$L_1({\mathfrak{p}\mathfrak{s}\mathfrak{l}}_{n\mathrm{\mid }n})$ from BRST reductions, associated varieties and nilpotent orbits

We verify a conjecture of Beem and the first author stating that a certain family of physically motivated BRST reductions of beta-gamma systems and free fermions is isomorphic to $L_1({\mathfrak{p}\mathfrak{s}\mathfrak{l}}_{n\mathrm{\mid }n})$, and that its associated variety is isomorphic as a Poisson variety to the minimal nilpotent orbit closure $\overline{{\mathbb{O}}_{\mathrm{m}\mathrm{i}\mathrm{n}}({\mathfrak{s}\mathfrak{l}}_n)}$. This shows in particular that $L_1({\mathfrak{p}\mathfrak{s}\mathfrak{l}}_{n\mathrm{\mid }n})$ is quasi-lisse. Combining this with other results in the literature (in particular work of Ballin et al.), this paper provides a concrete and important example of how one can extract two symplectic dual varieties from a rather well-known vertex operator algebra.