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 ℂ4r/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 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 \( \mathcal{N} \) = 8 theories for H3,4. 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 x SU(N)-k)/ℤN and (U(N)k x U(N)-k) /ℤk are actually equivalent.
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Tachikawa, Y., Zafrir, G. Reflection groups and 3d \( \mathcal{N} \)> 6 SCFTs. J. High Energ. Phys. 2019, 176 (2019). https://doi.org/10.1007/JHEP12(2019)176
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DOI: https://doi.org/10.1007/JHEP12(2019)176