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Microlocal Morse theory of wrapped Fukaya categories

Sheel Ganatra, John Pardon, Vivek Shende

TL;DR

This work generalizes the Nadler–Zaslow paradigm from infinitesimal to wrapped and stopped settings by establishing a precise equivalence between partially wrapped Fukaya categories and microlocal sheaf categories with prescribed microsupport. The authors develop a microlocal Morse framework and a doubling/antimicrolocalization strategy to reduce general Weinstein sectors to cotangent-bundle scenarios, enabling a uniform sheaf-theoretic description of wrapped objects. Key contributions include the fundamental equivalence Perf\mathcal{W}(T^*M,Λ)^{op} ≅ Sh_Λ(M)^c, a robust microlocal Morse theatre, and a suite of explicit computations (cotangent bundles, plumbings, Seidel-type categories) that illuminate mirror-symmetric phenomena in noncompact and singular settings. The results provide a powerful toolkit for translating wrapped-Fukaya data into microlocal sheaf calculations, with broad implications for homological mirror symmetry and toric/Weinstein geometry.

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.

Microlocal Morse theory of wrapped Fukaya categories

TL;DR

This work generalizes the Nadler–Zaslow paradigm from infinitesimal to wrapped and stopped settings by establishing a precise equivalence between partially wrapped Fukaya categories and microlocal sheaf categories with prescribed microsupport. The authors develop a microlocal Morse framework and a doubling/antimicrolocalization strategy to reduce general Weinstein sectors to cotangent-bundle scenarios, enabling a uniform sheaf-theoretic description of wrapped objects. Key contributions include the fundamental equivalence Perf\mathcal{W}(T^*M,Λ)^{op} ≅ Sh_Λ(M)^c, a robust microlocal Morse theatre, and a suite of explicit computations (cotangent bundles, plumbings, Seidel-type categories) that illuminate mirror-symmetric phenomena in noncompact and singular settings. The results provide a powerful toolkit for translating wrapped-Fukaya data into microlocal sheaf calculations, with broad implications for homological mirror symmetry and toric/Weinstein geometry.

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 and a subanalytic isotropic subset of its co-sphere bundle , we show that the partially wrapped Fukaya category of stopped at is equivalent to the category of compact objects in the unbounded derived category of sheaves on with microsupport inside . By an embedding trick, we also deduce a sheaf theoretic description of the wrapped Fukaya category of any Weinstein sector admitting a stable polarization.

Paper Structure

This paper contains 54 sections, 104 theorems, 107 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Weinstein manifolds and partially wrapped Fukaya categories
  3. Topological and sheaf theoretic interpretations
  4. Main results
  5. Acknowledgements
  6. Stratifications
  7. Generalities
  8. Microlocal approximation of constructible sets
  9. Cornering and conormals of constructible open sets
  10. Microlocal Morse categories
  11. A category for any Lambda
  12. Open inclusions
  13. Sheaf categories
  14. Categories of sheaves and functors between them
  15. Constructible sheaves
  16. ...and 39 more sections

Key Result

Theorem 1.1

Let $M$ be a real analytic manifold, and let $\Lambda \subseteq S^*M$ be a subanalytic closed isotropic subset. There there is a canonical equivalence of categories which carries the linking disk at any smooth Legendrian point $p\in\Lambda$ to a co-representative of the microstalk functor at $p\in\Lambda$ and carries the cotangent fiber at a point $p\in M$ not in the image of $\Lambda$ to a co-re

Figures (5)

  • Figure 1: A compact constructible set $X$ (left) and its outward cornering $X^{\underline\epsilon}$ (right).
  • Figure 2: The neighborhoods $X^{\underline\epsilon,\delta}$.
  • Figure 3: The isotopy from $U^{-\underline\epsilon,\delta+\eta}$ to $U^{-\underline\epsilon,\delta-\eta}$ to $U^{-\underline\epsilon}$.
  • Figure 4: The family of arcs $\gamma_\theta$ (top) and their smoothings $\tilde{\gamma}_\theta$ near the origin (bottom).
  • Figure 5: A standard cusp.

Theorems & Definitions (233)

  • Example
  • Theorem 1.1
  • Remark 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • ...and 223 more