Microlocal Morse theory of wrapped Fukaya categories

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 $\Lambda$ of its co-sphere bundle $S^{\ast }M$, we show that the partially wrapped Fukaya category of $T^{\ast }M$ stopped at $\Lambda$ is equivalent to the category of compact objects in the unbounded derived category of sheaves on $M$ with microsupport inside $\Lambda$. By an embedding trick, we also deduce a sheaf theoretic description of the wrapped Fukaya category of any Weinstein sector admitting a stable polarization.

arXiv comment87 pages, final version to appear in Annals of Mathematics

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Cited by in corpus (27)

• Mirror symmetry for very affine hypersurfaces
• Arboreal Singularities in Weinstein Skeleta
• Sheaf quantization in Weinstein symplectic manifolds
• Aspects of functoriality in homological mirror symmetry for toric varieties
• Homological mirror symmetry at large volume
• Perverse sheaves of categories and some applications
• Derived categories of character sheaves
• Sheaves via augmentations of Legendrian surfaces
• Continuous and coherent actions on wrapped Fukaya categories
• Fiber Floer cohomology and conormal stops
• Augmentations are sheaves for Legendrian graphs
• Mirror symmetry for Berglund-Hübsch Milnor fibers
• Nilpotent invariance of semi-topological K-theory of dg-algebras and the lattice conjecture
• Arboreal singularities from Lefschetz fibrations
• Mirror symmetry for perverse schobers from birational geometry
• Mirror Symmetry for Truncated Cluster Varieties
• Representations, sheaves, and Legendrian $(2,m)$ torus links
• Monodromy of monomially admissible Fukaya-Seidel categories mirror to toric varieties
• Knot Categorification from Mirror Symmetry, Part II: Lagrangians
• Local mirror symmetry via SYZ
• Gamma II for toric varieties from integrals on T-dual branes and homological mirror symmetry
• Representing the big tilting sheaves as holomorphic Morse branes
• Exotic families of symplectic manifolds with Milnor fibers of $ADE$-type
• Categorical non-properness in wrapped Floer theory
• Augmentations and sheaves for links
• Lagrangian cobordism functor in microlocal sheaf theory I
• Microsheaves from Hitchin fibers via Floer theory