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

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 $\mathrm{S}\mathrm{U}(2{)}_k$-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.