# Vertex algebras associated with hypertoric varieties

## TL;DR

This paper constructs a new family of vertex algebras linked to hypertoric varieties using BRST reduction, providing geometric localization and explicit conformal structures, and relates their Zhu algebras to quantizations of these varieties.

## Contribution

It introduces a novel construction of vertex algebras associated with hypertoric varieties via algebro-geometric methods and BRST reduction, connecting algebraic and geometric aspects.

## Key findings

- Vertex algebras are constructed for hypertoric varieties.
- Explicit conformal vectors are provided for certain cases.
- Zhu algebras realize filtered quantizations of hypertoric coordinate rings.

## Abstract

We construct a family of vertex algebras associated with a family of symplectic singularity/resolution, called hypertoric varieties. While the hypertoric varieties are constructed by a certain Hamiltonian reduction associated with a torus action, our vertex algebras are constructed by (semi-infinite) BRST reduction. The construction works algebro-geometrically and we construct sheaves of $\hbar$-adic vertex algebras over hypertoric varieties which localize the vertex algebras. We show when the vertex algebras are vertex operator algebras by giving explicit conformal vectors. We also show that the Zhu algebras of the vertex algebras, associative algebras associated with non-negatively graded vertex algebras, gives a certain family of filtered quantizations of the coordinate rings of the hypertoric varieties.