Does Fukaya see all symplectic topology?

Question

I recently had a debate with my friend about how much of symplectic topology is about Fukaya category. I thought that for the most part, symplectic topology is not about Fukaya category. Now, to prove that I am right I need a mathematical example of Fukaya category failing to discern some symplectic topology.

We know that according to certain definitions of Fukaya category, countably many embedded monotone Lagrangian tori in the projective plane (constructed by Vianna) are isomorphic objects of Fukaya category (if we also choose the local systems in a right way). These tori are not Hamiltonian isotopic, however.

There have been some works (I think Perutz--Sheridan among them) about Fukaya category which don't really prove it exists but just list some of its expected properties and then say "If there exists such a category, it also satisfies this and this".

The question is: is the fact that Fukaya category fails to see the distinction between Lagrangian tori a feature or a bug? Is it possible to prove that if the definition of Fukaya category of closed Fano manifold satisfies such-and-such widely expected properties, then there will be non-Hamiltonian isotopic tori defining isomorphic objects?

Answer 1

This answer just provides some general comments pertatining to the question asked in the title of the post. (When I first started graduate school, I was somewhat skeptical of Fukaya categories, and in fact went around asking people when one could show that Fukaya catgories can actually distinguish Lagrangians up to Hamiltonian isotopy. So I am quite sympathetic to the question.)

It is difficult to give a precise answer to how much symplectic topology Fukaya categories know, because there are many flavors of `the Fukaya category' and it seems likely that more will be created in the future. After all, Fukaya categories are a particular approach to packaging holomorphic curve counts in a way that is particularly amenable to homological algebra, and there are a lot of interesting curves that one may try to count!

The Vianna tori are actually an excellent example of Fukaya categories influencing more normal symplectic topology; I think I was once told that various people had tried to prove that all tori are isotopic to the clifford or chekanov torus until Vianna came up with his examples, which were, as far as I understand, motivated by mirror symmetry considerations. Vianna does not need the Fukaya category to distinguish his tori, but the superpotential, which is the invariant that Vianna used to distinguish his tori (and which Jonny Evans mentioned), was constructed by FOOO in their book on Fukaya categories -- it is part of the data of the curved Fukaya category. (I don't know when superpotentials of Lagrangian tori were first discussed and computed.) As discussed in the comments to Evans' answer, after uncurving the objects supported on these different tori often isomorphic in the Fukaya category over the Novikov field; but the curved Fukaya category is a core part of the Fukaya-categorical approach to symplectic topology.

It seems worth noting that in some sense the Fukaya category cannot even distinguish discern classical topology. For example, there are various flavors of Fukaya category that one can associate to a Lioville domain $M$, and most of them are zero if $M$ is a subcritical Weinstein domain. Yet, there are many subcritical weinstein domains which have non-symplectomorphic completions; for example, they may have different singular homologies; so the Fukaya category doesn't know about singular cohomology! However, subcritical Weinstein domains satisfy certain h-principles, and so one might say that they have no symplectic topology beyond their smooth topology. The question the boundary between flexibility (e.g. Weinstein domains satisfying an h-principle) and rigidity (Weinstein domains having interesting invariants which preclude them from satisfying h-principles) has long been a major strand of research and has been intertwined with improvements in Fukaya-categorical technology for at least a decade. For example Abouzaid-Seidel's "homologous recombination" paper used Fukaya categories as a very rich source of exotic examples of this sort, and there is later work involving "bulk deformed Fukaya categories" which can detect non-flexibility even when more usual Fukaya categories are empty.

Going to the other extreme, from the perspective of symplectic geometry, the symplectic ball of symplectic form $\omega$ is not the same as the symplectic ball with symplectic form $2\omega$; and indeed, ball-packing problems (e.g. how much of the symplectic ball in $C^n$ of radius 1 can I fill by $k$ equal-radius symplectic balls embedded disjointly via symplectomorphisms) are a classicl problem that is sensitive to this difference. There are some fukaya-categorical invariants which are also sensitive to phenomeon, which goes "beyond symplectic topology"; for example, one might say that symplectic cohomology over the novikov ring is part of the general structure of the fukaya category, and this object gives rise to many interesting quantitative invariants of a Liouville domain. (But I think that in the Fukaya category over the novikov ring, branes supported on Hamiltonian-isotopic Lagrangian submanifolds are not isomorphic, because the continuation maps have positive valuation; so one might complain that this invariant sees too much!). For more of this flavor, you might look at Kyler Siegel's recent paper on `higher symplectic capacities', which use some of the higher-algebraic structure on filtered symplectic cohomology, which is closely related to symplectic homology over the novikov ring, to build interesting quantitative invariants of Lioville domains. The understanding of what higher-algebraic structure exists on symplectic cohomology has been closely tied to the development of the theory of Fukaya categories. But note that Siegel uses the formalism of symplectic field theory, which is intimately related to the formalism of Fukaya categories in a way that is still being unraveled.

Yet, going back to the original statement about smooth topology, by being cleverer with holomorphic curves one can construct invariants which are tricky to extract from any Fukaya category. (Evans already mentioned several such invariants, but I think the following example is fun because it is so close to being of Fukaya-categorical flavor.) For example, Jingyu Zhao in `Periodic Symplectic Cohomologies' constructs a version of "Periodic Symplectic Cohomologies" which satisfies a localization theorem; namely, it is just the homology of the underlying manifold tensored with $\mathbb{Q}[[u]][u^-1]$, via a nontrivial isomorphism. One reason this is interesting because, as described two paragraphs ago, the flavors of Fukaya category that are typically studied for a Weinstein domain cannot possibly know about the homology of the Weinstein domain. Yet, Zhao's invariant is not a quantitative invariant, in the sense that it is an invariant of the Lioville domain after completion, so it's the same "kind" of invariant as the wrapped fukaya category. There is a variant of "periodic symplectic cohomology" that one can compute using the usual wrapped fukaya category of a Weinstein domain by some homological algebra, see Ganatra's preprint on the cyclic open-closed map, but it does not satisfy the localization theorem.

Anyway, we don't know how much symplectic topology Fukaya categories see, but the Fukaya categorical perspective has given rise to an immense number of invariants that see ever more symplectic topology. One must just keep asking "how much symplectic topology can be seen"? Go forth and classify Lagrangian tori in $CP^3$ up to Hamiltonian isotopy using any tools you have!

Answer 2

First of all, the Vianna tori are distinguished by the Fukaya category. To get an object of the category, you need to pick a torus and a rank 1 local system on it. The set of local systems making the torus into a nonzero object is an invariant which, I think, distinguishes these tori (it's the same as the critical set of the superpotential).

For something which goes beyond the Fukaya category, you could look at Mohammed Abouzaid's result on exotic spheres which uses higher dimensional moduli spaces of holomorphic curves

https://arxiv.org/abs/0812.4781

For a concrete statement about what Fukaya categories cannot see inspired by this result, Georgios Dimitroglou Rizell and I came up with some compactly supported symplectomorphisms of cotangent bundles of spheres which are not Hamiltonian isotopic to the identity through compactly supported things but which act trivially on the Fukaya category.

http://arxiv.org/abs/1407.3173

Answer 3

It is true that as long as we choose local systems/bounding cochains carefully, the Clifford torus and Chekanov torus equipped with their corresponding local systems are isomorphic in the Fukaya category. However, this doesn't mean one cannot distinguish Clifford/Chekanov tori using Lagrangian Floer theory.

We know that the wall crossing formula between Clifford and Chekanov tori is a birational map between moduli space of local systems $(\mathbb{C}^\times)^2 \rightarrow (\mathbb{C}^\times)^2$. In particular, they are not identical as charts in the moduli space of Lagrangian branes (although this is not rigorously defined). Hence the two Lagrangians defining the two $(\mathbb{C}^\times)^2$-charts cannot be Hamiltonian isotopic (otherwise the corresponding charts of local systems should be the same). To be more precise, if $L$ and $L'$ are Hamiltonian isotopic, then the branes equipped with trivial local systems should be the same.

If one translates the argument into microlocal sheaf theory, then this is just Proposition 6.1 in Shende-Treumann-Williams-Zaslow, where the moduli space of Lagrangian branes is replaced by moduli space of constructible sheaves with singular support on a certain Legendrian $\Lambda$. After attaching a Weinstein handle to $\Lambda$, one will get compact exact Lagrangians.

However, if you ask whether Fukaya categories see all geometry of Lagrangian submanifolds, the question would be tricky. I think in general we don't know the answer well.

For example, the nearby Lagrangian conjecture says that all compact exact Lagrangians in a contangent bundle $T^*Q$ (of a compact manifold $Q$) should be Hamiltonian isotopic to the zero section. According to Abouzaid (and Abouzaid-Kragh) every object in the compact Fukaya category should be isomorphic to the zero section. However, still we do not know if they are Hamiltonian isotopic. There are also analogous questions for general Weinstein manifolds which seem to be even harder.

Finally, the question whether Fukaya category sees all symplectic geometry of the ambient manifold is also tricky. Let's consider (one of) the simplest case(s) where the wrapped Fukaya category of a Weinstein manifold $\mathcal{W}(M)$ is zero. Does this imply that the Weinstein submanifold should be flexible (where all $h$-principles are satisfied and no geometric obstructions exist other than some algebraic topology data)? In general the answer is false. Maydansky first constructed a counterexample and after that Murphy-Siegel showed a more general result. However the tool they used was still some kind of Floer theoretic invariant (twisted symplectic cohomology).

What if the twisted symplectic cohomology is also trivial? In particular consider a Weinstein manifold is asymptotically $\mathbb{C}^n$, it should be diffeomorphic to $\mathbb{C}^n$ by Eliashberg-Floer-McDuff. According to Seidel its wrapped Fukaya category and symplectic cohomology should vanish. Yet we don't know if that manifold should be symplectomorphic to $\mathbb{C}^n$.

As far as I know, for Weinstein manifolds (or maybe even Liouville manifolds), all non-symplectomorphic manifolds are distinguished by either their classical invariants (e.g. their smooth topology or first Chern class) or Floer theoretic invariants. We probably don't know anything beyond that.