Bipartite field theories: from D-brane probes to scattering amplitudes View Full Text


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Article Info

DATE

2012-11-26

AUTHORS

Sebastián Franco

ABSTRACT

We introduce and initiate the investigation of a general class of 4d, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{N}=1$\end{document} quiver gauge theories whose Lagrangian is defined by a bipartite graph on a Riemann surface, with or without boundaries. We refer to such class of theories as Bipartite Field Theories (BFTs). BFTs underlie a wide spectrum of interesting physical systems, including: D3-branes probing toric Calabi-Yau 3-folds, their mirror configurations of D6-branes, cluster integrable systems in (0 + 1) dimensions and leading singularities in scattering amplitudes for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{N}=4$\end{document} SYM. While our discussion is fully general, we focus on models that are relevant for scattering amplitudes. We investigate the BFT perspective on graph modifications, the emergence of Calabi-Yau manifolds (which arise as the master and moduli spaces of BFTs), the translation between square moves in the graph and Seiberg duality and the identification of dual theories by means of the underlying Calabi-Yaus, the phenomenon of loop reduction and the interpretation of the boundary operator for cells in the positive Grassmannian as higgsing in the BFT. We develop a technique based on generalized Kasteleyn matrices that permits an efficient determination of the Calabi-Yau geometries associated to arbitrary graphs. Our techniques allow us to go beyond the planar limit by both increasing the number of boundaries of the graphs and the genus of the underlying Riemann surface. Our investigation suggests a central role for Calabi-Yau manifolds in the context of leading singularities, whose full scope is yet to be uncovered. More... »

PAGES

141

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    36 schema:description We introduce and initiate the investigation of a general class of 4d, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{N}=1$\end{document} quiver gauge theories whose Lagrangian is defined by a bipartite graph on a Riemann surface, with or without boundaries. We refer to such class of theories as Bipartite Field Theories (BFTs). BFTs underlie a wide spectrum of interesting physical systems, including: D3-branes probing toric Calabi-Yau 3-folds, their mirror configurations of D6-branes, cluster integrable systems in (0 + 1) dimensions and leading singularities in scattering amplitudes for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{N}=4$\end{document} SYM. While our discussion is fully general, we focus on models that are relevant for scattering amplitudes. We investigate the BFT perspective on graph modifications, the emergence of Calabi-Yau manifolds (which arise as the master and moduli spaces of BFTs), the translation between square moves in the graph and Seiberg duality and the identification of dual theories by means of the underlying Calabi-Yaus, the phenomenon of loop reduction and the interpretation of the boundary operator for cells in the positive Grassmannian as higgsing in the BFT. We develop a technique based on generalized Kasteleyn matrices that permits an efficient determination of the Calabi-Yau geometries associated to arbitrary graphs. Our techniques allow us to go beyond the planar limit by both increasing the number of boundaries of the graphs and the genus of the underlying Riemann surface. Our investigation suggests a central role for Calabi-Yau manifolds in the context of leading singularities, whose full scope is yet to be uncovered.
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    43 Calabi-Yau
    44 Calabi-Yau geometry
    45 Calabi-Yau manifolds
    46 D-brane probes
    47 Grassmannian
    48 Kasteleyn matrix
    49 Lagrangian
    50 Riemann surface
    51 SYM
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    56 boundaries
    57 boundary operators
    58 cells
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    60 class
    61 cluster integrable systems
    62 configuration
    63 context
    64 determination
    65 dimensions
    66 discussion
    67 dual theory
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    69 efficient determination
    70 emergence
    71 field theory
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    74 general class
    75 genus
    76 geometry
    77 graph
    78 graph modifications
    79 identification
    80 integrable systems
    81 interesting physical systems
    82 interpretation
    83 investigation
    84 limit
    85 loop reduction
    86 manifold
    87 matrix
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    89 mirror configuration
    90 model
    91 modification
    92 moves
    93 number
    94 number of boundaries
    95 operators
    96 perspective
    97 phenomenon
    98 physical systems
    99 planar limit
    100 positive Grassmannian
    101 probe
    102 reduction
    103 role
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    105 singularity
    106 spectra
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    109 surface
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    111 technique
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