# On finite symmetries and their gauging in two dimensions

## TL;DR

This paper generalizes the concept of gauging finite symmetries in two-dimensional theories from Abelian groups to non-Abelian fusion categories, revealing new dualities and symmetries in topological quantum field theories.

## Contribution

It extends the framework of symmetry gauging from groups to fusion categories and explores the resulting dualities and topological quantum field theories.

## Key findings

- Gauging non-Abelian symmetries leads to dual symmetries described by fusion categories.
- Gauging subgroups of anomalous finite groups can produce non-Abelian symmetries.
- The gauged theories are characterized as topological quantum field theories with dual categorical symmetries.

## Abstract

It is well-known that if we gauge a $\mathbb{Z}_n$ symmetry in two dimensions, a dual $\mathbb{Z}_n$ symmetry appears, such that re-gauging this dual $\mathbb{Z}_n$ symmetry leads back to the original theory. We describe how this can be generalized to non-Abelian groups, by enlarging the concept of symmetries from those defined by groups to those defined by unitary fusion categories. We will see that this generalization is also useful when studying what happens when a non-anomalous subgroup of an anomalous finite group is gauged: for example, the gauged theory can have non-Abelian group symmetry even when the original symmetry is an Abelian group. We then discuss the axiomatization of two-dimensional topological quantum field theories whose symmetry is given by a category. We see explicitly that the gauged version is a topological quantum field theory with a new symmetry given by a dual category.