# Reflection groups and 3d $\mathcal{N}\ge $ 6 SCFTs

## TL;DR

This paper classifies the moduli spaces of known 3d $
cal=8$ and $
cal=6$ SCFTs using reflection groups, suggesting new theories and revealing equivalences among existing models.

## Contribution

It identifies the structure of moduli spaces as reflection quotient spaces and proposes the existence of new $
cal=8$ theories related to sporadic reflection groups.

## Key findings

- Moduli spaces are of the form $C^{4r}/
G$ with $
G$ a reflection group.
- Two ABJM theories are shown to be equivalent.
- Evidence for two new $
cal=8$ theories related to $H_{3,4}$ reflection groups.

## Abstract

We point out that the moduli spaces of all known 3d $\mathcal{N}=$ 8 and $\mathcal{N}=$ 6 SCFTs, after suitable gaugings of finite symmetry groups, have the form $\mathbb{C}^{4r}/\Gamma$ where $\Gamma$ is a real or complex reflection group depending on whether the theory is $\mathcal{N}=$ 8 or $\mathcal{N}=$ 6, respectively. Real reflection groups are either dihedral groups, Weyl groups, or two sporadic cases $H_{3,4}$. Since the BLG theories and the maximally supersymmetric Yang-Mills theories correspond to dihedral and Weyl groups, it is strongly suggested that there are two yet-to-be-discovered 3d $\mathcal{N}=$ 8 theories for $H_{3,4}$. We also show that all known $\mathcal{N}=$ 6 theories correspond to complex reflection groups collectively known as $G(k,x,N)$. Along the way, we demonstrate that two ABJM theories $(SU(N)_k\times SU(N)_{-k})/\mathbb{Z}_N$ and $(U(N)_k\times U(N)_{-k})/\mathbb{Z}_k$ are actually equivalent.