A new quasi-lisse affine vertex algebra of type D4
Dražen Adamović, Ivana Vukorepa
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Source: Crossref
Published: Jul 8, 2026
DOI: 10.1142/s0219199726500628
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In this paper, 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 [O. Perše, A note on representations of some affine vertex algebras of type [Formula: see text], Glas. Mat. Ser. III 48(68) (2013) 81–90], and it was proved in [T. Arakawa and K. Kawasetsu, Quasi-lisse vertex algebras and modular linear differential equations, in Lie Groups, Geometry, and Representation Theory[Formula: see text] A Tribute to the Life and Work of Bertram Kostant (Springer, 2018), pp. 41–57] 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 paper, 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 405 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].
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