Journal of High Energy Physics

, 2019:96 | Cite as

Combinatorics of Wilson loops in \( \mathcal{N} \) 4 SYM theory

  • Wolfgang MückEmail author
Open Access
Regular Article - Theoretical Physics


The theory of Wilson loops for gauge theories with unitary gauge groups is formulated in the language of symmetric functions. The main objects in this theory are two generating functions, which are related to each other by the involution that exchanges an irreducible representation with its conjugate. Both of them contain all information about the Wilson loops in arbitrary representations as well as the correlators of multiply­ wound Wilson loops. This general framework is combined with the results of the Gaussian matrix model, which calculates the expectation values of -BPS circular Wilson loops in N = 4 Super-Yang-Mills theory. General, explicit, formulas for the connected correlators of multiply-wound Wilson loops in terms of the traces of symmetrized matrix products are obtained, as well as their inverses. It is shown that the generating functions for Wilson loops in mutually conjugate representations are related by a duality relation whenever they can be calculated by a Hermitian matrix model.


Matrix Models Wilson 't Hooft and Polyakov loops 1/N Expansion 


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]
    G. ‘t Hooft, A planar diagram theory for strong interactions, Nucl. Phys.B 72 (1974) 461 [INSPIRE].
  2. [2]
    E. Brézin, C. Itzykson, G. Parisi and J.B. Zuber, Planar diagrams, Commun. Math. Phys.59 (1978) 35 [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    C. Itzykson and J.B. Zuber, The planar approximation. 2, J. Math. Phys.21 (1980) 411 [INSPIRE].
  4. [4]
    J.M. Maldacena, Wilson loops in large N field theories, Phys. Rev. Lett.80 (1998) 4859 [hep-th/9803002] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    S.-J. Rey and J.-T. Yee, Macroscopic strings as heavy quarks in large N gauge theory and anti-de Sitter supergravity, Eur. Phys. J.C 22 (2001) 379 [hep-th/9803001] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  6. [6]
    N. Drukker, D.J. Gross and H. Ooguri, Wilson loops and minimal surfaces, Phys. Rev.D 60 (1999) 125006 [hep-th/9904191] [INSPIRE].ADSMathSciNetGoogle Scholar
  7. [7]
    J.A. Minahan and K. Zaremba, The Bethe ansatz for N = 4 super Yang-Mills, JHEP03 (2003) 013 [hep-th/0212208] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  8. [8]
    V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys.313 (2012) 71 [arXiv:0712.2824] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    V. Pestun et al., Localization techniques in quantum field theories, J. Phys.A 50 (2017) 440301 [arXiv:1608.02952] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  10. [10]
    N. Drukker and B. Fiol, All-genus calculation of Wilson loops using D-branes, JHEP02 (2005) 010 [hep-th/0501109] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  11. [11]
    S. Yamaguchi, Bubbling geometries for half BPS Wilson lines, Int. J. Mod . Phys.A 22 (2007) 1353 [hep-th/0601089] [INSPIRE].
  12. [12]
    S. Yamaguchi, Wilson loops of anti-symmetric representation and D5-branes, JHEP05 (2006) 037 [hep-th/0603208] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    J. Gomis and F. Passerini, Holographic Wilson loops, JHEP08 (2006) 074 [hep-th/0604007] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  14. [14]
    O. Lunin, On gravitational description of Wilson lines, JHEP06 (2006) 026 [hep-th/0604133] [INSPIRE].
  15. [15]
    J. Gomis and F. Passerini, Wilson loops as D3-branes, JHEP01 (2007) 097 [hep-th/0612022] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  16. [16]
    S. Förste, D. Ghoshal and S. Theisen, Stringy corrections to the Wilson loop in N = 4 super Yang-Mills theory, JHEP08 (1999) 013 [hep-th/9903042] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    N. Drukker, D.J. Gross and A.A. Tseytlin, Green-Schwarz string in AdS 5 x S 5: semiclassical partition function, JHEP04 (2000) 021 [hep-th/0001204] [INSPIRE].
  18. [18]
    G.W. Semenoff and K. Zaremba, More exact predictions of SUSYM for string theory, Nucl. Phys.B 616 (2001) 34 [hep-th/0106015] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    M. Kruczenski and A. Tirziu, Matching the circular Wilson loop with dual open string solution at 1-loop in strong coupling, JHEP05 (2008) 064 [arXiv:0803.0315] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    A. Faraggi and L.A. Pando Zayas, The spectrum of excitations of holographic Wilson loops, JHEP05 (2011) 018 [arXiv:1101.5145] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    A. Faraggi, W. Mueck and L.A. Pando Zayas, One-loop effective action of the holographic antisymmetric Wilson loop, Phys. Rev.D 85 (2012) 106015 [arXiv:1112.5028] [INSPIRE].ADSGoogle Scholar
  22. [22]
    A. Faraggi, J.T. Liu, L.A. Pando Zayas and G. Zhang, One-loop structure of higher rank Wilson loops in AdSjCFT, Phys. Lett.B 740 (2015) 218 [arXiv:1409.3187] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  23. [23]
    A. Faraggi, L.A. Pando Zayas, G.A. Silva and D. Trancanelli, Toward precision holography with supersymmetric Wilson loops, JHEP04 (2016) 053 [arXiv:1601.04708] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  24. [24]
    M. Horikoshi and K. Okuyama, a′-expansion of anti-symmetric Wilson loops in N = 4 SYM from Fermi gas, PTEP2016 (2016) 113B05 [arXiv:1607.01498] [INSPIRE].
  25. [25]
    V. Forini, A.A. Tseytlin and E. Vescovi, Perturbative computation of string one-loop corrections to Wilson loop minimal surfaces in AdS 5 x S 5 , JHEP03 (2017) 003 [arXiv:1702.02164] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  26. [26]
    J. Aguilera-Damia, A. Faraggi, L.A. Pando Zayas, V. Rathee and G.A. Silva, Toward precision holography in type IIA with Wilson loops, JHEP08 (2018) 044 [arXiv:1805.00859] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  27. [27]
    J. Aguilera-Damia, A. Faraggi, L.A. Pando Zayas, V. Rathee and G.A. Silva, Zeta-function regularization of holographic Wilson loops, Phys. Rev.D 98 (2018) 046011 [arXiv:1802.03016] [INSPIRE].
  28. [28]
    D. Medina-Rincon, Matching quantum string corrections and circular Wilson loops in AdS 4 x CP 3, JHEP08 (2019) 158 [arXiv:1907.02984] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  29. [29]
    M. David, R. De Leon Ardon, A. Faraggi, L.A. Pando Zayas and G.A. Silva, One-loop holography with strings in AdS 4 X CP 3, JHEP10 (2019) 070 [arXiv:1907.08590] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    J.K. Erickson, G.W. Semenoff and K. Zaremba, Wilson loops in N = 4 supersymmetric Yang-Mills theory, Nucl. Phys.B 582 (2000) 155 [hep-th/0003055] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  31. [31]
    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].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  32. [32]
    G. Akemann and P.H. Damgaard, Wilson loops in N = 4 supersymmetric Yang-Mills theory from random matrix theory, Phys. Lett.B 513 (2001) 179 [Erratum ibid.B 524 (2002) 400] [hep-th/0101225] [INSPIRE].
  33. [33]
    S.A. Hartnoll and S.P. Kumar, Higher rank Wilson loops from a matrix model, JHEP08 (2006) 026 [hep-th/0605027] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  34. [34]
    B. Fiol and G. Torrents, Exact results for Wilson loops in arbitrary representations, JHEP01 (2014) 020 [arXiv:1311.2058] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  35. [35]
    K. Okuyama and G.W. Semenoff, Wilson loops in N = 4 SYM and fermion droplets, JHEP06 (2006) 057 [hep-th/0604209] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  36. [36]
    X. Chen-Lin, Symmetric Wilson loops beyond leading order, SciPost Phys.1 (2016) 013 [arXiv:1610.02914] [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    J. Gordon, Antisymmetric Wilson loops in N = 4 SYM beyond the planar limit, JHEP01 (2018) 107 [arXiv:1708.05778] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  38. [38]
    K. Okuyama, Phase transition of anti-symmetric Wilson loops in N = 4 SYM, JHEP12 (2017) 125 [arXiv:1709.04166] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  39. [39]
    A.F. Canazas Garay, A. Faraggi and W. Mück, Antisymmetric Wilson loops in N = 4 SYM: from exact results to non-planar corrections, JHEP08 (2018) 149 [arXiv:1807.04052] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  40. [40]
    K. Okuyama, Connected correlator of 1/2 BPS Wilson loops in N = 4 SYM, JHEP10 (2018) 037 [arXiv:1808.10161] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  41. [41]
    B. Fiol, J. Martínez-Montoya and A. Rios Fukelman, Wilson loops in terms of color invariants, JHEP05 (2019) 202 [arXiv:1812.06890] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  42. [42]
    K. Zarembo, Supersymmetric Wilson loops, Nucl. Phys.B 643 (2002) 157 [hep-th/0205160] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  43. [43]
    N. Drukker, 1/4 BPS circular loops, unstable world-sheet instantons and the matrix model, JHEP09 (2006) 004 [hep-th/0605151] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  44. [44]
    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
  45. [45]
    N. Drukker, S. Giombi, R. Ricci and D. Trancanelli, Supersymmetric Wilson loops on S 3 , JHEP05 (2008) 017 [arXiv:0711.3226] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  46. [46]
    V. Forini, V. Giangreco M. Puletti, L. Griguolo, D. Seminara and E. Vescovi, Precision calculation of 1/4-BPS Wilson loops in AdS 5 X S 5, JHEP02 (2016) 105 [arXiv:1512.00841] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  47. [47]
    M. Billo, F. Galvagno and A. Lerda, BPS Wilson loops in generic conformal N = 2 SU(N) SYMtheories, JHEP08 (2019) 108 [arXiv:1906.07085] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  48. [48]
    A.F. Canazas Garay, A. Faraggi and W. Mück, Note on generating functions and connected correlators of l/2-BPS Wilson loops in N = 4 SYM theory, JHEP08 (2019) 149 [arXiv:1906.03816] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  49. [49]
    M. Marino, Chern-Simons theory, matrix models and topological strings, Int. Ser. Monogr. Phys.131 (2005) 1 [INSPIRE].MathSciNetzbMATHGoogle Scholar
  50. [50]
    I. Macdonald, Symmetric functions and Hall polynomials, second edition, Oxford University Press, Oxford, U.K. (1995).zbMATHGoogle Scholar
  51. [51]
  52. [52]
    H. Ooguri and C. Vafa, Knot invariants and topological strings, Nucl. Phys.B 577 (2000) 419 [hep-th/9912123] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  53. [53]
    J. Ambj¢rn, L. Chekhov, C.F. Kristjansen and Yu. Makeenko, Matrix model calculations beyond the spherical limit, Nucl. Phys.B 404 (1993) 127 [Erratum ibid.B 449 (1995) 681] [hep-th/9302014] [INSPIRE].
  54. [54]
    The Sage developers, Sage Math, the Sage Mathematics Software System (version 8.7),, (2019).
  55. [55]
    F.W.J. Olver et al. eds., NIST digital library of mathematical functions, release 1.0.22,, 15 March 2019.

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Dipartimento di Pisica “Ettore Pancini”Univermta degli Studi di Napoli “Federico II”NapoliItaly
  2. 2.Istituto Nazionale di Pimca NucleareSezione di NapoliNapoliItaly

Personalised recommendations