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

Tanmay Deshpande (School of Mathematics, Tata Institute of Fundamental Research, Mumbai, India)

Swarnava Mukhopadhyay (School of Mathematics, Tata Institute of Fundamental Research, Mumbai, India)

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.

Received 3 July 2021

Published 5 June 2023