Vertex algebras related to regular representations of $S{L}_2$

Abstract

We construct a family of potentially quasi-lisse (non-rational) vertex algebras, denoted by ${\mathcal{C}}_p$, $p\ge 2$, which are closely related to the vertex algebra of chiral differential operators on $SL(2)$ at level $-2+\frac{1}{p}$. We prove that for $p=3$, there is an isomorphism between ${\mathcal{C}}_3$ and the affine vertex algebra $L_{-5/3}({\mathfrak{g}}_2)$ from Deligne's series. Moreover, we also establish isomorphisms between ${\mathcal{C}}_4$ and ${\mathcal{C}}_5$ and certain affine $W$-algebras of types $F_4$ and $E_8$, respectively. In this way, we resolve the problem of decomposing certain conformal embeddings of affine vertex algebras into affine $W$-algebras. An important feature is that ${\mathcal{C}}_p$ is $\frac{1}{2}{\mathbb{Z}}_{\ge 0}$-graded with finite-dimensional graded subspaces and convergent characters. Therefore, for all $p\ge 2$, we show that the characters of ${\mathcal{C}}_p$ exhibit modularity, supporting the conjectural quasi-lisse property.

29 pages. v2: minor edits