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Conformal blocks for Galois covers of algebraic curves

Part of: Families, fibrations Groups and algebras in quantum theory Lie algebras and Lie superalgebras Curves

Published online by Cambridge University Press:  29 August 2023

Jiuzu Hong  and
Shrawan Kumar
Jiuzu Hong
Affiliation:
Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USA jiuzu@email.unc.edu
Shrawan Kumar
Affiliation:
Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USA shrawan@email.unc.edu
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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 the 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 $\mathscr {G}$ 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 $\mathscr {G}$-torsors on $\Sigma /\Gamma$ when the level $c$ is divisible by $|\Gamma |$ (establishing a conjecture due to Pappas and Rapoport).

Keywords

twisted affine Kac–Moody Lie algebrasconformal blockspropagation of vacuafactorization theoremHurwitz stackflat projective connectionparahoric Bruhat–Tits group schemesmoduli stack of bundles

MSC classification

Primary: 17B67: Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 17B68: Virasoro and related algebras 14H81: Relationships with physics 17B81: Applications to physics 14D21: Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)
Secondary: 14H60: Vector bundles on curves and their moduli 81R10: Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, $W$-algebras and other current algebras and their representations
Type
Research Article
Information
Compositio Mathematica , Volume 159 , Issue 10 , October 2023 , pp. 2191 - 2259
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Copyright
© 2023 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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