Some Rational Vertex Algebras

TL;DR

This paper classifies all irreducible modules for certain rational vertex algebras associated with symplectic affine Lie algebras and proves complete reducibility of modules in category O.

Contribution

It provides a complete classification of irreducible modules and establishes complete reducibility for modules in category O for these vertex algebras.

Findings

Complete set of irreducible modules identified.

Modules in category O are completely reducible.

The vertex algebras are shown to be rational.

Abstract

Let $L((n-\tfrac 3 2)\Lambda_0)$, $n \in \Bbb N$, be a vertex operator algebra associated to the irreducible highest weight module $L((n-\tfrac 3 2)\Lambda_0)$ for a symplectic affine Lie algebra. We find a complete set of irreducible modules for $L((n-\tfrac 3 2)\Lambda_0)$ and show that every module for $L((n-\tfrac 3 2)\Lambda_0)$ from the category $\Cal O$ is completely reducible.