Conformal embeddings of affine vertex algebras in minimal $W$-algebras I: structural results

TL;DR

This paper determines the specific levels at which affine vertex algebras embed conformally into minimal W-algebras for basic simple Lie superalgebras, revealing new structural insights and applications to realizations of affine vertex algebras.

Contribution

It provides a complete classification of conformal embeddings of affine vertex algebras into minimal W-algebras for a broad class of Lie superalgebras, including explicit realization results.

Findings

Conformal embeddings occur at specific levels related to the dual Coxeter number.

Identifies levels where the W-algebra does not collapse to its affine part.

Constructs realizations of affine vertex algebras inside tensor products involving W-algebras.

Abstract

We find all values of $k\in \mathbb C$, for which the embedding of the maximal affine vertex algebra in a simple minimal W-algebra $W_k(\mathfrak g,\theta)$ is conformal, where $\mathfrak g$ is a basic simple Lie superalgebra and $-\theta$ its minimal root. In particular, it turns out that if $W_k(\mathfrak g,\theta)$ does not collapse to its affine part, then the possible values of these $k$ are either $-\frac{2}{3} h^\vee$ or $-\frac{h^\vee-1}{2}$, where $h^\vee$ is the dual Coxeter number of $\mathfrak g$ for the normalization $(\theta,\theta)=2$. As an application of our results, we present a realization of simple affine vertex algebra $V_{-\tfrac{n+1}{2} } (sl(n+1))$ inside of the tensor product of the vertex algebra $W_{\tfrac{n-1}{2}} (sl(2| n), \theta)$ (also called the Bershadsky-Knizhnik algebra) with a lattice vertex algebra.