Vertex Operator Algebras Associated to Type G Affine Lie Algebras II

TL;DR

This paper investigates the structure of vertex operator algebras linked to affine Lie algebra of type G2 at specific levels, establishing the finiteness of irreducible modules in a certain category using explicit singular vector formulas.

Contribution

It provides the first explicit formula for singular vectors in these VOAs and proves the finiteness of irreducible modules in the category al, advancing understanding of their representation theory.

Findings

Finite number of irreducible modules in category al for the VOAs.

Explicit singular vector formulas derived for the algebra.

Finiteness established via A(V)-theory and module correspondence.

Abstract

We continue the study of the vertex operator algebra $L(k,0)$ associated to a type $G_2^{(1)}$ affine Lie algebra at admissible one-third integer levels, $k = -2 + m + \tfrac{i}{3}\ (m\in \mathbb{Z}_{\ge 0}, i = 1,2)$, initiated in \cite{AL}. Our main result is that there is a finite number of irreducible $L(k,0)$-modules from the category $\mathcal{O}$. The proof relies on the knowledge of an explicit formula for the singular vectors. After obtaining this formula, we are able to show that there are only finitely many irreducible $A(L(k,0))$-modules form the category $\mathcal{O}$. The main result then follows from the bijective correspondence in $A(V)$-theory.