Orbifold theory for vertex algebras and Galois correspondence

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