2019/09/24 by Tanmay Deshpande, Deshpande, Tanmay, Swarnava Mukhopadhyay +1 · 1 citation
Mathematics · #14D21 #14H10 #14H60 #17B67 #18D10 #81R10 #81T40 #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1909.10799
openalex publication_date 2019/09/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ℓ 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.