On $\mathbb{Z}/2\mathbb{Z}$ permutation gauging

TL;DR

This paper constructs a unitary $Z/2Z$ permutation gauging of a modular category, providing explicit formulas for the modular data and fusion rules, and explores implications for mapping class group representations.

Contribution

It explicitly constructs the $Z/2Z$ permutation gauging of a modular category and derives formulas for modular data and fusion rules, confirming conjectures and revealing new identities.

Findings

Provided explicit formula for modular data of gauged theory

Verified the conjectured fusion rules formula

Proved finiteness of certain mapping class group representations

Abstract

We explicitly construct a (unitary) $\mathbb{Z}/2\mathbb{Z}$ permutation gauging of a (unitary) modular category $\mathcal{C}$. In particular, the formula for the modular data of the gauged theory is provided in terms of modular data of $\mathcal{C}$, which provides positive evidence of the reconstruction program. Moreover as a direct consequence, the formula for the fusion rules is derived, verifying the conjectured formula of Edie-Michell-Jones-Plavnik. Our construction explicitly shows the genus-$0$ data of the gauged theory contains higher genus data of the original theory. As applications, we obtain an identity for the modular data that does not come from modular group relations, and we prove that representations of the symmetric mapping class group (associated to closed surfaces) coming from weakly group theoretical modular categories have finite images.