The small E8 instanton and the Kraft Procesi transition View Full Text


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Article Info

DATE

2018-07-16

AUTHORS

Amihay Hanany, Noppadol Mekareeya

ABSTRACT

One of the simplest (1, 0) supersymmetric theories in six dimensions lives on the world volume of one M5 brane at a D type singularity ℂ2/Dk. The low energy theory is given by an SQCD theory with Sp(k − 4) gauge group, a precise number of 2k flavors which is anomaly free, and a scale which is set by the inverse gauge coupling. The Higgs branch at finite coupling ℋf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathrm{\mathscr{H}}}_f $$\end{document} is a closure of a nilpotent orbit of D2k and develops many more flat directions as the inverse gauge coupling is set to zero (violating a standard lore that wrongly claims the Higgs branch remains classical). The quaternionic dimension grows by 29 for any k and the Higgs branch stops being a closure of a nilpotent orbit for k > 4, with an exception of k = 4 where it becomes minE8¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overline{{ \min}_{E_8}} $$\end{document}, the closure of the minimal nilpotent orbit of E8, thus having a rare phenomenon of flavor symmetry enhancement in six dimensions. Geometrically, the natural inclusion of ℋf⊂ℋ∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathrm{\mathscr{H}}}_f\subset {\mathrm{\mathscr{H}}}_{\infty } $$\end{document} fits into the Brieskorn Slodowy theory of transverse slices, and the transverse slice is computed to be minE8¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overline{{ \min}_{E_8}} $$\end{document} for any k > 3. This is identified with the well known small E8 instanton transition where 1 tensor multiplet is traded with 29 hypermultiplets, thus giving a physical interpretation to the geometric theory. By the analogy with the classical case, we call this the Kraft Procesi transition. More... »

PAGES

98

References to SciGraph publications

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    29 schema:description One of the simplest (1, 0) supersymmetric theories in six dimensions lives on the world volume of one M5 brane at a D type singularity ℂ2/Dk. The low energy theory is given by an SQCD theory with Sp(k − 4) gauge group, a precise number of 2k flavors which is anomaly free, and a scale which is set by the inverse gauge coupling. The Higgs branch at finite coupling ℋf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathrm{\mathscr{H}}}_f $$\end{document} is a closure of a nilpotent orbit of D2k and develops many more flat directions as the inverse gauge coupling is set to zero (violating a standard lore that wrongly claims the Higgs branch remains classical). The quaternionic dimension grows by 29 for any k and the Higgs branch stops being a closure of a nilpotent orbit for k > 4, with an exception of k = 4 where it becomes minE8¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overline{{ \min}_{E_8}} $$\end{document}, the closure of the minimal nilpotent orbit of E8, thus having a rare phenomenon of flavor symmetry enhancement in six dimensions. Geometrically, the natural inclusion of ℋf⊂ℋ∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathrm{\mathscr{H}}}_f\subset {\mathrm{\mathscr{H}}}_{\infty } $$\end{document} fits into the Brieskorn Slodowy theory of transverse slices, and the transverse slice is computed to be minE8¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overline{{ \min}_{E_8}} $$\end{document} for any k > 3. This is identified with the well known small E8 instanton transition where 1 tensor multiplet is traded with 29 hypermultiplets, thus giving a physical interpretation to the geometric theory. By the analogy with the classical case, we call this the Kraft Procesi transition.
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    36 D2k
    37 DK
    38 E8
    39 Higgs branch
    40 Kraft-Procesi transitions
    41 M5-branes
    42 SQCD theories
    43 analogy
    44 anomalies
    45 branches
    46 brane
    47 cases
    48 classical case
    49 closure
    50 coupling
    51 dimensions
    52 direction
    53 energy theory
    54 enhancement
    55 exception
    56 finite
    57 flat directions
    58 flavor
    59 gauge
    60 gauge couplings
    61 gauge group
    62 geometric theory
    63 group
    64 hypermultiplets
    65 inclusion
    66 instanton transitions
    67 instantons
    68 interpretation
    69 inverse gauge
    70 low-energy theory
    71 minimal nilpotent orbit
    72 multiplets
    73 natural inclusion
    74 nilpotent orbits
    75 number
    76 orbit
    77 phenomenon
    78 physical interpretation
    79 precise number
    80 quaternionic dimension
    81 rare phenomenon
    82 scale
    83 singularity
    84 slices
    85 small E8 instantons
    86 supersymmetric theories
    87 symmetry enhancement
    88 tensor multiplet
    89 theory
    90 transition
    91 transverse slices
    92 type singularity
    93 volume
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    96 schema:name The small E8 instanton and the Kraft Procesi transition
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