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 ${\mathcal{W}}_{\ell }$

Abstract and introduction rewritten, sections 2 and 4 expanded, minor corrections

References in corpus (2)

• Orbifolds and cosets of minimal $\mathcal{W}$-algebras
• Unitary W-algebras and three-dimensional higher spin gravities with spin one symmetry

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