Crossed modular categories and the Verlinde formula for twisted conformal blocks

Abstract

In this paper, we give a Verlinde formula for computing the ranks of the bundles of twisted conformal blocks associated with a simple Lie algebra equipped with an action of a finite group $Γ$ and a positive integral level $\ell$ under the assumption that "$Γ$ preserves a Borel". As a motivation for this Verlinde formula, we prove a categorical Verlinde formula which computes the fusion coefficients for any $Γ$-crossed modular fusion category as defined by Turaev. To relate these two versions of the Verlinde formula, we formulate the notion of a $Γ$-crossed modular functor and show that it is very closely related to the notion of a $Γ$-crossed modular fusion category. We compute the Atiyah algebra and prove (with same assumptions) that the bundles of $Γ$-twisted conformal blocks associated with a twisted affine Lie algebra define a $Γ$-crossed modular functor. Along the way, we prove equivalence between a $Γ$-crossed modular functor and its topological analogue. We then apply these results to derive the Verlinde formula for twisted conformal blocks. We also explicitly describe the crossed S-matrices that appear in the Verlinde formula for twisted conformal blocks.