On unitary vertex operator algebras

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 $c\ge rank(\mathfrak{g})+24((\rho \mathrm{\mid }{\rho }^{\vee })+\mathrm{\parallel }\rho \mathrm{\parallel }\mathrm{\parallel }{\rho }^{\vee }\mathrm{\parallel })$, 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 $osp(1\mathrm{\mid }2n)$ and $osp(2\mathrm{\mid }2n)$. It shows that $W_k(osp(1\mathrm{\mid }2n))$ is unitary for $c\ge n+\frac{1}{2}$ and gives an explicit bound for $osp(2\mathrm{\mid }2n)$. Next, we prove that if $\mathfrak{s}$ is an even $s{l}_2$-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 $s{l}_{qm}$ 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 $s{l}_4$ attached to the partition $2+2$ is unitary only for $k=-2$ and $k=-\frac{3}{2}$. 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 $s{l}_2$ subalgebra.