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-tobe-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.
Highlights
Introduction and summary1.1 Brief summaryOur aim in this paper is to demonstrate that 3d N = 8 and N = 6 superconformal field theories (SCFTs) can be usefully labeled by real and complex reflection groups, respectively
We will see below that, again after suitable finite gaugings, their moduli spaces are of the form C4N /Γ, where Γ is a complex reflection group known as G(k, p, N ), where p is a divisor of k
We can phrase our observation in a precise manner: For any N = 8 or N = 6 theory Q, one can pick a relative of Q which is ‘locally oldest’, so that its moduli space is given by C4r/Γ where Γ is a real or complex reflection group, depending on the number of supersymmetries
Summary
In 3d, known N = 8 theories are either the low-energy limit of an N = 8 super Yang-Mills, or a Bagger-Lambert-Gustavsson theory [1, 2].1 Their moduli spaces (after suitable finite gaugings) have the form C4N /Γ, where Γ is a Weyl group for the former, and a dihedral group for the latter. We will see below that, again after suitable finite gaugings, their moduli spaces are of the form C4N /Γ, where Γ is a complex reflection group known as G(k, p, N ), where p is a divisor of k This suggests us first that there is a strong possibility that there are two yet-to-bediscovered 3d N = 8 theories associated to H3 and H4.
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