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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
R.M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48 (1993) 3427 [gr-qc/9307038] [INSPIRE].
T. Jacobson, G. Kang and R.C. Myers, On black hole entropy, Phys. Rev. D 49 (1994) 6587 [gr-qc/9312023] [INSPIRE].
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].
T. Jacobson, G. Kang and R.C. Myers, Black hole entropy in higher curvature gravity, gr-qc/9502009 [INSPIRE].
A. Sen, Black hole entropy function and the attractor mechanism in higher derivative gravity, JHEP 09 (2005) 038 [hep-th/0506177] [INSPIRE].
A. Sen, Entropy function for heterotic black holes, JHEP 03 (2006) 008 [hep-th/0508042] [INSPIRE].
A. Sen, Quantum entropy function from AdS 2 /CF T 1 correspondence, Int. J. Mod. Phys. A 24 (2009) 4225 [arXiv:0809.3304] [INSPIRE].
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].
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].
D. Shih, A. Strominger and X. Yin, Recounting dyons in N = 4 string theory, JHEP 10 (2006) 087 [hep-th/0505094] [INSPIRE].
D. Gaiotto, Re-recounting dyons in N = 4 string theory, hep-th/0506249 [INSPIRE].
D. Shih and X. Yin, Exact black hole degeneracies and the topological string, JHEP 04 (2006) 034 [hep-th/0508174] [INSPIRE].
D.P. Jatkar and A. Sen, Dyon spectrum in CHL models, JHEP 04 (2006) 018 [hep-th/0510147] [INSPIRE].
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].
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].
J.R. David and A. Sen, CHL dyons and statistical entropy function from D1-D5 system, JHEP 11 (2006) 072 [hep-th/0605210] [INSPIRE].
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].
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].
A. Dabholkar and D. Gaiotto, Spectrum of CHL dyons from genus-two partition function, JHEP 12 (2007) 087 [hep-th/0612011] [INSPIRE].
A. Sen, Black hole entropy function, attractors and precision counting of microstates, Gen. Rel. Grav. 40 (2008) 2249 [arXiv:0708.1270] [INSPIRE].
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].
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].
A. Dabholkar, J. Gomes and S. Murthy, Counting all dyons in N = 4 string theory, JHEP 05 (2011) 059 [arXiv:0803.2692] [INSPIRE].
A. Sen, Arithmetic of quantum entropy function, JHEP 08 (2009) 068 [arXiv:0903.1477] [INSPIRE].
A. Dabholkar, J. Gomes, S. Murthy and A. Sen, Supersymmetric index from black hole entropy, JHEP 04 (2011) 034 [arXiv:1009.3226] [INSPIRE].
A. Sen, Arithmetic of N = 8 black holes, JHEP 02 (2010) 090 [arXiv:0908.0039] [INSPIRE].
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].
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].
S.N. Solodukhin, The conical singularity and quantum corrections to entropy of black hole, Phys. Rev. D 51 (1995) 609 [hep-th/9407001] [INSPIRE].
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].
D.V. Fursaev, Temperature and entropy of a quantum black hole and conformal anomaly, Phys. Rev. D 51 (1995) 5352 [hep-th/9412161] [INSPIRE].
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].
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].
R.K. Kaul and P. Majumdar, Logarithmic correction to the Bekenstein-Hawking entropy, Phys. Rev. Lett. 84 (2000) 5255 [gr-qc/0002040] [INSPIRE].
S. Carlip, Logarithmic corrections to black hole entropy from the Cardy formula, Class. Quant. Grav. 17 (2000) 4175 [gr-qc/0005017] [INSPIRE].
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].
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].
A. Medved, A comment on black hole entropy or does nature abhor a logarithm?, Class. Quant. Grav. 22 (2005) 133 [gr-qc/0406044] [INSPIRE].
D.N. Page, Hawking radiation and black hole thermodynamics, New J. Phys. 7 (2005) 203 [hep-th/0409024] [INSPIRE].
R. Banerjee and B.R. Majhi, Quantum tunneling beyond semiclassical approximation, JHEP 06 (2008) 095 [arXiv:0805.2220] [INSPIRE].
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].
B.R. Majhi, Fermion tunneling beyond semiclassical approximation, Phys. Rev. D 79 (2009) 044005 [arXiv:0809.1508] [INSPIRE].
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].
R. Aros, D. Diaz and A. Montecinos, Logarithmic correction to BH entropy as Noether charge, JHEP 07 (2010) 012 [arXiv:1003.1083] [INSPIRE].
S.N. Solodukhin, Entanglement entropy of round spheres, Phys. Lett. B 693 (2010) 605 [arXiv:1008.4314] [INSPIRE].
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].
S. Giombi, A. Maloney and X. Yin, One-loop partition functions of 3D gravity, JHEP 08 (2008) 007 [arXiv:0804.1773] [INSPIRE].
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].
M.R. Gaberdiel, R. Gopakumar and A. Saha, Quantum W -symmetry in AdS 3 , JHEP 02 (2011) 004 [arXiv:1009.6087] [INSPIRE].
P.B. Gilkey, Invariance theory, the heat equation and the Atiyah-Singer index theorem, Publish or Perish Inc., U.S.A. (1984).
D. Vassilevich, Heat kernel expansion: user’s manual, Phys. Rept. 388 (2003) 279 [hep-th/0306138] [INSPIRE].
A. Sen, Entropy function and AdS 2/CF T 1 correspondence, JHEP 11 (2008) 075 [arXiv:0805.0095] [INSPIRE].
C. Beasley, D. Gaiotto, M. Guica, L. Huang, A. Strominger, et al., Why ZBH = |Ztop|2, hep-th/0608021 [INSPIRE].
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].
A. Dabholkar, J. Gomes and S. Murthy, Quantum black holes, localization and the topological string, JHEP 06 (2011) 019 [arXiv:1012.0265] [INSPIRE].
A. Dabholkar, J. Gomes and S. Murthy, Localization & exact holography, arXiv:1111.1161 [INSPIRE].
A. Sen and C. Vafa, Dual pairs of type-II string compactification, Nucl. Phys. B 455 (1995) 165 [hep-th/9508064] [INSPIRE].
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].
J.R. David, On the dyon partition function in N = 2 theories, JHEP 02 (2008) 025 [arXiv:0711.1971] [INSPIRE].
H. Ooguri, A. Strominger and C. Vafa, Black hole attractors and the topological string, Phys. Rev. D 70 (2004) 106007 [hep-th/0405146] [INSPIRE].
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].
F. Denef and G.W. Moore, Split states, entropy enigmas, holes and halos, hep-th/0702146 [INSPIRE].
A. Sen, Logarithmic corrections to N = 2 black hole entropy: an infrared window into the microstates, arXiv:1108.3842 [INSPIRE].
S. Christensen and M. Duff, New gravitational index theorems and supertheorems, Nucl. Phys. B 154 (1979) 301 [INSPIRE].
S. Christensen and M. Duff, Quantizing gravity with a cosmological constant, Nucl. Phys. B 170 (1980) 480 [INSPIRE].
M. Duff and P. van Nieuwenhuizen, Quantum inequivalence of different field representations, Phys. Lett. B 94 (1980) 179 [INSPIRE].
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].
R. Camporesi, Harmonic analysis and propagators on homogeneous spaces, Phys. Rept. 196 (1990) 1 [INSPIRE].
R. Camporesi and A. Higuchi, Spectral functions and ζ functions in hyperbolic spaces, J. Math. Phys. 35 (1994) 4217 [INSPIRE].
R. Camporesi, The spinor heat kernel in maximally symmetric spaces, Commun. Math. Phys. 148 (1992) 283 [INSPIRE].
R. Camporesi and A. Higuchi, Arbitrary spin effective potentials in anti-de Sitter space-time, Phys. Rev. D 47 (1993) 3339 [INSPIRE].
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].
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].
J.R. David, On walls of marginal stability in N = 2 string theories, JHEP 08 (2009) 054 [arXiv:0905.4115] [INSPIRE].
S. Ferrara, J.A. Harvey, A. Strominger and C. Vafa, Second quantized mirror symmetry, Phys. Lett. B 361 (1995) 59 [hep-th/9505162] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1106.0080
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
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
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP11(2011)143