On unitary vertex operator algebras

Abstract

We prove that a conformal vertex algebra with a non-zero invariant Hermitian form is unitary as soon as it admits a non-zero positive energy module with a positive definite invariant Hermitian form. Using free field realizations, we deduce that the principal simple $W$-algebra $W_k(\mathfrak g)$ of a simple Lie algebra $\mathfrak g$ is unitary for central charge $c\ge rank(\mathfrak g)+24\bigl((ρ|ρ^\vee)+\|ρ\|\,\|ρ^\vee\|\bigr)$, and, if $\mathfrak g$ is simply laced, for all $c\ge rank(\mathfrak g)$. Among basic Lie superalgebras which are not Lie algebras, the free field argument applies to $osp(1|2n)$ and $osp(2|2n)$. It shows that $W_k(osp(1|2n))$ is unitary for $c\ge n+\frac12$ and gives an explicit bound for $osp(2|2n)$. Next, we prove that if $\mathfrak s$ is an even $sl_2$-subalgebra of a simple Lie algebra $\mathfrak g$ whose centralizer is not abelian, then the simple $W$-algebra $W_k(\mathfrak g,\mathfrak s)$ is unitary for at most finitely many $k$. We illustrate this on the $W$-algebras of $sl_{qm}$ attached to the nilpotents of Jordan type $q^m$: unitarity is possible only at two levels, and for $q=2$ it does occur at both. For instance, the $W$-algebra of $sl_4$ attached to the partition $2+2$ is unitary only for $k=-2$ and $k=-\frac32$. This fails for Lie superalgebras, as minimal $W$-algebras show, but still holds under stronger assumptions. Finally, we prove that a simple $W$-algebra $W_k(\mathfrak g,\mathfrak s)$ can be unitary for only finitely many $k$ if $\mathfrak g$ is a simple Lie algebra and $\mathfrak s$ is not a distinguished $sl_2$ subalgebra.