Vertex algebras related to regular representations of $SL_2$
We construct a family of potentially quasi-lisse (non-rational) vertex algebras, denoted by $\mathcal{C}_p$, $p \geq 2$, 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 $\frac{1}{2} \mathbb{Z}_{\geq 0}$-graded with finite-dimensional graded subspaces and convergent characters. Therefore, for all $p \geq 2$, we show that the characters of $\mathcal{C}_p$ exhibit modularity, supporting the conjectural quasi-lisse property.