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The Large Affine Hecke Category is Calabi-Yau

Yuji Okitani

Abstract

We show that the large affine Hecke category defines an oriented, fully extended topological field theory. More generally, we establish conditions under which ind-coherent convolution categories define such theories, analogously to known results for finite Hecke categories and $D$-modules.

The Large Affine Hecke Category is Calabi-Yau

Abstract

We show that the large affine Hecke category defines an oriented, fully extended topological field theory. More generally, we establish conditions under which ind-coherent convolution categories define such theories, analogously to known results for finite Hecke categories and -modules.
Paper Structure (23 sections, 13 theorems, 29 equations)

This paper contains 23 sections, 13 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Idea of the Proof
  3. Small vs Large
  4. Affine Character Sheaves
  5. Application to 3d Mirror Symmetry
  6. Organization of the Paper
  7. Acknowledgments
  8. Topological Quantum Field Theories
  9. Review of TQFTs
  10. Traces and Centers
  11. Oriented TQFTS
  12. Weakly Oriented TQFTS
  13. Specializing to the Morita 2-Category
  14. Ind-coherent Sheaves
  15. Assumptions on Stacks
  16. ...and 8 more sections

Key Result

Theorem 1.2

The affine Hecke category $\mathcal{H}^{\operatorname{aff}}$ is 2-dualizable with a natural Calabi-Yau structure, so that it defines a fully extended and oriented 2-dimensional TQFT. In particular, there is a natural identification of the center of $\mathcal{H}^{\operatorname{aff}}$ with its trace.

Theorems & Definitions (34)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4: ben-zviCharacterTheoryComplex2015*Theorem 3.18
  • Example 3.1
  • Remark 3.2
  • ...and 24 more