Symmetries of 2d TQFTs and Equivariant Verlinde Formulae for General Groups

TL;DR

This paper analyzes symmetries in 2d semisimple TQFTs, showing how they act and how to gauge them, leading to a generalized Verlinde formula with applications to Hitchin moduli spaces.

Contribution

It provides an explicit description of symmetry actions and gauging in 2d TQFTs, and generalizes the equivariant Verlinde formula for arbitrary Lie groups.

Findings

0-form symmetries act as permutations

1-form symmetries act by phases

Generalized Verlinde formula predicts Hitchin moduli space geometry

Abstract

We study (generalized) discrete symmetries of 2d semisimple TQFTs. These are 2d TQFTs whose fusion rules can be diagonalized. We show that, in this special basis, the 0-form symmetries always act as permutations while 1-form symmetries act by phases. This leads to an explicit description of the gauging of these symmetries. One application of our results is a generalization of the equivariant Verlinde formula to the case of general Lie groups. The generalized formula leads to many predictions for the geometry of Hitchin moduli spaces, which we explicitly check in several cases with low genus and SO(3) gauge group.