Twisted sectors for tensor product VOAs associated to permutation groups

Let V be a vertex operator algebra. It is shown that the categories of weak, admissible and ordinary g-twisted modules for the tensor product VOA V ⊗k are isomorphic to the categories of weak, admissible and ordinary V -modules respectively where g is a k cycle automorphism of V ⊗k. For arbitrary permutation automorphism g of V ⊗k the category of admissible g-twisted modules for V ⊗k is semi simple and the simple objects are determined if V is rational. The key result is a construction of the weak g-twisted V ⊗k-modules from weak V -modules.