Examples of Invertible Gauging via Orbifold Data, Zesting, and Equivariantisation

TL;DR

This paper explores methods for gauging invertible symmetries in 3D topological theories, illustrating their relations and applications through various examples including Dijkgraaf--Witten and Chern--Simons theories.

Contribution

It introduces a unified framework connecting equivariantisation, zesting, and orbifold constructions for gauging symmetries, including new Morita-equivalence results for orbifold data.

Findings

All $ extbf{Z}_2$-symmetries in $ extbf{D}( extbf{Z}_2)$ are analyzed.

Obstructions to gauging central symmetries in $ extbf{SU}(2)_k$ Chern--Simons theory are discussed.

Morita-equivalence of zested orbifold data is established.

Abstract

We study the gauging of invertible symmetries, particularly in 3 dimensions, using equivariantisation, $G$-crossed braided zesting, and the generalised orbifold construction. We discuss how these methods are related and illustrate them in various examples. We cover all $\mathbb{Z}_2$-symmetries in Dijkgraaf--Witten $\mathbb{Z}_2$-gauge theory $\mathcal{D}(\mathbb{Z}_2)$, the $\mathbb{Z}_2$-symmetries described by Tambara--Yamagami categories, and obstructions to gauging the central symmetry in Chern--Simons $\mathrm{SU}(2)_k$-gauge theory. We introduce zested orbifold data for symmetries related by zesting and show that the two associated orbifold data are Morita-equivalent, i.e.\ they have the same underlying surface defect.