Permutation orbifolds and simple current extensions

Abstract

In this article, we study permutation orbifolds and simple current extensions in the framework of vertex operator (super)algebras. We extend the construction of permutation-twisted modules for tensor products of vertex operator algebras to vertex operator superalgebras with $\frac12\mathbb Z$-grading, including the effect of the canonical involution. Using tensor category methods and simple current extensions, we build an induction theory for permutation-twisted modules associated with solvable automorphism groups, arising from semidirect products of simple current automorphisms and cyclic permutations. In particular, we describe the structure and classification of irreducible twisted modules in terms of stabilizer subgroups and associated projective representations, and determine their multiplicities explicitly. As applications, we illustrate the theory with explicit examples from code vertex operator algebras, lattice-type simple current extensions, and the Moonshine vertex operator algebra.