Uniform product of A_{g,n}(V) for an orbifold model V and G-twisted Zhu algebra

Abstract

Let V be a vertex operator algebra and G a finite automorphism group of V. For each g\in G and nonnegative rational number n\in {\mathbb Z}/|g|, a g-twisted Zhu algebra A_{g,n}(V) plays an important role in the theory of vertex operator algebras, but the given product in A_{g,n}(V) depends on the eigenspaces of g. We show that there is a uniform definition of products on V and we introduce a G-twisted Zhu algebra A_{G,n}(V) which covers all g-twisted Zhu algebras. Assume that V is simple and let {\cal S} be a finite set of inequivalent irreducible twisted V-modules which is closed under the action of G. There is a finite dimensional semisimple associative algebra {\cal A}_{\alpha}(G,{\cal S}) for a suitable 2-cocycle naturally determined by the G-action on {\cal S}. We show that a duality theorem of Schur-Weyl type holds for the actions of {\cal A}_{\alpha}(G,{\cal S}) and V^G on the direct sum of twisted V-modules in {\cal S} as an application of the theory of A_{G,n}(V). It follows as a natural consequence of the result that for any g\in G every irreducible g-twisted V-module is a completely reducible V^G-module.