---
address: 'Fachbereich Mathematik, Technische Universität Darmstadt, 64289 Darmstadt, Germany.'
author:
- 'Andrew R. Linshaw'
title: Invariant theory and the Heisenberg vertex algebra
---

[The invariant subalgebra $\cH^+$ of the Heisenberg vertex algebra $\cH$ under its automorphism group $\mathbb{Z}/2\mathbb{Z}$ was shown by Dong-Nagatomo to be a $\cW$-algebra of type $\cW(2,4)$. Similarly, the rank $n$ Heisenberg vertex algebra $\cH(n)$ has the orthogonal group $O(n)$ as its automorphism group, and we conjecture that $\cH(n)^{O(n)}$ is a $\cW$-algebra of type $\cW(2,4,6,\dots,n^2+3n)$. We prove our conjecture for $n=2$ and $n=3$, and we show that this conjecture implies that $\cH(n)^G$ is strongly finitely generated for any reductive group $G\subset O(n)$. ]{}

Introduction
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We call a vertex algebra $\cV$ [*strongly finitely generated*]{} if there exists a finite set of generators such that the collection of iterated Wick products of the generators and their derivatives spans $\cV$. This property has several important consequences, and in particular implies that both Zhu’s associative algebra $A(\cV)$, and Zhu’s commutative algebra $\cV / C_2(\cV)$, are finitely generated. By an [*invariant vertex algebra*]{}, we mean a subalgebra $\cV^G\subset \cV$, where $G$ is a group of automorphisms of $\cV$. It is our belief that if $\cV$ is a simple, strongly finitely generated 