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# Cambridge Journal of Mathematics

## Volume 11 (2023)

### Number 1

### Crossed modular categories and the Verlinde formula for twisted conformal blocks

Pages: 159 – 297

DOI: https://dx.doi.org/10.4310/CJM.2023.v11.n1.a2

#### Authors

#### 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 “$\Gamma$ preserves a Borel”. For $\Gamma = \mathbb{Z} / 2$ and double covers of $\mathbb{P}^1$, this formula was conjectured by Birke–Fuchs–Schweigert [**23**]. 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.

We relate these two versions of the Verlinde formula, by formulating 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 the same assumptions) that the bundles of $\Gamma$-twisted conformal blocks associated with a twisted affine Lie algebra define a $\Gamma$-crossed modular functor.

We also prove a useful criterion for rigidity of weakly fusion categories to deduce that the level $\ell$ $\Gamma$-twisted conformal blocks define a $\Gamma$-crossed modular fusion category. Along the way, we prove the 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 describe the $\mathrm{S}$-matrices of the $\Gamma$-crossed modular fusion categories associated with twisted conformal blocks.

#### Keywords

twisted conformal blocks, twisted Verlinde formula, twisted affine Lie algebras, crossed modular categories, crossed modular functors

#### 2010 Mathematics Subject Classification

Primary 14H60, 17B67. Secondary 18-xx, 32G34, 81T40.

This work was supported by the Department of Atomic Energy, India, under project no. 12-R&D-TFR-5.01-0500.

S.M. acknowledges the support of SERB, India (SRG/2019/000513).

Received 3 July 2021

Published 5 June 2023