A Hilbert theorem for vertex algebras

TL;DR

This paper proves that invariant subalgebras of certain vertex algebras under reductive group actions are strongly finitely generated, contrasting with their classical limits which often require infinitely many generators.

Contribution

It establishes strong finite generation for a broad class of invariant subalgebras using classical invariant theory and recent algebraic results.

Findings

Invariant subalgebras are strongly finitely generated in many cases.

Classical limits often need infinitely many generators and relations.

The results apply to eta ext{γ}-systems, bc-systems, and bc ext{γ}-systems.

Abstract

Given a simple vertex algebra A and a reductive group G of automorphisms of A, the invariant subalgebra A^G is strongly finitely generated in most examples where its structure is known. This phenomenon is subtle, and is generally not true of the classical limit of A^G, which often requires infinitely many generators and infinitely many relations to describe. Using tools from classical invariant theory, together with recent results on the structure of the W_{1+\infty} algebra, we establish the strong finite generation of a large family of invariant subalgebras of \beta\gamma-systems, bc-systems, and bc\beta\gamma-systems.