Invariant Hermitian forms on vertex algebras

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 $\frac{1}{2}\mathbb{Z}$-grading is compatible with parity, unless $W_k(\mathfrak{g},\theta /2)$ "collapses" to its affine subalgebra.

Latex file, 33 pages; minor revision. To appear in Communications in Contemporary Mathematics