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Betti Tate's thesis and the trace of perverse schobers

Benjamin Gammage, Justin Hilburn

TL;DR

The paper formulates and proves a Betti-analytic analogue of Tate's thesis, identifying the categorical trace of the 2-category of perverse schobers with an automorphic-to-spectral equivalence on Betti loop spaces. By combining a Betti geometric class field theory, a 3d mirror symmetry framework, and a spectral-trace result of Ben-Zvi–Nadler–Preygel, the authors establish the conjectured trace equivalence in the simplest nontrivial case, where the A-model trace of spherical adjunctions matches cosheaves on the Betti loop space. The construction hinges on a detailed analysis of cosheaves on placid ind-schemes, a stratified description of the Betti loop space, and explicit finite-type approximations leading to an equivalence $\mathsf{Sh}^!_\mathcal{S}(\mathfrak{L}\mathbb{C}) \simeq \mathsf{IndCoh}(\mathcal{L}_B(\mathbb{C}/\mathbb{C}^\times))$. This work provides a blueprint for extending trace-like invariants of 2-categories (such as the 2-category of perverse schobers) to loop-space sheaf theories, with implications for 3d- and 4d-topological field theories and holomorphic symplectic geometry.

Abstract

We propose a conjecture on the categorical trace of the 2-category of perverse schobers (expected to model the Fukaya-Fueter 2-category of a holomorphic symplectic space). By proving a Betti geometric version of Tate's thesis, and combining it with our previous 3d mirror symmetry equivalence and the Ben-Zvi--Nadler--Preygel result on spectral traces, we are able to establish our conjecture in the simplest interesting case.

Betti Tate's thesis and the trace of perverse schobers

TL;DR

The paper formulates and proves a Betti-analytic analogue of Tate's thesis, identifying the categorical trace of the 2-category of perverse schobers with an automorphic-to-spectral equivalence on Betti loop spaces. By combining a Betti geometric class field theory, a 3d mirror symmetry framework, and a spectral-trace result of Ben-Zvi–Nadler–Preygel, the authors establish the conjectured trace equivalence in the simplest nontrivial case, where the A-model trace of spherical adjunctions matches cosheaves on the Betti loop space. The construction hinges on a detailed analysis of cosheaves on placid ind-schemes, a stratified description of the Betti loop space, and explicit finite-type approximations leading to an equivalence . This work provides a blueprint for extending trace-like invariants of 2-categories (such as the 2-category of perverse schobers) to loop-space sheaf theories, with implications for 3d- and 4d-topological field theories and holomorphic symplectic geometry.

Abstract

We propose a conjecture on the categorical trace of the 2-category of perverse schobers (expected to model the Fukaya-Fueter 2-category of a holomorphic symplectic space). By proving a Betti geometric version of Tate's thesis, and combining it with our previous 3d mirror symmetry equivalence and the Ben-Zvi--Nadler--Preygel result on spectral traces, we are able to establish our conjecture in the simplest interesting case.
Paper Structure (12 sections, 13 theorems, 28 equations)

This paper contains 12 sections, 13 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. 2-categories and trace decategorification
  3. A-side traces
  4. Further remarks
  5. Categorical conventions
  6. Acknowledgements
  7. Proof of \ref{['mainthm:betti-tate']}
  8. Cosheaves on a placid ind-scheme
  9. Stratification of the loop space
  10. The Betti automorphic category
  11. The Betti spectral category
  12. Constructing a functor

Key Result

Theorem A

There is an equivalence between the category of (co)sheaves on $\mathfrak{L} \mathbb{C}$ with singular support in conormals to strata of $\mathcal{S}$ and the category of ind-coherent sheaves on $\mathcal{Y}.$ This equivalence intertwines the structure of module categories for the monoidal category whose automorphic and sp where $\mathsf{Loc}$ denotes the category of (possibly infinite-dimensiona

Theorems & Definitions (30)

  • Theorem A: "Betti Tate's thesis"
  • Theorem 1.1: GHMG
  • Theorem 1.2: BZNP*Theorem 1.2.12
  • Corollary 1.3
  • Conjecture A
  • Theorem B
  • Remark 1.4
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • ...and 20 more