Local systems of twisted vertex operators, vertex operator superalgebras and twisted modules

Abstract

We introduce the notion of ``local system of $\Bbb{Z}_{T}$-twisted vertex operators'' on a $\Bbb{Z}_{2}$-graded vector space $M$, generalizing the notion of local system of vertex operators [Li]. First, we prove that any local system of $\Bbb{Z}_{T}$-twisted vertex operators on $M$ has a vertex superalgebra structure with an automorphism $σ$ of order $T$ with $M$ as a $σ$-twisted module. Then we prove that for a vertex (operator) superalgebra $V$ with an automorphism $σ$ of order $T$, giving a $σ$-twisted $V$-module $M$ is equivalent to giving a vertex (operator) superalgebra homomorphism from $V$ to some local system of $\Bbb{Z}_{T}$-twisted vertex operators on $M$. As applications, we study the twisted modules for vertex operator (super)algebras associated to some well-known infinite-dimensional Lie (super)algebras and we prove the complete reducibility of $\Bbb{Z}_{T}$-twisted modules for vertex operator algebras associated to standard modules for an affine Lie algebra.