Twisted sectors for tensor product vertex operator algebras associated to permutation groups

Abstract

Let V be a vertex operator algebra, and for k a positive integer, let g be a k-cycle permutation of the vertex operator algebra V^{\otimes k}. We prove that the categories of weak, weak admissible and ordinary g-twisted modules for the tensor product vertex operator algebra V^{\otimes k} are isomorphic to the categories of weak, weak admissible and ordinary V-modules, respectively. The main result is an explicit construction of the weak g-twisted V^{\otimes k}-modules from weak V-modules. For an arbitrary permutation automorphism g of V^{\otimes k} the category of weak admissible g-twisted modules for V^{\otimes k$ is semisimple and the simple objects are determined if V is rational. In addition, we extend these results to the more general setting of γg-twisted V^{\otimes k}-modules for γa general automorphism of V acting diagonally on V^{\otimes k} and a g a permutation automorphism of V^{\otimes k}.