[Paper Review] Microlocal Morse theory of wrapped Fukaya categories

Sheel Ganatra, John Pardon+1 more|arXiv (Cornell University)|Sep 24, 2018

Homotopy and Cohomology in Algebraic Topology57 references47 citations

TL;DR

The paper proves an equivalence between partially wrapped Fukaya categories of cotangent bundles stopped at a subanalytic isotropic set and compact objects in unbounded derived categories of sheaves with microsupport in that stop, and extends to stably polarized Weinstein sectors via a microlocal framework.

ABSTRACT

The Nadler--Zaslow correspondence famously identifies the finite-dimensional Floer homology groups between Lagrangians in cotangent bundles with the finite-dimensional Hom spaces between corresponding constructible sheaves. We generalize this correspondence to incorporate the infinite-dimensional spaces of morphisms 'at infinity', given on the Floer side by Reeb trajectories (also known as "wrapping") and on the sheaf side by allowing unbounded infinite rank sheaves which are categorically compact. When combined with existing sheaf theoretic computations, our results confirm many new instances of homological mirror symmetry. More precisely, given a real analytic manifold $M$ and a subanalytic isotropic subset $Λ$ of its co-sphere bundle $S^*M$, we show that the partially wrapped Fukaya category of $T^*M$ stopped at $Λ$ is equivalent to the category of compact objects in the unbounded derived category of sheaves on $M$ with microsupport inside $Λ$. By an embedding trick, we also deduce a sheaf theoretic description of the wrapped Fukaya category of any Weinstein sector admitting a stable polarization.

Motivation & Objective

• Motivate the study of wrapped Fukaya categories in non-compact settings and their relation to sheaf theory.
• Generalize Nadler–Zaslow to include infinite-dimensional morphism spaces at infinity (wrapping).
• Establish a precise equivalence between wrapped Fukaya categories stopped at a stop and compact objects in microsupported sheaf categories.

Proposed method

• Define and work with partially wrapped Fukaya categories W(X, Lambda) stopped at Lambda.
• Use stratifications, microlocal sheaf theory, and microlocal Morse theory to model objects and morphisms.
• Prove an axiom-driven equivalence by comparing functors on Lambda and their behavior under inclusions (Lambda subset Lambda').
• Relate linking disks and microstalks via microlocal Morse descriptions and wrapping exact triangles.
• Extend to real-analytic Liouville manifolds X with subanalytic isotropic stops and stable polarizations to obtain a functor to microlocal sheaf categories.

Experimental results

Research questions

• RQ1 Can the partially wrapped Fukaya category stopped at a subanalytic isotropic stop Lambda be fully described in terms of microlocal sheaf theory?
• RQ2 Does Perf W(T* M, Lambda)^{op} identify with the compact objects in Sh_Lambda(M)?
• RQ3 How does the wrap/stop operation correspond to microlocal categorical operations on the sheaf side?
• RQ4 Can the cotangent bundle case be extended to general stably polarized Weinstein sectors via a doubling/antimicrolocalization approach?
• RQ5 What are the implications for homological mirror symmetry in terms of microlocal sheaves and wrapped Floer theory?

Key findings

• There is a canonical equivalence Perf W(T* M, Lambda)^{op} ≃ Sh_Lambda(M)^{c} for real analytic M and subanalytic isotropic Lambda inside S^*M.
• The equivalence sends linking disks at smooth Legendrian points to co-representatives of microstalk functors, and cotangent fibers to co-representatives of stalk functors.
• The result provides a sheaf-theoretic description of wrapped Fukaya categories via compact objects in unbounded derived categories with microsupport in Lambda.
• The approach relies on axiomatizing Lambda ↦ Perf W(T* M, Lambda)^{op} and matching it with Lambda ↦ Sh_Lambda(M)^{c} under inclusions, leveraging Whitney stratifications and microlocal Morse theory.
• Theorem 1.4 extends the equivalence to general stable-polarization Liouville manifolds, embedding Perf W(X, Lambda)^{op} into μ sh_{c_{X, Lambda}}(c_{X, Lambda})^{c} and becoming an equivalence under Weinstein/homological cocores.
• The framework provides a path to reinterpret many computations in mirror symmetry as microlocal sheaf category calculations.