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

TL;DR

This paper constructs singular vectors in affine vertex algebras of type D, analyzes their quotient structures, and identifies unique irreducible modules, advancing understanding of their representation theory.

Contribution

It introduces a series of singular vectors for affine vertex algebras of type D and studies the structure of their quotients and modules, including the case of D4.

Findings

Singular vectors constructed at specific levels for type D affine vertex algebras.

The quotient algebra at level 1 has a unique irreducible module for D4.

The maximal ideal is generated by three singular vectors.

Abstract

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.