Vertex Operator Algebras Associated to Type G Affine Lie Algebras II

Details Vertex Operator Algebras Associated to Type G Affine Lie Algebras II Vertex Operator Algebras Associated to Type G Affine Lie Algebras II

2011-12-29

arxiv.org/abs/1112.6289v1

Mathematics

Representation Theory

28 pages, 1 table

Scientific paper

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.

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