VOAs labelled by complex reflection groups and 4d SCFTs

Abstract

We define and study a class of $\mathcal{N}=2$ vertex operator algebras ${\mathcal{W}}_{\mathsf{G}}$ 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 $\mathsf{G}$. If $\mathsf{G}$ is a Coxeter group, the $\mathcal{N}=2$ super Virasoro algebra enhances to the (small) $\mathcal{N}=4$ superconformal algebra. With the exception of $\mathsf{G}={\mathbb{Z}}_2$, 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 ${\mathcal{W}}_{\mathsf{G}}$ in terms of rank$(\mathsf{G})$ $\beta \gamma bc$ ghost systems, generalizing a construction of Adamovic for the $\mathcal{N}=4$ algebra at $c=-9$. If $\mathsf{G}$ is a Weyl group, ${\mathcal{W}}_{\mathsf{G}}$ 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 $\mathsf{G}$. More generally, if $\mathsf{G}$ is a crystallographic complex reflection group, ${\mathcal{W}}_{\mathsf{G}}$ is conjecturally associated to an $\mathcal{N}=3$ $4d$ superconformal field theory. The free-field realization allows to determine the elusive `$R$-filtration' of ${\mathcal{W}}_{\mathsf{G}}$, and thus to recover the full Macdonald index of the parent $4d$ theory

70 pages

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