Kontsevich cosheaf conjecture roughly states that wrapped Fukaya category can be recovered from local information on the Lagrangian skeleton. What are some reasons why would one believe it? I believe it looks most reasonable in the microlocal approach to Fukaya category but maybe there is some intuition for other models of Fukaya category as well.
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6$\begingroup$ Some naive reasons why it's reasonable to expect local information on the skeleton to know everything 1. It's true for cotangent bundles and represents a nice categorical enhancement of many historical results about cotangent bundles, starting with Viterbo/Abbondandolo-Schwarz results that symplectic homology of a cotangent bundle is the homology of the loopspace of the zero section. 2. Liouville flow retracts a Weinstein manifold onto its skeleton, so everything about symplectomorphism topology of the completion should be determined by an infinitesimal neighborhood of the skeleton. $\endgroup$– Jonny EvansCommented Feb 5, 2019 at 20:03
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3$\begingroup$ Maybe more speculative: 3. In mirror symmetry, you could think of R^n as a torus with very large radius (like how R is a circle with very large radius). When taking the dual Lagrangian torus fibration, a non-compact R^n fibre should therefore be dual to a point. You see this in recent work by Lekili and Polishchuk, where they find mirrors to punctured surfaces which are nodal curves: the structure sheaf at the node is mirror to a wrapped Lagrangian brane which is is a non-compact R going off to the puncture... $\endgroup$– Jonny EvansCommented Feb 5, 2019 at 20:04
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3$\begingroup$ ...So what is mirror to the Lagrangian torus fibration of the cotangent bundle by cotangent fibres? The zero section itself. Quite how you figure out that the derived category should be replaced by the category of microlocal sheaves, I don't know. $\endgroup$– Jonny EvansCommented Feb 5, 2019 at 20:04
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1$\begingroup$ In fact, Kontsevich explains the motivation for the conjecture in his own words in his paper "Symplectic geometry of homological algebra". $\endgroup$– Jonny EvansCommented Feb 5, 2019 at 21:47
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2$\begingroup$ @Jonny: your comments are great, and would deserve to be turned into an answer! $\endgroup$– DamienCCommented Dec 29, 2020 at 10:26
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On the suggestion of DamienC, I'm converting my comments into an answer. I didn't do this before because I don't really know the answer to the deeper question of why the microlocal sheaf category is the correct thing to give the Fukaya category (and I mostly just say things which are either obvious or wild speculation).
Some naive reasons why it's reasonable to expect local information on the skeleton to know everything:
It's true for cotangent bundles and represents a nice categorical enhancement of many historical results about cotangent bundles, starting with Viterbo/Abbondandolo-Schwarz results that symplectic homology of a cotangent bundle is the homology of the loopspace of the zero section.
Liouville flow retracts a Weinstein manifold onto its skeleton, so everything about symplectic topology of the completion should be determined by the germ of the manifold along the skeleton.
More speculatively:
- In mirror symmetry, you could think of $\mathbf{R}^n$ as a torus with very large radius (like how $\mathbf{R}$ is a circle with very large radius). When taking the dual Lagrangian torus fibration, a non-compact $\mathbf{R}^n$ fibre should therefore be dual to a point (circle with very small radius). You see examples of this in recent work by Lekili and Polishchuk (https://arxiv.org/abs/1705.06023), where they find mirrors to punctured surfaces which are nodal curves: the structure sheaf at the node is mirror to a wrapped Lagrangian brane which is a non-compact $\mathbf{R}$ going off to the puncture. So what is mirror to the Lagrangian $\mathbf{R}^n$-fibration of the cotangent bundle by cotangent fibres? The zero section itself. Quite how you figure out that the derived category should be replaced by the category of microlocal sheaves, I don't know.
In fact, Kontsevich explains the motivation for the conjecture in his own words in his paper "Symplectic geometry of homological algebra": https://www.ihes.fr/~maxim/TEXTS/Symplectic_AT2009.pdf
Edit^2: The following comments are misguided but I'll leave them here to give context to John Pardon's clarifying (and very illuminating) comments below.
Edit: I should add that the conjecture isn't going to be true if you allow arbitrary skeleta, only skeleta with mild (arboreal) singularities. In a talk by Daniel Alvarez-Gavela the other week, he mentioned the following example to see why this is true.
Take a Legendrian knot in the sphere (boundary of the ball) and attach a Weinstein handle along it. For a suitable choices of Liouville vector field, the skeleton of the resulting handlebody is just the core of the handle union the cone on the Legendrian. Topologically this is just homeomorphic to a sphere (but it is very singular at the cone point). The skeleton therefore doesn't depend on the knot, but the Fukaya category of the handlebody certainly does. You need to "arborealise" the singularity before you get a skeleton to which the Kontsevich conjecture can possibly apply.
answered Dec 30, 2020 at 19:17
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1$\begingroup$ The validity of the conjecture does not depend on having an arboreal skeleton. For example, the descent result here arxiv.org/abs/1809.03427 doesn't even make reference to the skeleton (or take the comparison with microlocal sheaves from arxiv.org/abs/1809.08807 and note that compact objects in microlocal sheaves form a cosheaf of categories). In the example of a ball with a single critical handle attached, there is a simple two-element sectorial cover to which the descent result applies. $\endgroup$ Commented Dec 31, 2020 at 21:46
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1$\begingroup$ What the example of a ball union a single critical handle does show is that the Kontsevich--Nadler sheaf of categories depends on more than just the topology of the core (the costalk at the cone point will be the partially wrapped Fukaya category of the ball stopped at the attaching locus, whereas everywhere else the costalk will simply be Perf Z). $\endgroup$ Commented Dec 31, 2020 at 21:47
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1$\begingroup$ In fact, even in the case of an arboreal skeleton, the cosheaf depends on more than just the local topology of the skeleton. For example, at the simplest arboreal singularity (the A2 "tripod" singularity, i.e. a trivalent vertex) the category depends on the cyclic ordering of the three incident edges induced by the ambient symplectic structure. $\endgroup$ Commented Dec 31, 2020 at 21:49
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$\begingroup$ Thanks for the clarifying comments, John! $\endgroup$ Commented Jan 1, 2021 at 10:01