holds:

(8)

At the beginning of this subsection we called the Floer cohomology an “invariant”, but

we did not specify what it is invariant with respect to. In fact, is built so that there is

a canonical isomorphism for a Hamiltonian diﬀeomorphism, and

this isomorphism was the key ingredient in the proof of this result.

Remark 2.2.We should think of this result as saying that deforming Lagrangians by Hamiltonian

vector ﬁelds is a less ﬂexible operation than we might expect from purely diﬀerential-topological

considerations. Indeed, the normal and tangent bundles are isomorphic (an -compatible

almost complex structure, as introduced later in this subsection, deﬁnes such an isomorphism).

Choose a vector ﬁeld Xwhose zeroes are isolated and have index . On one hand, the

Poincaré–Hopf index theorem implies that the sum of the indices of the zeroes of is equal to the

Euler characteristic . On the other hand, our identiﬁcation allows us to interpret

this same sum as the signed intersection number of with a transverse pushoﬀ of itself. We can

summarize this reasoning in the following inequality:

(9)

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FUNCTORIALITYINCATEGORICALSYMPLECTICGEOMETRYMOHAMMEDABOUZAIDANDNATHANIELBOTTMANAbstract.Categoricalsymplecticgeometryisthestudyofarichcollectionofinvariantsofsym-plecticmanifolds,includingtheFukayaA1-category,Floercohomology,andsymplecticcohomol-ogy.BeginningwithworkofWehrheimandWoodwardinthelate200...

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