Moduli spaces of anti-invariant vector bundles and twisted conformal blocks

Zelaci, Hacen

doi:10.48550/arxiv.1711.08296

Cited by 1 publication

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“…Starting with the work of Pappas-Rapoport [77,78], the moduli stack of Bruhat-Tits torsor G associated to a pair (G, Γ ⊆ Aut(G)) has been studied by several authors [12,49]. Results of [52] (with some restrictions on level) and [99] (order two automorphisms of SL(r)) connect twisted conformal blocks with global sections of line bundles on Bun G . Pappas-Rapoport [77,78] ask if there is a Verlinde formula for these spaces of non-abelian theta functions or twisted conformal blocks?…”

Section: Introductionmentioning

confidence: 99%

Crossed modular categories and the Verlinde formula for twisted conformal blocks

Deshpande¹,

Mukhopadhyay²

2019

Preprint

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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. Contents 1. Introduction 2.1. Rigidity in weakly fusion categories 2.2. Braided crossed categories 3. Twisted Kac-Moody Lie algebras and their representations 3.1. Twisted affine Lie algebras 3.2. Twisted affine Kac-Moody Lie algebras 3.3. Twisted affine Lie algebras as Kac-Moody Lie algebras 3.4. Modules for twisted affine Lie algebras 4. Sheaf of Twisted conformal blocks 4.1. Family of pointed Γ-curves 4.2. Coordinate free highest weight integrable modules 4.3. Properties of twisted Vacua 5. Atiyah algebra of the twisted WZW connection on M Γ g,n

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