Journal of High Energy Physics

, 2012:141 | Cite as

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

  • Sebastián Franco
Open Access


We introduce and initiate the investigation of a general class of 4d, \(\mathcal{N}=1\) 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 \(\mathcal{N}=4\) 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.


Supersymmetric gauge theory Duality in Gauge Field Theories 


  1. [1]
    S. Franco, A. Hanany, K.D. Kennaway, D. Vegh and B. Wecht, Brane dimers and quiver gauge theories, JHEP 01 (2006) 096 [hep-th/0504110] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  2. [2]
    B. Feng, Y.-H. He, K.D. Kennaway and C. Vafa, Dimer models from mirror symmetry and quivering amoebae, Adv. Theor. Math. Phys. 12 (2008) 489 [hep-th/0511287] [INSPIRE].MathSciNetMATHGoogle Scholar
  3. [3]
    A. Goncharov and R. Kenyon, Dimers and cluster integrable systems, arXiv:1107.5588 [INSPIRE].
  4. [4]
    N. Arkani-Hamed, J. Bourjaily, F. Cachazo, A. Goncharov, A. Postnikov and J. Trnka, to appear.Google Scholar
  5. [5]
    E. Witten, Perturbative gauge theory as a string theory in twistor space, Commun. Math. Phys. 252 (2004) 189 [hep-th/0312171] [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  6. [6]
    F. Cachazo, P. Svrček and E. Witten, MHV vertices and tree amplitudes in gauge theory, JHEP 09 (2004) 006 [hep-th/0403047] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    R. Britto, F. Cachazo and B. Feng, New recursion relations for tree amplitudes of gluons, Nucl. Phys. B 715 (2005) 499 [hep-th/0412308] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  8. [8]
    R. Britto, F. Cachazo, B. Feng and E. Witten, Direct proof of tree-level recursion relation in Yang-Mills theory, Phys. Rev. Lett. 94 (2005) 181602 [hep-th/0501052] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  9. [9]
    A. Brandhuber, P. Heslop and G. Travaglini, A Note on dual superconformal symmetry of the N = 4 super Yang-Mills S-matrix, Phys. Rev. D 78 (2008) 125005 [arXiv:0807.4097] [INSPIRE].MathSciNetADSGoogle Scholar
  10. [10]
    N. Arkani-Hamed, F. Cachazo and J. Kaplan, What is the simplest quantum field theory?, JHEP 09 (2010) 016 [arXiv:0808.1446] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  11. [11]
    Z. Bern, L.J. Dixon and D.A. Kosower, On-shell methods in perturbative QCD, Annals Phys. 322 (2007) 1587 [arXiv:0704.2798] [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  12. [12]
    C. Berger et al., An automated implementation of on-shell methods for one-loop amplitudes, Phys. Rev. D 78 (2008) 036003 [arXiv:0803.4180] [INSPIRE].ADSGoogle Scholar
  13. [13]
    Z. Bern, L.J. Dixon and D.A. Kosower, Two-loop g → gg splitting amplitudes in QCD, JHEP 08 (2004) 012 [hep-ph/0404293] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  14. [14]
    Z. Bern, J. Carrasco, H. Johansson and D. Kosower, Maximally supersymmetric planar Yang-Mills amplitudes at five loops, Phys. Rev. D 76 (2007) 125020 [arXiv:0705.1864] [INSPIRE].MathSciNetADSGoogle Scholar
  15. [15]
    J. Drummond, J. Henn, V. Smirnov and E. Sokatchev, Magic identities for conformal four-point integrals, JHEP 01 (2007) 064 [hep-th/0607160] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  16. [16]
    L.F. Alday and J.M. Maldacena, Gluon scattering amplitudes at strong coupling, JHEP 06 (2007) 064 [arXiv:0705.0303] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  17. [17]
    J.M. Drummond, J.M. Henn and J. Plefka, Yangian symmetry of scattering amplitudes in N = 4 super Yang-Mills theory, JHEP 05 (2009) 046 [arXiv:0902.2987] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  18. [18]
    N. Arkani-Hamed, F. Cachazo, C. Cheung and J. Kaplan, A duality for the S matrix, JHEP 03 (2010) 020 [arXiv:0907.5418] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  19. [19]
    S. Fomin and A. Zelevinsky, Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002) 497.MathSciNetMATHCrossRefGoogle Scholar
  20. [20]
    K.D. Kennaway, Brane tilings, Int. J. Mod. Phys. A 22 (2007) 2977 [arXiv:0706.1660] [INSPIRE].MathSciNetADSGoogle Scholar
  21. [21]
    S. Franco and G. Torroba, work in progress.Google Scholar
  22. [22]
    A. Hanany and D. Vegh, Quivers, tilings, branes and rhombi, JHEP 10 (2007) 029 [hep-th/0511063] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  23. [23]
    O. Aharony and A. Hanany, Branes, superpotentials and superconformal fixed points, Nucl. Phys. B 504 (1997) 239 [hep-th/9704170] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  24. [24]
    O. Aharony, A. Hanany and B. Kol, Webs of (p, q) five-branes, five-dimensional field theories and grid diagrams, JHEP 01 (1998) 002 [hep-th/9710116] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    N.C. Leung and C. Vafa, Branes and toric geometry, Adv. Theor. Math. Phys. 2 (1998) 91 [hep-th/9711013] [INSPIRE].MathSciNetMATHGoogle Scholar
  26. [26]
    A. Postnikov, Total positivity, Grassmannians and networks, math/0609764.
  27. [27]
    N. Seiberg, Electric-magnetic duality in supersymmetric nonAbelian gauge theories, Nucl. Phys. B 435 (1995) 129 [hep-th/9411149] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  28. [28]
    K.A. Intriligator and N. Seiberg, Lectures on supersymmetric gauge theories and electric-magnetic duality, Nucl. Phys. Proc. Suppl. 45BC (1996) 1 [hep-th/9509066] [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  29. [29]
    R. de Mello Koch and S. Ramgoolam, From matrix models and quantum fields to Hurwitz space and the absolute Galois group, arXiv:1002.1634 [INSPIRE].
  30. [30]
    V. Jejjala, S. Ramgoolam and D. Rodriguez-Gomez, Toric CFTs, permutation triples and Belyi pairs, JHEP 03 (2011) 065 [arXiv:1012.2351] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  31. [31]
    A. Hanany and R.-K. Seong, Brane tilings and specular duality, JHEP 08 (2012) 107 [arXiv:1206.2386] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    S. Franco, Dimer models, integrable systems and quantum Teichmüller space, JHEP 09 (2011) 057 [arXiv:1105.1777] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  33. [33]
    R. Eager, S. Franco and K. Schaeffer, Dimer models and integrable systems, JHEP 06 (2012) 106 [arXiv:1107.1244] [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    A. Amariti, D. Forcella and A. Mariotti, Integrability on the master space, JHEP 06 (2012) 053 [arXiv:1203.1616] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    S. Franco, D. Galloni and Y.-H. He, Towards the continuous limit of cluster integrable systems, JHEP 09 (2012) 020 [arXiv:1203.6067] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    E.I. Buchbinder and F. Cachazo, Two-loop amplitudes of gluons and octa-cuts in N = 4 super Yang-Mills, JHEP 11 (2005) 036 [hep-th/0506126] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  37. [37]
    F. Cachazo, Sharpening the leading singularity, arXiv:0803.1988 [INSPIRE].
  38. [38]
    F. Cachazo, M. Spradlin and A. Volovich, Leading singularities of the two-loop six-particle MHV amplitude, Phys. Rev. D 78 (2008) 105022 [arXiv:0805.4832] [INSPIRE].ADSGoogle Scholar
  39. [39]
    M. Spradlin, A. Volovich and C. Wen, Three-loop leading singularities and BDS ansatz for five particles, Phys. Rev. D 78 (2008) 085025 [arXiv:0808.1054] [INSPIRE].ADSGoogle Scholar
  40. [40]
    J. Kaplan, Unraveling L(n,k): grassmannian kinematics, JHEP 03 (2010) 025 [arXiv:0912.0957] [INSPIRE].ADSCrossRefGoogle Scholar
  41. [41]
    S. Franco and D. Vegh, Moduli spaces of gauge theories from dimer models: Proof of the correspondence, JHEP 11 (2006) 054 [hep-th/0601063] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  42. [42]
    D. Forcella, A. Hanany, Y.-H. He and A. Zaffaroni, The master space of N = 1 gauge theories, JHEP 08 (2008) 012 [arXiv:0801.1585] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  43. [43]
    A. Postnikov, D. Speyer and L. Williams, Matching polytopes, toric geometry and the non-negative part of the Grassmannian, arXiv:0706.2501.
  44. [44]
    A. Hanany and K.D. Kennaway, Dimer models and toric diagrams, hep-th/0503149 [INSPIRE].
  45. [45]
    S. Franco, A. Hanany, J. Park and D. Rodriguez-Gomez, Towards M 2-brane theories for generic toric singularities, JHEP 12 (2008) 110 [arXiv:0809.3237] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  46. [46]
    S. Franco, I.R. Klebanov and D. Rodriguez-Gomez, M 2-branes on orbifolds of the cone over Q 1,1,1, JHEP 08 (2009) 033 [arXiv:0903.3231] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  47. [47]
    M. Aganagic, A stringy origin of M 2 brane Chern-Simons theories, Nucl. Phys. B 835 (2010) 1 [arXiv:0905.3415] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  48. [48]
    F. Benini, C. Closset and S. Cremonesi, Chiral flavors and M2-branes at toric CY 4 singularities, JHEP 02 (2010) 036 [arXiv:0911.4127] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  49. [49]
    S. Franco et al., Gauge theories from toric geometry and brane tilings, JHEP 01 (2006) 128 [hep-th/0505211] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  50. [50]
    S. Franco, D. Galloni and R.K. Seong, New directions in bipartite field theories, to appear.Google Scholar
  51. [51]
    D. Forcella, A. Hanany and A. Zaffaroni, Master space, Hilbert series and Seiberg duality, JHEP 07 (2009) 018 [arXiv:0810.4519] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  52. [52]
    S. Franco et al., work in progress.Google Scholar
  53. [53]
    B. Feng, A. Hanany and Y.-H. He, D-brane gauge theories from toric singularities and toric duality, Nucl. Phys. B 595 (2001) 165 [hep-th/0003085] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  54. [54]
    C.E. Beasley and M.R. Plesser, Toric duality is Seiberg duality, JHEP 12 (2001) 001 [hep-th/0109053] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  55. [55]
    B. Feng, A. Hanany, Y.-H. He and A.M. Uranga, Toric duality as Seiberg duality and brane diamonds, JHEP 12 (2001) 035 [hep-th/0109063] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  56. [56]
    B. Feng, S. Franco, A. Hanany and Y.-H. He, Symmetries of toric duality, JHEP 12 (2002) 076 [hep-th/0205144] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  57. [57]
    D. Xie, M. Yamazaki and M. Yamazaki, Network and Seiberg duality, JHEP 09 (2012) 036 [arXiv:1207.0811] [INSPIRE].ADSCrossRefGoogle Scholar

Copyright information

© SISSA 2012

Authors and Affiliations

  1. 1.Theory Group, SLAC National Accelerator LaboratoryMenlo ParkU.S.A.
  2. 2.Institute for Particle Physics Phenomenology, Department of PhysicsDurham UniversityDurhamU.K.

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