Invariant Theory and the Heisenberg Vertex Algebra

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