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Non-supersymmetric Wilson loop in \( \mathcal{N} \) = 4 SYM and defect 1d CFT

  • Matteo Beccaria
  • Simone Giombi
  • Arkady A. Tseytlin
Open Access
Regular Article - Theoretical Physics

Abstract

Following Polchinski and Sully (arXiv:1104.5077), we consider a generalized Wilson loop operator containing a constant parameter ζ in front of the scalar coupling term, so that ζ = 0 corresponds to the standard Wilson loop, while ζ = 1 to the locally supersymmetric one. We compute the expectation value of this operator for circular loop as a function of ζ to second order in the planar weak coupling expansion in \( \mathcal{N} \) = 4 SYM theory. We then explain the relation of the expansion near the two conformal points ζ = 0 and ζ = 1 to the correlators of scalar operators inserted on the loop. We also discuss the AdS5 × S5 string 1-loop correction to the strong-coupling expansion of the standard circular Wilson loop, as well as its generalization to the case of mixed boundary conditions on the five-sphere coordinates, corresponding to general ζ. From the point of view of the defect CFT1 defined on the Wilson line, the ζ-dependent term can be seen as a perturbation driving a RG flow from the standard Wilson loop in the UV to the supersymmetric Wilson loop in the IR. Both at weak and strong coupling we find that the logarithm of the expectation value of the standard Wilson loop for the circular contour is larger than that of the supersymmetric one, which appears to be in agreement with the 1d analog of the F-theorem.

Keywords

AdS-CFT Correspondence Wilson ’t Hooft and Polyakov loops Supersymmetric Gauge Theory 

Notes

Open Access

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References

  1. [1]
    J.M. Maldacena, Wilson loops in large N field theories, Phys. Rev. Lett. 80 (1998) 4859 [hep-th/9803002] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  2. [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].ADSCrossRefMATHGoogle Scholar
  3. [3]
    L.F. Alday and J. Maldacena, Comments on gluon scattering amplitudes via AdS/CFT, JHEP 11 (2007) 068 [arXiv:0710.1060] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    J. Polchinski and J. Sully, Wilson Loop Renormalization Group Flows, JHEP 10 (2011) 059 [arXiv:1104.5077] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    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].MathSciNetMATHGoogle Scholar
  6. [6]
    S. Giombi, R. Roiban and A.A. Tseytlin, Half-BPS Wilson loop and AdS 2 /CFT 1, Nucl. Phys. B 922 (2017) 499 [arXiv:1706.00756] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  7. [7]
    A.M. Polyakov, Gauge Fields as Rings of Glue, Nucl. Phys. B 164 (1980) 171 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  8. [8]
    V.S. Dotsenko and S.N. Vergeles, Renormalizability of Phase Factors in the Nonabelian Gauge Theory, Nucl. Phys. B 169 (1980) 527 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  9. [9]
    J.-L. Gervais and A. Neveu, The Slope of the Leading Regge Trajectory in Quantum Chromodynamics, Nucl. Phys. B 163 (1980) 189 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  10. [10]
    I. Ya. Arefeva, Quantum contour field equations, Phys. Lett. B 93 (1980) 347.Google Scholar
  11. [11]
    H. Dorn, Renormalization of Path Ordered Phase Factors and Related Hadron Operators in Gauge Field Theories, Fortsch. Phys. 34 (1986) 11 [INSPIRE].ADSMathSciNetGoogle Scholar
  12. [12]
    R.M. Marinho and L. Boanerges Peixoto, Charge renormalization of the Yang-Mills theory up to fourth order using dimensional regularization, Nuovo Cim. A 97 (1987) 148 [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    K. Zarembo, Localization and AdS/CFT Correspondence, J. Phys. A 50 (2017) 443011 [arXiv:1608.02963] [INSPIRE].ADSMathSciNetMATHGoogle Scholar
  17. [17]
    I.R. Klebanov, S.S. Pufu and B.R. Safdi, F-Theorem without Supersymmetry, JHEP 10 (2011) 038 [arXiv:1105.4598] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    L. Fei, S. Giombi, I.R. Klebanov and G. Tarnopolsky, Generalized F -Theorem and the ϵ Expansion, JHEP 12 (2015) 155 [arXiv:1507.01960] [INSPIRE].ADSMathSciNetMATHGoogle Scholar
  19. [19]
    R.C. Myers and A. Sinha, Seeing a c-theorem with holography, Phys. Rev. D 82 (2010) 046006 [arXiv:1006.1263] [INSPIRE].ADSGoogle Scholar
  20. [20]
    H. Casini and M. Huerta, On the RG running of the entanglement entropy of a circle, Phys. Rev. D 85 (2012) 125016 [arXiv:1202.5650] [INSPIRE].ADSGoogle Scholar
  21. [21]
    S. Giombi and I.R. Klebanov, Interpolating between a and F , JHEP 03 (2015) 117 [arXiv:1409.1937] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    I. Affleck and A.W.W. Ludwig, Universal noninteger ’ground state degeneracy’ in critical quantum systems, Phys. Rev. Lett. 67 (1991) 161 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    D. Friedan and A. Konechny, On the boundary entropy of one-dimensional quantum systems at low temperature, Phys. Rev. Lett. 93 (2004) 030402 [hep-th/0312197] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  24. [24]
    D. Young, Wavy Line Wilson Loops in the AdS/CFT Correspondence, MSc Thesis, British Columbia University, Vancouver U.S.A. (2003).Google Scholar
  25. [25]
    M.S. Bianchi, L. Griguolo, M. Leoni, A. Mauri, S. Penati and D. Seminara, The quantum 1/2 BPS Wilson loop in \( \mathcal{N} \) = 4 Chern-Simons-matter theories, JHEP 09 (2016) 009 [arXiv:1606.07058] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  26. [26]
    J.L. Cardy, Is There a c Theorem in Four-Dimensions?, Phys. Lett. B 215 (1988) 749 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  27. [27]
    B. Jantzen, New proofs for the two Barnes lemmas and an additional lemma, J. Math. Phys. 54 (2013) 012304 [arXiv:1211.2637] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  28. [28]
    M. Czakon, Automatized analytic continuation of Mellin-Barnes integrals, Comput. Phys. Commun. 175 (2006) 559 [hep-ph/0511200] [INSPIRE].
  29. [29]
    W.N. Bailey, Generalized hypergeometric series, Cambridge University Press, Cambridge U.K. (1935).MATHGoogle Scholar
  30. [30]
    A.M. Polyakov and V.S. Rychkov, Gauge field strings duality and the loop equation, Nucl. Phys. B 581 (2000) 116 [hep-th/0002106] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  31. [31]
    N. Drukker and S. Kawamoto, Small deformations of supersymmetric Wilson loops and open spin-chains, JHEP 07 (2006) 024 [hep-th/0604124] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  32. [32]
    M. Sakaguchi and K. Yoshida, Holography of Non-relativistic String on AdS 5 × S 5, JHEP 02 (2008) 092 [arXiv:0712.4112] [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    N. Drukker and V. Forini, Generalized quark-antiquark potential at weak and strong coupling, JHEP 06 (2011) 131 [arXiv:1105.5144] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  34. [34]
    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, JHEP 06 (2012) 048 [arXiv:1202.4455] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  35. [35]
    J. Maldacena, private communication (2017).Google Scholar
  36. [36]
    M. Kim, N. Kiryu, S. Komatsu and T. Nishimura, Structure Constants of Defect Changing Operators on the 1/2 BPS Wilson Loop, JHEP 12 (2017) 055 [arXiv:1710.07325] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  37. [37]
    M. Kim and N. Kiryu, Structure constants of operators on the Wilson loop from integrability, JHEP 11 (2017) 116 [arXiv:1706.02989] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  38. [38]
    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
  39. [39]
    N. Drukker, D.J. Gross and A.A. Tseytlin, Green-Schwarz string in AdS 5 × S 5 : Semiclassical partition function, JHEP 04 (2000) 021 [hep-th/0001204] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  40. [40]
    E.I. Buchbinder and A.A. Tseytlin, 1/N correction in the D3-brane description of a circular Wilson loop at strong coupling, Phys. Rev. D 89 (2014) 126008 [arXiv:1404.4952] [INSPIRE].ADSGoogle Scholar
  41. [41]
    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].ADSMathSciNetCrossRefGoogle Scholar
  42. [42]
    J.L. Miramontes and J.M. Sanchez de Santos, Are there infrared problems in the 2 − d nonlinear σ-models?, Phys. Lett. B 246 (1990) 399 [INSPIRE].ADSCrossRefGoogle Scholar
  43. [43]
    T. Hartman and L. Rastelli, Double-trace deformations, mixed boundary conditions and functional determinants in AdS/CFT, JHEP 01 (2008) 019 [hep-th/0602106] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  44. [44]
    D.E. Diaz and H. Dorn, Partition functions and double-trace deformations in AdS/CFT, JHEP 05 (2007) 046 [hep-th/0702163] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  45. [45]
    S. Giombi, I.R. Klebanov, S.S. Pufu, B.R. Safdi and G. Tarnopolsky, AdS Description of Induced Higher-Spin Gauge Theory, JHEP 10 (2013) 016 [arXiv:1306.5242] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  46. [46]
    M. Beccaria, X. Bekaert and A.A. Tseytlin, Partition function of free conformal higher spin theory, JHEP 08 (2014) 113 [arXiv:1406.3542] [INSPIRE].ADSCrossRefGoogle Scholar
  47. [47]
    S. Giombi, C. Sleight and M. Taronna, Spinning AdS Loop Diagrams: Two Point Functions, arXiv:1708.08404 [INSPIRE].
  48. [48]
    S. Giombi, R. Ricci, R. Roiban and A.A. Tseytlin, Quantum dispersion relations for excitations of long folded spinning superstring in AdS 5 × S 5, JHEP 01 (2011) 128 [arXiv:1011.2755] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  49. [49]
    C. Kristjansen and Y. Makeenko, More about One-Loop Effective Action of Open Superstring in AdS 5 × S 5, JHEP 09 (2012) 053 [arXiv:1206.5660] [INSPIRE].ADSCrossRefGoogle Scholar
  50. [50]
    R. Bergamin and A.A. Tseytlin, Heat kernels on cone of AdS 2 and k-wound circular Wilson loop in AdS 5 × S 5 superstring, J. Phys. A 49 (2016) 14LT01 [arXiv:1510.06894] [INSPIRE].
  51. [51]
    V. Forini, A.A. Tseytlin and E. Vescovi, Perturbative computation of string one-loop corrections to Wilson loop minimal surfaces in AdS 5 × S 5, JHEP 03 (2017) 003 [arXiv:1702.02164] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  52. [52]
    C.G. Callan Jr. and Z. Gan, Vertex Operators in Background Fields, Nucl. Phys. B 272 (1986) 647 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  53. [53]
    A.A. Tseytlin, Conformal Anomaly in Two-Dimensional σ-model on Curved Background and Strings, Phys. Lett. B 178 (1986) 34 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  54. [54]
    H.P. McKean and I.M. Singer, Curvature and eigenvalues of the Laplacian, J. Diff. Geom. 1 (1967) 43 [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  55. [55]
    G. Kennedy, R. Critchley and J.S. Dowker, Finite Temperature Field Theory with Boundaries: Stress Tensor and Surface Action Renormalization, Annals Phys. 125 (1980) 346 [INSPIRE].ADSCrossRefGoogle Scholar
  56. [56]
    E.S. Fradkin and A.A. Tseytlin, On quantized string models, Annals Phys. 143 (1982) 413 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  57. [57]
    J.S. Dowker, The Hybrid spectral problem and Robin boundary conditions, J. Phys. A 38 (2005) 4735 [math/0409442] [INSPIRE].
  58. [58]
    M. Billò, V. Gonçalves, E. Lauria and M. Meineri, Defects in conformal field theory, JHEP 04 (2016) 091 [arXiv:1601.02883] [INSPIRE].ADSMathSciNetMATHGoogle Scholar
  59. [59]
    D. Correa, J. Maldacena and A. Sever, The quark anti-quark potential and the cusp anomalous dimension from a TBA equation, JHEP 08 (2012) 134 [arXiv:1203.1913] [INSPIRE].ADSCrossRefGoogle Scholar
  60. [60]
    N. Drukker, Integrable Wilson loops, JHEP 10 (2013) 135 [arXiv:1203.1617] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  61. [61]
    M.S. Bianchi, G. Giribet, M. Leoni and S. Penati, The 1/2 BPS Wilson loop in ABJ(M) at two loops: The details, JHEP 10 (2013) 085 [arXiv:1307.0786] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  62. [62]
    L. Griguolo, G. Martelloni, M. Poggi and D. Seminara, Perturbative evaluation of circular 1/2 BPS Wilson loops in N = 6 Super Chern-Simons theories, JHEP 09 (2013) 157 [arXiv:1307.0787] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Dipartimento di Matematica e Fisica Ennio De GiorgiUniversità del Salento & INFNLecceItaly
  2. 2.Department of PhysicsPrinceton UniversityPrincetonU.S.A.
  3. 3.Blackett LaboratoryImperial CollegeLondonU.K.
  4. 4.Lebedev InstituteMoscowRussia

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