Arc spaces and the vertex algebra commutant problem

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.

Small correction in Theorem 4.6, reference added

References in corpus (2)

• Jet schemes and invariant theory
• A Hilbert theorem for vertex algebras

Cited by in corpus (9)

• Jet schemes and invariant theory
• Cosets of free field algebras via arc spaces
• Standard monomials and invariant theory for arc spaces I: general linear group
• Standard monomials and invariant theory for arc spaces III: special linear group
• Vector bundles induced from jet schemes
• The global sections of chiral de Rham complexes on compact Ricci-flat Kähler manifolds II
• Invariant subalgebras of the small $\mathcal{N}=4$ superconformal algebra
• The global sections of chiral de Rham complexes on compact Ricci-flat Kähler manifolds
• The global sections of the chiral de Rham complex on a Kummer surface