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

We construct and classify $(1 \; 2 \; \cdots \; k)$-twisted $V^{\otimes k}$-modules for $k$ even and $V$ a vertex operator superalgebra. In particular, we show that the category of weak $(1 \; 2 \; \cdots \; k)$-twisted $V^{\otimes k}$-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 is 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.