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Exact 1/N expansion of Wilson loop correlators in \( \mathcal{N} \) = 4 Super-Yang-Mills theory

A preprint version of the article is available at arXiv.

Abstract

Supersymmetric circular Wilson loops in \( \mathcal{N} \) = 4 Super-Yang-Mills theory are discussed starting from their Gaussian matrix model representations. Previous results on the generating functions of Wilson loops are reviewed and extended to the more general case of two different loop contours, which is needed to discuss coincident loops with opposite orientations. A combinatorial formula representing the connected correlators of multiply wound Wilson loops in terms of the matrix model solution is derived. Two new results are obtained on the expectation value of the circular Wilson loop, the expansion of which into a series in 1/N and to all orders in the ’t Hooft coupling λ was derived by Drukker and Gross about twenty years ago. The connected correlators of two multiply wound Wilson loops with arbitrary winding numbers are calculated as a series in 1/N. The coefficient functions are derived not only as power series in λ, but also to all orders in λ by expressing them in terms of the coefficients of the Drukker and Gross series. This provides an efficient way to calculate the 1/N series, which can probably be generalized to higher-point correlators.

References

  1. J. M. Maldacena, Wilson loops in large N field theories, Phys. Rev. Lett. 80 (1998) 4859 [hep-th/9803002] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  2. 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].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  3. N. Drukker, D. J. Gross and H. Ooguri, Wilson loops and minimal surfaces, Phys. Rev. D 60 (1999) 125006 [hep-th/9904191] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  4. G. ’t Hooft, A Planar Diagram Theory for Strong Interactions, Nucl. Phys. B 72 (1974) 461 [INSPIRE].

  5. E. Brézin, C. Itzykson, G. Parisi and J. B. Zuber, Planar Diagrams, Commun. Math. Phys. 59 (1978) 35 [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  6. C. Itzykson and J. B. Zuber, The Planar Approximation. 2., J. Math. Phys. 21 (1980) 411 [INSPIRE].

  7. N. Drukker and B. Fiol, All-genus calculation of Wilson loops using D-branes, JHEP 02 (2005) 010 [hep-th/0501109] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  8. S. Yamaguchi, Bubbling geometries for half BPS Wilson lines, Int. J. Mod. Phys. A 22 (2007) 1353 [hep-th/0601089] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  9. S. Yamaguchi, Wilson loops of anti-symmetric representation and D5-branes, JHEP 05 (2006) 037 [hep-th/0603208] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  10. J. Gomis and F. Passerini, Holographic Wilson Loops, JHEP 08 (2006) 074 [hep-th/0604007] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  11. O. Lunin, On gravitational description of Wilson lines, JHEP 06 (2006) 026 [hep-th/0604133] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  12. J. Gomis and F. Passerini, Wilson Loops as D3-branes, JHEP 01 (2007) 097 [hep-th/0612022] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  13. S. Förste, D. Ghoshal and S. Theisen, Stringy corrections to the Wilson loop in N = 4 superYang-Mills theory, JHEP 08 (1999) 013 [hep-th/9903042] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  14. N. Drukker, D. J. Gross and A. A. Tseytlin, Green-Schwarz string in AdS5 × S5: Semiclassical partition function, JHEP 04 (2000) 021 [hep-th/0001204] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  15. G. W. Semenoff and K. Zarembo, More exact predictions of SUSYM for string theory, Nucl. Phys. B 616 (2001) 34 [hep-th/0106015] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  16. M. Kruczenski and A. Tirziu, Matching the circular Wilson loop with dual open string solution at 1-loop in strong coupling, JHEP 05 (2008) 064 [arXiv:0803.0315] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  17. A. Faraggi and L. A. Pando Zayas, The Spectrum of Excitations of Holographic Wilson Loops, JHEP 05 (2011) 018 [arXiv:1101.5145] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  18. 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].

    ADS  Article  Google Scholar 

  19. C. Kristjansen and Y. Makeenko, More about One-Loop Effective Action of Open Superstring in AdS5 × S5, JHEP 09 (2012) 053 [arXiv:1206.5660] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  20. A. Faraggi, J. T. Liu, L. A. Pando Zayas and G. Zhang, One-loop structure of higher rank Wilson loops in AdS/CFT, Phys. Lett. B 740 (2015) 218 [arXiv:1409.3187] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  21. A. Faraggi, L. A. Pando Zayas, G. A. Silva and D. Trancanelli, Toward precision holography with supersymmetric Wilson loops, JHEP 04 (2016) 053 [arXiv:1601.04708] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  22. M. Horikoshi and K. Okuyama, α′-expansion of Anti-Symmetric Wilson Loops in \( \mathcal{N} \) = 4 SYM from Fermi Gas, PTEP 2016 (2016) 113B05 [arXiv:1607.01498] [INSPIRE].

  23. V. Forini, A. A. Tseytlin and E. Vescovi, Perturbative computation of string one-loop corrections to Wilson loop minimal surfaces in AdS5 × S5, JHEP 03 (2017) 003 [arXiv:1702.02164] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  24. 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].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  25. V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  26. V. Pestun et al., Localization techniques in quantum field theories, J. Phys. A 50 (2017) 440301 [arXiv:1608.02952] [INSPIRE].

    MathSciNet  MATH  Article  Google Scholar 

  27. K. Zarembo, Localization and AdS/CFT Correspondence, J. Phys. A 50 (2017) 443011 [arXiv:1608.02963] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  28. 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].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  29. 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].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  30. 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. 524 (2002) 400] [hep-th/0101225] [INSPIRE].

  31. S. A. Hartnoll and S. P. Kumar, Higher rank Wilson loops from a matrix model, JHEP 08 (2006) 026 [hep-th/0605027] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  32. B. Fiol and G. Torrents, Exact results for Wilson loops in arbitrary representations, JHEP 01 (2014) 020 [arXiv:1311.2058] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  33. J. Ambjørn, L. Chekhov, C. F. Kristjansen and Y. Makeenko, Matrix model calculations beyond the spherical limit, Nucl. Phys. B 404 (1993) 127 [Erratum ibid. 449 (1995) 681] [hep-th/9302014] [INSPIRE].

  34. K. Okuyama and G. W. Semenoff, Wilson loops in N = 4 SYM and fermion droplets, JHEP 06 (2006) 057 [hep-th/0604209] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  35. X. Chen-Lin, Symmetric Wilson Loops beyond leading order, SciPost Phys. 1 (2016) 013 [arXiv:1610.02914] [INSPIRE].

    ADS  Article  Google Scholar 

  36. J. Gordon, Antisymmetric Wilson loops in \( \mathcal{N} \) = 4 SYM beyond the planar limit, JHEP 01 (2018) 107 [arXiv:1708.05778] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  37. K. Okuyama, Phase Transition of Anti-Symmetric Wilson Loops in \( \mathcal{N} \) = 4 SYM, JHEP 12 (2017) 125 [arXiv:1709.04166] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  38. K. Okuyama, Connected correlator of 1/2 BPS Wilson loops in \( \mathcal{N} \) = 4 SYM, JHEP 10 (2018) 037 [arXiv:1808.10161] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  39. M. Beccaria and A. Hasan, On topological recursion for Wilson loops in \( \mathcal{N} \) = 4 SYM at strong coupling, JHEP 04 (2021) 194 [arXiv:2102.12322] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  40. B. Fiol, J. Martínez-Montoya and A. Rios Fukelman, Wilson loops in terms of color invariants, JHEP 05 (2019) 202 [arXiv:1812.06890] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  41. A. F. Canazas Garay, A. Faraggi and W. Mück, Antisymmetric Wilson loops in \( \mathcal{N} \) = 4 SYM: from exact results to non-planar corrections, JHEP 08 (2018) 149 [arXiv:1807.04052] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  42. M. Beccaria and A. A. Tseytlin, On the structure of non-planar strong coupling corrections to correlators of BPS Wilson loops and chiral primary operators, JHEP 01 (2021) 149 [arXiv:2011.02885] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  43. 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].

    ADS  Article  Google Scholar 

  44. M. Mariño, Chern-Simons theory, matrix models, and topological strings, Int. Ser. Monogr. Phys. 131 (2005) 1 [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  45. A. F. Canazas Garay, A. Faraggi and W. Mück, Note on generating functions and connected correlators of 1/2-BPS Wilson loops in \( \mathcal{N} \) = 4 SYM theory, JHEP 08 (2019) 149 [arXiv:1906.03816] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  46. W. Mück, Combinatorics of Wilson loops in \( \mathcal{N} \) = 4 SYM theory, JHEP 11 (2019) 096 [arXiv:1908.11582] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  47. K. Okuyama, Spectral form factor and semi-circle law in the time direction, JHEP 02 (2019) 161 [arXiv:1811.09988] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  48. I. Macdonald, Symmetric Functions and Hall Polynomials, Oxford University Press, 2nd edition (1995).

  49. A. Lascoux, Symmetric functions, https://www.emis.de/journals/SLC/wpapers/s68vortrag/ALCoursSf2.pdf.

  50. I. M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V. S. Retakh and J.-Y. Thibon, Noncommutative symmetric functions, hep-th/9407124 [INSPIRE].

  51. H. Ooguri and C. Vafa, Knot invariants and topological strings, Nucl. Phys. B 577 (2000) 419 [hep-th/9912123] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  52. M. L. Mehta, A Method of Integration Over Matrix Variables, Commun. Math. Phys. 79 (1981) 327 [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  53. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, Academic Press, New York, 5th edition (1994).

    MATH  Google Scholar 

  54. F. W. J. Olver eds., NIST Digital Library of Mathematical Functions, Release 1.0.22 of 2019-03-15 [http://dlmf.nist.gov/].

  55. T. Agoh and K. Dilcher, Convolution Identities for Stirling Numbers of the First Kind, Integers 10 (2010) 101.

    MathSciNet  MATH  Article  Google Scholar 

  56. The Sage Developers, SageMath, the Sage Mathematics Software System (Version 9.0), (2020) [10.5281/zenodo.593563] [https://www.sagemath.org].

  57. E. Rainville, Special Functions, Mac Millan, New York (1960).

    MATH  Google Scholar 

  58. E. Rainville, The contiguous function relations for pFq with appliactions to Bateman’s \( {J}_n^{u,v} \) and Rice’s Hn(ζ, p, v), Bull. Am. Math. Soc. 51 (1945) 714.

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Mück, W. Exact 1/N expansion of Wilson loop correlators in \( \mathcal{N} \) = 4 Super-Yang-Mills theory. J. High Energ. Phys. 2021, 1 (2021). https://doi.org/10.1007/JHEP07(2021)001

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  • DOI: https://doi.org/10.1007/JHEP07(2021)001

Keywords

  • 1/N Expansion
  • Wilson
  • ’t Hooft and Polyakov loops
  • AdS-CFT Correspondence