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

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 ${\mathbb{C}}^{4r}/\hat{I}“$ where $\hat{I}“$ 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.

37 pages