Abstract. 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 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.

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Motivated by the study of indecomposable, nonsimple modules for a vertex operator algebra V , we study the relationship between various types of V -modules and modules for the higher level Zhu algebras for V , denoted A n (V ), for n ∈ N, first introduced by Dong, Li, and Mason in 1998. We resolve some issues that arise in a few theorems previously presented when these algebras were first introduced, and give examples illustrating the need for certain modifications of the statements of those theorems. We establish that whether or not A n−1 (V ) is isomorphic to a direct summand of A n (V ) affects the types of indecomposable V -modules which can be constructed by inducing from an A n (V )-module, and in particular whether there are V -modules induced from A n (V )-modules that were not already induced by A 0 (V ). We give some characterizations of the V -modules that can be constructed from such inducings, in particular as regards their singular vectors. To illustrate these results, we discuss two examples of A 1 (V ): when V is the vertex operator algebra associated to either the Heisenberg algebra or the Virasoro algebra. For these two examples, we show how the structure of A 1 (V ) in relationship to A 0 (V ) determines what types of indecomposable V -modules can be induced from a module for the level zero versus level one Zhu algebras. We construct a family of indecomposable modules for the Virasoro vertex operator algebra that are logarithmic modules and are not highest weight modules.

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We conjecture that the category of permutation-twisted modules for a multi-fold tensor product vertex operator superalgebra and a cyclic permutation of even order is isomorphic to the category of parity-twisted modules for the underlying vertex operator superalgebra. This conjecture is based on our observations of the cyclic permutation-twisted modules for free fermions as we discuss in this work, as well as previous work of the first author constructing and classifying permutation-twisted modules for tensor product vertex operator superalgebras and a permutation of odd order. In addition, we observe that the transposition isomorphism for two free fermions corresponds to a lift of the −1 isometry of the integral lattice vertex operator superalgebra corresponding to two free fermions under boson-fermion correspondence. We conjecture that all even order cyclic permutation automorphisms of free fermions can be realized as lifts of lattice isometries under boson-fermion correspondence. We discuss the role of parity stability in the construction of these twisted modules and prove that in general, parity-unstable weak twisted modules for a vertex operator superalgebras come in pairs that form orthogonal invariant subspaces of parity-stable weak twisted modules, clarifying their role in many other settings.