Extremality, Holography and Coarse Graining

Conference paper
Part of the Springer Proceedings in Physics book series (SPPHY, volume 142)

Abstract

I discuss some of the concepts at the crossroads of gravitational thermodynamics, holography and quantum mechanics. First, the origin of gravitational thermodynamics due to coarse graining of quantum information is exemplified using the half-BPS sector of \(\mathcal{N}\,=\,4\) SYM and its LLM description in type IIB supergravity.

Keywords

Black Hole Young Diagram Conformal Field Theory Extremal Black Hole Giant Graviton 
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.

References

  1. 1.
    L.F. Abbott, S. Deser, Nucl. Phys. B 195, 76 (1982). V. Iyer, R.M. Wald, Phys. Rev. D 50, 846 (1994). I.M. Anderson, C.G. Torre, Phys. Rev. Lett. 77, 4109 (1996).C.G. Torre, Local cohomology in field theory with applications to the Einstein equations. arXiv:hep-th/9706092. G. Barnich, F. Brandt, M. Henneaux, Commun. Math. Phys. 174, 57 (1995). G. Barnich, F. Brandt, M. Henneaux, Phys. Rep. 338, 439 (2000)Google Scholar
  2. 2.
    A.J. Amsel, G.T. Horowitz, D. Marolf, M.M. Roberts, J. High Energy Phys. 0909, 044 (2009)Google Scholar
  3. 3.
    N. Arkani-Hamed, S. Dubovsky, A. Nicolis, E. Trincherini, G. Villadoro, J. High Energy Phys. 0705, 055 (2007)Google Scholar
  4. 4.
    T. Azeyanagi, N. Ogawa, S. Terashima, Emergent AdS 3 in the zero entropy extremal black Holes. arXiv:1010.4291 (hep-th). Y. Matsuo, T. Nishioka, New near-horizon limit in Kerr/CFT. arXiv:1010.4549 (hep-th). T. Azeyanagi, N. Ogawa, S. Terashima, On non-chiral extension of Kerr/CFT. (arXiv:1102.3423 (hep-th))Google Scholar
  5. 5.
    H. Bacry, A. Grossmann, J. Zak, Phys. Rev. B 12, 1118 (1975)Google Scholar
  6. 6.
    V. Balasubramanian, B. Czech, Quantitative approaches to information recovery from black holes. (arXiv:1102.3566 (hep-th))Google Scholar
  7. 7.
    V. Balasubramanian, R. Gopakumar, F. Larsen, Nucl. Phys. B 526, 415 (1998)Google Scholar
  8. 8.
    V. Balasubramanian, P. Kraus, A.E. Lawrence, Phys. Rev. D 59, 046003 (1999)Google Scholar
  9. 9.
    V. Balasubramanian, M. Berkooz, A. Naqvi, M.J. Strassler, J. High Energy Phys. 0204, 034 (2002)Google Scholar
  10. 10.
    V. Balasubramanian, A. Naqvi, J. Simon, J. High Energy Phys. 0408, 023 (2004)Google Scholar
  11. 11.
    V. Balasubramanian, V. Jejjala, J. Simon, Int. J. Mod. Phys. D14, 2181–2186 (2005)Google Scholar
  12. 12.
    V. Balasubramanian, P. Kraus, M. Shigemori, Class. Quantum Gravity 22, 4803 (2005)Google Scholar
  13. 13.
    V. Balasubramanian, J. de Boer, V. Jejjala, J. Simon, J. High Energy Phys. 0512, 006 (2005)Google Scholar
  14. 14.
    V. Balasubramanian, B. Czech, K. Larjo, J. Simon, J. High Energy Phys. 0611, 001 (2006)Google Scholar
  15. 15.
    V. Balasubramanian, D. Marolf, M. Rozali, Gen. Relativ. Gravit. 38, 1529 (2006). (Int. J. Mod. Phys. D 15, 2285 (2006))Google Scholar
  16. 16.
    V. Balasubramanian, B. Czech, K. Larjo, D. Marolf, J. Simon, J. High Energy Phys. 0712, 067 (2007)Google Scholar
  17. 17.
    V. Balasubramanian, J. de Boer, S. El-Showk, I. Messamah, Class. Quantum Gravity 25, 214004 (2008)Google Scholar
  18. 18.
    V. Balasubramanian, B. Czech, V.E. Hubeny, K. Larjo, M. Rangamani, J. Simon, Gen. Relativ. Gravit. 40, 1863 (2008)Google Scholar
  19. 19.
    V. Balasubramanian, J. de Boer, V. Jejjala, J. Simon, J. High Energy Phys. 0805, 067 (2008)Google Scholar
  20. 20.
    V.  Balasubramanian, J. de Boer, M. Sheikh-Jabbari, J. Simón, J. High Energy Phys. 1002, 017 (2010)Google Scholar
  21. 21.
    V. Balasubramanian, J. Parsons, S.F. Ross, Class. Quantum Gravity 28, 045004 (2011)Google Scholar
  22. 22.
    M. Banados, C. Teitelboim, J. Zanelli, Phys. Rev. Lett. 69, 1849 (1992)Google Scholar
  23. 23.
    M. Banados, M. Henneaux, C. Teitelboim, J. Zanelli, Phys. Rev. D 48, 1506 (1993)Google Scholar
  24. 24.
    T. Banks, W. Fischler, S.H. Shenker, L. Susskind, Phys. Rev. D55, 5112–5128 (1997)Google Scholar
  25. 25.
    J.M. Bardeen, G.T. Horowitz, Phys. Rev. D 60, 104030 (1999)Google Scholar
  26. 26.
    J.M. Bardeen, B. Carter, S.W. Hawking, Commun. Math. Phys. 31, 161 (1973)Google Scholar
  27. 27.
    V. Bargmann, P. Butera, L. Girardello, J.R. Klauder, Rep. Math. Phys. 2, 221 (1971)Google Scholar
  28. 28.
    G. Barnich, Class. Quantum Gravity 20, 3685 (2003)Google Scholar
  29. 29.
    G. Barnich, F. Brandt, Nucl. Phys. B 633, 3 (2002)Google Scholar
  30. 30.
    G. Barnich, G. Compere, J. Math. Phys. 49, 042901 (2008)Google Scholar
  31. 31.
    B. Bates, F. Denef, Exact solutions for supersymmetric stationary black hole composites. arXiv:hep-th/0304094Google Scholar
  32. 32.
    M. Becker, S. Cremonini, W. Schulgin, J. High Energy Phys. 1009, 022 (2010). M. Becker, S. Cremonini, W. Schulgin, J. High Energy Phys. 1102, 007 (2011)Google Scholar
  33. 33.
    K. Behrndt, A.H. Chamseddine, W.A. Sabra, Phys. Lett. B442, 97–101 (1998). K. Behrndt, M. Cvetic, W.A. Sabra, Nucl. Phys. B553, 317–332 (1999)Google Scholar
  34. 34.
    J.D. Bekenstein, Phys. Rev. D 7, 2333 (1973)Google Scholar
  35. 35.
    I. Bena, N.P. Warner, Phys. Rev. D 74, 066001 (2006)Google Scholar
  36. 36.
    I. Bena, N.P. Warner, Lect. Notes Phys. 755, 1 (2008)Google Scholar
  37. 37.
    I. Bena, C.-W. Wang, N.P. Warner, J. High Energy Phys. 0611, 042 (2006)Google Scholar
  38. 38.
    I. Bena, S. Giusto, C. Ruef, N.P. Warner, J. High Energy Phys. 0911, 089 (2009)Google Scholar
  39. 39.
    I. Bena, S. Giusto, C. Ruef, N.P. Warner, J. High Energy Phys. 1003, 047 (2010)Google Scholar
  40. 40.
    I. Bena, N. Bobev, S. Giusto, C. Ruef, N.P. Warner, J. High Energy Phys. 1103, 022 (2011)Google Scholar
  41. 41.
    D. Berenstein, J. High Energy Phys. 0407, 018 (2004)Google Scholar
  42. 42.
    D. Berenstein, J. High Energy Phys. 0601, 125 (2006)Google Scholar
  43. 43.
    P. Berglund, E.G. Gimon, T.S. Levi, J. High Energy Phys. 0606, 007 (2006)Google Scholar
  44. 44.
    L. Boltzmann, Kais. Akad. Wiss. Wien Math. Naturwiss. Classe 76, 373–435 (1877). L. Boltzmann, Kais. Akad. Wiss. Wien Math. Naturwiss. Classe 66, 275–370 (1872)Google Scholar
  45. 45.
    R. Bousso, J. High Energy Phys. 9907, 004 (1999)Google Scholar
  46. 46.
    R. Bousso, Rev. Mod. Phys. 74, 825–874 (2002)Google Scholar
  47. 47.
    D. Brecher, A. Chamblin, H.S. Reall, Nucl. Phys. B 607, 155 (2001)Google Scholar
  48. 48.
    I. Bredberg, T. Hartman, W. Song, A. Strominger, J. High Energy Phys. 1004, 019 (2010)Google Scholar
  49. 49.
    I. Bredberg, C. Keeler, V. Lysov, A. Strominger, Cargese lectures on the Kerr/CFT correspondence. arXiv:1103.2355 (hep-th)Google Scholar
  50. 50.
    J.D. Brown, M. Henneaux, Commun. Math. Phys. 104, 207 (1986)Google Scholar
  51. 51.
    P. Calabrese, J.L. Cardy, J. Stat. Mech. 0406, P06002 (2004)Google Scholar
  52. 52.
    J.L. Cardy, Nucl. Phys. B 270, 186 (1986)Google Scholar
  53. 53.
    A. Castro, F. Larsen, J. High Energy Phys. 0912, 037 (2009)Google Scholar
  54. 54.
    A. Castro, D. Grumiller, F. Larsen, R. McNees, J. High Energy Phys. 0811, 052 (2008)Google Scholar
  55. 55.
    A. Castro, C. Keeler, F. Larsen, J. High Energy Phys. 1007, 033 (2010)Google Scholar
  56. 56.
    A. Castro, A. Maloney, A. Strominger, Phys. Rev. D 82, 024008 (2010)Google Scholar
  57. 57.
    D.D.K. Chow, M. Cvetic, H. Lu, C.N. Pope, Phys. Rev. D 79, 084018 (2009)Google Scholar
  58. 58.
    B.D. Chowdhury, S.D. Mathur, Class. Quantum Gravity 25, 135005 (2008). B.D. Chowdhury, S.D. Mathur, Class. Quantum Gravity 25, 225021 (2008). B.D. Chowdhury, S.D. Mathur, Class. Quantum Gravity 26, 035006 (2009)Google Scholar
  59. 59.
    G. Compere, Symmetries and conservation laws in Lagrangian gauge theories with applications to the mechanics of black holes and to gravity in three dimensions. arXiv:0708.3153 (hep-th)Google Scholar
  60. 60.
    G. Compere, K. Murata, T. Nishioka, J. High Energy Phys. 0905, 077 (2009)Google Scholar
  61. 61.
    G. Compere, W. Song, A. Virmani, Microscopics of extremal Kerr from spinning M5 branes. (arXiv:1010.0685 (hep-th))Google Scholar
  62. 62.
    S. Corley, A. Jevicki, S. Ramgoolam, Adv. Theor. Math. Phys. 5, 809–839 (2002)Google Scholar
  63. 63.
    O. Coussaert, M. Henneaux, Self-dual solutions of 2 + 1 Einstein gravity with a negative cosmological constant. arXiv:hep-th/9407181Google Scholar
  64. 64.
    C. Crnkovic, E. Witten, Covariant description Of canonical formalism in geometrical theories, in Three Hundred Years of Gravitation, ed. by S.W. Hawking, W. Israel (Cambridge University Press, Cambridge/New York, 1987), pp. 676–684Google Scholar
  65. 65.
    M. Cvetic, F. Larsen, J. High Energy Phys. 0909, 088 (2009)Google Scholar
  66. 66.
    M. Cvetic, H. Lu, C.N. Pope, Nucl. Phys. B 545, 309 (1999)Google Scholar
  67. 67.
    G. Dall’Agata, S. Giusto, C. Ruef, J. High Energy Phys. 1102, 074 (2011)Google Scholar
  68. 68.
    J. de Boer, S. El-Showk, I. Messamah, D.V.d. Bleeken, Quantizing N  = 2 multicenter solutions. arXiv:0807.4556 (hep-th)Google Scholar
  69. 69.
    J. de Boer, S. El-Showk, I. Messamah, D.V.d. Bleeken, A bound on the entropy of supergravity? arXiv:0906.0011 (hep-th)Google Scholar
  70. 70.
    J. de Boer, M.M. Sheikh-Jabbari, J. Simon, Near horizon limits of massless BTZ and their CFT duals. arXiv:1011.1897 (hep-th)Google Scholar
  71. 71.
    L. D’Errico, W. Mueck, R. Pettorino, J. High Energy Phys. 0705, 063 (2007)Google Scholar
  72. 72.
    R. de Mello Koch, N. Ives, M. Stephanou, Phys. Rev. D79, 026004 (2009)Google Scholar
  73. 73.
    R. de Mello Koch, T.K. Dey, N. Ives, M. Stephanou, J. High Energy Phys. 0908, 083 (2009)Google Scholar
  74. 74.
    F. Denef, G.W. Moore, Split states, entropy enigmas, holes and halos. (hep-th/0702146 (HEP-TH))Google Scholar
  75. 75.
    A. Dhar, G. Mandal, N.V. Suryanarayana, J. High Energy Phys. 0601, 118 (2006)Google Scholar
  76. 76.
    O.J.C. Dias, H.S. Reall, J.E. Santos, J. High Energy Phys. 0908, 101 (2009)Google Scholar
  77. 77.
    R. Dijkgraaf, J.M. Maldacena, G.W. Moore, E.P. Verlinde, A black hole farey tail. arXiv: hep-th/0005003Google Scholar
  78. 78.
    A. Einstein, Ann. Phys. 49, 769 (1916). (Ann. Phys. 14, 517 (2005))Google Scholar
  79. 79.
    R. Fareghbal, C.N. Gowdigere, A.E. Mosaffa, M.M. Sheikh-Jabbari, J. High Energy Phys. 0808, 070 (2008)Google Scholar
  80. 80.
    R. Fareghbal, C.N. Gowdigere, A.E. Mosaffa, M.M. Sheikh-Jabbari, arXiv:0805.0203 (hep-th)Google Scholar
  81. 81.
    J.M. Figueroa-O’Farrill, J. Simon, Adv. Theor. Math. Phys. 8, 217 (2004)Google Scholar
  82. 82.
    V.P. Frolov, K.S. Thorne, Phys. Rev. D39, 2125 (1989)Google Scholar
  83. 83.
    W.D. Goldberger, J. High Energy Phys. 0903, 069 (2009)Google Scholar
  84. 84.
    L. Grant, L. Maoz, J. Marsano, K. Papadodimas, V.S. Rychkov, J. High Energy Phys. 0508, 025 (2005)Google Scholar
  85. 85.
    S.S. Gubser, I.R. Klebanov, A.M. Polyakov, Phys. Lett. B 428, 105 (1998)Google Scholar
  86. 86.
    M. Guica, A. Strominger, J. High Energy Phys. 1102, 010 (2011)Google Scholar
  87. 87.
    M. Guica, T. Hartman, W. Song, A. Strominger, Phys. Rev. D 80, 124008 (2009)Google Scholar
  88. 88.
    R.K. Gupta, A. Sen, J. High Energy Phys. 0904, 034 (2009)Google Scholar
  89. 89.
    T. Hartman, A. Strominger, J. High Energy Phys. 0904, 026 (2009)Google Scholar
  90. 90.
    T. Hartman, K. Murata, T. Nishioka, A. Strominger, J. High Energy Phys. 0904, 019 (2009)Google Scholar
  91. 91.
    S.W. Hawking, Commun. Math. Phys. 43, 199 (1975) (Erratum-ibid. 46, 206 (1976))Google Scholar
  92. 92.
    S.W. Hawking, Phys. Rev. D14, 2460–2473 (1976)Google Scholar
  93. 93.
    S.W. Hawking, G.F.R. Ellis, The Large Scale Structure of Space-Time (Cambridge University Press, Cambridge, 1973)Google Scholar
  94. 94.
    S.W. Hawking, R. Penrose, Proc. R. Soc. Lond. A314, 529–548 (1970)Google Scholar
  95. 95.
    I. Heemskerk, J. Polchinski, Holographic and Wilsonian renormalization groups. arXiv:1010.1264 (hep-th) T. Faulkner, H. Liu, M. Rangamani, Integrating out geometry: Holographic Wilsonian RG and the membrane paradigm. arXiv:1010.4036 (hep-th)Google Scholar
  96. 96.
    S. Hellerman, J. Polchinski, Phys. Rev. D59, 125002 (1999)Google Scholar
  97. 97.
    C.P. Herzog, J. Phys. A 42, 343001 (2009). S.A. Hartnoll, Class. Quantum Gravity 26, 224002 (2009). G.T. Horowitz, Introduction to holographic superconductors. arXiv:1002.1722 (hep-th).J. McGreevy, Adv. High Energy Phys. 2010, 723105 (2010).T. Faulkner, H. Liu, J. McGreevy, D. Vegh, Emergent quantum criticality, Fermi surfaces, and AdS2. arXiv:0907.2694 (hep-th)Google Scholar
  98. 98.
    M. Hillery, R.F. O’Connell, M.O. Scully, E.P. Wigner, Phys. Rep. 106(3), 121–167 (1984)Google Scholar
  99. 99.
    C.M. Hull, P.K. Townsend, Nucl. Phys. B 438, 109 (1995)Google Scholar
  100. 100.
    V. Jejjala, S. Nampuri, J. High Energy Phys. 1002, 088 (2010)Google Scholar
  101. 101.
    V. Jejjala, O. Madden, S.F. Ross, G. Titchener, Phys. Rev. D71, 124030 (2005)Google Scholar
  102. 102.
    J. Kinney, J.M. Maldacena, S. Minwalla, S. Raju, Commun. Math. Phys. 275, 209 (2007)Google Scholar
  103. 103.
    H.K. Kunduri, J. Lucietti, H.S. Reall, Class. Quantum Gravity 24, 4169 (2007)Google Scholar
  104. 104.
    P. Kraus, H. Ooguri, S. Shenker, Phys. Rev. D 67, 124022 (2003). L. Fidkowski, V. Hubeny, M. Kleban, S. Shenker, J. High Energy Phys. 0402, 014 (2004). T.S. Levi, S.F. Ross, Phys. Rev. D 68, 044005 (2003).V. Balasubramanian, T.S. Levi, Phys. Rev. D 70, 106005 (2004).D. Brecher, J. He, M. Rozali, J. High Energy Phys. 0504, 004 (2005).G. Festuccia, H. Liu, J. High Energy Phys. 0604, 044 (2006).B. Freivogel, V.E. Hubeny, A. Maloney, R. Myers, M. Rangamani, S. Shenker, J. High Energy Phys. 0603, 007 (2006).K. Maeda, M. Natsuume, T. Okamura, Phys. Rev. D 74, 046010 (2006).A. Hamilton, D. Kabat, G. Lifschytz, D.A. Lowe. arXiv:hep-th/0612053Google Scholar
  105. 105.
    H. Lin, O. Lunin, J. Maldacena, J. High Energy Phys. 0410, 025 (2004)Google Scholar
  106. 106.
    F. Loran, H. Soltanpanahi, Class. Quantum Gravity 26, 155019 (2009)Google Scholar
  107. 107.
    H. Lu, J. Mei, C. N. Pope, J. High Energy Phys. 0904, 054 (2009). T. Azeyanagi, N. Ogawa, S. Terashima, J. High Energy Phys. 0904, 061 (2009)Google Scholar
  108. 108.
    O. Lunin, S.D. Mathur, Nucl. Phys. B 623, 342 (2002)Google Scholar
  109. 109.
    O. Lunin, S.D. Mathur, Phys. Rev. Lett. 88, 211303 (2002)Google Scholar
  110. 110.
    O. Lunin, J.M. Maldacena, L. Maoz, Gravity solutions for the D1-D5 system with angular momentum. arXiv:hep-th/0212210Google Scholar
  111. 111.
    J.M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998). (Int. J. Theor. Phys. 38, 1113 (1999))Google Scholar
  112. 112.
    J.M. Maldacena, J. High Energy Phys. 0304, 021 (2003). L. Dyson, M. Kleban, L. Susskind, J. High Energy Phys. 0210, 011 (2002). L. Dyson, J. Lindesay, L. Susskind, J. High Energy Phys. 0208, 045 (2002). N. Goheer, M. Kleban, L. Susskind, J. High Energy Phys. 0307, 056 (2003). D. Birmingham, I. Sachs, S.N. Solodukhin, Phys. Rev. D 67, 104026 (2003). J.L.F. Barbon, E. Rabinovici, J. High Energy Phys. 0311, 047 (2003). J.L.F. Barbon, E. Rabinovici, Fortschr. Phys. 52, 642–649 (2004). J.L.F. Barbon, E. Rabinovici, Topology change and unitarity in quantum black hole dynamics. arXiv:hep-th/0503144. M. Kleban, M. Porrati, R. Rabadan, J. High Energy Phys. 0410, 030 (2004)Google Scholar
  113. 113.
    J.M. Maldacena, H. Ooguri, J. Math. Phys. 42, 2929–2960 (2001)Google Scholar
  114. 114.
    J.M. Maldacena, J. Michelson, A. Strominger, J. High Energy Phys. 9902, 011 (1999)Google Scholar
  115. 115.
    J.M. Maldacena, H. Ooguri, J. Son, J. Math. Phys. 42, 2961–2977 (2001). S. Hemming, E. Keski-Vakkuri, P. Kraus, J. High Energy Phys. 0210, 006 (2002)Google Scholar
  116. 116.
    J. Maldacena, D. Martelli, Y. Tachikawa, J. High Energy Phys. 0810, 072 (2008)Google Scholar
  117. 117.
    G. Mandal, J. High Energy Phys. 0508, 052 (2005)Google Scholar
  118. 118.
    L. Maoz, V.S. Rychkov, J. High Energy Phys. 0508, 096 (2005)Google Scholar
  119. 119.
    E.J. Martinec, W. McElgin, J. High Energy Phys. 0204, 029 (2002)Google Scholar
  120. 120.
    S.D. Mathur, Fortschr. Phys. 53, 793 (2005)Google Scholar
  121. 121.
    S.D. Mathur, Class. Quantum Gravity 23, R115 (2006)Google Scholar
  122. 122.
    S.D. Mathur, Lect. Notes Phys. 769, 3–48 (2009)Google Scholar
  123. 123.
    S. D. Mathur, Class. Quantum Gravity 26, 224001 (2009)Google Scholar
  124. 124.
    S.D. Mathur, Fuzzballs and the information paradox: a summary and conjectures. arXiv:0810.4525 (hep-th)Google Scholar
  125. 125.
    S.D. Mathur, The information paradox and the infall problem. (arXiv:1012.2101 (hep-th))Google Scholar
  126. 126.
    S.D. Mathur, C.J. Plumberg, Correlations in Hawking radiation and the infall problem. (arXiv:1101.4899 (hep-th))Google Scholar
  127. 127.
    J. McGreevy, L. Susskind, N. Toumbas, J. High Energy Phys. 0006, 008 (2000)Google Scholar
  128. 128.
    R.C. Myers, J. High Energy Phys. 9912, 022 (1999)Google Scholar
  129. 129.
    R.C. Myers, O. Tafjord, J. High Energy Phys. 0111, 009 (2001)Google Scholar
  130. 130.
    J. Navarro-Salas, P. Navarro, Nucl. Phys. B579, 250–266 (2000). M. Cadoni, S. Mignemi, Phys. Lett. B490, 131–135 (2000)Google Scholar
  131. 131.
    D.N. Page, Phys. Rev. Lett. 71, 3743 (1993)Google Scholar
  132. 132.
    R. Penrose, Phys. Rev. Lett. 14, 57–59 (1965)Google Scholar
  133. 133.
    A.M. Perelomov, Teor. Mat. Fiz. 6, 213 (1971)Google Scholar
  134. 134.
    J. Polchinski, Phys. Rev. Lett. 75, 4724 (1995)Google Scholar
  135. 135.
    V.S. Rychkov, J. High Energy Phys. 0601, 063 (2006)Google Scholar
  136. 136.
    S. Ryu, T. Takayanagi, Phys. Rev. Lett. 96, 181602 (2006). S. Ryu, T. Takayanagi, J. High Energy Phys. 0608, 045 (2006). S. Ryu, T. Takayanagi, J. Phys. A A42, 504008(2009)Google Scholar
  137. 137.
    N. Seiberg, Phys. Rev. Lett. 79, 3577 (1997)Google Scholar
  138. 138.
    A. Sen, J. High Energy Phys. 0509, 038 (2005)Google Scholar
  139. 139.
    A. Sen, Gen. Relativ. Gravit. 40, 2249 (2008)Google Scholar
  140. 140.
    A. Sen, J. High Energy Phys. 0811, 075 (2008)Google Scholar
  141. 141.
    A. Sen, Int. J. Mod. Phys. A 24, 4225 (2009)Google Scholar
  142. 142.
    A. Sen, J. High Energy Phys. 0908, 068 (2009)Google Scholar
  143. 143.
    A. Sen, J. High Energy Phys. 1005, 097 (2010)Google Scholar
  144. 144.
    C.E. Shannon, Bell Syst. Tech. J. 27, 379–423 (1948)Google Scholar
  145. 145.
    J. Simon, Phys. Rev. D 81, 024003 (2010)Google Scholar
  146. 146.
    K. Skenderis, M. Taylor, Phys. Rep. 467, 117 (2008)Google Scholar
  147. 147.
    A. Strominger, J. High Energy Phys. 9802, 009 (1998)Google Scholar
  148. 148.
    A. Strominger, J. High Energy Phys. 9901, 007 (1999)Google Scholar
  149. 149.
    A. Strominger, C. Vafa, Phys. Lett. B 379, 99 (1996)Google Scholar
  150. 150.
    L. Susskind, J. Math. Phys. 36, 6377 (1995)Google Scholar
  151. 151.
    Y. Takayama, A. Tsuchiya, J. High Energy Phys. 0510, 004 (2005)Google Scholar
  152. 152.
    G. ’t Hooft, Dimensional reduction in quantum gravity. arXiv:gr-qc/9310026Google Scholar
  153. 153.
    J.A. Wheeler, Phys. Rev. 97, 511 (1955). J.A. Wheeler, Ann. Phys. 2, 604 (1957)Google Scholar
  154. 154.
    E. Witten, Nucl. Phys. B 443, 85 (1995)Google Scholar
  155. 155.
    E. Witten, Adv. Theor. Math. Phys. 2, 253 (1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.School of MathematicsUniversity of EdinburghEdinburghUK

Personalised recommendations