Holographic calculations of Euclidean Wilson loop correlator in Euclidean anti-de Sitter space

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Regular Article - Theoretical Physics
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Abstract

The correlation functions of two or more Euclidean Wilson loops of various shapes in Euclidean anti-de Sitter space are computed by considering the minimal area surfaces connecting the loops. The surfaces are parametrized by Riemann theta functions associated with genus three hyperelliptic Riemann surfaces. In the case of two loops, the distance L by which they are separated can be adjusted by continuously varying a specific branch point of the auxiliary Riemann surface. When L is much larger than the characteristic size of the loops, then the loops are approximately regarded as local operators and their correlator as the correlator of two local operators. Similarly, when a loop is very small compared to the size of another loop, the small loop is considered as a local operator corresponding to a light supergravity mode.

Keywords

AdS-CFT Correspondence Long strings 

Notes

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References

  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].MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  3. [3]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    J.M. Maldacena, Wilson loops in large-N field theories, Phys. Rev. Lett. 80 (1998) 4859 [hep-th/9803002] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle 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].ADSCrossRefMATHGoogle 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.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
  8. [8]
    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
  9. [9]
    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
  10. [10]
    R. Ishizeki, M. Kruczenski and S. Ziama, Notes on Euclidean Wilson loops and Riemann theta functions, Phys. Rev. D 85 (2012) 106004 [arXiv:1104.3567] [INSPIRE].ADSGoogle Scholar
  11. [11]
    M. Kruczenski and S. Ziama, Wilson loops and Riemann theta functions II, JHEP 05 (2014) 037 [arXiv:1311.4950] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    M. Kruczenski, Wilson loops and minimal area surfaces in hyperbolic space, JHEP 11 (2014) 065 [arXiv:1406.4945] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    M. Babich and A. Bobenko, Willmore Tori with umbilic lines and minimal surfaces in hyperbolic space, Duke Math. J. 72 (1993) 151.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    E.D. Belokolos, A.I. Bobenko, V.Z. Enol’skii, A.R. Its and V.B. Matveev, Algebro-Geometric Approach to Nonlinear Integrable Equations, Springer-Verlag series in Non-linear Dynamics, Springer-Verlag, Heidelberg Germany (1994).Google Scholar
  15. [15]
    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
  16. [16]
    D.J. Gross and H. Ooguri, Aspects of large-N gauge theory dynamics as seen by string theory, Phys. Rev. D 58 (1998) 106002 [hep-th/9805129] [INSPIRE].ADSMathSciNetGoogle Scholar
  17. [17]
    K. Zarembo, Wilson loop correlator in the AdS/CFT correspondence, Phys. Lett. B 459 (1999) 527 [hep-th/9904149] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    J. Nian and H.J. Pirner, Wilson loop-loop correlators in AdS/QCD, Nucl. Phys. A 833 (2010) 119 [arXiv:0908.1330] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    B.A. Burrington and L.A. Pando Zayas, Phase transitions in Wilson loop correlator from integrability in global AdS, Int. J. Mod. Phys. A 27 (2012) 1250001 [arXiv:1012.1525] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    L.F. Alday, E.I. Buchbinder and A.A. Tseytlin, Correlation function of null polygonal Wilson loops with local operators, JHEP 09 (2011) 034 [arXiv:1107.5702] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    L.F. Alday, B. Eden, G.P. Korchemsky, J. Maldacena and E. Sokatchev, From correlation functions to Wilson loops, JHEP 09 (2011) 123 [arXiv:1007.3243] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    L.F. Alday, P. Heslop and J. Sikorowski, Perturbative correlation functions of null Wilson loops and local operators, JHEP 03 (2013) 074 [arXiv:1207.4316] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    K. Pohlmeyer, Integrable Hamiltonian Systems and Interactions Through Quadratic Constraints, Commun. Math. Phys. 46 (1976) 207 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    A. Jevicki and K. Jin, Moduli dynamics of AdS 3 strings, JHEP 06 (2009) 064 [arXiv:0903.3389] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    M. Kruczenski, Spiky strings and single trace operators in gauge theories, JHEP 08 (2005) 014 [hep-th/0410226] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  26. [26]
    A. Irrgang and M. Kruczenski, Rotating Wilson loops and open strings in AdS3, J. Phys. A 46 (2013) 075401 [arXiv:1210.2298] [INSPIRE].ADSMATHGoogle Scholar
  27. [27]
    L.F. Alday and J. Maldacena, Null polygonal Wilson loops and minimal surfaces in Anti-de-Sitter space, JHEP 11 (2009) 082 [arXiv:0904.0663] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  28. [28]
    R. Miranda, Graduate Studies in Mathematics. Vol. 5: Algebraic Curves and Riemann Surfaces, AMS Press, Providence Germany (1995).Google Scholar
  29. [29]
    C. Birkenhake and H. Lange, Grundlehren der mathematischen Wissenschaften. Vol. 302: Complex Abelian Varieties, Springer-Verlag, Heidelberg Germany (2004).Google Scholar
  30. [30]
    D. Mumford, C. Musili, M. Nort,E. Previato and M. Stillman, Tata Lectures in Theta I & II, Modern Birkhäuser Classics, Birkhäuser, Boston U.S.A. (2007).CrossRefGoogle Scholar
  31. [31]
    J.D. Fay, Lectures Notes in Mathematics. Vol. 352: Theta Functions on Riemann Surfaces, Springer-Verlag, Berlin Germany (1973).CrossRefGoogle Scholar
  32. [32]
    H.F. Baker, Abels Theorem and the Allied Theory, Including the Theory of the Theta Functions, Cambridge University Press, Cambridge U.K. (1897).MATHGoogle Scholar
  33. [33]
    H.M. Farkas and I. Kra, Graduate Texts in Mathematics. Vol. 71: Riemann Surfaces, second edition, Springer-Verlag, Heidelberg Germany (1991).Google Scholar
  34. [34]
    U. Görtz and T. Wedhorn, Algebraic Geometry I: Schemes. With Examples and Exercises, Advanced Lectures in Mathematics, Springer Vieweg Teubner Verlag, Berlin Germany (2010).CrossRefMATHGoogle Scholar
  35. [35]
    G. Harder, Aspects of Mathematics. Vol. 35: Lectures on Algebraic Geometry I: Sheaves, Cohomology of Sheaves, and Applications to Riemann Surfaces, Springer Vieweg Teubner Verlag, Berlin Germany (2011).MATHGoogle Scholar
  36. [36]
    A.I. Bobenko and C. Klein, Lecture Notes in Mathematics. Vol. 2013: Computational Approach to Riemann Surfaces, Springer-Verlag, Berlin Germany (2011).Google Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  1. 1.Department of Physics and AstronomyUniversity of KentuckyLexingtonUnited States

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