Conformal blocks for Galois covers of algebraic curves

Abstract

We study the spaces of twisted conformal blocks attached to a $\Gamma$-curve $\Sigma$ with marked $\Gamma$-orbits and an action of $\Gamma$ on a simple Lie algebra $\mathfrak{g}$, where $\Gamma$ is a finite group. We prove that if $\Gamma$ stabilizes a Borel subalgebra of $\mathfrak{g}$, then Propagation Theorem and Factorization Theorem hold. We endow a flat projective connection on the sheaf of twisted conformal blocks attached to a smooth family of pointed $\Gamma$-curves; in particular, it is locally free. We also prove that the sheaf of twisted conformal blocks on the stable compactification of Hurwitz stack is locally free. Let $\mathcal{G}$ be the parahoric Bruhat-Tits group scheme on the quotient curve $\Sigma /\Gamma$ obtained via the $\Gamma$-invariance of Weil restriction associated to $\Sigma$ and the simply-connected simple algebraic group $G$ with Lie algebra $\mathfrak{g}$. We prove that the space of twisted conformal blocks can be identified with the space of generalized theta functions on the moduli stack of quasi-parabolic $\mathcal{G}$-torsors on $\Sigma /\Gamma$ when the level $c$ is divisible by $|\Gamma |$ (establishing a conjecture due to Pappas-Rapoport).

arXiv commentThis new version of the paper fixes an error in the statement of Lemma 8.3 in the published version of the paper

References in corpus (5)

• The action of outer automorphisms on bundles of chiral blocks
• From WZW models to Modular Functors
• Fusion ring revisited
• Conformal blocks attached to twisted groups
• Conformal blocks, Verlinde formula and diagram automorphisms

Cited by in corpus (4)

• Conformal blocks attached to twisted groups
• Conformal blocks, Verlinde formula and diagram automorphisms
• Twisted Moduli Spaces and Duistermaat-Heckman Measures
• Twisted Conjugation on Connected Simple Lie Groups and Twining Characters