Constructing modular categories from orbifold data

Abstract

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