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Sheaf quantization in Weinstein symplectic manifolds

David Nadler, Vivek Shende

TL;DR

This work develops a comprehensive microlocal-sheaf theoretic framework for Weinstein symplectic manifolds, constructing a category \\mathfrak{Sh}(X) via microlocalization and defining a microlocal specialization that produces objects from exact Lagrangians. Central technical advances include a full faithfulness result for nearby cycles under gapped specialization and an antimicrolocalization theory that realizes microsheaves as localizations of ordinary sheaves, enabling a robust specialization functor. The authors further develop the embedding trick, collars, ribbons, and relative doubling to quantize Legendrians and Lagrangians and to establish invariance under Weinstein deformations, with Maslov data governing descent and monodromy. The framework connects with wrapped Fukaya categories in stable polarization regimes, providing a sheaf-theoretic handle on quantization that is compatible with polarizations, relative settings, and Weinstein pairs, and it yields a versatile toolkit for comparing microlocal and Fukaya-type invariants in symplectic topology.

Abstract

Using the microlocal theory of sheaves, we associate a category to each Weinstein manifold. By constructing a microlocal specialization functor, we show that exact Lagrangians give objects in our category, and that the category is invariant under Weinstein homotopy.

Sheaf quantization in Weinstein symplectic manifolds

TL;DR

This work develops a comprehensive microlocal-sheaf theoretic framework for Weinstein symplectic manifolds, constructing a category \\mathfrak{Sh}(X) via microlocalization and defining a microlocal specialization that produces objects from exact Lagrangians. Central technical advances include a full faithfulness result for nearby cycles under gapped specialization and an antimicrolocalization theory that realizes microsheaves as localizations of ordinary sheaves, enabling a robust specialization functor. The authors further develop the embedding trick, collars, ribbons, and relative doubling to quantize Legendrians and Lagrangians and to establish invariance under Weinstein deformations, with Maslov data governing descent and monodromy. The framework connects with wrapped Fukaya categories in stable polarization regimes, providing a sheaf-theoretic handle on quantization that is compatible with polarizations, relative settings, and Weinstein pairs, and it yields a versatile toolkit for comparing microlocal and Fukaya-type invariants in symplectic topology.

Abstract

Using the microlocal theory of sheaves, we associate a category to each Weinstein manifold. By constructing a microlocal specialization functor, we show that exact Lagrangians give objects in our category, and that the category is invariant under Weinstein homotopy.

Paper Structure

This paper contains 43 sections, 83 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Propagation and displacement
  3. Positively displaceable from legendrians
  4. Displaceability, Reeb chords, and gappedness
  5. Microlocal criterion for comparing mixed restrictions
  6. Relative microsupport
  7. Microsupport estimates
  8. Relative microsupport
  9. Microsupport of nearby cycles
  10. Nearby cycles and relative external Hom
  11. Stalks of nearby cycles
  12. The first arrow
  13. The second arrow
  14. On the full faithfulness of nearby cycles
  15. Microsheaves
  16. ...and 28 more sections

Key Result

Theorem 1.1

Let $(V, \xi)$ be a contact manifold, and $C$ a stable presentable symmetric monoidal category. A null-homotopy of the $C$-valued Maslov obstruction $V \to B^2 Pic(C)$ determines a sheaf of $C$-linear stable categories $\mu sh$ on $V$, locally isomorphic to microsheaves on a cosphere bundle of the s

Theorems & Definitions (221)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Remark 2.5
  • Lemma 2.6
  • proof
  • ...and 211 more