# Has anything precise been written about the Fukaya category and Lagrangian skeletons?

At some point in this past year, some Fukaya people I know got very excited about the Fukaya categories of symplectic manifolds with "Lagrangian skeletons." As I understand it, a Lagrangian skeleton is a union of Lagrangian submanifolds which a symplectic manifold retracts to. One good example would be the zero-section of a cotangent bundle, but there are others; for example, the exceptional fiber of the crepant resolution of $\mathbb C^2/\Gamma$ for $\Gamma$ a finite subgroup of $SL(2,\mathbb C)$. From the rumors I've heard, apparently there's some connection between the geometry of the skeleton and the Fukaya category of the symplectic manifold; this is understood well in the case of a cotangent bundle from work of Nadler and Nadler-Zaslow

I'm very interested in the Fukaya categories of some manifolds like this, but the only thing I've actually seen written on the subject is Paul Seidel's moderately famous picture of Kontsevich carpet-bombing his research program

which may be amusing, but isn't very mathematically rigorous. Google searching hasn't turned up much, so I was wondering if any of you have anything to suggest.

I noticed this question has been bumped up to the front page, and the most recent answer is about 8 years old: the subject has moved on since then, and more has been written. Here is my understanding of some of the recent developments.

Caveat: This whole area has undergone rapid progress in the last few years, and I am not working on this question, so what I say is probably not up-to-date (even as I write, never mind in the future).

Kontsevich's idea was that you can relate the Fukaya category of a Weinstein manifold to the microlocal sheaf category of its skeleton. As far as I know, a proof of this conjecture is work in progress by Ganatra-Pardon-Shende (one preliminary part of which is already available).

I think the idea of their proof is roughly the following:

• You prove that the Fukaya category has a co-sheaf property, which means it can be computed locally first on some subsets and then the answer can be glued together using homotopy colimits. For this, you need suitable functors relating the Fukaya category of a subset to the Fukaya category of the whole manifold. This is complicated by the fact that your "subsets" might not be very nicely embedded inthe whole manifold: for example, if your ambient manifold is $T^*M$ then you want to allow subsets like $T^*M'$ where $M'\subset M$ is a codimension zero submanifold with boundary (that's because you ultimately want to work locally on the skeleton). That is problematic because the Liouville vector field for $T^*M'$ and the Liouville vector field for $TM$ don't match up nicely. The first GPS paper constructs these categories and functors for "Liouville sectors" (a suitably broad class of inclusions, related to Sylvan's notion of stops and partially wrapped Floer homology). I think the proof of the co-sheaf property is still ongoing work?

• Now you compute the local pieces of the Fukaya category and show that they agree with the microlocal sheaf categories; since both have co-sheaf gluing, you get the same global answers.

The second part relies on some local computations of Fukaya categories. Nadler has introduced the notion of "arboreal skeleton" which is a skeleton with certain "generic" singularities. For example, trivalent graphs in dimension 1; trivalent graph times interval or cone on 1-skeleton of a tetrahedron in dimension 2; etc. He computes the microlocal sheaf category for these; I'm not sure if the corresponding partially-wrapped Fukaya categories have been calculated in all cases yet. Finally, you want to show that any Weinstein manifold has an arboreal skeleton: Starkston has some results in this direction which may represent the state of the art.

Leaving this aside for a moment, there are also special cases where the Konstevich conjecture/local-to-global results for the Fukaya category has been established independently of this general program. These include (but again, I'm probably missing some):

I also attended the Seidel talk that people are referring to. I should have my notes somewhere; if I succeed in finding them and if they contain anything not already in Sheel Ganatra's notes, I'll scan them and post them.

Edit: Kontsevich spoke about this last week in Miami. Check out these notes from the talk. Find the abstract here.

• Good spot, Kevin! – Tim Perutz Jan 10 '10 at 16:33
• If only I could vote you up again... – S. Carnahan Jan 27 '10 at 3:23
• Since the two links mentioned in the first paragraph no longer work, I have added version from Internet Archive. It seems the same text is available here - and this link is still working: ihes.fr/~maxim/TEXTS/Symplectic_AT2009.pdf (just in case, here is link to the Wayback Machine). – Martin Sleziak Aug 30 '18 at 12:41

For brief, precise, and hopefully readable descriptions of the current state of the art, I suggest:

this illustrated introduction (section 1.1, 1.2, 1.3)

and

this list of applications (section 6)

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A comment re. Jonny's nice answer: there was indeed a time when that was the envisioned strategy of proof. However our present approach does not require the arborealization. Because: now we know that the Fukaya and microlocal categories associated to any (singular) Legendrian in a cosphere bundle agree.

Update (11/11/2019): In the updated version of GPS2, we prove descent for general sectorial covers. This is in my view better than the assertion that the Fukaya category is a cosheaf over a skeleton. In particular, one could deduce the existence of said cosheaf by an (unwritten) purely geometric argument that an open cover of the skeleton lifts to a sectorial cover of the symplectic manifold. However, the notion of sectorial cover is more flexible, as it is not tied to a particular skeleton; in a certain sense it captures all skeleta at once.

The virtue of the cosheaf on the skeleton is that it is not some arbitrary cosheaf of categories, but in fact is the Kashiwara-Schapira stack of microlocal sheaf theory. (Historical note: this assertion goes beyond the original conjecture of Kontsevich, and is probably best attributed to Nadler.) As mentioned above, GPS3 is the required local calculation. In principle, one could use GPS2 to glue, but actually there is a short-cut for most Weinstein manifolds of interest in mirror symmetry and geometric representation theory. The reason is that these manifolds typically have trivial stable symplectic normal bundle, so one can use the embedding trick without messing about with descent from a Lagrangian Grassmannian bundle. This reduces the problem to the cotangent comparison of GPS3, provided one has certain full faithfulness of embedding results in both partially wrapped Floer theory and in microlocal sheaf theory.

Full faithful embeddings come from a certain doubling construction, which was originally introduced in sheaf theory by Guillermou, it is described in e.g. section 11.4 of his omnibus. Guillermou assumes there that the relevant skeleton is smooth, but similar ideas give a construction in general and will appear in my upcoming paper with Nadler. On the wrapped Floer theory side the construction is Example 8.6 in GPS2. Combining the above one learns that for a Weinstein manifold with stably trivial symplectic normal bundle, the wrapped Fukaya category is equivalent to the category of microlocal sheaves on any Lagrangian skeleton.

(Most likely the `embedding trick' also works to establish the result in the general case, but this requires a bit more complicated argument.)

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Finally I want to clear up what seems to be a common confusion.

The cosheaf property along the Lagrangian skeleton, or equivalently the gluing of Fukaya categories of Liouville sectors, is sometimes mirror to the fact that the functor category of coherent sheaves carries colimits to colimits.

There is a different sort of local-to-global principle which the Fukaya category should satisfy, mirror to various descent statements in algebraic geometry; i.e. the fact that category of coherent sheaves is a sheaf in various topologies. This was proposed by Seidel, and established for Fukaya categories of noncompact surfaces by Lee. It is unknown to me whether there is an explicit general formulation of this principle.

Note that the former procedure is local over the Lagrangian skeleton, and the latter is local over e.g. the tropical skeleton. I do not know any actual relation between Lagrangian and tropical skeleta, although one can imagine that one can produce the former from some sort of combinatorial Morse work on the latter.

Similarly I do not know a general relation between the two sorts of locality, except that one can sometimes use the former to prove the latter. E.g., one can read this paper as using Lagrangian skeleton locality to offload the hard work of proving mirror-to-affine-locality to a competent sheaf theorist.

On this topic I've only seen Ganatra's notes from Paul's Talbot talk (see Scott's answer). An aspect of this, that a Fukaya category can behave in a sheafy way, is part of Nadler's proof that "microlocal branes are constructible sheaves".

The most relevant reference I can think of for the Fukaya category of the crepant resolution of $\mathbb{C}^2/\Gamma$ is Abouzaid's paper about Floer theory in plumbings.

Bad form as it may be, I'll ask a reciprocal question in this answer box. Suppose you knew that the Fukaya category of this crepant resolution was the constructible derived category on the exceptional fibre (was that what you had in mind?). What could one deduce? Would there then be a quiver presentation? A reasonable description of Hochschild cohomology?

I asked Paul a couple months ago, and he said that nothing has been written, but there may be notes floating around from some talks Kontsevich gave in France last March. There are a few examples given in the last talk at Talbot on the bottom of this page. I was unable to work out a precise mathematical statement from the information given, but I'm not a symplectic geometer.

Not sure if this is helpful to you, but Bing came upon this paper by Ruan which I'm thinking you likely have already seen and "get."

The Fukaya category of symplectic neighborhood of a non-Hausdorff manifold