Invariant Hermitian forms on vertex algebras

TL;DR

This paper investigates invariant Hermitian forms on conformal vertex algebras and their modules, establishing existence results and conditions for unitarity, especially in the context of minimal simple W-algebras.

Contribution

It proves the existence of non-zero invariant Hermitian forms on any W-algebra and characterizes when these forms can be unitary in minimal simple W-algebras.

Findings

Invariant Hermitian forms exist on arbitrary W-algebras.

Unitarity of forms in minimal W-algebras depends on grading and parity compatibility.

Certain W-algebras collapse to affine subalgebras, affecting unitarity.

Abstract

We study invariant Hermitian forms on a conformal vertex algebra and on their (twisted) modules. We establish existence of a non-zero invariant Hermitian form on an arbitrary $W$-algebra. We show that for a minimal simple $W$-algebra $W_k(\mathfrak g,\theta/2)$ this form can be unitary only when its $\tfrac{1}{2}\mathbb Z$-grading is compatible with parity, unless $W_k(\mathfrak g,\theta/2)$ "collapses" to its affine subalgebra.