Functoriality in categorical symplectic geometry

By Mohammed Abouzaid and Nathaniel Bottman

Abstract

Categorical symplectic geometry is the study of a rich collection of invariants of symplectic manifolds, including the Fukaya -category, Floer cohomology, and symplectic cohomology. 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 and 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.

1. Introduction

A symplectic manifold is a smooth even-dimensional manifold , together with a 2-form that is closed () and nondegenerate 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:

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

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

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

with (this amounts 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 , and may be expressed directly in terms of coordinates on a standard affine chart of projective space as

Via the complex geometry result that Kählerness is preserved by restriction to complex submanifolds, 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: 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 pseudo-holomorphic curves

Unlike Riemannian geometry, symplectic geometry has no local symplectic invariants as a consequence of Darboux’s theorem Reference Dar82. Below, we state this theorem in combination with Weinstein’s Lagrangian neighborhood theorem Reference Wei71, which is the analogous result for the local geometry near a Lagrangian submanifold. 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 neighborhood which is symplectomorphic to a neighborhood 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 .

A 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 Reference Gro85, Mikhail Gromov opened the floodgates to a wide variety of rigidity results by showing that pseudo-holomorphic curves can be used to probe the geometry of a symplectic manifold. Consider, for instance, the following result.

Theorem 1.4 (Reference Gro85, Theorem 0.4.A).

For any closed embedded Lagrangian , there exists a nonconstant 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

vanishes pointwise on the domain. We can easily deduce Corollary 1.5, 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.

We prove the contrapositive. Define by

and note that is a primitive of . Note also that since vanishes on , defines a class in . Fix a holomorphic disk , whose existence is guaranteed by Theorem 1.4, and denote by the restriction of to . By Stokes’s theorem, we have

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 the fact that is nonconstant. This shows that cannot be zero.

One of Gromov’s key insights in Reference Gro85 is that one can use similar ideas even in the case of symplectic 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 .

Gromov showed that the moduli space of -holomorphic maps from a compact Riemann surface to a closed symplectic manifold representing a fixed class in and with boundary on a Lagrangian admits a natural compactification. 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 group associated to a pair and of appropriate⁠Footnote1 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.

1

This 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.

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 developed. It plays a central role in Maxim Kontsevich’s homological mirror symmetry (HMS) conjecture Reference 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 invariant of called the derived category of coherent sheaves on . A great deal of work has gone 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. Lagrangian correspondences, and functorial properties of pseudo-holomorphic curve invariants

The purpose of this paper is to explain several related answers to the following question.

What functorial properties are enjoyed by Floer cohomology, the Fukaya -category, and other symplectic invariants defined by counting pseudo-holomorphic curves?

This question has been the subject of active study since the mid-2000s by Ma’u, Wehrheim, and Woodward; Lekili and Lipyanskiy; Fukaya; and the second author. In the years leading up to that, it was just below the surface in symplectic geometry, for instance, in the work of Seidel and Smith Reference SS06 and Perutz Reference Per07, and in conjectures due to Seidel about extending his exact triangle for Dehn twists (see §1.2.2). Even earlier than that, several authors considered the more basic question of what we should mean by a morphism between symplectic manifolds in the context of Fourier integral operators and quantization. In the following subsubsections we discuss some basic aspects of the boxed question and briefly describe our plan for the current paper.

1.2.1. What is a morphism between symplectic manifolds?

If we want to address the boxed question, we first need to figure out what the “morphisms from to should be. An obvious candidate would be the smooth maps with . This condition implies that is an immersion, which is too restrictive a condition to produce a useful notion of functoriality.

To find a more satisfactory notion of morphism, we follow Weinstein’s symplectic creed Reference Wei82, which holds that “Everything is a Lagrangian submanifold.” One instance of this phenomenon is that the graph of a symplectomorphism is a Lagrangian

i.e., a Lagrangian correspondence from to . On the other hand, the class of Lagrangian correspondences is much richer than that of symplectomorphisms or even symplectic immersions, and so Lagrangian correspondences form a good notion of morphisms between symplectic manifolds. We will often denote a Lagrangian correspondence as an arrow .

Remark 1.6.

The study of Lagrangian correspondences goes back at least to work of Hörmander Reference H71 and Sniatycki and Tulczyjew Reference ST73. Hörmander considered homogeneous canonical relations, i.e., Lagrangian correspondences that are canonical and avoid . At nearly the same time, Sniatycki and Tulczyjew considered symplectic relations, i.e., isotropic submanifolds of . Guillemin and Sternberg Reference GS79 and Weinstein Reference Wei81 developed the subject further by pursuing quantization questions. All these authors also considered the question of when Lagrangian correspondences can be composed, a topic which we discuss below.

Given two Lagrangian correspondences , we can form their geometric composition,

When the intersection defining the fiber product is clean, is an immersed Lagrangian in . (This can be found in Reference GS05, Theorem 4. It was known by the early 1980s as a consequence of work of Hörmander that was later improved by Guillemin and Weinstein.) This cleanness hypothesis is not strong enough for the analysis of pseudo-holomorphic quilts that we will discuss in §3. With a view toward that section, we define two hypotheses on and :

If the intersection is transverse, we say that and have immersed composition. This implies that is an embedded submanifold of , and that restricts to a Lagrangian immersion of into .

If and have immersed composition and moreover is an embedding, we say that and have embedded composition.

These hypotheses were introduced by Wehrheim and Woodward (see, e.g., Reference WW10a, §1). (Their definition of embedded composition requires only that the restriction of be injective, which is equivalent to our definition in the case that is compact.)

It is very natural to try to form a category in which the objects are symplectic manifolds and the morphisms are Lagrangian correspondences. This idea was pursued by Weinstein Reference Wei81 and Guillemin and Sternberg Reference GS79. This does not literally lead to a category because geometric composition does not necessarily produce even a submanifold. Wehrheim and Woodward addressed this issue by defining a generalized Lagrangian correspondence from to to be a sequence

where each is a Lagrangian correspondence. We can form the formal composition of and by simply concatenating sequences.

In Reference WW10a, Wehrheim and Woodward constructed a 2-category whose objects are certain symplectic manifolds, and where the 1-morphisms are generalized Lagrangian correspondences. In §4, we will describe the second author’s ongoing project to define a chain-level version of this 2-category, which is intended to provide a complete answer to the boxed question near the beginning of §1.2 in the case of the Fukaya category.

1.2.2. Sources of Lagrangian correspondences

There are a variety of sources of Lagrangian correspondences. In the following list, we provide several examples.

Graphs of local symplectomorphisms. Suppose that and have the same dimension and that satisfies . Taking the graph of yields a Lagrangian correspondence . Note that the equality implies that is an immersion, hence a local diffeomorphism.

Representation spaces, following Reference WW15a. Suppose that is a three-dimensional bordism between two surfaces and . Suppose that is a tangle with components , …, , and denote the ends in by . Given the additional data of and conjugacy classes , …, in , we define

where is a small loop around . Similarly, we define and to be spaces of -representations of subject to conditions on loops around the points in . In good situations, naturally have the structure of compact symplectic manifolds, cf. Reference WW15a, Proposition 3.6. (This is related to the fact that can be identified with moduli spaces of flat principal bundles with restricted holonomy. The symplectic nature of these spaces is a rich topic, going back to work of Atiyah and Bott Reference AB83 and Goldman Reference Gol84.) Moreover, suppose that is an elementary bordism-with-tangle, which essentially says that it cannot be divided into two pieces in a topologically interesting way. Then under certain conditions on the conjugacy classes , is a smooth Lagrangian correspondence, cf. Reference WW15a, Lemma 3.5. A consequence of this is that if we allow to not necessarily be elementary, we can divide it into elementary pieces and thereby produce a generalized Lagrangian correspondence from to .

In §5.3, we describe a closely related construction, and an application to constructing 3-manifold invariants. The ideas we described in the previous paragraph can also be used to construct tangle invariants, as in Reference WW15a.

Fibered coisotropics. Suppose that is an embedded submanifold. For any point , we define the symplectic complement by

We say that is coisotropic if for every . The subspaces define a foliation of , called the isotropic foliation.

Suppose that is coisotropic with the additional property that there is a locally trivial fibration whose fibers are connected, and the fibers of are the leaves of the isotropic foliation. Then is a fibered coisotropic. carries a symplectic form satisfying the defining inequality . An immediate consequence of this equality is that the embedding is Lagrangian, i.e., defines a Lagrangian correspondence . This is a very general construction; for instance, the upcoming two sources of Lagrangian correspondences are both special cases of fibered coisotropics.

Symplectic reduction. Suppose that is a compact Lie group acting by symplectomorphisms on . In the presence of certain additional data, we say that acts in Hamiltonian fashion on , and we can form the symplectic quotient . Associated to this construction is the moment correspondence . We treat this topic in detail in §3.9.

Lefschetz–Bott vanishing cycles. A symplectic Lefschetz–Bott fibration is, approximately, the data of

a smooth proper map , where is an oriented surface,

an almost complex structure (resp., a complex structure) on (resp., on ) near the critical set (resp., the critical values), and

a closed 2-form on ,

subject to the requirements that

is pseudo-holomorphic near ,

is nondegenerate near ,

is nondegenerate on ,

is an embedded submanifold with finitely many components, and

the Hessian of is nondegenerate in the normal directions to ,

along with some additional technical conditions that the interested reader can find in Reference WW16, Definition 2.1. This is a widely applicable setting, considered by Seidel in unpublished notes from 1998 and developed by Perutz in Reference Per07. In the special case that is discrete, this is called a symplectic Lefschetz fibration, which is the setting of Seidel’s treatment Reference Sei08 of symplectic Picard–Lefschetz theory.

Suppose that , and that is a critical value. Set to be the critical points lying over . Given a regular value nearby and a path connecting and , we define to be the points in the smooth fiber that flow to via a suitable notion of parallel transport. Then is a coisotropic that fibers over , whose fibers are necessarily spheres. By our discussion of fibered coisotropics above, there is therefore a Lagrangian correspondence . This construction is a fundamental ingredient in Wehrheim and Woodward’s exact triangle for fibered Dehn twists (discussed in §5.1) and Seidel and Smith’s symplectic Khovanov homology (discussed in §5.2).

1.2.3. The plan for this paper

The core of this paper is our description of Ma’u, Wehrheim, and Woodward’s construction of functors

between extended Fukaya categories—given Lagrangian correspondences—in §3; and of the second author’s ongoing construction of the symplectic -category in §4, which extends and enhances the Ma’u, Wehrheim, and Woodward functors. To set the stage, we rapidly introduce Floer cohomology and the Fukaya category in §2, and then the quilted Floer theory of Wehrheim and Woodward in §3.5. Finally, in §5, we will survey a variety of applications of the theory of pseudo-holomorphic quilts, including an announcement in §5.7 of new work of the first and second author that aims to apply the Barr–Beck theorem to functors between Fukaya categories. Throughout this paper, we will emphasize the following principle.

The operadic principle in symplectic geometry. The algebraic nature of a symplectic invariant defined by counting rigid pseudo-holomorphic 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 Reference Fuk17. Specifically, he associates functors to Lagrangian correspondences under very general hypotheses. Fukaya used quilts to accomplish this, but he took a quite different approach from the one taken by Wehrheim and Woodward. See §3.11 for an account of Fukaya’s work.

2. Floer cohomology, the Fukaya -category, and the operadic principle

Default geometric hypotheses: In §§2.12.2, we assume our symplectic manifolds and Lagrangians are closed and aspherical, i.e., they 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. Our aim is to introduce these building blocks concisely, and with an eye toward the functorial constructions that we will focus on in later sections. For other approaches to the same material, we recommend Reference Aur14 and Reference Smi15.

2.1. Floer cohomology

Given two Lagrangians and in a symplectic manifold, their Lagrangian 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 with the property that . By Cartan’s magic formula, is symplectic if and only if is closed:

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 (Reference Ad78, Appendix 9, Reference Ad65), which typically referred to as the Arnold–Givental conjecture.

Theorem 2.1 ( case of Reference Flo88, Theorem 1).

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:

At the beginning of §2.1 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 (one can see this by using the duality between and induced by ). Choose a vector field whose zeroes are isolated and have index . On one hand, the Poincaré–Hopf index theorem implies that the sum of the indices of the zeros 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 inequality,

2.1.1. The definition of

Fix transversely intersecting Lagrangians and . In this subsubsection, we will sketch the definition of . is the homology of a chain complex whose chain group is generated freely by intersection points

Given a base field , the chain group