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

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 $$\mathcal{W}$^{\ell} = $\mathcal{W}$^{\ell - 3/2} (\mathfrak{s}\mathfrak{l}3, f{\theta})$\mathrm{.}Thesimplequotientof$$\mathcal{W}$^{\ell}$isdenotedby$$\mathcal{W}$_{\ell}$,andfor$\ell$ a positive integer, ${\mathcal{W}}_{\mathrm{\ell }}$ is known to be $C_2$-cofinite and rational. We prove that for all positive integers $\mathrm{\ell }$, ${\mathcal{W}}_{\mathrm{\ell }}$ contains a rank one lattice vertex algebra $V_L$, and that the coset ${\mathcal{C}}_{\mathrm{\ell }}=\text{Com}(V_L,{\mathcal{W}}_{\mathrm{\ell }})$ is isomorphic to the principal, rational $\mathcal{W}(\mathfrak{s}{\mathfrak{l}}_{2\mathrm{\ell }})$-algebra at level $(2\mathrm{\ell }+3)\mathrm{/}(2\mathrm{\ell }+1)-2\mathrm{\ell }$. 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 ${\mathcal{W}}_{\mathrm{\ell }}$