Abstract
Non-perturbative quantum corrections to supersymmetric black hole entropy often involve nontrivial number-theoretic phases called Kloosterman sums. We show how these sums can be obtained naturally from the functional integral of supergravity in asymptotically AdS 2 space for a class of black holes. They are essentially topological in origin and correspond to charge-dependent phases arising from the various gauge and gravitational Chern-Simons terms and boundary Wilson lines evaluated on Dehn-filled solid 2-torus. These corrections are essential to obtain an integer from supergravity in agreement with the quantum degeneracies, and reveal an intriguing connection between topology, number theory, and quantum gravity. We give an assessment of the current understanding of quantum entropy of black holes.
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References
A. Sen, Quantum entropy function from AdS 2 /CFT 1 correspondence, Int. J. Mod. Phys. A 24 (2009) 4225 [arXiv:0809.3304] [INSPIRE].
A. Sen, Entropy function and AdS 2 /CFT 1 correspondence, JHEP 11 (2008) 075 [arXiv:0805.0095] [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, JHEP 04 (2013) 062 [arXiv:1111.1161] [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].
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].
M. Eichler and D. Zagier, The Theory of Jacobi Forms, Birkhäuser, (1985).
A. Dabholkar, S. Murthy and D. Zagier, Quantum Black Holes, Wall Crossing and Mock Modular Forms, arXiv:1208.4074 [INSPIRE].
L.C. Jeffrey, Chern-Simons-Witten invariants of lens spaces and torus bundles and the semiclassical approximation, Commun. Math. Phys. 147 (1992) 563 [INSPIRE].
E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121 (1989) 351 [INSPIRE].
H. Rademacher, Lectures on elementary number theory, Robert E. Krieger Publishing Company, (1964).
R. Dijkgraaf, J.M. Maldacena, G.W. Moore and E.P. Verlinde, A black hole farey tail, hep-th/0005003 [INSPIRE].
J. Manschot and G.W. Moore, A modern fareytail, Commun. Num. Theor. Phys. 4 (2010) 103 [arXiv:0712.0573] [INSPIRE].
G. Lopes Cardoso, B. de Wit and T. Mohaupt, Deviations from the area law for supersymmetric black holes, Fortsch. Phys. 48 (2000) 49 [hep-th/9904005] [INSPIRE].
A. Dabholkar and J.A. Harvey, Nonrenormalization of the superstring tension, Phys. Rev. Lett. 63 (1989) 478 [INSPIRE].
A. Dabholkar, G.W. Gibbons, J.A. Harvey and F. Ruiz Ruiz, Superstrings and solitons, Nucl. Phys. B 340 (1990) 33 [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].
B. de Wit, J.W. van Holten and A. Van Proeyen, Transformation rules of N = 2 supergravity multiplets, Nucl. Phys. B 167 (1980) 186 [INSPIRE].
B. de Wit, P.G. Lauwers and A. Van Proeyen, Lagrangians of N = 2 supergravity-matter systems, Nucl. Phys. B 255 (1985) 569 [INSPIRE].
B. de Wit, J.W. van Holten and A. Van Proeyen, Structure of N = 2 supergravity, Nucl. Phys. B 184 (1981) 77 [Erratum ibid. B 222 (1983) 516] [INSPIRE].
G. Lopes Cardoso, B. de Wit and T. Mohaupt, Macroscopic entropy formulae and nonholomorphic corrections for supersymmetric black holes, Nucl. Phys. B 567 (2000) 87 [hep-th/9906094] [INSPIRE].
T. Mohaupt, Black hole entropy, special geometry and strings, Fortsch. Phys. 49 (2001) 3 [hep-th/0007195] [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].
R.K. Gupta and S. Murthy, All solutions of the localization equations for N = 2 quantum black hole entropy, JHEP 02 (2013) 141 [arXiv:1208.6221] [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].
J. Gomes, Quantum entropy and exact 4D/5D connection, JHEP 01 (2015) 109 [arXiv:1305.2849] [INSPIRE].
B. de Wit and S. Katmadas, Near-horizon analysis of D = 5 BPS black holes and rings, JHEP 02 (2010) 056 [arXiv:0910.4907] [INSPIRE].
A. Sen, Arithmetic of N = 8 black holes, JHEP 02 (2010) 090 [arXiv:0908.0039] [INSPIRE].
J.M. Maldacena and A. Strominger, AdS 3 black holes and a stringy exclusion principle, JHEP 12 (1998) 005 [hep-th/9804085] [INSPIRE].
J. de Boer, M.C.N. Cheng, R. Dijkgraaf, J. Manschot and E. Verlinde, A Farey Tail for Attractor Black Holes, JHEP 11 (2006) 024 [hep-th/0608059] [INSPIRE].
A. Maloney and E. Witten, Quantum gravity partition functions in three dimensions, JHEP 02 (2010) 029 [arXiv:0712.0155] [INSPIRE].
M. Bañados, C. Teitelboim and J. Zanelli, The black hole in three-dimensional space-time, Phys. Rev. Lett. 69 (1992) 1849 [hep-th/9204099] [INSPIRE].
A. Strominger, AdS 2 quantum gravity and string theory, JHEP 01 (1999) 007 [hep-th/9809027] [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].
A. Sen, Arithmetic of quantum entropy function, JHEP 08 (2009) 068 [arXiv:0903.1477] [INSPIRE].
P. Kirk and E. Klassen, Chern-Simons invariants of 3-manifolds and representation spaces of knot groups, Math. Ann. 287 (1990) 343.
J. Hansen and P. Kraus, Generating charge from diffeomorphisms, JHEP 12 (2006) 009 [hep-th/0606230] [INSPIRE].
A. Dabholkar, J. Gomes, S. Murthy and A. Sen, Supersymmetric index from black hole entropy, JHEP 04 (2011) 034 [arXiv:1009.3226] [INSPIRE].
J.M. Maldacena, A. Strominger and E. Witten, Black hole entropy in M-theory, JHEP 12 (1997) 002 [hep-th/9711053] [INSPIRE].
S. Gukov, Three-dimensional quantum gravity, Chern-Simons theory and the A polynomial, Commun. Math. Phys. 255 (2005) 577 [hep-th/0306165] [INSPIRE].
A. Castro, N. Lashkari and A. Maloney, A de Sitter Farey Tail, Phys. Rev. D 83 (2011) 124027 [arXiv:1103.4620] [INSPIRE].
C. Beasley, Localization for Wilson Loops in Chern-Simons Theory, Adv. Theor. Math. Phys. 17 (2013) 1 [arXiv:0911.2687] [INSPIRE].
J. Kallen, Cohomological localization of Chern-Simons theory, JHEP 08 (2011) 008 [arXiv:1104.5353] [INSPIRE].
A. Dabholkar, F. Denef, G.W. Moore and B. Pioline, Precision counting of small black holes, JHEP 10 (2005) 096 [hep-th/0507014] [INSPIRE].
A. Sen, U-duality Invariant Dyon Spectrum in type II on T 6, JHEP 08 (2008) 037 [arXiv:0804.0651] [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].
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].
A. Dabholkar, J. Gomes and S. Murthy, Counting all dyons in N = 4 string theory, JHEP 05 (2011) 059 [arXiv:0803.2692] [INSPIRE].
L. Rozansky, A Contribution to the trivial connection to Jones polynomial and Witten’s invariant of 3 − D manifolds. 1., Commun. Math. Phys. 175 (1996) 275 [hep-th/9401061] [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].
F. Denef and G.W. Moore, Split states, entropy enigmas, holes and halos, JHEP 11 (2011) 129 [hep-th/0702146] [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].
S. Murthy and V. Reys, Quantum black hole entropy and the holomorphic prepotential of N = 2 supergravity, JHEP 10 (2013) 099 [arXiv:1306.3796] [INSPIRE].
A. Sen, Logarithmic corrections to N = 2 black hole entropy: an infrared window into the microstates, arXiv:1108.3842 [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].
S. Murthy and V. Reys, Functional determinants, index theorems, and exact quantum black hole entropy, in preparation.
R. Gupta, Y. Ito and I. Jeon, in preparation.
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Dabholkar, A., Gomes, J. & Murthy, S. Nonperturbative black hole entropy and Kloosterman sums. J. High Energ. Phys. 2015, 74 (2015). https://doi.org/10.1007/JHEP03(2015)074
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DOI: https://doi.org/10.1007/JHEP03(2015)074