Constructing modular categories from orbifold data

Abstract

The notion of an orbifold datum $A$ in a modular fusion category $\mathcal{C}$ was introduced as part of a generalised orbifold construction for Reshetikhin–Turaev TQFTs by Carqueville, Runkel, and Schaumann in 2018. In this paper, given a simple orbifold datum $A$ in $\mathcal{C}$, we introduce a ribbon category ${\mathcal{C}}_A$ and show that it is again a modular fusion category. The definition of ${\mathcal{C}}_A$ is motivated by properties of Wilson lines in the generalised orbifold. We analyse two examples in detail: (i) when $A$ is given by a simple commutative $\Delta$-separable Frobenius algebra $A$ in $\mathcal{C}$; (ii) when $A$ is an orbifold datum in $\mathcal{C}=\mathrm{Vect}$, built from a spherical fusion category $\mathcal{S}$. We show that, in case (i), ${\mathcal{C}}_A$ is ribbon-equivalent to the category of local modules of $A$, and, in case (ii), to the Drinfeld centre of $\mathcal{S}$. The category $\mathcal{C}{C}_A$ thus unifies these two constructions into a single algebraic setting.