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Integral Transforms for Finite Gauge Theory

Jackson Van Dyke

TL;DR

The paper addresses the quantization of $\\pi$-finite spaces in dimension three and analyzes how symmetries act on the resulting TQFTs at two levels: projectively on the full phase space and linearly on a fixed Lagrangian. It introduces a twice-categorified analogue of Blattner-Kostant-Sternberg kernels to describe projective actions of orthogonal symmetries on the Dijkgraaf-Witten-type theory $\\sigma_{BL}^3$, while Aut$(L)$ yields coherent linear actions, linking to metaplectic-type phenomena for finite abelian metric groups. The framework relies on a $3$-functor formalism for spans of $\\pi$-finite groupoids, enabling canonical bimodule-categorical descriptions of these symmetries and their obstructions, including an anomaly captured by a $4$-dimensional bulk theory in the orthogonal case. Connections are drawn to Crane-Yetter, Reshetikhin-Turaev/Turaev-Viro theories, and relative Langlands-type structures, illustrating how gauging symmetries of finite phase-space data produces enriched $3$-manifold invariants with potential physical and mathematical applications. Overall, the work establishes a precise correspondence between higher-categorical quantization of finite spaces and the interplay of linear vs. projective symmetry actions in $3$-dimensional topological gauge theories.

Abstract

This paper shows that quantization of $π$-finite spaces, as a functor out of a higher category of spans, is equivariant in two ways: Symmetries of a given polarization/Lagrangian always induce coherent symmetries of the quantization. On the other hand, symmetries of the entire phase space a priori only induce projective symmetries, with an invertible once-categorified theory, the anomaly theory, encoding the projectivity. We give projective symmetries of three-dimensional finite gauge theories a concrete description via a twice-categorified analogue of Blattner-Kostant-Sternberg kernels and the associated integral transforms, such as the Fourier transform. This establishes an analogy between certain instances of the $π$-finite quantization procedure considered herein and the geometric quantization of a symplectic vector space.

Integral Transforms for Finite Gauge Theory

TL;DR

The paper addresses the quantization of -finite spaces in dimension three and analyzes how symmetries act on the resulting TQFTs at two levels: projectively on the full phase space and linearly on a fixed Lagrangian. It introduces a twice-categorified analogue of Blattner-Kostant-Sternberg kernels to describe projective actions of orthogonal symmetries on the Dijkgraaf-Witten-type theory , while Aut yields coherent linear actions, linking to metaplectic-type phenomena for finite abelian metric groups. The framework relies on a -functor formalism for spans of -finite groupoids, enabling canonical bimodule-categorical descriptions of these symmetries and their obstructions, including an anomaly captured by a -dimensional bulk theory in the orthogonal case. Connections are drawn to Crane-Yetter, Reshetikhin-Turaev/Turaev-Viro theories, and relative Langlands-type structures, illustrating how gauging symmetries of finite phase-space data produces enriched -manifold invariants with potential physical and mathematical applications. Overall, the work establishes a precise correspondence between higher-categorical quantization of finite spaces and the interplay of linear vs. projective symmetry actions in -dimensional topological gauge theories.

Abstract

This paper shows that quantization of -finite spaces, as a functor out of a higher category of spans, is equivariant in two ways: Symmetries of a given polarization/Lagrangian always induce coherent symmetries of the quantization. On the other hand, symmetries of the entire phase space a priori only induce projective symmetries, with an invertible once-categorified theory, the anomaly theory, encoding the projectivity. We give projective symmetries of three-dimensional finite gauge theories a concrete description via a twice-categorified analogue of Blattner-Kostant-Sternberg kernels and the associated integral transforms, such as the Fourier transform. This establishes an analogy between certain instances of the -finite quantization procedure considered herein and the geometric quantization of a symplectic vector space.
Paper Structure (6 sections, 1 theorem, 1 equation)

This paper contains 6 sections, 1 theorem, 1 equation.

Table of Contents

  1. Introduction
  2. Relationship with other work
  3. Three-functor formalisms
  4. Outline
  5. Acknowledgements
  6. Orthogonal symmetries

Key Result

Theorem 1

The projective action of $\mathop{\mathrm{O}}\nolimits\left(A = L\oplus L^\vee\right)$ on $\sigma_{BL}^3$ can be described via explicit bimodule categories. Explicitly, any element of $\mathop{\mathrm{O}}\nolimits\left(A\right)$ exchanging $L$ with $L^\vee$ acts on $\sigma_{BL}^3$ via a Fourier-type

Theorems & Definitions (1)

  • Theorem