Arc spaces and the vertex algebra commutant problem

TL;DR

This paper explores the structure of commutants in vertex algebras, linking them to arc space invariants, and provides explicit finite generators for specific algebraic systems relevant in conformal field theory.

Contribution

It introduces new finite generating sets for commutants in vertex algebras where the subalgebra is affine, expanding understanding of their algebraic structure.

Findings

Finite generating sets identified for specific vertex algebra examples.

Connections established between commutants and arc space invariants.

Results applicable to coset conformal field theories.

Abstract

Given a vertex algebra $\mathcal{V}$ and a subalgebra $\mathcal{A}\subset \mathcal{V}$, the commutant $\text{Com}(\mathcal{A},\mathcal{V})$ is the subalgebra of $\mathcal{V}$ which commutes with all elements of $\mathcal{A}$. This construction is analogous to the ordinary commutant in the theory of associative algebras, and is important in physics in the construction of coset conformal field theories. When $\mathcal{A}$ is an affine vertex algebra, $\text{Com}(\mathcal{A},\mathcal{V})$ is closely related to rings of invariant functions on arc spaces. We find strong finite generating sets for a family of examples where $\mathcal{A}$ is affine and $\mathcal{V}$ is a $\beta\gamma$-system, $bc$-system, or $bc\beta\gamma$-system.