Reflection groups and 3d N documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$ mathcal{N} $$end{document} & 6 SCFTs

摘要

A bstract We point out that the moduli spaces of all known 3d N documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$ mathcal{N} $$end{document} = 8 and N documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$ mathcal{N} $$end{document} = 6 SCFTs, after suitable gaugings of finite symmetry groups, have the form ?~( 4r ) / Γ where Γ is a real or complex reflection group depending on whether the theory is N documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$ mathcal{N} $$end{document} = 8 or N documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$ mathcal{N} $$end{document} = 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 documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$ mathcal{N} $$end{document} = 8 theories for H_(3,4). We also show that all known N documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$ mathcal{N} $$end{document} = 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 ))/?_( N )and (U( N )_( k )x U( N )_( -k )) / ?_( k )are actually equivalent.