Orbifold theory for vertex algebras and Galois correspondence

2023/02/19 by Dong, Chongying, Ren, Li, Yang, Chao · 1 citation
#17B69 #FOS: Mathematics #Quantum Algebra (math.QA)

paper · doi:10.48550/arxiv.2302.09474

Abstract

Let V be a simple vertex algebra of countable dimension, G be a finite automorphism group of V and σ be a central element of G. Assume that \cal S is a finite set of inequivalent irreducible σ-twisted V-modules such that \cal S is invariant under the action of G. Then there is a finite dimensional semisimple associative algebra \cal A_α(G,\cal S) for a suitable 2-cocycle α naturally determined by the G-action on \cal S such that (\cal A_α(G,\cal S),V^G) form a dual pair on the sum \cal M of σ-twisted V-modules in \cal S in the sense that (1) the actions of \cal A_α(G,\cal S) and V^G on \cal M commute, (2) each irreducible \cal A_α(G,\cal S)-module appears in \cal M, (3) the multiplicity space of each irreducible \cal A_α(G,\cal S)-module is an irreducible V^G-module, (4) the multiplicitiy spaces of different irreducible \cal A_α(G,\cal S)-modules are inequivalent V^G-modules. As applications, every irreducible σ-twisted V-module is a direct sum of finitely many irreducible V^G-modules and irreducible V^G-modules appearing in different G-orbits are inequivalent. This result generalizes many previous ones. We also establish a bijection between subgroups of G and subalgebras of V containing V^G.

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