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

Understand

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 $G(k,x,N)$.