BPS wilson loops in generic conformal \( \mathcal{N} \) = 2 SU(N) SYM theories

  • M. BillòEmail author
  • F. Galvagno
  • A. Lerda
Open Access
Regular Article - Theoretical Physics


We consider the 1/2 BPS circular Wilson loop in a generic \( \mathcal{N} \) = 2 SU(N) SYM theory with conformal matter content. We study its vacuum expectation value, both at finite N and in the large-N limit, using the interacting matrix model provided by localization results. We single out some families of theories for which the Wilson loop vacuum expectation values approaches the \( \mathcal{N} \) = 4 result in the large-N limit, in agreement with the fact that they possess a simple holographic dual. At finite N and in the generic case, we explicitly compare the matrix model result with the field-theory perturbative expansion up to order g8 for the terms proportional to the Riemann value ζ (5), finding perfect agreement. Organizing the Feynman diagrams as suggested by the structure of the matrix model turns out to be very convenient for this computation.


Extended Supersymmetry Wilson ’t Hooft and Polyakov loops Supersymmetric Gauge Theory Conformal Field Theory 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited


  1. [1]
    J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys.38 (1999) 1113 [hep-th/9711200] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  2. [2]
    J.K. Erickson, G.W. Semenoff and K. Zarembo, Wilson loops in N = 4 supersymmetric Yang-Mills theory, Nucl. Phys.B 582 (2000) 155 [hep-th/0003055] [INSPIRE].CrossRefADSMathSciNetzbMATHGoogle Scholar
  3. [3]
    D.E. Berenstein, R. Corrado, W. Fischler and J.M. Maldacena, The operator product expansion for Wilson loops and surfaces in the large N limit, Phys. Rev.D 59 (1999) 105023 [hep-th/9809188] [INSPIRE].ADSMathSciNetGoogle Scholar
  4. [4]
    N. Drukker and D.J. Gross, An exact prediction of N = 4 SUSYM theory for string theory, J. Math. Phys.42 (2001) 2896 [hep-th/0010274] [INSPIRE].CrossRefADSMathSciNetzbMATHGoogle Scholar
  5. [5]
    G.W. Semenoff and K. Zarembo, More exact predictions of SUSYM for string theory, Nucl. Phys.B 616 (2001) 34 [hep-th/0106015] [INSPIRE].CrossRefADSMathSciNetzbMATHGoogle Scholar
  6. [6]
    V. Pestun and K. Zarembo, Comparing strings in AdS 5 × S 5to planar diagrams: An example, Phys. Rev.D 67 (2003) 086007 [hep-th/0212296] [INSPIRE].ADSGoogle Scholar
  7. [7]
    K. Zarembo, Supersymmetric Wilson loops, Nucl. Phys.B 643 (2002) 157 [hep-th/0205160] [INSPIRE].CrossRefADSMathSciNetzbMATHGoogle Scholar
  8. [8]
    N. Drukker, 1/4 BPS circular loops, unstable world-sheet instantons and the matrix model, JHEP09 (2006) 004 [hep-th/0605151] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  9. [9]
    G.W. Semenoff and D. Young, Exact 1/4 BPS Loop: Chiral primary correlator, Phys. Lett.B 643 (2006) 195 [hep-th/0609158] [INSPIRE].CrossRefADSMathSciNetzbMATHGoogle Scholar
  10. [10]
    S. Giombi, R. Ricci and D. Trancanelli, Operator product expansion of higher rank Wilson loops from D-branes and matrix models, JHEP10 (2006) 045 [hep-th/0608077] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  11. [11]
    N. Drukker, S. Giombi, R. Ricci and D. Trancanelli, Wilson loops: From four-dimensional SYM to two-dimensional YM, Phys. Rev.D 77 (2008) 047901 [arXiv:0707.2699] [INSPIRE].ADSGoogle Scholar
  12. [12]
    N. Drukker, S. Giombi, R. Ricci and D. Trancanelli, More supersymmetric Wilson loops, Phys. Rev.D 76 (2007) 107703 [arXiv:0704.2237] [INSPIRE].ADSMathSciNetGoogle Scholar
  13. [13]
    N. Drukker, S. Giombi, R. Ricci and D. Trancanelli, Supersymmetric Wilson loops on S 3, JHEP05 (2008) 017 [arXiv:0711.3226] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  14. [14]
    J. Gomis, S. Matsuura, T. Okuda and D. Trancanelli, Wilson loop correlators at strong coupling: From matrices to bubbling geometries, JHEP08 (2008) 068 [arXiv:0807.3330] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  15. [15]
    A. Bassetto, L. Griguolo, F. Pucci and D. Seminara, Supersymmetric Wilson loops at two loops, JHEP06 (2008) 083 [arXiv:0804.3973] [INSPIRE].
  16. [16]
    S. Giombi and V. Pestun, Correlators of local operators and 1/8 BPS Wilson loops on S 2from 2d YM and matrix models, JHEP10 (2010) 033 [arXiv:0906.1572] [INSPIRE].CrossRefADSzbMATHGoogle Scholar
  17. [17]
    A. Bassetto, L. Griguolo, F. Pucci, D. Seminara, S. Thambyahpillai and D. Young, Correlators of supersymmetric Wilson-loops, protected operators and matrix models in N = 4 SYM, JHEP08 (2009) 061 [arXiv:0905.1943] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  18. [18]
    A. Bassetto, L. Griguolo, F. Pucci, D. Seminara, S. Thambyahpillai and D. Young, Correlators of supersymmetric Wilson loops at weak and strong coupling, JHEP03 (2010) 038 [arXiv:0912.5440] [INSPIRE].CrossRefADSMathSciNetzbMATHGoogle Scholar
  19. [19]
    S. Giombi and V. Pestun, Correlators of Wilson Loops and Local Operators from Multi-Matrix Models and Strings in AdS, JHEP01 (2013) 101 [arXiv:1207.7083] [INSPIRE].
  20. [20]
    M. Bonini, L. Griguolo and M. Preti, Correlators of chiral primaries and 1/8 BPS Wilson loops from perturbation theory, JHEP09 (2014) 083 [arXiv:1405.2895] [INSPIRE].
  21. [21]
    A. Kapustin, Wilson-t Hooft operators in four-dimensional gauge theories and S-duality, Phys. Rev.D 74 (2006) 025005 [hep-th/0501015] [INSPIRE].ADSMathSciNetGoogle Scholar
  22. [22]
    D.M. McAvity and H. Osborn, Energy momentum tensor in conformal field theories near a boundary, Nucl. Phys.B 406 (1993) 655 [hep-th/9302068] [INSPIRE].CrossRefADSMathSciNetzbMATHGoogle Scholar
  23. [23]
    D.M. McAvity and H. Osborn, Conformal field theories near a boundary in general dimensions, Nucl. Phys.B 455 (1995) 522 [cond-mat/9505127] [INSPIRE].
  24. [24]
    M. Billò, V. Gonçalves, E. Lauria and M. Meineri, Defects in conformal field theory, JHEP04 (2016) 091 [arXiv:1601.02883] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  25. [25]
    M. Cooke, A. Dekel and N. Drukker, The Wilson loop CFT: Insertion dimensions and structure constants from wavy lines, J. Phys.A 50 (2017) 335401 [arXiv:1703.03812] [INSPIRE].
  26. [26]
    M. Kim, N. Kiryu, S. Komatsu and T. Nishimura, Structure Constants of Defect Changing Operators on the 1/2 BPS Wilson Loop, JHEP12 (2017) 055 [arXiv:1710.07325] [INSPIRE].CrossRefADSMathSciNetzbMATHGoogle Scholar
  27. [27]
    S. Giombi and S. Komatsu, More Exact Results in the Wilson Loop Defect CFT: Bulk-Defect OPE, Nonplanar Corrections and Quantum Spectral Curve, J. Phys.A 52 (2019) 125401 [arXiv:1811.02369] [INSPIRE].ADSGoogle Scholar
  28. [28]
    V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys.313 (2012) 71 [arXiv:0712.2824] [INSPIRE].CrossRefADSMathSciNetzbMATHGoogle Scholar
  29. [29]
    S.-J. Rey and T. Suyama, Exact Results and Holography of Wilson Loops in N = 2 Superconformal (Quiver) Gauge Theories, JHEP01 (2011) 136 [arXiv:1001.0016] [INSPIRE].CrossRefADSMathSciNetzbMATHGoogle Scholar
  30. [30]
    F. Passerini and K. Zarembo, Wilson Loops in N = 2 Super-Yang-Mills from Matrix Model, JHEP09 (2011) 102 [Erratum ibid.10 (2011) 065] [arXiv:1106.5763] [INSPIRE].CrossRefADSMathSciNetzbMATHGoogle Scholar
  31. [31]
    J.G. Russo and K. Zarembo, Localization at Large N, in Proceedings, 100th anniversary of the birth of I.Ya. Pomeranchuk (Pomeranchuk 100): Moscow, Russia, June 56, 2013, pp. 287–311, 2014, arXiv:1312.1214 [INSPIRE].
  32. [32]
    B. Fiol, B. Garolera and G. Torrents, Probing \( \mathcal{N} \) = 2 superconformal field theories with localization, JHEP01 (2016) 168 [arXiv:1511.00616] [INSPIRE].CrossRefADSGoogle Scholar
  33. [33]
    M. Baggio, V. Niarchos and K. Papadodimas, tt equations, localization and exact chiral rings in 4d \( \mathcal{N} \) = 2 SCFTs, JHEP02 (2015) 122 [arXiv:1409.4212] [INSPIRE].
  34. [34]
    M. Baggio, V. Niarchos and K. Papadodimas, Exact correlation functions in SU(2) \( \mathcal{N} \) = 2 superconformal QCD, Phys. Rev. Lett.113 (2014) 251601 [arXiv:1409.4217] [INSPIRE].CrossRefADSzbMATHGoogle Scholar
  35. [35]
    E. Gerchkovitz, J. Gomis and Z. Komargodski, Sphere Partition Functions and the Zamolodchikov Metric, JHEP11 (2014) 001 [arXiv:1405.7271] [INSPIRE].
  36. [36]
    M. Baggio, V. Niarchos and K. Papadodimas, On exact correlation functions in SU(N) \( \mathcal{N} \) = 2 superconformal QCD,JHEP11(2015) 198 [arXiv:1508.03077] [INSPIRE].CrossRefADSMathSciNetzbMATHGoogle Scholar
  37. [37]
    M. Baggio, V. Niarchos, K. Papadodimas and G. Vos, Large-N correlation functions in \( \mathcal{N} \) = 2 superconformal QCD,JHEP01(2017) 101 [arXiv:1610.07612] [INSPIRE].CrossRefADSzbMATHGoogle Scholar
  38. [38]
    E. Gerchkovitz, J. Gomis, N. Ishtiaque, A. Karasik, Z. Komargodski and S.S. Pufu, Correlation Functions of Coulomb Branch Operators, JHEP01 (2017) 103 [arXiv:1602.05971] [INSPIRE].CrossRefADSMathSciNetzbMATHGoogle Scholar
  39. [39]
    D. Rodriguez-Gomez and J.G. Russo, Large N Correlation Functions in Superconformal Field Theories, JHEP06 (2016) 109 [arXiv:1604.07416] [INSPIRE].CrossRefADSMathSciNetzbMATHGoogle Scholar
  40. [40]
    D. Rodriguez-Gomez and J.G. Russo, Operator mixing in large N superconformal field theories on S 4and correlators with Wilson loops, JHEP12 (2016) 120 [arXiv:1607.07878] [INSPIRE].CrossRefADSMathSciNetzbMATHGoogle Scholar
  41. [41]
    M. Billó, F. Fucito, A. Lerda, J.F. Morales, Ya.S. Stanev and C. Wen, Two-point Correlators in N = 2 Gauge Theories, Nucl. Phys.B 926 (2018) 427 [arXiv:1705.02909] [INSPIRE].
  42. [42]
    M. Billó, F. Galvagno, P. Gregori and A. Lerda, Correlators between Wilson loop and chiral operators in \( \mathcal{N} \) = 2 conformal gauge theories, JHEP03 (2018) 193 [arXiv:1802.09813] [INSPIRE].CrossRefzbMATHGoogle Scholar
  43. [43]
    M. Billó, F. Fucito, G.P. Korchemsky, A. Lerda and J.F. Morales, Two-point correlators in non-conformal \( \mathcal{N} \) = 2 gauge theories, JHEP05 (2019) 199 [arXiv:1901.09693] [INSPIRE].CrossRefADSzbMATHGoogle Scholar
  44. [44]
    D. Correa, J. Henn, J. Maldacena and A. Sever, An exact formula for the radiation of a moving quark in N = 4 super Yang-Mills, JHEP06 (2012) 048 [arXiv:1202.4455] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  45. [45]
    D. Correa, J. Maldacena and A. Sever, The quark anti-quark potential and the cusp anomalous dimension from a TBA equation, JHEP08 (2012) 134 [arXiv:1203.1913] [INSPIRE].CrossRefADSGoogle Scholar
  46. [46]
    A. Lewkowycz and J. Maldacena, Exact results for the entanglement entropy and the energy radiated by a quark, JHEP05 (2014) 025 [arXiv:1312.5682] [INSPIRE].CrossRefADSzbMATHGoogle Scholar
  47. [47]
    B. Fiol, E. Gerchkovitz and Z. Komargodski, Exact Bremsstrahlung Function in N = 2 Superconformal Field Theories, Phys. Rev. Lett.116 (2016) 081601 [arXiv:1510.01332] [INSPIRE].CrossRefADSGoogle Scholar
  48. [48]
    V. Mitev and E. Pomoni, Exact Bremsstrahlung and Effective Couplings, JHEP06 (2016) 078 [arXiv:1511.02217] [INSPIRE].CrossRefADSMathSciNetzbMATHGoogle Scholar
  49. [49]
    M. Bonini, L. Griguolo, M. Preti and D. Seminara, Bremsstrahlung function, leading Lüscher correction at weak coupling and localization, JHEP02 (2016) 172 [arXiv:1511.05016] [INSPIRE].CrossRefADSzbMATHGoogle Scholar
  50. [50]
    L. Bianchi, M. Lemos and M. Meineri, Line Defects and Radiation in \( \mathcal{N} \) = 2 Conformal Theories, Phys. Rev. Lett.121 (2018) 141601 [arXiv:1805.04111] [INSPIRE].CrossRefADSGoogle Scholar
  51. [51]
    C. Gomez, A. Mauri and S. Penati, The Bremsstrahlung function of \( \mathcal{N} \) = 2 SCQCD, JHEP03 (2019) 122 [arXiv:1811.08437] [INSPIRE].CrossRefADSGoogle Scholar
  52. [52]
    R. Andree and D. Young, Wilson Loops in N = 2 Superconformal Yang-Mills Theory, JHEP09 (2010) 095 [arXiv:1007.4923] [INSPIRE].CrossRefADSMathSciNetzbMATHGoogle Scholar
  53. [53]
    E. Pomoni and C. Sieg, From N = 4 gauge theory to N = 2 conformal QCD: three-loop mixing of scalar composite operators, arXiv:1105.3487 [INSPIRE].
  54. [54]
    E. Pomoni, Integrability in N = 2 superconformal gauge theories, Nucl. Phys.B 893 (2015) 21 [arXiv:1310.5709] [INSPIRE].CrossRefADSMathSciNetzbMATHGoogle Scholar
  55. [55]
    I.G. Koh and S. Rajpoot, Finite N = 2 extended supersymmetric field theories, Phys. Lett.135B (1984) 397 [INSPIRE].CrossRefADSGoogle Scholar
  56. [56]
    I.P. Ennes, C. Lozano, S.G. Naculich and H.J. Schnitzer, Elliptic models, type IIB orientifolds and the AdS/CFT correspondence, Nucl. Phys.B 591 (2000) 195 [hep-th/0006140] [INSPIRE].CrossRefADSMathSciNetzbMATHGoogle Scholar
  57. [57]
    A. Bourget, D. Rodriguez-Gomez and J.G. Russo, A limit for large R-charge correlators in \( \mathcal{N} \) = 2 theories, JHEP05(2018) 074 [arXiv:1803.00580] [INSPIRE].CrossRefADSzbMATHGoogle Scholar
  58. [58]
    M. Beccaria, On the large R-charge \( \mathcal{N} \) = 2 chiral correlators and the Toda equation, JHEP02 (2019) 009 [arXiv:1809.06280] [INSPIRE].CrossRefADSGoogle Scholar
  59. [59]
    M. Beccaria, Double scaling limit of N = 2 chiral correlators with Maldacena-Wilson loop, JHEP02 (2019) 095 [arXiv:1810.10483] [INSPIRE].CrossRefADSMathSciNetzbMATHGoogle Scholar
  60. [60]
    M.T. Grisaru, W. Siegel and M. Roček, Improved Methods for Supergraphs, Nucl. Phys.B 159 (1979) 429 [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  61. [61]
    A. Bourget, D. Rodriguez-Gomez and J.G. Russo, Universality of Toda equation in \( \mathcal{N} \) = 2 superconformal field theories, JHEP02 (2019) 011 [arXiv:1810.00840] [INSPIRE].CrossRefADSzbMATHGoogle Scholar
  62. [62]
    A.K. Cyrol, M. Mitter and N. Strodthoff, FormTracerA Mathematica Tracing Package Using FORM, Comput. Phys. Commun.219 (2017) 346 [arXiv:1610.09331] [INSPIRE].CrossRefADSzbMATHGoogle Scholar
  63. [63]
    A. Armoni, M. Shifman and G. Veneziano, From superYang-Mills theory to QCD: Planar equivalence and its implications, in From fields to strings: Circumnavigating theoretical physics. Ian Kogan memorial collection (3 volume set), M. Shifman, A. Vainshtein and J. Wheater, eds., pp. 353–444, (2004), hep-th/0403071 [INSPIRE].
  64. [64]
    D. Anselmi, J. Erlich, D.Z. Freedman and A.A. Johansen, Positivity constraints on anomalies in supersymmetric gauge theories, Phys. Rev.D 57 (1998) 7570 [hep-th/9711035] [INSPIRE].ADSMathSciNetGoogle Scholar
  65. [65]
    S. Kovacs, A perturbative reanalysis of N = 4 supersymmetric Yang-Mills theory, Int. J. Mod. Phys.A 21 (2006) 4555 [hep-th/9902047] [INSPIRE].CrossRefADSMathSciNetzbMATHGoogle Scholar
  66. [66]
    S.A. Larin, F.V. Tkachov and J.A.M. Vermaseren, The FORM version of MINCER, NIKHEF-H-91-18 (1991) [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Dipartimento di FisicaUniversità di TorinoTorinoItaly
  2. 2.I.N.F.N. — sezione di TorinoTorinoItaly
  3. 3.Dipartimento di Scienze e Innovazione TecnologicaUniversità del Piemonte OrientaleAlessandriaItaly

Personalised recommendations