Cosets of Bershadsky–Polyakov algebras and rational ${\mathcal W}$-algebras of type A

Abstract

The Bershadsky–Polyakov algebra is the ${\mathcal W}$-algebra associated to ${\mathfrak s}{\mathfrak l}_3$ with its minimal nilpotent element $f_{\theta }$. For notational convenience we define ${\mathcal W}^{\ell } = {\mathcal W}^{\ell - 3/2} ({\mathfrak s}{\mathfrak l}_3, f_{\theta })$. The simple quotient of ${\mathcal W}^{\ell }$ is denoted by ${\mathcal W}_{\ell }$, and for $\ell$ a positive integer, ${\mathcal W}_{\ell }$ is known to be $C_2$-cofinite and rational. We prove that for all positive integers $\ell$, ${\mathcal W}_{\ell }$ contains a rank one lattice vertex algebra $V_L$, and that the coset ${\mathcal C}_{\ell } = \text {Com}(V_L, {\mathcal W}_{\ell })$ is isomorphic to the principal, rational ${\mathcal W}({\mathfrak s}{\mathfrak l}_{2\ell })$-algebra at level $(2\ell +3)/(2\ell +1) -2\ell$. This was conjectured in the physics literature over 20 years ago. As a byproduct, we construct a new family of rational, $C_2$-cofinite vertex superalgebras from ${\mathcal W}_{\ell }$.