Quantum Galois theory for finite groups

Abstract

Dong and Mason [DM1] initiated a systematic research for a vertex operator algebra with a finite automorphism group, which is referred to as the “operator content of orbifold models” by physicists [DVVV]. The purpose of this paper is to extend one of their main results. We will assume that the reader is familiar with the vertex operator algebras (VOA), see [B],[FLM]. Throughout this paper, V denotes a simple vertex operator algebra, G is a finite automorphism group of V , C denotes the complex number field, and Z denotes rational integers. Let H be a subgroup of G and Irr(G) denote the set of all irreducible CG-characters. In their paper [DM1], they studied the sub VOA V H = {v ∈ V : h(v) = v for all h ∈ H} of H-invariants and the subspace V χ on which G acts according to χ ∈ Irr(G). Especially, they conjectured the following Galois correspondence between sub VOAs of V and subgroups of G and proved it for an Abelian or dihedral group G [DM1, Theorem 1] and later for nilpotent groups [DM2], which is an origin of their title of [DM1].