Black hole/string transition for the small Schwarzschild black hole of AdS 5 × S5 and critical unitary matrix models

  • L. Álvarez-GauméEmail author
  • P. Basu
  • M. Mariño
  • S.R. Wadia
Regular Article - Theoretical Physics


In this paper we discuss the black hole–string transition of the small Schwarzschild black hole of AdS 5×S5 using the AdS/CFT correspondence at finite temperature. The finite temperature gauge theory effective action, at weak and strong coupling, can be expressed entirely in terms of constant Polyakov lines which are SU(N) matrices. In showing this we have taken into account that there are no Nambu–Goldstone modes associated with the fact that the 10-dimensional black hole solution sits at a point in S5. We show that the phase of the gauge theory in which the eigenvalue spectrum has a gap corresponds to supergravity saddle points in the bulk theory. We identify the third order N=∞ phase transition with the black hole–string transition. This singularity can be resolved using a double scaling limit in the transition region where the large N expansion is organized in terms of powers of N-2/3. The N=∞ transition now becomes a smooth crossover in terms of a renormalized string coupling constant, reflecting the physics of large but finite N. Multiply wound Polyakov lines condense in the crossover region. We also discuss the implications of our results for the resolution of the singularity of the lorenztian section of the small Schwarzschild black hole.


Black Hole Gauge Theory Partition Function Saddle Point Matrix Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    L. Susskind, Some speculations about black hole entropy in string theory, arXiv:hep-th/9309145Google Scholar
  2. 2.
    G.T. Horowitz, J. Polchinski, Phys. Rev. D 55, 6189 (1997)MathSciNetCrossRefADSGoogle Scholar
  3. 3.
    A. Sen, Mod. Phys. Lett. A 10, 2081 (1995)CrossRefADSGoogle Scholar
  4. 4.
    M.J. Bowick, L. Smolin, L.C.R. Wijewardhana, Gen. Relat. Grav. 19, 113 (1987)MathSciNetCrossRefGoogle Scholar
  5. 5.
    M.J. Bowick, L. Smolin, L.C.R. Wijewardhana, Phys. Rev. Lett. 56, 424 (1986)CrossRefADSGoogle Scholar
  6. 6.
    A. Dabholkar, Phys. Rev. Lett. 94, 241301 (2005)MathSciNetCrossRefADSGoogle Scholar
  7. 7.
    A. Dabholkar, R. Kallosh, A. Maloney, JHEP 0412, 059 (2004)MathSciNetCrossRefADSGoogle Scholar
  8. 8.
    A. Dabholkar, F. Denef, G.W. Moore, B. Pioline, JHEP 0510, 096 (2005)MathSciNetCrossRefADSGoogle Scholar
  9. 9.
    T. Mohaupt, arXiv:hep-th/0512048Google Scholar
  10. 10.
    A. Giveon, D. Kutasov, E. Rabinovici, A. Sever, Nucl. Phys. B 719, 3 (2005)MathSciNetCrossRefADSGoogle Scholar
  11. 11.
    A. Giveon, D. Kutasov, JHEP 0601, 120 (2006)MathSciNetCrossRefADSGoogle Scholar
  12. 12.
    A. Giveon, E. Rabinovici, A. Sever, JHEP 0307, 055 (2003)MathSciNetCrossRefADSGoogle Scholar
  13. 13.
    D. Kutasov, Accelerating branes and the string/black hole transition, arXiv:hep-th/0509170Google Scholar
  14. 14.
    Y. Nakayama, K.L. Panigrahi, S.J. Rey, H. Takayanagi, JHEP 0501, 052 (2005)MathSciNetCrossRefADSGoogle Scholar
  15. 15.
    Y. Nakayama, S.J. Rey, Y. Sugawara, JHEP 0509, 020 (2005)MathSciNetCrossRefADSGoogle Scholar
  16. 16.
    L. Álvarez-Gaumé, C. Gomez, H. Liu, S. Wadia, Phys. Rev. D 71, 124023 (2005)MathSciNetCrossRefADSGoogle Scholar
  17. 17.
    B. Sundborg, Nucl. Phys. B 573, 349 (2000)zbMATHMathSciNetCrossRefADSGoogle Scholar
  18. 18.
    A.M. Polyakov, Int. J. Mod. Phys. A 17S1, 119 (2002)MathSciNetCrossRefGoogle Scholar
  19. 19.
    O. Aharony, J. Marsano, S. Minwalla, K. Papadodimas, M. Van Raamsdonk, Phys. Rev. D 71, 125018 (2005)MathSciNetCrossRefADSGoogle Scholar
  20. 20.
    H. Liu, Fine structure of Hagedorn transitions, arXiv:hep-th/0408001Google Scholar
  21. 21.
    M. Spradlin, A. Volovich, Nucl. Phys. B 711, 199 (2005)MathSciNetADSGoogle Scholar
  22. 22.
    J. Hallin, D. Persson, Phys. Lett. B 429, 232 (1998)CrossRefADSGoogle Scholar
  23. 23.
    S.W. Hawking, D.N. Page, Commun. Math. Phys. 87, 577 (1983)MathSciNetCrossRefADSGoogle Scholar
  24. 24.
    D.J. Gross, E. Witten, Phys. Rev. D 21, 446 (1980)CrossRefADSGoogle Scholar
  25. 25.
    S. Wadia, A Study of U(N) Lattice Gauge Theory In Two-Dimensions, EFI-79/44-CHICAGOGoogle Scholar
  26. 26.
    S.R. Wadia, Phys. Lett. B 93, 403 (1980)MathSciNetCrossRefADSGoogle Scholar
  27. 27.
    V.E. Hubeny, M. Rangamani, JHEP 0205, 027 (2002)MathSciNetCrossRefADSGoogle Scholar
  28. 28.
    G.T. Horowitz, V.E. Hubeny, JHEP 0006, 031 (2000)MathSciNetCrossRefADSGoogle Scholar
  29. 29.
    E. Witten, Adv. Theor. Math. Phys. 2, 505 (1998)zbMATHMathSciNetGoogle Scholar
  30. 30.
    P. Basu, S.R. Wadia, Phys. Rev. D 73, 045022 (2006)MathSciNetCrossRefADSGoogle Scholar
  31. 31.
    Y.Y. Goldschmidt, J. Math. Phys. 21, 1842 (1980)CrossRefADSGoogle Scholar
  32. 32.
    V. Periwal, D. Shevitz, Phys. Rev. Lett. 64, 1326 (1990)CrossRefADSGoogle Scholar
  33. 33.
    I.R. Klebanov, J. Maldacena, N. Seiberg, arXiv:hep-th/0309168Google Scholar
  34. 34.
    V. Periwal, D. Shevitz, Nucl. Phys. B 344, 731 (1990)MathSciNetCrossRefADSGoogle Scholar
  35. 35.
    D. Yamada, L.G. Yaffe, arXiv:hep-th/0602074Google Scholar
  36. 36.
    J.M. Maldacena, JHEP 0304, 021 (2003)MathSciNetCrossRefADSGoogle Scholar
  37. 37.
    P. Kraus, H. Ooguri, S. Shenker, Phys. Rev. D 67, 124022 (2003)MathSciNetCrossRefADSGoogle Scholar
  38. 38.
    L. Gervais, B. Sakita, Phys. Rev. D 11, 2943 (1975)CrossRefADSGoogle Scholar
  39. 39.
    G. Mandal, Mod. Phys. Lett. A 5, 1147 (1990)zbMATHMathSciNetCrossRefADSGoogle Scholar
  40. 40.
    L. Fidkowski, V. Hubeny, M. Kleban, S. Shenker, JHEP 0402, 014 (2004)MathSciNetCrossRefADSGoogle Scholar
  41. 41.
    G. Festuccia, H. Liu, Excursions beyond the horizon: Black hole singularities in Yang–Mills theories. I, arXiv:hep-th/0506202Google Scholar
  42. 42.
    S. Hawking, Black holes and the information paradox, Prepared for GR17: 17th International Conference on General Relativity and Gravitation, Dublin, Ireland, 18–24 July 2004Google Scholar
  43. 43.
    C. Crnkovic, M.R. Douglas, G.W. Moore, Nucl. Phys. B 360, 507 (1991)MathSciNetCrossRefADSGoogle Scholar
  44. 44.
    C. Crnkovic, M.R. Douglas, G.W. Moore, Int. J. Mod. Phys. A 7, 7693 (1992)zbMATHMathSciNetCrossRefADSGoogle Scholar
  45. 45.
    I.R. Klebanov, A. Hashimoto, Nucl. Phys. B 434, 264 (1995)zbMATHMathSciNetCrossRefADSGoogle Scholar
  46. 46.
    O. Aharony, J. Marsano, S. Minwalla, K. Papadodimas, M. Van Raamsdonk, Adv. Theor. Math. Phys. 8, 603 (2004)zbMATHMathSciNetGoogle Scholar
  47. 47.
    O. Aharony, S. Minwalla, T. Wiseman, arXiv:hep-th/0507219Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • L. Álvarez-Gaumé
    • 1
    Email author
  • P. Basu
    • 2
  • M. Mariño
    • 1
  • S.R. Wadia
    • 2
  1. 1.CERNGenevaSwitzerland
  2. 2.Tata Institute of Fundamental ResearchMumbaiIndia

Personalised recommendations