VOAs labelled by complex reflection groups and 4d SCFTs

We define and study a class of $\mathcal{N}=2$ vertex operator algebras $\mathcal{W}_{\mathcal{\mathsf{G}}}$ labelled by complex reflection groups. They are extensions of the $\mathcal{N}=2$ super Virasoro algebra obtained by introducing additional generators, in correspondence with the invariants of the complex reflection group $\mathcal{\mathsf{G}}$. If $\mathcal{\mathsf{G}}$ is a Coxeter group, the $\mathcal{N}=2$ super Virasoro algebra enhances to the (small) $\mathcal{N}=4$ superconformal algebra. With the exception of $\mathcal{\mathsf{G}} = \mathbb{Z}_2$, which corresponds to just the $\mathcal{N}=4$ algebra, these are non-deformable VOAs that exist only for a specific negative value of the central charge. We describe a free-field realization of $\mathcal{W}_{\mathcal{\mathsf{G}}}$ in terms of rank$(\mathcal{\mathsf{G}})$ $\beta \gamma bc$ ghost systems, generalizing a construction of Adamovic for the $\mathcal{N}=4$ algebra at $c = -9$. If $\mathcal{\mathsf{G}}$ is a Weyl group, $\mathcal{W}_{\mathcal{\mathsf{G}}}$ is believed to coincide with the $\mathcal{N}=4$ VOA that arises from the four-dimensional super Yang-Mills theory whose gauge algebra has Weyl group $\mathcal{\mathsf{G}}$. More generally, if $\mathcal{\mathsf{G}}$ is a crystallographic complex reflection group, $\mathcal{W}_{\mathcal{\mathsf{G}}}$ is conjecturally associated to an $\mathcal{N}=3$ $4d$ superconformal field theory. The free-field realization allows to determine the elusive `$R$-filtration' of $\mathcal{W}_{\mathcal{\mathsf{G}}}$, and thus to recover the full Macdonald index of the parent $4d$ theory