Vertex Operator Algebras Associated to Type G Affine Lie Algebras

In this paper, we study representations of the vertex operator algebra $L(k,0)$ at one-third admissible levels $k=-5\mathrm{/}3,-4\mathrm{/}3,-2\mathrm{/}3$ for the affine algebra of type $G_2^{(1)}$. We first determine singular vectors and then obtain a description of the associative algebra $A(L(k,0))$ using the singular vectors. We then prove that there are only finitely many irreducible $A(L(k,0))$-modules from the category $\mathcal{O}$. Applying the $A(V)$-theory, we prove that there are only finitely many irreducible weak $L(k,0)$-modules from the category $\mathcal{O}$ and that such an $L(k,0)$-module is completely reducible. Our result supports the conjecture made by Adamovi{'c} and Milas in \cite{AM}.