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

Abstract

Abstract We point out that the moduli spaces of all known 3d $$ \mathcal{N} $$ N = 8 and $$ \mathcal{N} $$ 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 $$ \mathcal{N} $$ N = 8 or $$ \mathcal{N} $$ N = 6, respectively. Real reflection groups are either dihedral groups, Weyl groups, or two sporadic cases H3,4 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 $$ \mathcal{N} $$ N = 8 theories for H3,4. We also show that all known $$ \mathcal{N} $$ 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.