VOAs labelled by complex reflection groups and 4d SCFTs

TL;DR

This work constructs a broad family of ${\cal N}=2$ vertex operator algebras ${\cal W}_{\mathsf G}$ labeled by complex reflection groups, extending the ${\cal N}=2$ SCA by generators tied to the invariant ring of the group. A central technical advance is a uniform free-field realization in terms of rank$(\mathsf G)$ copies of the ${\beta}{\gamma}{bc}$ system, enabling explicit OPEs, the closure of the algebra, and a natural Poisson map to the classical invariant ring $\mathscr R_{\mathsf G}$. For Coxeter (Weyl) groups, the enhancement to small ${\cal N}=4$ SCA and the identification ${\cal W}_{\mathrm{Weyl}(\mathfrak g)} = \chi[\mathrm{SYM}_{\mathfrak g}]$ connect the 2d VOA to 4d ${\cal N}=4$ theories, while crystallographic complex reflection groups correspond to ${\cal N}=3$ 4d SCFTs, with the VOAs capturing their Higgs/C Coulomb branch data through the associated variety. The authors define and study the elusive ${\cal R}$-filtration from free fields, demonstrate its equivalence to the 4d $R$-filtration in known cases, and show how the refined vacuum character encodes the Macdonald index, with Hall-Littlewood limits matching HL rings of parent 4d theories. The paper provides detailed free-field realizations and OPEs for a range of examples, including $A_1$, $I_2(p)$, $A_3$, $B_3$, $H_3$, and $D_4$, along with a discussion of screening operators and null relations, highlighting a rich landscape of ${\cal N}=2$ and ${\cal N}=4$ VOAs connected to 4d physics. These results offer a unified framework to study VOAs arising from 4d ${\cal N}\ge3$ SCFTs and furnish practical tools for extracting 4d data from 2d algebras.

Abstract

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$(\mathcal{\mathsf{G}})$ $βγbc$ 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

Figures (1)

• Figure 1: Euler-Venn diagram depicting the relations among complex reflection groups, crystallographic complex reflection groups, Coxeter groups, and Weyl groups.