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

Abstract

We find all values of k∈C, for which the embedding of the maximal affine vertex algebra in a simple minimal W-algebra Wk(g,θ) is conformal, where g is a basic simple Lie superalgebra and −θ its minimal root. In particular, it turns out that if Wk(g,θ) does not collapse to its affine part, then the possible values of these k are either −23h∨ or −h∨−12, where h∨ is the dual Coxeter number of g for the normalization (θ,θ)=2. As an application of our results, we present a realization of simple affine vertex algebra V−n+12(sl(n+1)) inside the tensor product of the vertex algebra Wn−12(sl(2|n),θ) (also called the Bershadsky–Knizhnik algebra) with a lattice vertex algebra.