Crossed modular categories and the Verlinde formula for twisted conformal blocks

TL;DR

This paper develops a Verlinde formula for twisted conformal blocks associated with Lie algebras under finite group actions, connecting categorical and geometric approaches, and explicitly describing the involved S-matrices.

Contribution

It introduces a Verlinde formula for twisted conformal blocks and establishes a categorical framework relating $ ext{Gamma}$-crossed modular categories and functors.

Findings

Derived a Verlinde formula for twisted conformal blocks.

Proved the equivalence between $ ext{Gamma}$-crossed modular functors and topological analogues.

Explicitly described the crossed S-matrices in the Verlinde formula.

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 $\Gamma$ and a positive integral level $\ell$ under the assumption that "$\Gamma$ preserves a Borel". As a motivation for this Verlinde formula, we prove a categorical Verlinde formula which computes the fusion coefficients for any $\Gamma$-crossed modular fusion category as defined by Turaev. To relate these two versions of the Verlinde formula, we formulate the notion of a $\Gamma$-crossed modular functor and show that it is very closely related to the notion of a $\Gamma$-crossed modular fusion category. We compute the Atiyah algebra and prove (with same assumptions) that the bundles of $\Gamma$-twisted conformal blocks associated with a twisted affine Lie algebra define a $\Gamma$-crossed modular functor. Along the way, we prove equivalence between a $\Gamma$-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.