Simple current extensions beyond semi-simplicity

TL;DR

This paper investigates simple current extensions of vertex operator algebras (VOAs), establishing conditions under which these extensions are either VOAs or super VOAs, and exploring their properties beyond semi-simplicity.

Contribution

It proves that simple current extensions of order two are either VOAs or super VOAs, extending to arbitrary order and providing criteria for lifting indecomposable objects.

Findings

Extensions are either VOAs or super VOAs based on categorical dimension.

Extension criteria depend on conformal dimensions.

Examples include non-rational, $C_2$-cofinite simple current extensions.

Abstract

Let V be a simple VOA and consider a representation category of V that is a vertex tensor category in the sense of Huang-Lepowsky. In particular, this category is a braided tensor category. Let J be an object in this category that is a simple current of order two of either integer or half-integer conformal dimension. We prove that $V\oplus J$ is either a VOA or a super VOA. If the representation category of V is in addition ribbon, then the categorical dimension of J decides this parity question. Combining with Carnahan's work, we extend this result to simple currents of arbitrary order. Our next result is a simple sufficient criterion for lifting indecomposable objects that only depends on conformal dimensions. Several examples of simple current extensions that are $C_2$-cofinite and non-rational are then given and induced modules listed.