Fusion Rules for $\mathbb{Z}/2\mathbb{Z}$ Permutation Gauging

TL;DR

This paper derives explicit fusion rules for gauging the $\

Contribution

It provides formulas for fusion rules of extensions and equivariantizations after gauging $\

Findings

Fusion rules expressed in terms of original category $\

Formulas involve modular data of $\

Applications to quantum groups at roots of unity

Abstract

In this note, we examine the gauging of the $\mathbb{Z}/2\mathbb{Z}$ permutation action on the tensor square of a modular tensor category. When $\mathcal{C}$ has no nontrivial invertible objects, we provide formulas for the fusion rules of both the extensions, expressed in terms of the fusion rules of $\mathcal{C}$, and the subsequent equivariantizations, which additionally requires the modular data of $\mathcal{C}$. We discuss several examples related to quantum groups at roots of unity.