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Logarithmic corrections to \( \mathcal{N} = {4} \) and \( \mathcal{N} = {8} \) black hole entropy: a one loop test of quantum gravity

Abstract

We compute logarithmic corrections to the entropy of supersymmetric extremal black holes in \( \mathcal{N} = {4} \) and \( \mathcal{N} = {8} \) supersymmetric string theories and find results in perfect agreement with the microscopic results. In particular these logarithmic corrections vanish for quarter BPS black holes in \( \mathcal{N} = {4} \)supersymmetric theories, but has a finite coefficient for 1/8 BPS black holes in the \( \mathcal{N} = {8} \) supersymmetric theory. On the macroscopic side these computations require evaluating the one loop determinant of massless fields around the near horizon geometry, and include, in particular, contributions from dynamical four dimensional gravitons propagating in the loop. Thus our analysis provides a test of one loop quantum gravity corrections to the black hole entropy, or equivalently of the AdS 2/CF T 1 correspondence. We also extend our analysis to \( \mathcal{N} = {2} \)supersymmetric STU model and make a prediction for the logarithmic correction to the black hole entropy in that theory.

References

  1. [1]

    R.M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48 (1993) 3427 [gr-qc/9307038] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  2. [2]

    T. Jacobson, G. Kang and R.C. Myers, On black hole entropy, Phys. Rev. D 49 (1994) 6587 [gr-qc/9312023] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  3. [3]

    V. Iyer and R.M. Wald, Some properties of Noether charge and a proposal for dynamical black hole entropy, Phys. Rev. D 50 (1994) 846 [gr-qc/9403028] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  4. [4]

    T. Jacobson, G. Kang and R.C. Myers, Black hole entropy in higher curvature gravity, gr-qc/9502009 [INSPIRE].

  5. [5]

    A. Sen, Black hole entropy function and the attractor mechanism in higher derivative gravity, JHEP 09 (2005) 038 [hep-th/0506177] [INSPIRE].

    Article  ADS  Google Scholar 

  6. [6]

    A. Sen, Entropy function for heterotic black holes, JHEP 03 (2006) 008 [hep-th/0508042] [INSPIRE].

    Article  ADS  Google Scholar 

  7. [7]

    A. Sen, Quantum entropy function from AdS 2 /CF T 1 correspondence, Int. J. Mod. Phys. A 24 (2009) 4225 [arXiv:0809.3304] [INSPIRE].

    ADS  Google Scholar 

  8. [8]

    R. Dijkgraaf, E.P. Verlinde and H.L. Verlinde, Counting dyons in N = 4 string theory, Nucl. Phys. B 484 (1997) 543 [hep-th/9607026] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  9. [9]

    G. Lopes Cardoso, B. de Wit, J. Kappeli and T. Mohaupt, Asymptotic degeneracy of dyonic N =4 string states and black hole entropy, JHEP 12(2004) 075 [hep-th/0412287] [INSPIRE].

    ADS  Google Scholar 

  10. [10]

    D. Shih, A. Strominger and X. Yin, Recounting dyons in N = 4 string theory, JHEP 10 (2006) 087 [hep-th/0505094] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  11. [11]

    D. Gaiotto, Re-recounting dyons in N = 4 string theory, hep-th/0506249 [INSPIRE].

  12. [12]

    D. Shih and X. Yin, Exact black hole degeneracies and the topological string, JHEP 04 (2006) 034 [hep-th/0508174] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  13. [13]

    D.P. Jatkar and A. Sen, Dyon spectrum in CHL models, JHEP 04 (2006) 018 [hep-th/0510147] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  14. [14]

    J.R. David, D.P. Jatkar and A. Sen, Product representation of dyon partition function in CHL models, JHEP 06 (2006) 064 [hep-th/0602254] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  15. [15]

    A. Dabholkar and S. Nampuri, Spectrum of dyons and black holes in CHL orbifolds using Borcherds lift, JHEP 11 (2007) 077 [hep-th/0603066] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  16. [16]

    J.R. David and A. Sen, CHL dyons and statistical entropy function from D1-D5 system, JHEP 11 (2006) 072 [hep-th/0605210] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  17. [17]

    J.R. David, D.P. Jatkar and A. Sen, Dyon spectrum in N = 4 supersymmetric type II string theories, JHEP 11 (2006) 073 [hep-th/0607155] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  18. [18]

    J.R. David, D.P. Jatkar and A. Sen, Dyon spectrum in generic N = 4 supersymmetric Z N orbifolds, JHEP 01 (2007) 016 [hep-th/0609109] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  19. [19]

    A. Dabholkar and D. Gaiotto, Spectrum of CHL dyons from genus-two partition function, JHEP 12 (2007) 087 [hep-th/0612011] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  20. [20]

    A. Sen, Black hole entropy function, attractors and precision counting of microstates, Gen. Rel. Grav. 40 (2008) 2249 [arXiv:0708.1270] [INSPIRE].

    Article  MATH  ADS  Google Scholar 

  21. [21]

    S. Banerjee, A. Sen and Y.K. Srivastava, Generalities of quarter BPS dyon partition function and dyons of torsion two, JHEP 05 (2008) 101 [arXiv:0802.0544] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  22. [22]

    S. Banerjee, A. Sen and Y.K. Srivastava, Partition functions of torsion > 1 dyons in heterotic string theory on T 6, JHEP 05 (2008) 098 [arXiv:0802.1556] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  23. [23]

    A. Dabholkar, J. Gomes and S. Murthy, Counting all dyons in N = 4 string theory, JHEP 05 (2011) 059 [arXiv:0803.2692] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  24. [24]

    A. Sen, Arithmetic of quantum entropy function, JHEP 08 (2009) 068 [arXiv:0903.1477] [INSPIRE].

    Article  ADS  Google Scholar 

  25. [25]

    A. Dabholkar, J. Gomes, S. Murthy and A. Sen, Supersymmetric index from black hole entropy, JHEP 04 (2011) 034 [arXiv:1009.3226] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  26. [26]

    A. Sen, Arithmetic of N = 8 black holes, JHEP 02 (2010) 090 [arXiv:0908.0039] [INSPIRE].

    Article  ADS  Google Scholar 

  27. [27]

    M. Cvetič and D. Youm, Dyonic BPS saturated black holes of heterotic string on a six torus, Phys. Rev. D 53 (1996) 584 [hep-th/9507090] [INSPIRE].

    ADS  Google Scholar 

  28. [28]

    M. Cvetič and A.A. Tseytlin, Solitonic strings and BPS saturated dyonic black holes, Phys. Rev. D 53 (1996) 5619 [Erratum ibid. D 55 (1997) 3907] [hep-th/9512031] [INSPIRE].

    ADS  Google Scholar 

  29. [29]

    S.N. Solodukhin, The conical singularity and quantum corrections to entropy of black hole, Phys. Rev. D 51 (1995) 609 [hep-th/9407001] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  30. [30]

    S.N. Solodukhin, On ‘nongeometric’ contribution to the entropy of black hole due to quantum corrections, Phys. Rev. D 51 (1995) 618 [hep-th/9408068] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  31. [31]

    D.V. Fursaev, Temperature and entropy of a quantum black hole and conformal anomaly, Phys. Rev. D 51 (1995) 5352 [hep-th/9412161] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  32. [32]

    R.B. Mann and S.N. Solodukhin, Conical geometry and quantum entropy of a charged Kerr black hole, Phys. Rev. D 54 (1996) 3932 [hep-th/9604118] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  33. [33]

    R.B. Mann and S.N. Solodukhin, Universality of quantum entropy for extreme black holes, Nucl. Phys. B 523 (1998) 293 [hep-th/9709064] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  34. [34]

    R.K. Kaul and P. Majumdar, Logarithmic correction to the Bekenstein-Hawking entropy, Phys. Rev. Lett. 84 (2000) 5255 [gr-qc/0002040] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  35. [35]

    S. Carlip, Logarithmic corrections to black hole entropy from the Cardy formula, Class. Quant. Grav. 17 (2000) 4175 [gr-qc/0005017] [INSPIRE].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  36. [36]

    T. Govindarajan, R. Kaul and V. Suneeta, Logarithmic correction to the Bekenstein-Hawking entropy of the BTZ black hole, Class. Quant. Grav. 18 (2001) 2877 [gr-qc/0104010] [INSPIRE].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  37. [37]

    K.S. Gupta and S. Sen, Further evidence for the conformal structure of a Schwarzschild black hole in an algebraic approach, Phys. Lett. B 526 (2002) 121 [hep-th/0112041] [INSPIRE].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  38. [38]

    A. Medved, A comment on black hole entropy or does nature abhor a logarithm?, Class. Quant. Grav. 22 (2005) 133 [gr-qc/0406044] [INSPIRE].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  39. [39]

    D.N. Page, Hawking radiation and black hole thermodynamics, New J. Phys. 7 (2005) 203 [hep-th/0409024] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  40. [40]

    R. Banerjee and B.R. Majhi, Quantum tunneling beyond semiclassical approximation, JHEP 06 (2008) 095 [arXiv:0805.2220] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  41. [41]

    R. Banerjee and B.R. Majhi, Quantum tunneling, trace anomaly and effective metric, Phys. Lett. B 674 (2009) 218 DOI:dx.doi.org [arXiv:0808.3688] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  42. [42]

    B.R. Majhi, Fermion tunneling beyond semiclassical approximation, Phys. Rev. D 79 (2009) 044005 [arXiv:0809.1508] [INSPIRE].

    ADS  Google Scholar 

  43. [43]

    R.-G. Cai, L.-M. Cao and N. Ohta, Black holes in gravity with conformal anomaly and logarithmic term in black hole entropy, JHEP 04 (2010) 082 [arXiv:0911.4379] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  44. [44]

    R. Aros, D. Diaz and A. Montecinos, Logarithmic correction to BH entropy as Noether charge, JHEP 07 (2010) 012 [arXiv:1003.1083] [INSPIRE].

    Article  ADS  Google Scholar 

  45. [45]

    S.N. Solodukhin, Entanglement entropy of round spheres, Phys. Lett. B 693 (2010) 605 [arXiv:1008.4314] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  46. [46]

    S. Banerjee, R.K. Gupta and A. Sen, Logarithmic corrections to extremal black hole entropy from quantum entropy function, JHEP 03 (2011) 147 [arXiv:1005.3044] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  47. [47]

    S. Giombi, A. Maloney and X. Yin, One-loop partition functions of 3D gravity, JHEP 08 (2008) 007 [arXiv:0804.1773] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  48. [48]

    J.R. David, M.R. Gaberdiel and R. Gopakumar, The heat kernel on AdS 3 and its applications, JHEP 04 (2010) 125 [arXiv:0911.5085] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  49. [49]

    M.R. Gaberdiel, R. Gopakumar and A. Saha, Quantum W -symmetry in AdS 3 , JHEP 02 (2011) 004 [arXiv:1009.6087] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  50. [50]

    P.B. Gilkey, Invariance theory, the heat equation and the Atiyah-Singer index theorem, Publish or Perish Inc., U.S.A. (1984).

    MATH  Google Scholar 

  51. [51]

    D. Vassilevich, Heat kernel expansion: user’s manual, Phys. Rept. 388 (2003) 279 [hep-th/0306138] [INSPIRE].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  52. [52]

    A. Sen, Entropy function and AdS 2/CF T 1 correspondence, JHEP 11 (2008) 075 [arXiv:0805.0095] [INSPIRE].

    Article  ADS  Google Scholar 

  53. [53]

    C. Beasley, D. Gaiotto, M. Guica, L. Huang, A. Strominger, et al., Why ZBH = |Ztop|2, hep-th/0608021 [INSPIRE].

  54. [54]

    N. Banerjee, S. Banerjee, R.K. Gupta, I. Mandal and A. Sen, Supersymmetry, localization and quantum entropy function, JHEP 02 (2010) 091 [arXiv:0905.2686] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  55. [55]

    A. Dabholkar, J. Gomes and S. Murthy, Quantum black holes, localization and the topological string, JHEP 06 (2011) 019 [arXiv:1012.0265] [INSPIRE].

    Article  ADS  Google Scholar 

  56. [56]

    A. Dabholkar, J. Gomes and S. Murthy, Localization & exact holography, arXiv:1111.1161 [INSPIRE].

  57. [57]

    A. Sen and C. Vafa, Dual pairs of type-II string compactification, Nucl. Phys. B 455 (1995) 165 [hep-th/9508064] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  58. [58]

    A. Gregori, C. Kounnas and P. Petropoulos, Nonperturbative triality in heterotic and type-II N =2 strings,Nucl. Phys. B 553 (1999) 108 [hep-th/9901117] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  59. [59]

    J.R. David, On the dyon partition function in N = 2 theories, JHEP 02 (2008) 025 [arXiv:0711.1971] [INSPIRE].

    Article  ADS  Google Scholar 

  60. [60]

    H. Ooguri, A. Strominger and C. Vafa, Black hole attractors and the topological string, Phys. Rev. D 70 (2004) 106007 [hep-th/0405146] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  61. [61]

    G. Cardoso, B. de Wit and S. Mahapatra, Subleading and non-holomorphic corrections to N =2 BPS black hole entropy, JHEP 02 (2009) 006 [arXiv:0808.2627][INSPIRE].

    Article  ADS  Google Scholar 

  62. [62]

    F. Denef and G.W. Moore, Split states, entropy enigmas, holes and halos, hep-th/0702146 [INSPIRE].

  63. [63]

    A. Sen, Logarithmic corrections to N = 2 black hole entropy: an infrared window into the microstates, arXiv:1108.3842 [INSPIRE].

  64. [64]

    S. Christensen and M. Duff, New gravitational index theorems and supertheorems, Nucl. Phys. B 154 (1979) 301 [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  65. [65]

    S. Christensen and M. Duff, Quantizing gravity with a cosmological constant, Nucl. Phys. B 170 (1980) 480 [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  66. [66]

    M. Duff and P. van Nieuwenhuizen, Quantum inequivalence of different field representations, Phys. Lett. B 94 (1980) 179 [INSPIRE].

    ADS  Google Scholar 

  67. [67]

    S. Christensen, M. Duff, G. Gibbons and M. Roček, Vanishing one loop β-function in gauged N >4 supergravity, Phys. Rev. Lett. 45 (1980) 161 [INSPIRE].

    Article  ADS  Google Scholar 

  68. [68]

    R. Camporesi, Harmonic analysis and propagators on homogeneous spaces, Phys. Rept. 196 (1990) 1 [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  69. [69]

    R. Camporesi and A. Higuchi, Spectral functions and ζ functions in hyperbolic spaces, J. Math. Phys. 35 (1994) 4217 [INSPIRE].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  70. [70]

    R. Camporesi, The spinor heat kernel in maximally symmetric spaces, Commun. Math. Phys. 148 (1992) 283 [INSPIRE].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  71. [71]

    R. Camporesi and A. Higuchi, Arbitrary spin effective potentials in anti-de Sitter space-time, Phys. Rev. D 47 (1993) 3339 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  72. [72]

    R. Camporesi and A. Higuchi, On the eigen functions of the Dirac operator on spheres and real hyperbolic spaces, J. Geom. Phys. 20 (1996) 1 [gr-qc/9505009] [INSPIRE].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  73. [73]

    G. Cardoso, J. David, B. de Wit and S. Mahapatra, The mixed black hole partition function for the STU model, JHEP 12 (2008) 086 [arXiv:0810.1233] [INSPIRE].

    Article  ADS  Google Scholar 

  74. [74]

    J.R. David, On walls of marginal stability in N = 2 string theories, JHEP 08 (2009) 054 [arXiv:0905.4115] [INSPIRE].

    Article  ADS  Google Scholar 

  75. [75]

    S. Ferrara, J.A. Harvey, A. Strominger and C. Vafa, Second quantized mirror symmetry, Phys. Lett. B 361 (1995) 59 [hep-th/9505162] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

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Correspondence to Shamik Banerjee.

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ArXiv ePrint: 1106.0080

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Banerjee, S., Gupta, R.K., Mandal, I. et al. Logarithmic corrections to \( \mathcal{N} = {4} \) and \( \mathcal{N} = {8} \) black hole entropy: a one loop test of quantum gravity. J. High Energ. Phys. 2011, 143 (2011). https://doi.org/10.1007/JHEP11(2011)143

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Keywords

  • Black Holes in String Theory
  • Superstrings and Heterotic Strings