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Microsheaf composition of Lagrangian correspondences

Wenyuan Li, David Nadler, Vivek Shende

TL;DR

This work extends the Kashiwara–Schapira microsheaf framework from cotangent bundles to Weinstein manifolds by defining conic microsheaf quantizations $\mu sh_W$ and constructing functors from immersed Lagrangian correspondences. It proves that, under suitable Legendrian/displaceability hypotheses, microsheaf kernels compose in a way that commutes with the geometric composition of Lagrangians, and shows compatibility with the existing sheaf-quantization diagram via a new family gappedness criterion for commuting with nearby cycles. The paper develops a comprehensive toolkit—doubling, Künneth, duality, and microlocal nearby cycles—and demonstrates three key applications: Viterbo transfers, sheaf quantization of contact isotopies, and Hamiltonian group actions on microsheaf categories, with connections to wrapped Fukaya categories. A central achievement is the gapped composition framework, which provides precise conditions under which tensor and Hom compositions of kernels commute with nearby cycles, enabling robust, functorial quantization of Lagrangian correspondences in a broad symplectic setting. Collectively, these results advance a 2-categorical, sheaf-theoretic understanding of Lagrangian correspondences and their quantizations, with potential implications for symplectic topology, representation theory, and topological field theories.

Abstract

In exact symplectic manifolds whose Liouville flow is gradientlike for a proper Morse function, one can associate conic microsheaves to eventually conic exact Lagrangians. Here we study how this 'microsheaf quantization' interacts with composition of Lagrangian correspondences. In particular: these operations commute when the composition is embedded. As an illustration, we show that Lie groups of exact symplectomorphisms act on microsheaf categories. The key technical advance is a version 'in families' of the gappedness criterion for commuting nearby cycles past tensor or Hom.

Microsheaf composition of Lagrangian correspondences

TL;DR

This work extends the Kashiwara–Schapira microsheaf framework from cotangent bundles to Weinstein manifolds by defining conic microsheaf quantizations and constructing functors from immersed Lagrangian correspondences. It proves that, under suitable Legendrian/displaceability hypotheses, microsheaf kernels compose in a way that commutes with the geometric composition of Lagrangians, and shows compatibility with the existing sheaf-quantization diagram via a new family gappedness criterion for commuting with nearby cycles. The paper develops a comprehensive toolkit—doubling, Künneth, duality, and microlocal nearby cycles—and demonstrates three key applications: Viterbo transfers, sheaf quantization of contact isotopies, and Hamiltonian group actions on microsheaf categories, with connections to wrapped Fukaya categories. A central achievement is the gapped composition framework, which provides precise conditions under which tensor and Hom compositions of kernels commute with nearby cycles, enabling robust, functorial quantization of Lagrangian correspondences in a broad symplectic setting. Collectively, these results advance a 2-categorical, sheaf-theoretic understanding of Lagrangian correspondences and their quantizations, with potential implications for symplectic topology, representation theory, and topological field theories.

Abstract

In exact symplectic manifolds whose Liouville flow is gradientlike for a proper Morse function, one can associate conic microsheaves to eventually conic exact Lagrangians. Here we study how this 'microsheaf quantization' interacts with composition of Lagrangian correspondences. In particular: these operations commute when the composition is embedded. As an illustration, we show that Lie groups of exact symplectomorphisms act on microsheaf categories. The key technical advance is a version 'in families' of the gappedness criterion for commuting nearby cycles past tensor or Hom.

Paper Structure

This paper contains 52 sections, 148 theorems, 427 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Sample applications
  3. Viterbo functor
  4. Sheaf quantization of contact isotopy
  5. Hamiltonian action and equivariant microsheaves
  6. Sufficiently Legendrian subsets
  7. Contact isotopies
  8. Positive displaceable from fibers
  9. Self-displaceability
  10. Recollections on nearby cycles
  11. Microsupport estimates
  12. Commuting nearby cycles
  13. Nearby cycles and exterior tensor products
  14. Global sections of nearby cycles and tensor products
  15. Recollections on microsheaves
  16. ...and 37 more sections

Key Result

Proposition 2.1.1

Given a Liouville subdomain embedding of Weinstein manifolds $i_{01}: W_0 \hookrightarrow W_1$, the image of the constant rank one local system under is the functor $i_{01*}: \operatorname{\mu sh}_{\mathfrak{c}_{W_0}}(\mathfrak{c}_{W_0}) \to \operatorname{\mu sh}_{\mathfrak{c}_{W_1}}(\mathfrak{c}_{W_1})$.

Figures (8)

  • Figure 1: Part of the diagram of non-proper base changes that appears in the commutativity of the parametric nearby cycle functors, where the natural transformation in each square is induced by adjunction of six functors.
  • Figure 2: Part of the diagram of that appears in the commutativity of the exterior tensor products and nearby cycle functors, where the natural transformation in each square is induced by adjunctions of six functors.
  • Figure 3: The self gapped family of Legendrians $\Lambda_t \subset S^*_{\eta>0}\mathbb{R}^2$ that has non-gapped doubling in Example \ref{['ex: gapped collar']}. The figure on the left and middle are the Lagrangian projections of the Legendrians $\Lambda_t$ and doublings $(\Lambda_t)_{\pm\epsilon}$ where the Reeb chords are the double points. The figure on the right is the front projection of the doubling $(\Lambda_t)_{\pm\epsilon}$.
  • Figure 4: Let the base be $M_1 = M_2 = \mathbb{R}$ and the Legendrians be $\Lambda_1 = \Lambda_2 = \{(0, 1)\} \subset S^*\mathbb{R}$. The figure on the left shows the front projection of $(\Lambda_1)_{\cup,s} \times (\Lambda_2)_{\cup,s}$, $(\Lambda_1 \,\widehat{\times}\, \Lambda_2)_{\cup,s'}$ on the base $\mathbb{R}^2$. The figure on the right shows the $U$-shape Lagrangian filling $\Lambda_1 \times \cup_{-s,s}$ and $\Lambda_2 \times \cup_{-s,s}$ into $T^*\mathbb{R}$.
  • Figure 5: Let $W = pt$ and $L \subset W \times \mathbb{R}$ be a Legendrian consisting of two points whose Reeb chord has length less than $\epsilon$. The figure on the bottom shows the Lagrangians $L \times \cup_{0,+\infty}$, $\mathfrak{c}_W^L = (\mathfrak{c}_{W} \times \mathbb{R}) \cup (L \times \mathbb{R}_{>0})$ and $\mathfrak{c}_W \times \sqcup_{0,+\infty} \subset W \times T^*\mathbb{R}$, and the figure on the top shows the front projection of the doublings $(\underline{L} \times \cup_{0,+\infty})_{\cup,\epsilon}$, $(\underline{\mathfrak{c}}{}_{W}^L)_{\cup,\epsilon}$, and $(\underline{\mathfrak{c}}{}_{W} \times \sqcup_{0,+\infty})_{\cup,\epsilon} \subset S^*\mathbb{R}^2$ under the contact embedding onto the base $\mathbb{R}^2$.
  • ...and 3 more figures

Theorems & Definitions (328)

  • Proposition 2.1.1
  • proof
  • Corollary 2.1.2
  • proof
  • Proposition 2.2.1
  • proof
  • Remark 2.2.2
  • Definition 2.2.3
  • Corollary 2.2.4
  • proof
  • ...and 318 more