Resurgence of the dressing phase for AdS5 × S5

Open Access
Regular Article - Theoretical Physics

Abstract

We discuss the resummation of the strong coupling asymptotic expansion of the dressing phase of the AdS5 × S5 superstring. The dressing phase proposed by Beisert, Eden and Staudacher can be recovered from a modified Borel-Ecalle resummation of this asymptotic expansion only by completing it with new, non-perturbative and exponentially suppressed terms that can be organized into different sectors labelled by an instanton-like number. We compute the contribution to the dressing phase coming from the sum over all the instanton sectors and show that it satisfies the homogeneous crossing symmetry equation. We comment on the semiclassical origin of the non-perturbative terms from the world-sheet theory point of view even though their precise explanation remains still quite mysterious.

Keywords

Nonperturbative Effects Integrable Field Theories AdS-CFT Correspondence 

References

  1. [1]
    G. Arutyunov and S. Frolov, Foundations of the AdS 5 × S 5 superstring: I, J. Phys. A 42 (2009) 254003 [arXiv:0901.4937] [INSPIRE].ADSMathSciNetMATHGoogle Scholar
  2. [2]
    N. Beisert et al., Review of AdS/CFT integrability: an overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    G. Arutyunov and S. Frolov, String hypothesis for the AdS 5 × S 5 mirror, JHEP 03 (2009) 152 [arXiv:0901.1417] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    G. Arutyunov and S. Frolov, Thermodynamic Bethe ansatz for the AdS 5 × S 5 mirror model, JHEP 05 (2009) 068 [arXiv:0903.0141] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    D. Bombardelli, D. Fioravanti and R. Tateo, Thermodynamic Bethe ansatz for planar AdS/CFT: a proposal, J. Phys. A 42 (2009) 375401 [arXiv:0902.3930] [INSPIRE].ADSMathSciNetMATHGoogle Scholar
  6. [6]
    N. Gromov, V. Kazakov and P. Vieira, Exact spectrum of anomalous dimensions of planar N = 4 supersymmetric Yang-Mills theory, Phys. Rev. Lett. 103 (2009) 131601 [arXiv:0901.3753] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  7. [7]
    N. Gromov, V. Kazakov, A. Kozak and P. Vieira, Exact spectrum of anomalous dimensions of planar N = 4 supersymmetric Yang-Mills theory: TBA and excited states, Lett. Math. Phys. 91 (2010) 265 [arXiv:0902.4458] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    N. Gromov, V. Kazakov, S. Leurent and D. Volin, Quantum spectral curve for planar \( \mathcal{N} \) = 4 super-Yang-Mills theory, Phys. Rev. Lett. 112 (2014) 011602 [arXiv:1305.1939] [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    G. Arutyunov, S. Frolov and M. Staudacher, Bethe ansatz for quantum strings, JHEP 10 (2004) 016 [hep-th/0406256] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  10. [10]
    R.A. Janik, The AdS 5 × S 5 superstring worldsheet S-matrix and crossing symmetry, Phys. Rev. D 73 (2006) 086006 [hep-th/0603038] [INSPIRE].ADSMathSciNetGoogle Scholar
  11. [11]
    R. Hernandez and E. Lopez, Quantum corrections to the string Bethe ansatz, JHEP 07 (2006) 004 [hep-th/0603204] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  12. [12]
    G. Arutyunov and S. Frolov, On AdS 5 × S 5 string S-matrix, Phys. Lett. B 639 (2006) 378 [hep-th/0604043] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    N. Beisert, R. Hernandez and E. Lopez, A crossing-symmetric phase for AdS 5 × S 5 strings, JHEP 11 (2006) 070 [hep-th/0609044] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  14. [14]
    N. Beisert, B. Eden and M. Staudacher, Transcendentality and crossing, J. Stat. Mech. (2007) P01021 [hep-th/0610251] [INSPIRE].
  15. [15]
    N. Dorey, D.M. Hofman and J.M. Maldacena, On the singularities of the magnon S-matrix, Phys. Rev. D 76 (2007) 025011 [hep-th/0703104] [INSPIRE].ADSMathSciNetMATHGoogle Scholar
  16. [16]
    G. Arutyunov and S. Frolov, The dressing factor and crossing equations, J. Phys. A 42 (2009) 425401 [arXiv:0904.4575] [INSPIRE].ADSMathSciNetMATHGoogle Scholar
  17. [17]
    D. Volin, Minimal solution of the AdS/CFT crossing equation, J. Phys. A 42 (2009) 372001 [arXiv:0904.4929] [INSPIRE].MathSciNetMATHGoogle Scholar
  18. [18]
    P. Vieira and D. Volin, Review of AdS/CFT integrability, chapter III.3: the dressing factor, Lett. Math. Phys. 99 (2012) 231 [arXiv:1012.3992] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    S. Frolov, Konishi operator at intermediate coupling, J. Phys. A 44 (2011) 065401 [arXiv:1006.5032] [INSPIRE].ADSMathSciNetMATHGoogle Scholar
  20. [20]
    C. Gomez and R. Hernandez, Integrability and non-perturbative effects in the AdS/CFT correspondence, Phys. Lett. B 644 (2007) 375 [hep-th/0611014] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    R.A. Janik and T. Lukowski, Wrapping interactions at strong coupling: the giant magnon, Phys. Rev. D 76 (2007) 126008 [arXiv:0708.2208] [INSPIRE].ADSGoogle Scholar
  22. [22]
    D. Fioravanti and M. Rossi, On the commuting charges for the highest dimension SU(2) operator in planar \( \mathcal{N} \) = 4 SYM, JHEP 08 (2007) 089 [arXiv:0706.3936] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  23. [23]
    J. Ecalle, Les fonctions resurgentes, vol. I-III, Publications Mathématiques d’Orsay (1981).Google Scholar
  24. [24]
    G. Arutyunov, S. Frolov and A. Petkou, Perturbative and instanton corrections to the OPE of CPOs in \( \mathcal{N} \) = 4 SYM 4, Nucl. Phys. B 602 (2001) 238 [Erratum ibid. B 609 (2001) 540] [hep-th/0010137] [INSPIRE].
  25. [25]
    L.F. Alday and G.P. Korchemsky, Revisiting instanton corrections to the Konishi multiplet, JHEP 12 (2016) 005 [arXiv:1605.06346] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    B. Basso, G.P. Korchemsky and J. Kotanski, Cusp anomalous dimension in maximally supersymmetric Yang-Mills theory at strong coupling, Phys. Rev. Lett. 100 (2008) 091601 [arXiv:0708.3933] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  27. [27]
    B. Basso and G.P. Korchemsky, Nonperturbative scales in AdS/CFT, J. Phys. A 42 (2009) 254005 [arXiv:0901.4945] [INSPIRE].ADSMathSciNetMATHGoogle Scholar
  28. [28]
    I. Aniceto, The resurgence of the cusp anomalous dimension, J. Phys. A 49 (2016) 065403 [arXiv:1506.03388] [INSPIRE].ADSMathSciNetMATHGoogle Scholar
  29. [29]
    D. Dorigoni and Y. Hatsuda, Resurgence of the cusp anomalous dimension, JHEP 09 (2015) 138 [arXiv:1506.03763] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  30. [30]
    S.S. Gubser, I.R. Klebanov and A.M. Polyakov, A semiclassical limit of the gauge/string correspondence, Nucl. Phys. B 636 (2002) 99 [hep-th/0204051] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  31. [31]
    L.F. Alday and J.M. Maldacena, Comments on operators with large spin, JHEP 11 (2007) 019 [arXiv:0708.0672] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  32. [32]
    G.V. Dunne and M. Ünsal, Resurgence and dynamics of O(N ) and Grassmannian sigma models, JHEP 09 (2015) 199 [arXiv:1505.07803] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  33. [33]
    D.E. Berenstein, J.M. Maldacena and H.S. Nastase, Strings in flat space and pp waves from \( \mathcal{N} \) =4 super Yang-Mills, JHEP 04 (2002) 013 [hep-th/0202021] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  34. [34]
    T. Klose, T. McLoughlin, R. Roiban and K. Zarembo, Worldsheet scattering in AdS 5 × S 5, JHEP 03 (2007) 094 [hep-th/0611169] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  35. [35]
    L. Bianchi, V. Forini and B. Hoare, Two-dimensional S-matrices from unitarity cuts, JHEP 07 (2013) 088 [arXiv:1304.1798] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  36. [36]
    B. Hoare, R. Roiban and A.A. Tseytlin, On deformations of AdS n × S n supercosets, JHEP 06 (2014) 002 [arXiv:1403.5517] [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    F. Delduc, M. Magro and B. Vicedo, An integrable deformation of the AdS 5 × S 5 superstring action, Phys. Rev. Lett. 112 (2014) 051601 [arXiv:1309.5850] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  38. [38]
    F. Delduc, M. Magro and B. Vicedo, Derivation of the action and symmetries of the q-deformed AdS 5 × S 5 superstring, JHEP 10 (2014) 132 [arXiv:1406.6286] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  39. [39]
    B. Hoare, T.J. Hollowood and J.L. Miramontes, q-deformation of the AdS 5 × S 5 superstring S-matrix and its relativistic limit, JHEP 03 (2012) 015 [arXiv:1112.4485] [INSPIRE].
  40. [40]
    G. Arutyunov, M. de Leeuw and S.J. van Tongeren, The quantum deformed mirror TBA I, JHEP 10 (2012) 090 [arXiv:1208.3478] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  41. [41]
    G. Arutyunov, M. de Leeuw and S.J. van Tongeren, The quantum deformed mirror TBA II, JHEP 02 (2013) 012 [arXiv:1210.8185] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  42. [42]
    G. Arutyunov, M. de Leeuw and S.J. van Tongeren, The exact spectrum and mirror duality of the (AdS 5 × S 5)η superstring, Theor. Math. Phys. 182 (2015) 23 [Teor. Mat. Fiz. 182 (2014) 28] [arXiv:1403.6104] [INSPIRE].
  43. [43]
    S.J. van Tongeren, Integrability of the AdS 5 × S 5 superstring and its deformations, J. Phys. A 47 (2014) 433001 [arXiv:1310.4854] [INSPIRE].MathSciNetMATHGoogle Scholar
  44. [44]
    R. Borsato, O. Ohlsson Sax, A. Sfondrini, B. Stefanski Jr. and A. Torrielli, Dressing phases of AdS 3 /CFT 2, Phys. Rev. D 88 (2013) 066004 [arXiv:1306.2512] [INSPIRE].ADSGoogle Scholar
  45. [45]
    P. Sundin and L. Wulff, The complete one-loop BMN S-matrix in AdS 3 × S 3 × T 4, JHEP 06 (2016) 062 [arXiv:1605.01632] [INSPIRE].ADSCrossRefGoogle Scholar
  46. [46]
    R. Borsato, O. Ohlsson Sax, A. Sfondrini, B. Stefanski Jr. and A. Torrielli, On the dressing factors, Bethe equations and Yangian symmetry of strings on AdS 3 × S 3 × T 4, J. Phys. A 50 (2017) 024004 [arXiv:1607.00914] [INSPIRE].ADSGoogle Scholar
  47. [47]
    M.P. Heller, R.A. Janik and P. Witaszczyk, Hydrodynamic gradient expansion in gauge theory plasmas, Phys. Rev. Lett. 110 (2013) 211602 [arXiv:1302.0697] [INSPIRE].ADSCrossRefGoogle Scholar
  48. [48]
    M.P. Heller and M. Spalinski, Hydrodynamics beyond the gradient expansion: resurgence and resummation, Phys. Rev. Lett. 115 (2015) 072501 [arXiv:1503.07514] [INSPIRE].ADSCrossRefGoogle Scholar
  49. [49]
    G. Basar and G.V. Dunne, Hydrodynamics, resurgence and transasymptotics, Phys. Rev. D 92 (2015) 125011 [arXiv:1509.05046] [INSPIRE].ADSGoogle Scholar
  50. [50]
    I. Aniceto and M. Spalinski, Resurgence in extended hydrodynamics, Phys. Rev. D 93 (2016) 085008 [arXiv:1511.06358] [INSPIRE].ADSGoogle Scholar
  51. [51]
    J.G. Russo, A note on perturbation series in supersymmetric gauge theories, JHEP 06 (2012) 038 [arXiv:1203.5061] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  52. [52]
    E. Delabaere and F. Pham, Resurgent methods in semi-classical asymptotics, Ann. Inst. Henri Poincaré 71 (1999) 1.MathSciNetMATHGoogle Scholar
  53. [53]
    G.A. Edgar, Transseries for beginners, Real Anal. Exchange 35 (2009) 253 [arXiv:0801.4877].MathSciNetMATHGoogle Scholar
  54. [54]
    I. Aniceto and R. Schiappa, Nonperturbative ambiguities and the reality of resurgent transseries, Commun. Math. Phys. 335 (2015) 183 [arXiv:1308.1115] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  55. [55]
    R. Couso-Santamaría, R. Schiappa and R. Vaz, Finite N from resurgent large N , Annals Phys. 356 (2015) 1 [arXiv:1501.01007] [INSPIRE].
  56. [56]
    A.P. Prudnikov, Y.A. Brychkov and O.I. Marichev, Integrals and series. Vol. 3: More special functions, Gordon and Breach, New York U.S.A. (1989).Google Scholar
  57. [57]
    O.I. Marichev, A method of calculating integrals of special functions. Theory and tables of formulas (in Russian), Nauka i Tekhnika, Minsk U.S.S.R. (1978).Google Scholar
  58. [58]
    C.M. Bender and T.T. Wu, Anharmonic oscillator. II. A study of perturbation theory in large order, Phys. Rev. D 7 (1973) 1620 [INSPIRE].
  59. [59]
    J.C. Collins and D.E. Soper, Large order expansion in perturbation theory, Annals Phys. 112 (1978) 209 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  60. [60]
    N.M. Temme, Special functions: an introduction to the classical functions of mathematical physics, John Wiley & Sons Inc. (1996).Google Scholar
  61. [61]
    F.W. Olver, D.W. Lozier, R.F. Boisvert and C.W. Clark, NIST handbook of mathematical functions, Cambridge University Press, Cambridge U.K. (2010).MATHGoogle Scholar
  62. [62]
    R. Paris and D. Kaminski, Asymptotics and Mellin-Barnes integrals, Cambridge University Press, Cambridge U.K. (2001).CrossRefMATHGoogle Scholar

Copyright information

© The Author(s) 2017

Authors and Affiliations

  • Gleb Arutyunov
    • 1
    • 2
  • Daniele Dorigoni
    • 3
  • Sergei Savin
    • 1
    • 2
  1. 1.Institut für Theoretische PhysikUniversität HamburgHamburgGermany
  2. 2.Zentrum für Mathematische PhysikUniversität HamburgHamburgGermany
  3. 3.Centre for Particle Theory & Department of Mathematical SciencesDurham UniversityDurhamU.K.

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