A New Quasi-Lisse Affine Vertex Algebra of Type <i>D</i> ₄

We consider a family of potential quasi-lisse affine vertex algebras [Formula: see text] at levels [Formula: see text]. In the case [Formula: see text], the irreducible [Formula: see text]–modules were classified in [28], and it was proved in [11] that [Formula: see text] is a quasi-lisse vertex algebra. We conjecture that [Formula: see text] is quasi-lisse for every [Formula: see text], and that it contains a unique irreducible ordinary module. In this article we prove this conjecture for [Formula: see text], by using mostly computational methods. We show that the maximal ideal in the universal affine vertex algebra [Formula: see text] is generated by three singular vectors of conformal weight six. The explicit formulas were obtained using software. Then we apply Zhu’s theory and classify all irreducible [Formula: see text]–modules. It turns out that [Formula: see text] has [Formula: see text] irreducible modules in the category [Formula: see text], but a unique irreducible ordinary module. Finally, we prove that [Formula: see text] is quasi-lisse by showing that its associated variety is contained in the nilpotent cone of [Formula: see text]. We also prove that the associated variety [Formula: see text] is [Formula: see text], the Zariski closure of the subregular nilpotent orbit in [Formula: see text].