Abstract
We consider the AdS 2 /CFT 1 holographic correspondence near the horizon of big four-dimensional black holes preserving four supersymmetries in toroidally compactified Type-II string theory. The boundary partition function of CFT 1 is given by the known quantum degeneracies of these black holes. The bulk partition function is given by a functional integral over string fields in AdS 2. Using recent results on localization we reduce the infinite-dimensional functional integral to a finite number of ordinary integrals over a space of localizing instantons. Under reasonable assumptions about the relevant terms in the effective action, these integrals can be evaluated exactly to obtain a bulk partition function. It precisely reproduces all terms in the exact Rademacher expansion of the boundary partition function as nontrivial functions of charges except for the Kloosterman sum which can in principle follow from an analysis of phases in the background of orbifolded instantons. Our results can be regarded as a step towards proving ‘exact holography’ in that the bulk and boundary partition functions computed independently agree for finite charges. Since the bulk partition function defines the quantum entropy of the black hole, our results enable the evaluation of perturbative as well as nonperturbative quantum corrections to the Bekenstein-Hawking-Wald entropy of these black holes.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J.D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333 [INSPIRE].
S. Hawking, Particle creation by black holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206-206] [INSPIRE].
R.M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48 (1993) 3427 [gr-qc/9307038] [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, Entropy Function and AdS 2 /CF T 1 Correspondence, JHEP 11 (2008) 075 [arXiv:0805.0095] [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].
E. Witten, Topological quantum field theory, Commun. Math. Phys. 117 (1988) 353 [INSPIRE].
E. Witten, Mirror manifolds and topological field theory, hep-th/9112056 [INSPIRE].
E. Witten, The N matrix model and gauged WZW models, Nucl. Phys. B 371 (1992) 191 [INSPIRE].
A.S. Schwarz and O. Zaboronsky, Supersymmetry and localization, Commun. Math. Phys. 183 (1997) 463 [hep-th/9511112] [INSPIRE].
O.V. Zaboronsky, Dimensional reduction in supersymmetric field theories, hep-th/9611157 [INSPIRE].
A. Dabholkar, J. Gomes and S. Murthy, Quantum black holes, localization and the topological string, JHEP 06 (2011) 019 [arXiv:1012.0265] [INSPIRE].
G. ’t Hooft, Dimensional reduction in quantum gravity, gr-qc/9310026 [INSPIRE].
L. Susskind, The World as a hologram, J. Math. Phys. 36 (1995) 6377 [hep-th/9409089] [INSPIRE].
J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [hep-th/9711200] [INSPIRE].
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].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
A. Castro, D. Grumiller, F. Larsen and R. McNees, Holographic description of AdS 2 black holes, JHEP 11 (2008) 052 [arXiv:0809.4264] [INSPIRE].
A. Sen, Arithmetic of quantum entropy function, JHEP 08 (2009) 068 [arXiv:0903.1477] [INSPIRE].
S. Elitzur, G.W. Moore, A. Schwimmer and N. Seiberg, Remarks on the canonical quantization of the Chern-Simons-Witten theory, Nucl. Phys. B 326 (1989) 108 [INSPIRE].
R. Dijkgraaf, J.M. Maldacena, G.W. Moore and E.P. Verlinde, A black hole Farey tail, hep-th/0005003 [INSPIRE].
A. Giveon, M. Porrati and E. Rabinovici, Target space duality in string theory, Phys. Rept. 244 (1994) 77 [hep-th/9401139] [INSPIRE].
D. Gaiotto, A. Strominger and X. Yin, New connections between 4 − D and 5 − D black holes, JHEP 02 (2006) 024 [hep-th/0503217] [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].
S. Banerjee and A. Sen, Duality orbits, dyon spectrum and gauge theory limit of heterotic string theory on T 6, JHEP 03 (2008) 022 [arXiv:0712.0043] [INSPIRE].
S. Banerjee and A. Sen, S-duality action on discrete T-duality invariants, JHEP 04 (2008) 012 [arXiv:0801.0149] [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].
J.M. Maldacena, G.W. Moore and A. Strominger, Counting BPS black holes in toroidal Type II string theory, hep-th/9903163 [INSPIRE].
D. Shih, A. Strominger and X. Yin, Counting dyons in N = 8 string theory, JHEP 06 (2006) 037 [hep-th/0506151] [INSPIRE].
A. Sen, N = 8 dyon partition function and walls of marginal stability, JHEP 07 (2008) 118 [arXiv:0803.1014] [INSPIRE].
M. Eichler and D. Zagier, The Theory of Jacobi Forms, Birkhäuser, Berlin Germany (1985).
K. Bringmann and S. Murthy, The positivity of black hole degeneracies, in preparation.
A. Sen, How Do Black Holes Predict the Sign of the Fourier Coefficients of Siegel Modular Forms?, Gen. Rel. Grav. 43 (2011) 2171 [arXiv:1008.4209] [INSPIRE].
A. Dabholkar, J. Gomes, S. Murthy and A. Sen, Supersymmetric index from black hole entropy, JHEP 04 (2011) 034 [arXiv:1009.3226] [INSPIRE].
N. Banerjee, I. Mandal and A. Sen, Black hole hair removal, JHEP 07 (2009) 091 [arXiv:0901.0359] [INSPIRE].
D.P. Jatkar, A. Sen and Y.K. Srivastava, Black hole hair removal: non-linear analysis, JHEP 02 (2010) 038 [arXiv:0907.0593] [INSPIRE].
H. Rademacher, Lectures on Elementary Number Theory, Robert E. Krieger Publishing Co., Malabar U.S.A. (1964).
J. Manschot and G.W. Moore, A Modern Farey Tail, Commun. Num. Theor. Phys. 4 (2010) 103 [arXiv:0712.0573] [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].
B. de Wit, J. van Holten and A. Van Proeyen, Transformation Rules of N = 2 Supergravity Multiplets, Nucl. Phys. B 167 (1980) 186 [INSPIRE].
B. de Wit, P. Lauwers and A. Van Proeyen, Lagrangians of N = 2 supergravity-matter systems, Nucl. Phys. B 255 (1985) 569 [INSPIRE].
B. de Wit, J. van Holten and A. Van Proeyen, Structure of N = 2 supergravity, Nucl. Phys. B 184 (1981) 77 [Erratum ibid. B 222 (1983) 516] [INSPIRE].
T. Mohaupt, Black hole entropy, special geometry and strings, Fortsch. Phys. 49 (2001) 3 [hep-th/0007195] [INSPIRE].
V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].
J. Gomis, T. Okuda and D. Trancanelli, Quantum ’t Hooft operators and S-duality in N = 4 super Yang-Mills, Adv. Theor. Math. Phys. 13 (2009) 1941 [arXiv:0904.4486] [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].
C. Beasley et al., Why Z BH = |Z top|2, hep-th/0608021 [INSPIRE].
F. Denef and G.W. Moore, Split states, entropy enigmas, holes and halos, JHEP 11 (2011) 129 [hep-th/0702146] [INSPIRE].
G. Lopes Cardoso, B. de Wit, J. Kappeli and T. Mohaupt, Stationary BPS solutions in N =2 supergravity with R 2 interactions, JHEP 12 (2000) 019 [hep-th/0009234] [INSPIRE].
G. Gibbons and S. Hawking, Action integrals and partition functions in quantum gravity, Phys. Rev. D 15 (1977) 2752 [INSPIRE].
D. Harlow, J. Maltz and E. Witten, Analytic continuation of Liouville theory, JHEP 12 (2011) 071 [arXiv:1108.4417] [INSPIRE].
S.R. Das, S. Naik and S.R. Wadia, Quantization of the Liouville Mode and String Theory, Mod. Phys. Lett. A 4 (1989) 1033 [INSPIRE].
J. Polchinski, String theory. Vol. 1: An introduction to the bosonic string, Cambridge University Press, Cambridge U.K. (1998).
A. Sen, Logarithmic Corrections to N = 2 Black Hole Entropy: An Infrared Window into the Microstates, arXiv:1108.3842 [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. Banerjee, R.K. Gupta, I. Mandal and A. Sen, Logarithmic corrections to N = 4 and N =8 black hole entropy: a one loop test of quantum gravity, JHEP 11 (2011) 143 [arXiv:1106.0080] [INSPIRE].
G. Lopes Cardoso, B. de Wit, J. Kappeli and T. Mohaupt, Black hole partition functions and duality, JHEP 03 (2006) 074 [hep-th/0601108] [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].
G. Cardoso, B. de Wit and S. Mahapatra, BPS black holes, the Hesse potential and the topological string, JHEP 06 (2010) 052 [arXiv:1003.1970] [INSPIRE].
N. Banerjee, D.P. Jatkar and A. Sen, Asymptotic expansion of the N = 4 dyon degeneracy, JHEP 05 (2009) 121 [arXiv:0810.3472] [INSPIRE].
S. Murthy and B. Pioline, A Farey tale for N = 4 dyons, JHEP 09 (2009) 022 [arXiv:0904.4253] [INSPIRE].
A. Dabholkar, J. Gomes and S. Murthy, Localization and non-perturbative effects, work in progress.
A. Sen, State operator correspondence and entanglement in AdS 2 /CF T 1, Entropy 13 (2011) 1305 [arXiv:1101.4254] [INSPIRE].
B. de Wit, S. Katmadas and M. van Zalk, New supersymmetric higher-derivative couplings: Full N = 2 superspace does not count!, JHEP 01 (2011) 007 [arXiv:1010.2150] [INSPIRE].
A. Galperin, E. Ivanov, V. Ogievetsky and E. Sokatchev, Harmonic superspace, Cambridge University Press, Cambridge U.K. (2004).
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1111.1161
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Dabholkar, A., Gomes, J. & Murthy, S. Localization & exact holography. J. High Energ. Phys. 2013, 62 (2013). https://doi.org/10.1007/JHEP04(2013)062
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2013)062