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.