Constructing modular categories from orbifold data

Abstract

In Carqueville et al., arXiv:1809.01483, the notion of an orbifold datum $\mathbb{A}$ in a modular fusion category $\mathcal{C}$ was introduced as part of a generalised orbifold construction for Reshetikhin-Turaev TQFTs. In this paper, given a simple orbifold datum $\mathbb{A}$ in $\mathcal{C}$, we introduce a ribbon category $\mathcal{C}_{\mathbb{A}}$ and show that it is again a modular fusion category. The definition of $\mathcal{C}_{\mathbb{A}}$ is motivated by properties of Wilson lines in the generalised orbifold. We analyse two examples in detail: (i) when $\mathbb{A}$ is given by a simple commutative $\Delta$-separable Frobenius algebra $A$ in $\mathcal{C}$; (ii) when $\mathbb{A}$ is an orbifold datum in $\mathcal{C} = \operatorname{Vect}$, built from a spherical fusion category $\mathcal{S}$. We show that in case (i), $\mathcal{C}_{\mathbb{A}}$ is ribbon-equivalent to the category of local modules of $A$, and in case (ii), to the Drinfeld centre of $\mathcal{S}$. The category $\mathcal{C}_{\mathbb{A}}$ thus unifies these two constructions into a single algebraic setting.