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In this paper, we give a Verlinde formula for computing the ranks of thebundles of twisted conformal blocks associated with a simple Lie algebraequipped with an action of a finite group $\Gamma$ and a positive integrallevel $\ell$ under the assumption that "$\Gamma$ preserves a Borel". As amotivation for this Verlinde formula, we prove a categorical Verlinde formulawhich computes the fusion coefficients for any $\Gamma$-crossed modular fusioncategory as defined by Turaev. To relate these two versions of the Verlindeformula, we formulate the notion of a $\Gamma$-crossed modular functor and showthat it is very closely related to the notion of a $\Gamma$-crossed modularfusion category. We compute the Atiyah algebra and prove (with sameassumptions) that the bundles of $\Gamma$-twisted conformal blocks associatedwith a twisted affine Lie algebra define a $\Gamma$-crossed modular functor.Along the way, we prove equivalence between a $\Gamma$-crossed modular functorand its topological analogue. We then apply these results to derive theVerlinde formula for twisted conformal blocks. We also explicitly describe thecrossed S-matrices that appear in the Verlinde formula for twisted conformalblocks.

Crossed modular categories and the Verlinde formula for twisted conformal blocks