Invariant Hermitian forms on vertex algebras

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 [Formula: see text]-algebra. We show that for a minimal simple [Formula: see text]-algebra [Formula: see text] this form can be unitary only when its [Formula: see text]-grading is compatible with parity, unless [Formula: see text] “collapses” to its affine subalgebra.