$L_1(\mathfrak{psl}_{n|n})$ from BRST reductions, associated varieties and nilpotent orbits

TL;DR

This paper confirms a conjecture linking BRST reductions of certain quantum systems to a specific vertex algebra, showing its associated variety matches a known nilpotent orbit closure, thus illustrating symplectic duality.

Contribution

It proves the isomorphism between BRST reductions and the vertex algebra $L_1(rak{psl}_{n|n})$, and identifies its associated variety as a minimal nilpotent orbit closure.

Findings

Confirmed the isomorphism between BRST reductions and $L_1(rak{psl}_{n|n})$

Identified the associated variety as the minimal nilpotent orbit closure

Established $L_1(rak{psl}_{n|n})$ as a quasi-lisse vertex algebra

Abstract

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 $L_1(\mathfrak{psl}_{n|n})$, and that its associated variety is isomorphic as a Poisson variety to the minimal nilpotent orbit closure $\overline{\mathbb{O}_{\mathrm{min}}(\mathfrak{sl}_n)}$. This shows in particular that $L_1(\mathfrak{psl}_{n|n})$ 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.