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

Authors: Dražen Adamović, Victor G. Kac, Pierluigi Möseneder Frajria, Paolo Papi, Ozren Perše

Journal of Algebra,
Volume 500,
2018,
Pages 117-152,
ISSN 0021-8693,
https://doi.org/10.1016/j.jalgebra.2016.12.005
(http://www.sciencedirect.com/science/article/pii/S0021869316304604)

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 $V_{-\frac{n-1}{2}}(sl(n+1))$ inside the tensor product of the vertex algebra $W_{-\frac{n-1}{2}}(sl(2|n),\theta)$ (also called the Bershadsky–Knizhnik algebra) with a lattice vertex algebra.

MSC: primary 17B69, secondary 17B20, 17B65

Keywords: Vertex algebra; Virasoro (=conformal) vector; Conformal embedding; Conformal level; Collapsing level