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

TL;DR

The work proposes a unified labeling of 3d $\mathcal{N}\ge 6$ SCFTs by reflection groups, showing that moduli spaces naturally take the form $\mathbb{C}^{4r}/\Gamma$ after finite gaugings. It differentiates between real reflection groups (for $\mathcal{N}=8$) and complex reflection groups, notably $G(k,p,N)$ (for $\mathcal{N}=6$), and demonstrates this labeling through detailed moduli-space analyses of ABJM/ABJ(M), BLG, and SYM theories. A key result is the equivalence between $\bigl(U(N)_k\times U(N)_{-k}\bigr)/\mathbb{Z}_k$ and $\bigl(SU(N)_k\times SU(N)_{-k}\bigr)/\mathbb{Z}_N$, supported by index computations and a path-integral argument exploiting 1-form symmetries. The paper also forecasts two yet-to-be-discovered $\mathcal{N}=8$ theories associated with $H_3$ and $H_4$, and calls for exploration of exceptional complex reflection groups in the $\mathcal{N}=6$ landscape. Overall, the approach links high-symmetry 3d SCFTs to a reflection-group structure, offering a structural organizing principle and concrete checks via moduli spaces and indices.

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 $\mathbb{C}^{4r}/Γ$ where $Γ$ 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.