Arithmetic-Geometric Correspondence of Character Stacks via Topological Quantum Field Theory

TL;DR

This paper develops a Topological Quantum Field Theory framework that links arithmetic and geometric methods to study cohomological invariants of G-character stacks, revealing deep correspondences and simplifying calculations.

Contribution

It introduces a novel TQFT-based arithmetic-geometric correspondence that generalizes classical character formulas and connects geometric and arithmetic approaches.

Findings

Established a natural transformation linking arithmetic and geometric TQFTs.

Provided new insights into the character tables of finite groups.

Simplified geometric calculations for upper triangular matrices.

Abstract

In this paper, we introduce Topological Quantum Field Theories (TQFTs) generalizing the arithmetic computations done by Hausel and Rodr\'iguez-Villegas and the geometric construction done by Logares, Mu\~noz, and Newstead to study cohomological invariants of $G$-representation varieties and $G$-character stacks. We show that these TQFTs are related via a natural transformation that we call the 'arithmetic-geometric correspondence' generalizing the classical formula of Frobenius on the irreducible characters of a finite group. We use this correspondence to extract some information on the character table of finite groups using the geometric TQFT, and vice versa, we greatly simplify the geometric calculations in the case of upper triangular matrices by lifting its irreducible characters to the geometric setting.