Constructing modular categories from orbifold data (Q2697604)

Summary: 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 by Carqueville, Runkel, and Schaumann in 2018. 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.