Geometric engineering, mirror symmetry and 6d10→4dN=2 View Full Text


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Article Info

DATE

2015-11-18

AUTHORS

Michele Del Zotto, Cumrun Vafa, Dan Xie

ABSTRACT

We study compactification of 6 dimensional (1,0) theories on T2. We use geometric engineering of these theories via F-theory and employ mirror symmetry technology to solve for the effective 4d N=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N}=2 $$\end{document} geometry for a large number of the (1,0) theories including those associated with conformal matter. Using this we show that for a given 6d theory we can obtain many inequivalent 4d N=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N}=2 $$\end{document} SCFTs. Some of these respect the global symmetries of the 6d theory while others exhibit SL(2, ℤ) duality symmetry inherited from global diffeomorphisms of the T2. This construction also explains the 6d origin of moduli space of 4d affine ADE quiver theories as flat ADE connections on T2. Among the resulting 4dN=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N}=2 $$\end{document} CFTs we find theories whose vacuum geometry is captured by an LG theory (as opposed to a curve or a local CY geometry). We obtain arbitrary genus curves of class S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{S} $$\end{document} with punctures from toroidal compactification of (1, 0) SCFTs where the curve of the class S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{S} $$\end{document} theory emerges through mirror symmetry. We also show that toroidal compactification of the little string version of these theories can lead to class S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{S} $$\end{document} theories with no punctures on arbitrary genus Riemann surface. More... »

PAGES

123

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    30 schema:description We study compactification of 6 dimensional (1,0) theories on T2. We use geometric engineering of these theories via F-theory and employ mirror symmetry technology to solve for the effective 4d N=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N}=2 $$\end{document} geometry for a large number of the (1,0) theories including those associated with conformal matter. Using this we show that for a given 6d theory we can obtain many inequivalent 4d N=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N}=2 $$\end{document} SCFTs. Some of these respect the global symmetries of the 6d theory while others exhibit SL(2, ℤ) duality symmetry inherited from global diffeomorphisms of the T2. This construction also explains the 6d origin of moduli space of 4d affine ADE quiver theories as flat ADE connections on T2. Among the resulting 4dN=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N}=2 $$\end{document} CFTs we find theories whose vacuum geometry is captured by an LG theory (as opposed to a curve or a local CY geometry). We obtain arbitrary genus curves of class S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{S} $$\end{document} with punctures from toroidal compactification of (1, 0) SCFTs where the curve of the class S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{S} $$\end{document} theory emerges through mirror symmetry. We also show that toroidal compactification of the little string version of these theories can lead to class S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{S} $$\end{document} theories with no punctures on arbitrary genus Riemann surface.
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    41 arbitrary genus Riemann surface
    42 class
    43 compactification
    44 conformal matter
    45 connection
    46 construction
    47 curves
    48 diffeomorphisms
    49 dimensional theory
    50 duality symmetry
    51 effective 4D
    52 engineering
    53 genus Riemann surfaces
    54 genus curves
    55 geometric engineering
    56 geometry
    57 global diffeomorphism
    58 global symmetry
    59 large number
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    63 number
    64 origin
    65 puncture
    66 quiver theories
    67 space
    68 string version
    69 surface
    70 symmetry
    71 technology
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