Permutation-twisted modules for even order cycles acting on tensor product vertex operator superalgebras

We construct and classify (1 2 ⋯ k)-twisted V^⊗k- modules for k even and V a vertex operator superalgebra. In particular, we show that the category of weak (1 2 ⋯ k)-twisted V^⊗k-modules for k even is isomorphic to the category of weak parity-twisted V-modules. This result shows that in the case of a cyclic permutation of even order, the construction and classification of permutation-twisted modules for tensor product vertex operator superalgebras are fundamentally different than in the case of a cyclic permutation of odd order, as previously constructed and classified by the first author. In particular, in the even order case it is the parity-twisted V-modules that play the significant role in place of the untwisted V-modules that play the significant role in the odd order case.