On irreducibility of modules of Whittaker type: twisted modules and nonabelian orbifolds

TL;DR

This paper extends irreducibility results for modules of vertex superalgebras to nonabelian orbifolds, demonstrating conditions under which twisted modules remain irreducible in orbifold subalgebras, with applications to various algebraic structures.

Contribution

It generalizes previous irreducibility theorems to nonabelian orbifolds of vertex superalgebras, including twisted modules and Whittaker modules, broadening the scope of these results.

Findings

Irreducibility of twisted modules under nonabelian orbifold actions

Conditions ensuring modules remain irreducible in orbifold subalgebras

Applications to modules of Whittaker type in various vertex superalgebras

Abstract

In arXiv:1811.04649, we extended the Dong-Mason theorem on irreducibility of modules for cyclic orbifold vertex algebras to the entire category weak modules and applied this result to Whittaker modules. In this paper we present further generalizations of these results for nonabelian orbifolds of vertex operator superalgebras. Let $V$ be a vertex superalgebra with a countable dimension and let $G$ be a finite subgroup of $\mathrm{Aut}(V)$. Assume that $h\in Z(G)$ where $Z(G)$ is the center of the group $G$. For any irreducible $h$-twisted (weak) $V$-module $M$, we prove that if $M\not\cong g\circ M$ for all $g\in G$ then $M$ is also irreducible as $V^G$-module. We also apply this result to examples and give irreducibility of modules of Whittaker type for orbifolds of Neveu-Schwarz vertex superalgebras, Heisenberg vertex algebras, Virasoro vertex operator algebra and Heisenberg-Virasoro vertex algebra.