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,
Pages 117-152,
ISSN 0021-8693,

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