Invariant theory and the Heisenberg vertex algebra

TL;DR

This paper investigates the structure of invariant subalgebras within Heisenberg vertex algebras under automorphism groups, proposing a conjecture about their W-algebra types and proving it for specific cases.

Contribution

It formulates a conjecture on the structure of invariant subalgebras of Heisenberg vertex algebras and proves it for n=2 and n=3, advancing understanding of their algebraic properties.

Findings

Proved the conjecture for n=2 and n=3 cases.

Established that invariants under reductive groups are strongly finitely generated.

Connected the structure of invariant subalgebras to W-algebra classifications.

Abstract

The invariant subalgebra H^+ of the Heisenberg vertex algebra H under its automorphism group Z/2Z was shown by Dong-Nagatomo to be a W-algebra of type W(2,4). Similarly, the rank n Heisenberg vertex algebra H(n) has the orthogonal group O(n) as its automorphism group, and we conjecture that H(n)^{O(n)} is a W-algebra of type 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 H(n)^G is strongly finitely generated for any reductive group G\subset O(n).