Reflection groups and 3d N > 6 SCFTs

We point out that the moduli spaces of all known 3d N = 8 and N = 6 SCFTs, after suitable gaugings of finite symmetry groups, have the form ℂ^4r/Γ where Γ is a real or complex reflection group depending on whether the theory is N = 8 or 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 N = 8 theories for H_3,4. We also show that all known 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 x SU(_N)_-k)/ℤ_N and (U(N)_k x U(N)_-k) /ℤ_k are actually equivalent.