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A note on representations of some affine vertex algebras of type $D$

In this note we construct a series of singular vectors in universal affine vertex operator algebras associated to $D_{; ; \ell}; ; ^{; ; (1)}; ; $ of levels $n-\ell+1$, for $n \in \Z _{; ; >0}; ; $. For $n=1$, we study the representation theory of the quotient vertex operator algebra modulo the ideal generated by that singular vector. In the case $\ell =4$, we show that the adjoint module is the unique irreducible ordinary module for simple vertex operator algebra $L_{; ; D_{; ; 4}; ; }; ; (-2, 0)$. We also show that the maximal ideal in associated universal affine vertex algebra is generated by three singular vectors.

vertex operator algebra; affine Kac-Moody algebra; Zhu's algebra

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