Vertex algebras related to regular representations of $SL_2$

TL;DR

This paper constructs a family of vertex algebras related to $SL(2)$, establishing isomorphisms with known algebras and demonstrating their modularity, which advances understanding of their structure and conformal embeddings.

Contribution

It introduces the $oxed{ ext{family of vertex algebras }oldsymbol{oxed{ ext{}} ext{ }oxed{ ext{}}}}$ related to $SL(2)$, proves isomorphisms with affine and ${W}$-algebras, and shows their characters are modular.

Findings

$oldsymbol{oxed{ ext{}} ext{ }oxed{ ext{}}}}$ are potentially quasi-lisse with finite-dimensional graded subspaces.

Established isomorphisms between $oldsymbol{oxed{ ext{}} ext{ }oxed{ ext{}}}}$ and affine ${W}$-algebras of types $F_4$ and $E_8$.

Characters of $oldsymbol{oxed{ ext{}} ext{ }oxed{ ext{}}}}$ exhibit modularity for all $p geq 2$.

Abstract

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.