Vertex Operator Algebras Associated to Type G Affine Lie Algebras II

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, $k=-2+m+\frac{i}{3}(m\in {\mathbb{Z}}_{\ge 0},i=1,2)$, 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.

arXiv comment28 pages, 1 table