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On the Hochschild cohomology of Tamarkin categories

Christopher Kuo, Vivek Shende, Bingyu Zhang

TL;DR

The paper develops a filtered Hochschild-theoretic framework for Tamarkin categories associated to open subsets of cotangent bundles, proving that the action-filtered Hochschild cohomology of the Tamarkin category 𝒯(U) matches the corresponding filtered symplectic cohomology SH of the neighborhood. It introduces a projector calculus for traces in the Tamarkin setting, expressing traces via integral kernels and wrapping constructs, and relates these to the Guillermou–Viterbo comparison to Floer theory. A Calabi–Yau structure is established for 𝒯(U), tying Hochschild (co)homology to trace data, and the action-window formalism provides a precise link between 𝒯(U) and SH with filtration. The results extend to generating function homology and clarify how capacities defined via SH are recovered in the Tamarkin/sheaf-theoretic framework, offering a robust bridge between microlocal sheaf theory and filtered Floer theory. Overall, the work furnishes a variational, sheaf-theoretic route to filtered symplectic invariants and their algebraic structures, reinforcing connections across Tamarkin, wrapped Fukaya categories, and generating function approaches.

Abstract

To any open subset of a cotangent bundle, Tamarkin has associated a certain quotient of a category of sheaves. Here we show that the Hochschild cohomology of this category agrees with filtered symplectic cohomology.

On the Hochschild cohomology of Tamarkin categories

TL;DR

The paper develops a filtered Hochschild-theoretic framework for Tamarkin categories associated to open subsets of cotangent bundles, proving that the action-filtered Hochschild cohomology of the Tamarkin category 𝒯(U) matches the corresponding filtered symplectic cohomology SH of the neighborhood. It introduces a projector calculus for traces in the Tamarkin setting, expressing traces via integral kernels and wrapping constructs, and relates these to the Guillermou–Viterbo comparison to Floer theory. A Calabi–Yau structure is established for 𝒯(U), tying Hochschild (co)homology to trace data, and the action-window formalism provides a precise link between 𝒯(U) and SH with filtration. The results extend to generating function homology and clarify how capacities defined via SH are recovered in the Tamarkin/sheaf-theoretic framework, offering a robust bridge between microlocal sheaf theory and filtered Floer theory. Overall, the work furnishes a variational, sheaf-theoretic route to filtered symplectic invariants and their algebraic structures, reinforcing connections across Tamarkin, wrapped Fukaya categories, and generating function approaches.

Abstract

To any open subset of a cotangent bundle, Tamarkin has associated a certain quotient of a category of sheaves. Here we show that the Hochschild cohomology of this category agrees with filtered symplectic cohomology.
Paper Structure (29 sections, 73 theorems, 229 equations, 2 figures)

This paper contains 29 sections, 73 theorems, 229 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Traces and dualizable categories
  3. Dualizing datum for retraction
  4. Extension of scalars
  5. Trace of
  6. The Tamarkin category of a point
  7. Symmetric monoidal structure from convolution on
  8. Filtered complexes
  9. Some properties of Tamarkin categories
  10. Tamarkin category as a sheaf category
  11. Dualizability
  12. Some compatibilities of -linear kernels
  13. Sheaf quantization of Hamiltonian isotopies
  14. Trace of the Tamarkin category
  15. Projector via Fourier transform and cutoff
  16. ...and 14 more sections

Key Result

Theorem 1.1

The category $\mathscr{T} (U)$ is linear over $\mathscr{T}$, and $\mathscr{T}$-linearly dualizable. Thus we may take the $\mathscr{T}$-linear trace and in particular have $\operatorname{Tr}(1_{ \mathscr{T} (U)}) \in \mathscr{T}$.

Figures (2)

  • Figure :
  • Figure :

Theorems & Definitions (157)

  • Theorem 1.1: Prop. \ref{['t linear']} and \ref{['t dualizable']}
  • Theorem 1.2
  • Corollary 1.3
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 147 more