Black hole entropy function, attractors and precision counting of microstates. (English) Zbl 1153.83007

Sen, Ashoke

Gen. Relativ. Gravitation 40, No. 11, 2249-2431 (2008).

A black hole is called extremal, if the two zeroes of the essential metric component coincide: For the Reissner-Nordström black hole, one has \(g_{00} = (1-a/r)\cdot (1-b/r) \) with certain positive constants \(a\) and \(b\) encoding charge and mass of the black hole. For \(a=b\), the hole is called extremal. For this case, several calculations are easier to perform, e.g. the entropy of the black hole.
From the paper’s introduction: “These lecture notes describe recent progress in our understanding of the attractor mechanism and entropy of extremal black holes based on the entropy function formalism. They also describe the precise computation of the microscopic degeneracy of a class of quarter BPS dyons in \({\mathcal{N}=4}\) supersymmetric string theories, and compare the statistical entropy of these dyons, expanded in inverse powers of electric and magnetic charges, with a similar expansion of the corresponding black hole entropy. This comparison is extended to include the contribution to the entropy from multi-centred black holes as well.”
The paper has 8 appendices and a reference list of 264 items.

Reviewer: Hans-Jürgen Schmidt (Potsdam)

MSC:

83-02	Research exposition (monographs, survey articles) pertaining to relativity and gravitational theory
83C57	Black holes
83E30	String and superstring theories in gravitational theory

Keywords:

extremal black holes

References:

[1]	Strominger, A., Vafa, C.: Microscopic origin of the Bekenstein–Hawking entropy. Phys. Lett. B 379, 99 (1996) [arXiv:hep-th/9601029] · Zbl 1376.83026
[2]	Ooguri, H., Strominger, A., Vafa, C.: Black hole attractors and the topological string. arXiv:hep-th/0405146
[3]	Sen, A.: Black hole entropy function and the attractor mechanism in higher derivative gravity. JHEP 0509, 038 (2005) [arXiv:hep-th/0506177]
[4]	Sen, A.: Entropy function for heterotic black holes. JHEP 0603, 008 (2006) [arXiv:hep-th/0508042] · Zbl 1226.83037
[5]	Wald, R.M.: Black hole entropy in the Noether charge. Phys. Rev. D 48, 3427 (1993) [arXiv:gr-qc/9307038] · Zbl 0942.83512
[6]	David, J.R., Sen, A.: CHL dyons and statistical entropy function from D1–D5 system. arXiv:hep-th/0605210
[7]	David, J.R., Jatkar, D.P., Sen, A.: Dyon spectrum in N = 4 supersymmetric type II string theories. arXiv:hep-th/0607155
[8]	David, J.R., Jatkar, D.P., Sen, A.: Dyon spectrum in generic N = 4 supersymmetric Z(N) orbifolds. arXiv:hep-th/0609109
[9]	Dijkgraaf, R., Verlinde, E.P., Verlinde, H.L.: Counting dyons in N = 4 string theory. Nucl. Phys. B 484, 543 (1997) [arXiv:hep-th/9607026] · Zbl 0925.81230
[10]	Jatkar, D.P., Sen, A.: Dyon spectrum in CHL models. JHEP 0604, 018 (2006) [arXiv:hep-th/0510147]
[11]	David, J.R., Jatkar, D.P., Sen, A.: Product representation of dyon partition function in CHL models. JHEP 0606, 064 (2006) [arXiv:hep-th/0602254]
[12]	Dabholkar, A., Nampuri, S.: Spectrum of dyons and black holes in CHL orbifolds using Borcherds lift. arXiv:hep-th/0603066 · Zbl 1245.81158
[13]	Shih, D., Strominger, A., Yin, X.: Recounting dyons in N = 4 string theory. arXiv:hep-th/0505094
[14]	Gaiotto, D.: Re-recounting dyons in N = 4 string theory. arXiv:hep-th/0506249
[15]	Shih, D., Yin, X.: Exact black hole degeneracies and the topological string. arXiv:hep-th/0508174
[16]	Shih, D., Strominger, A., Yin, X.: Counting dyons in N = 8 string theory. arXiv:hep-th/0506151
[17]	Dabholkar, A., Gaiotto, D.: Spectrum of CHL dyons from genus-two partition function. arXiv:hep-th/0612011 · Zbl 1246.81229
[18]	Cardoso, G.L., de Wit, B., Kappeli, J., Mohaupt, T.: Asymptotic degeneracy of dyonic N = 4 string states and black hole entropy. JHEP 0412, 075 (2004) [arXiv:hep-th/0412287]
[19]	Kraus, P., Larsen, F.: Microscopic black hole entropy in theories with higher derivatives. arXiv:hep-th/0506176
[20]	Kraus, P., Larsen, F.: Holographic gravitational anomalies. arXiv:hep-th/0508218
[21]	Kraus, P., Larsen, F.: Partition functions and elliptic genera from supergravity. arXiv:hep-th/0607138
[22]	Kraus, P.: Lectures on black holes and the AdS(3)/CFT(2) correspondence. arXiv:hep-th/0609074
[23]	Kunduri, H.K., Lucietti, J., Reall, H.S.: Near-horizon symmetries of extremal black holes. arXiv:0705.4214 [hep-th] · Zbl 1205.83047
[24]	Astefanesei, D., Goldstein, K., Jena, R.P., Sen, A., Trivedi, S.P.: Rotating attractors. JHEP 0610, 058 (2006) [arXiv:hep-th/0606244]
[25]	Jacobson, T., Kang, G., Myers, R.C.: On black hole entropy. Phys. Rev. D 49, 6587 (1994) [arXiv:gr-qc/9312023] · Zbl 0862.53068
[26]	Iyer, V., Wald, R.M.: Some properties of Noether charge and a proposal for dynamical black hole entropy. Phys. Rev. D 50, 846 (1994) [arXiv:gr-qc/9403028]
[27]	Jacobson, T., Kang, G., Myers, R.C.: Black hole entropy in higher curvature gravity. arXiv:gr-qc/9502009
[28]	Goldstein, K., Iizuka, N., Jena, R.P., Trivedi, S.P.: Non-supersymmetric attractors. Phys. Rev. D 72, 124021 (2005) [arXiv:hep-th/0507096]
[29]	Ferrara, S., Kallosh, R., Strominger, A.: N = 2 extremal black holes. Phys. Rev. D 52, 5412 (1995) [arXiv:hep-th/9508072]
[30]	Strominger, A.: Macroscopic entropy of N = 2 extremal black holes. Phys. Lett. B 383, 39 (1996) [arXiv:hep-th/9602111] · Zbl 0955.83507
[31]	Ferrara, S., Kallosh, R.: Supersymmetry and attractors. Phys. Rev. D 54, 1514 (1996) [arXiv:hep-th/9602136] · Zbl 1171.83329
[32]	Dabholkar, A., Sen, A., Trivedi, S.: Black hole microstates and attractor without supersymmetry. arXiv:hep-th/0611143
[33]	Breckenridge, J.C., Myers, R.C., Peet, A.W., Vafa, C.: D-branes and spinning black holes. Phys. Lett. B 391, 93 (1997) [arXiv:hep-th/9602065] · Zbl 0956.83064
[34]	Kallosh, R., Rajaraman, A., Wong, W.K.: Supersymmetric rotating black holes and attractors. Phys. Rev. D 55, 3246 (1997) [arXiv:hep-th/9611094]
[35]	Li, W., Strominger, A.: Supersymmetric probes in a rotating 5D attractor. arXiv:hep-th/0605139 · Zbl 1246.83205
[36]	Bardeen, J.M., Horowitz, G.T.: The extreme Kerr throat geometry: a vacuum analog of AdS(2) x S(2). Phys. Rev. D 60, 104030 (1999) [arXiv:hep-th/9905099]
[37]	Rasheed, D.: The rotating dyonic black holes of Kaluza–Klein theory. Nucl. Phys. B454, 379–401 (1995) [arXiv:hep-th/9505038] · Zbl 0925.83066
[38]	Cvetic, M., Youm, D.: Entropy of non-extreme charged rotating black holes in string theory. Phys. Rev. D 54, 2612 (1996) [arXiv:hep-th/9603147]
[39]	Matos, T., Mora, C.: Stationary dilatons with arbitrary electromagnetic field. Class. Quant. Grav. 14, 2331–2340 (1997) [arXiv:hep-th/9610013] · Zbl 0877.53063
[40]	Larsen, F.: Rotating Kaluza–Klein black holes. Nucl. Phys. B575 211–230, (2000) [arXiv:hep-th/9909102] · Zbl 0992.83068
[41]	Emparan, R., Reall, H.S.: A rotating black ring in five dimensions. Phys. Rev. Lett. 88, 101101 (2002) [arXiv:hep-th/0110260] · Zbl 1015.83018
[42]	Elvang, H., Emparan, R., Mateos, D., Reall, H.S.: Supersymmetric black rings and three-charge supertubes. Phys. Rev. D 71, 024033 (2005) [arXiv:hep-th/0408120]
[43]	Tachikawa, Y.: Black hole entropy in the presence of Chern–Simons terms. Class. Quant. Grav. 24, 737 (2007) [arXiv:hep-th/0611141] · Zbl 1170.83424
[44]	Sahoo, B., Sen, A.: Alpha’ corrections to extremal dyonic black holes in heterotic string theory. arXiv:hep-th/0608182
[45]	Sahoo, B., Sen, A.: BTZ black hole with Chern–Simons and higher derivative terms. JHEP 0607, 008 (2006) [arXiv:hep-th/0601228]
[46]	Deser, S., Jackiw, R., Templeton, S.: Topologically massive gauge theories. Ann. Phys. 140, 372 (1982) [Erratum-ibid. 185, 406.1988 APNYA,281,409 (1988 APNYA,281,409-449.2 000)]
[47]	Deser S., Jackiw R. and Templeton S. (1982). Three-dimensional massive gauge theories. Phys. Rev. Lett. 48: 975
[48]	Guralnik, G., Iorio, A., Jackiw, R., Pi, S.Y.: Dimensionally reduced gravitational Chern–Simons term and its kink. Annals Phys. 308, 222 (2003) [arXiv:hep-th/0305117] · Zbl 1037.83012
[49]	Hull, C.M., Townsend, P.K.: Unity of superstring dualities. Nucl. Phys. B 438, 109 (1995) [arXiv:hep-th/9410167] · Zbl 1052.83532
[50]	Duff, M.J.: Strong/weak coupling duality from the dual string. Nucl. Phys. B 442, 47 (1995) [arXiv:hep-th/9501030] · Zbl 0990.81661
[51]	Witten, E.: String theory dynamics in various dimensions. Nucl. Phys. B 443, 85 (1995) [arXiv:hep-th/9503124] · Zbl 0990.81663
[52]	Sen, A.: String–string duality conjecture in six-dimensions and charged solitonic strings. Nucl. Phys. B 450, 103 (1995) [arXiv:hep-th/9504027] · Zbl 0982.81520
[53]	Harvey, J.A., Strominger, A.: The heterotic string is a soliton. Nucl. Phys. B 449, 535 (1995) [Erratum-ibid. B 458, 456 (1996)] [arXiv:hep-th/9504047] · Zbl 1076.81555
[54]	Sen, A., Vafa, C.: Dual pairs of type II string compactification. Nucl. Phys. B 455, 165 (1995) [arXiv:hep-th/9508064] · Zbl 0925.81160
[55]	Chaudhuri, S., Hockney, G., Lykken, J.D.: Maximally supersymmetric string theories in D < 10. Phys. Rev. Lett. 75, 2264 (1995) [arXiv:hep-th/9505054] · Zbl 1020.81763
[56]	Chaudhuri, S., Polchinski, J.: Moduli space of CHL strings. Phys. Rev. D 52, 7168 (1995) [arXiv:hep-th/9506048]
[57]	Vafa, C., Witten, E.: Dual string pairs with N = 1 and N = 2 supersymmetry in four dimensions. Nucl. Phys. Proc. Suppl. 46, 225 (1996) [arXiv:hep-th/9507050] · Zbl 0957.81590
[58]	Chaudhuri, S., Lowe, D.A.: Type IIA heterotic duals with maximal supersymmetry. Nucl. Phys. B 459, 113 (1996) [arXiv:hep-th/9508144] · Zbl 1003.81523
[59]	Aspinwall, P.S.: Some relationships between dualities in string theory. Nucl. Phys. Proc. Suppl. 46, 30 (1996) [arXiv:hep-th/9508154] · Zbl 0957.81599
[60]	Maharana, J., Schwarz, J.H.: Noncompact symmetries in string theory. Nucl. Phys. B 390, 3 (1993) [arXiv:hep-th/9207016]
[61]	Sen, A.: Strong–weak coupling duality in four-dimensional string theory. Int. J. Mod. Phys. A 9, 3707 (1994) [arXiv:hep-th/9402002] · Zbl 0985.81635
[62]	Gross D.J. and Perry M.J. (1983). Magnetic monopoles in Kaluza–Klein theories. Nucl. Phys. B 226: 29
[63]	Sorkin R.d. (1983). Kaluza–Klein monopole. Phys. Rev. Lett. 51: 87
[64]	Gauntlett, J.P., Harvey, J.A., Liu, J.T.: Magnetic monopoles in string theory. Nucl. Phys. B 409, 363 (1993) [arXiv:hep-th/9211056] · Zbl 1043.81653
[65]	Zwiebach B. (1985). Curvature squared terms and string theories. Phys. Lett. B 156: 315
[66]	Harvey, J.A., Moore, G.W.: Fivebrane instantons and R**2 couplings in N = 4 string theory. Phys. Rev. D 57, 2323 (1998) [arXiv:hep-th/9610237]
[67]	Gregori, A., Kiritsis, E., Kounnas, C., Obers, N.A., Petropoulos, P.M., Pioline, B.: R**2 corrections and non-perturbative dualities of N = 4 string ground states. Nucl. Phys. B 510, 423 (1998) [arXiv:hep-th/9708062] · Zbl 0953.81080
[68]	Bershadsky, M., Cecotti, S., Ooguri, H., Vafa, C.: Holomorphic anomalies in topological field theories. Nucl. Phys. B 405, 279 (1993) [arXiv:hep-th/9302103] · Zbl 0908.58074
[69]	Antoniadis, I., Gava, E., Narain, K.S., Taylor, T.R.: Topological amplitudes in string theory. Nucl. Phys. B 413, 162 (1994) [arXiv:hep-th/9307158] · Zbl 1007.81522
[70]	Bershadsky, M., Cecotti, S., Ooguri, H., Vafa, C.: Kodaira–Spencer theory of gravity and exact results for quantum string amplitudes. Commun. Math. Phys. 165, 311 (1994) [arXiv:hep-th/9309140] · Zbl 0815.53082
[71]	Maldacena, J.M., Strominger, A., Witten, E.: Black hole entropy in M-theory. JHEP 9712, 002 (1997) [arXiv:hep-th/9711053] · Zbl 0951.83034
[72]	Cardoso, G.L., de Wit, B., Mohaupt, T.: Macroscopic entropy formulae and non-holomorphic corrections for supersymmetric black holes. Nucl. Phys. B 567, 87 (2000) [arXiv:hep-th/9906094] · Zbl 0951.81039
[73]	Sahoo, B., Sen, A.: Higher derivative corrections to non-supersymmetric extremal black holes in N = 2 supergravity. JHEP 0609, 029 (2006) [arXiv:hep-th/0603149]
[74]	Van Proeyen A., Wit B. and Holten J.W. (1980). Transformation rules of N = 2 supergravity multiplets. Nucl. Phys. B 167: 186
[75]	Bergshoeff E., Roo M. and Wit B. (1981). Extended conformal supergravity. Nucl. Phys. B 182: 173
[76]	de Wit, B., van Holten, J.W., Van Proeyen, A.: Structure of N = 2 supergravity. Nucl. Phys. B 184, 77 (1981) [Erratum-ibid. B 222, 516 (1983)]
[77]	Van Proeyen A. and Wit B. (1984). Potentials and symmetries of general gauged N = 2 supergravity–Yang-Mills models. Nucl. Phys. B 245: 89
[78]	Lauwers P.G., Van Proeyen A. and Wit B. (1985). Lagrangians of N = 2 supergravity–matter systems. Nucl. Phys. B 255: 569
[79]	Philippe R., Van Proeyen A. and Wit B. (1983). The improved tensor multiplet in N = 2 supergravity. Nucl. Phys. B 219: 143
[80]	Lauwers P.G., Philippe R., Su S.Q., Van Proeyen A. (1984). Gauge and matter fields coupled to N = 2 supergravity. Phys. Lett. B 134: 37
[81]	de Wit, B.: N = 2 electric–magnetic duality in a chiral background. Nucl. Phys. Proc. Suppl. 49, 191 (1996) [arXiv:hep-th/9602060] · Zbl 0957.81693
[82]	de Wit, B.: N = 2 symplectic reparametrizations in a chiral background. Fortsch. Phys. 44, 529 (1996) [arXiv:hep-th/9603191] · Zbl 1043.81721
[83]	Mohaupt, T.: Black hole entropy, special geometry and strings. Fortsch. Phys. 49, 3 (2001) [arXiv:hep-th/0007195] · Zbl 0985.83001
[84]	Behrndt, K., Cardoso, G.L., de Wit, B., Lust, D., Mohaupt, T., Sabra, W.A.: Higher-order black-hole solutions in N = 2 supergravity and Calabi–Yau string backgrounds. Phys. Lett. B 429, 289 (1998) [arXiv:hep-th/9801081] · Zbl 1355.83023
[85]	Cardoso, G.L., de Wit, B., Mohaupt, T.: Corrections to macroscopic supersymmetric black-hole entropy. Phys. Lett. B 451, 309 (1999) [arXiv:hep-th/9812082] · Zbl 1058.83516
[86]	Cardoso, G.L., de Wit, B., Mohaupt, T.: Deviations from the area law for supersymmetric black holes. Fortsch. Phys. 48, 49 (2000) [arXiv:hep-th/9904005] · Zbl 0952.83034
[87]	Cardoso, G.L., de Wit, B., Mohaupt, T.: Area law corrections from state counting and supergravity. Class. Quant. Grav. 17, 1007 (2000) [arXiv:hep-th/9910179] · Zbl 0952.81025
[88]	Cardoso, G.L., de Wit, B., Kappeli, J., Mohaupt, T.: Stationary BPS solutions in N = 2 supergravity with R**2 interactions. JHEP 0012, 019 (2000) [arXiv:hep-th/0009234]
[89]	Cardoso, G.L., de Wit, B., Kappeli, J., Mohaupt, T.: Examples of stationary BPS solutions in N = 2 supergravity theories with R**2-interactions. Fortsch. Phys. 49, 557 (2001) [arXiv:hep-th/0012232] · Zbl 0978.83046
[90]	Behrndt, K., Cardoso, G.L., Mahapatra, S.: Exploring the relation between 4D and 5D BPS solutions. Nucl. Phys. B 732, 200 (2006) [arXiv:hep-th/0506251] · Zbl 1192.83062
[91]	Dabholkar A. and Harvey J.A. (1989). Nonrenormalization of the superstring tension. Phys. Rev. Lett. 63: 478
[92]	Dabholkar A., Gibbons G.W., Harvey J.A. and Ruiz Ruiz F. (1990). Superstrings and solitons. Nucl. Phys. B 340: 33
[93]	Sen, A.: Extremal black holes and elementary string states. Mod. Phys. Lett. A 10, 2081 (1995) [arXiv:hep-th/9504147]
[94]	Peet, A.W.: Entropy and supersymmetry of D-dimensional extremal electric black holes versus string states. Nucl. Phys. B 456, 732 (1995) [arXiv:hep-th/9506200] · Zbl 0925.83038
[95]	Sen, A.: Black holes and elementary string states in N = 2 supersymmetric string theories. JHEP 9802, 011 (1998) [arXiv:hep-th/9712150] · Zbl 0955.83025
[96]	Dabholkar, A.: Exact counting of black hole microstates. arXiv:hep-th/0409148
[97]	Sen, A.: Stretching the horizon of a higher dimensional small black hole. JHEP 0507, 073 (2005) [arXiv:hep-th/0505122]
[98]	Dabholkar, A., Kallosh, R., Maloney, A.: A stringy cloak for a classical singularity. arXiv:hep-th/0410076
[99]	Sen, A.: How does a fundamental string stretch its horizon?. arXiv:hep-th/0411255
[100]	Hubeny, V., Maloney, A., Rangamani, M.: String-corrected black holes. arXiv:hep-th/0411272
[101]	Bak, D., Kim, S., Rey, S.J.: Exactly soluble BPS black holes in higher curvature N = 2 supergravity. arXiv:hep-th/0501014
[102]	Dabholkar, A., Iizuka, N., Iqubal, A., Sen, A., Shigemori, M.: Spinning strings as small black rings. arXiv:hep-th/0611166
[103]	Iizuka, N., Shigemori, M.: A note on D1–D5-J system and 5D small black ring. JHEP 0508, 100 (2005) [arXiv:hep-th/0506215]
[104]	Dabholkar, A., Iizuka, N., Iqubal, A., Shigemori, M.: Precision microstate counting of small black rings. Phys. Rev. Lett. 96, 071601 (2006) [arXiv:hep-th/0511120]
[105]	Giveon, A., Kutasov, D.: Fundamental strings and black holes. JHEP 0701, 071 (2007) [arXiv:hep-th/0611062] · Zbl 1200.81123
[106]	Dabholkar, A., Murthy, S.: Fundamental superstrings as holograms. arXiv:0707.3818
[107]	Johnson, C.: Heterotic coset models of microscopic strings and black holes. arXiv:0707.4303
[108]	Lapan, J.M., Simons, A., Strominger, A.: Fundamental superstrings as holograms. arXiv:0708.0016
[109]	Banados, M., Teitelboim, C., Zanelli, J.: The black hole in three-dimensional space–time. Phys. Rev. Lett. 69, 1849 (1992) [arXiv:hep-th/9204099]
[110]	Sfetsos, K., Skenderis, K.: Microscopic derivation of the Bekenstein–Hawking entropy formula for non-extremal black holes. Nucl. Phys. B 517, 179 (1998) [arXiv:hep-th/9711138] · Zbl 0920.53048
[111]	Strominger, A.: Black hole entropy from near-horizon microstates. JHEP 9802, 009 (1998) [arXiv:hep-th/9712251] · Zbl 0955.83010
[112]	Skenderis, K.: Black holes and branes in string theory. Lect. Notes Phys. 541, 325 (2000) [arXiv:hep-th/9901050] · Zbl 0973.83054
[113]	Brown J.D. and Henneaux M. (1986). Central charges in the canonical realization of asymptotic symmetries: an example from three-dimensional gravity. Commun. Math. Phys. 104: 207 · Zbl 0584.53039
[114]	Morales, J.F., Samtleben, H.: Entropy function and attractors for AdS black holes. JHEP 0610, 074 (2006) [arXiv:hep-th/0608044]
[115]	Cardoso, G.L., Oberreuter, J.M., Perz, J.: Entropy function for rotating extremal black holes in very special geometry. arXiv:hep-th/0701176
[116]	Saida, H., Soda, J.: Statistical entropy of BTZ black hole in higher curvature gravity. Phys. Lett. B 471, 358 (2000) [arXiv:gr-qc/9909061] · Zbl 0974.83030
[117]	Maldacena, J.M., Strominger, A.: AdS(3) black holes, a stringy exclusion principle. JHEP 9812, 005 (1998) [arXiv:hep-th/9804085] · Zbl 0951.83019
[118]	Solodukhin, S.N.: Holography with gravitational Chern–Simons. Phys. Rev. D 74, 024015 (2006) [arXiv:hep-th/0509148]
[119]	Achucarro, A., Ortiz, M.E.: Relating black holes in two-dimensions and three-dimensions. Phys. Rev. D 48, 3600 (1993) [arXiv:hep-th/9304068]
[120]	Boonstra, H.J., Peeters, B., Skenderis, K.: Brane intersections, anti-de Sitter spacetimes, dual superconformal theories. Nucl. Phys. B 533, 127 (1998) [arXiv:hep-th/9803231] · Zbl 0956.81060
[121]	Maldacena, J.M.: The large N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys. 2, 231 (1998) [Int. J. Theor. Phys. 38, 1113 (1999)] [arXiv:hep-th/9711200] · Zbl 0969.81047
[122]	Gubser, S.S., Klebanov, I.R., Polyakov, A.M.: Gauge theory correlators from non-critical string theory. Phys. Lett. B 428, 105 (1998) [arXiv:hep-th/9802109] · Zbl 1355.81126
[123]	Witten, E.: Anti-de Sitter space, holography. Adv. Theor. Math. Phys. 2, 253 (1998) [arXiv:hep-th/9802150] · Zbl 0914.53048
[124]	Henningson, M., Skenderis, K.: The holographic Weyl anomaly. JHEP 9807, 023 (1998) [arXiv:hep-th/9806087] · Zbl 0958.81083
[125]	Witten, E.: Three-dimensional gravity revisited. arXiv:0706.3359 [hep-th]
[126]	David, J.R., Sahoo, B., Sen, A.: AdS 3, black holes, higher derivative corrections. arXiv:0705.0735 [hep-th]
[127]	Nishimura, M., Tanii, Y.: Super Weyl anomalies in the AdS/CFT correspondence. Int. J. Mod. Phys. A 14, 3731 (1999) [arXiv:hep-th/9904010] · Zbl 0989.81116
[128]	David, J.R.: Anti-de Sitter gravity associated with the supergroup SU(1,1|2) x SU(1,1|2). Mod. Phys. Lett. A 14, 1143 (1999) [arXiv:hep-th/9904068]
[129]	Achucarro A. and Townsend P.K. (1986). A Chern–simons action for three-dimensional Anti-de sitter supergravity theories. Phys. Lett. B 180: 89
[130]	Achucarro A. and Townsend P.K. (1989). Extended supergravities in d = (2 + 1) as Chern–Simons theories. Phys. Lett. B 229: 383
[131]	Giacomini, A., Troncoso, R., Willison, S.: Three-dimensional supergravity reloaded. arXiv:hep-th/0610077 · Zbl 1117.83112
[132]	Witten E. (1988). (2+1)-dimensional gravity as an exactly soluble system. Nucl. Phys. B 311: 46 · Zbl 1258.83032
[133]	Exirifard, G.: The world-sheet corrections to dyons in the heterotic theory. arXiv:hep-th/0607094 · Zbl 1162.83014
[134]	Castro, A., Davis, J.L., Kraus, P., Larsen, F.: 5D attractors with higher derivatives. JHEP 0704, 091 (2007) [arXiv:hep-th/0702072]
[135]	Castro, A., Davis, J.L., Kraus, P., Larsen, F.: 5D black holes, strings with higher derivatives. arXiv:hep-th/0703087
[136]	Alishahiha, M.: On R**2 corrections for 5D black holes. arXiv:hep-th/0703099 · Zbl 1326.83047
[137]	Hanaki, K., Ohashi, K., Tachikawa, Y.: Supersymmetric completion of an R 2 term in five-dimensional supergravity. Prog. Theor. Phys. 117, 533 (2007) [arXiv:hep-th/0611329] · Zbl 1183.83103
[138]	Metsaev R. and Tseytlin A. (1987). Nucl. Phys. B 293: 385
[139]	Hull C.M. and Townsend P.K. (1987). The two loop beta function for sigma models with torsion. Phys. Lett. B 191: 115
[140]	Denef, F.: Supergravity flows, D-brane stability. JHEP 0008, 050 (2000) [arXiv:hep-th/0005049] · Zbl 0990.83553
[141]	Denef, F.: On the correspondence between D-branes, stationary supergravity solutions of type II Calabi–Yau compactifications, arXiv:hep-th/0010222
[142]	Denef, F., Greene, B.R., Raugas, M.: Split attractor flows, the spectrum of BPS D-branes on the quintic. JHEP 0105, 012 (2001) [arXiv:hep-th/0101135]
[143]	Denef, F.: Quantum quivers, Hall/hole halos. JHEP 0210, 023 (2002) [arXiv:hep-th/0206072]
[144]	Bates, B., Denef, F.: Exact solutions for supersymmetric stationary black hole composites. arXiv: hep-th/0304094 · Zbl 1306.83040
[145]	Denef, F., Moore, G.W.: Split states, entropy enigmas, holes, halos. arXiv:hep-th/0702146 · Zbl 1306.81213
[146]	Denef, F., Moore, G.W.: How many black holes fit on the head of a pin?. arXiv:0705.2564 [hep-th]
[147]	Diaconescu, E., Moore, G.W.: Crossing the Wall: Branes vs. Bundles. arXiv:0706.3193 [hep-th]
[148]	Breckenridge, J.C., Lowe, D.A., Myers, R.C., Peet, A.W., Strominger, A., Vafa, C.: Macroscopic, microscopic entropy of near-extremal spinning black holes. Phys. Lett. B 381, 423 (1996) [arXiv: hep-th/9603078]
[149]	Bershadsky, M., Vafa, C., Sadov, V.: D-Branes, topological field theories. Nucl. Phys. B 463, 420 (1996) [arXiv:hep-th/9511222] · Zbl 1004.81560
[150]	Cheng, M.C.N., Verlinde, E.: Dying dyons don’t count. arXiv:0706.2363 [hep-th]
[151]	Banerjee, N., Jatkar, D.P., Sen, A.: Adding charges to N = 4 dyons. arXiv:0705.1433 [hep-th]
[152]	Bergman, O.: Three-pronged strings, 1/4 BPS states in N = 4 super-Yang-Mills theory. Nucl. Phys. B 525, 104 (1998) [arXiv:hep-th/9712211] · Zbl 1031.81587
[153]	Moore, G.W.: Les Houches lectures on strings, arithmetic. arXiv:hep-th/0401049
[154]	Sen, A.: Two centered black holes, N = 4 dyon spectrum. arXiv:0705.3874 [hep-th]
[155]	Sen, A.: Kaluza–Klein dyons in string theory. Phys. Rev. Lett. 79, 1619 (1997) [arXiv:hep-th/ 9705212] · Zbl 0898.58064
[156]	Eguchi T., Gilkey P.B. and Hanson A.J. (1980). Gravitation, Gauge theories and differential geometry. Phys. Rept. 66: 213
[157]	Brill D. (1964). Phys. Rev. B 133: 845
[158]	Pope C.N. (1978). Axial vector anomalies, the index theorem in charged Schwarzschild, Taub–Nut spaces. Nucl. Phys. B 141: 432
[159]	Gauntlett, J.P., Kim, N., Park, J., Yi, P.: Monopole dynamics, BPS dyons in N = 2 super-Yang-Mills theories. Phys. Rev. D 61, 125012 (2000) [arXiv:hep-th/9912082]
[160]	Gaiotto, D., Strominger, A., Yin, X.: New connections between 4D, 5D black holes. JHEP 0602, 024 (2006) [arXiv:hep-th/0503217]
[161]	Russo, J.G., Susskind, L.: Asymptotic level density in heterotic string theory, rotating black holes. Nucl. Phys. B 437, 611 (1995) [arXiv:hep-th/9405117] · Zbl 1052.81597
[162]	David, J.R., Mandal, G., Wadia, S.R.: Microscopic formulation of black holes in string theory. Phys. Rept. 369, 549 (2002) [arXiv:hep-th/0203048] · Zbl 0998.83032
[163]	Dijkgraaf, R., Moore, G.W., Verlinde, E.P., Verlinde, H.L.: Elliptic genera of symmetric products, second quantized strings. Commun. Math. Phys. 185, 197 (1997) [arXiv:hep-th/9608096] · Zbl 0872.32006
[164]	Nekrasov, N., Schwarz, A.S.: Instantons on noncommutative R**4, (2,0) superconformal six dimensional theory. Commun. Math. Phys. 198, 689 (1998) [arXiv:hep-th/9802068] · Zbl 0923.58062
[165]	Seiberg, N., Witten, E.: String theory, noncommutative geometry. JHEP 9909, 032 (1999) [arXiv: hep-th/9908142] · Zbl 0957.81085
[166]	Myers, R.C.: Dielectric-branes. JHEP 9912, 022 (1999) [arXiv:hep-th/9910053] · Zbl 0958.81091
[167]	Dabholkar, A., Gaiotto, D., Nampuri, S.: Comments on the spectrum of CHL dyons. arXiv:hep-th/ 0702150
[168]	Sen, A.: Walls of marginal stability, dyon spectrum in N = 4 supersymmetric string theories. arXiv:hep-th/0702141
[169]	Sen, A.: Rare decay modes of quarter BPS dyons” arXiv:0707.1563 [hep-th]
[170]	Mukherjee, A., Mukhi, S., Nigam, R.: Dyon death eaters. arXiv:0707.3035 [hep-th] · Zbl 1168.83338
[171]	Cvetic, M., Youm, D.: Dyonic BPS saturated black holes of heterotic string on a six torus. Phys. Rev. D 53, 584 (1996) [arXiv:hep-th/9507090]
[172]	Duff, M.J., Liu, J.T., Rahmfeld, J.: Four-dimensional string–string–string triality. Nucl. Phys. B 459, 125 (1996) [arXiv:hep-th/9508094] · Zbl 0925.81167
[173]	Cardoso, G.L., de Wit, B., Kappeli, J., Mohaupt, T.: Black hole partition functions, duality. arXiv: hep-th/0601108
[174]	Sen, A.: Black holes, elementary strings, holomorphic anomaly. arXiv:hep-th/0502126
[175]	Dabholkar, A., Denef, F., Moore, G.W., Pioline, B.: Exact, asymptotic degeneracies of small black holes. arXiv:hep-th/0502157
[176]	Dabholkar, A., Denef, F., Moore, G.W., Pioline, B.: Precision counting of small black holes. JHEP 0510, 096 (2005) [arXiv:hep-th/0507014]
[177]	Balasubramanian, V., Kraus, P., Shigemori, M.: Massless black holes, black rings as effective geometries of the D1–D5 system. Class. Quant. Grav. 22, 4803 (2005) [arXiv:hep-th/0508110] · Zbl 1078.83018
[178]	Kallosh, R.: New attractors. JHEP 0512, 022 (2005) [arXiv:hep-th/0510024]
[179]	Tripathy, P.K., Trivedi, S.P.: Non-supersymmetric attractors in string theory. JHEP 0603, 022 (2006) [arXiv:hep-th/0511117] · Zbl 1226.81234
[180]	Prester, P.: Lovelock type gravity, small black holes in heterotic string theory. JHEP 0602, 039 (2006) [arXiv:hep-th/0511306]
[181]	Mohaupt, T.: Strings, higher curvature corrections, black holes. arXiv:hep-th/0512048 · Zbl 1158.83004
[182]	Goldstein, K., Jena, R.P., Mandal, G., Trivedi, S.P.: A C-function for non-supersymmetric attractors. JHEP 0602, 053 (2006) [arXiv:hep-th/0512138]
[183]	Alishahiha, M., Ebrahim, H.: Non-supersymmetric attractors, entropy function. JHEP 0603, 003 (2006) [arXiv:hep-th/0601016] · Zbl 1226.83055
[184]	Sinha, A., Suryanarayana, N.V.: Extremal single-charge small black holes: Entropy function analysis. Class. Quant. Grav. 23, 3305 (2006) [arXiv:hep-th/0601183] · Zbl 1101.83322
[185]	Kallosh, R., Sivanandam, N., Soroush, M.: The non-BPS black hole attractor equation. JHEP 0603, 060 (2006) [arXiv:hep-th/0602005] · Zbl 1226.83032
[186]	Chandrasekhar, B., Parvizi, S., Tavanfar, A., Yavartanoo, H.: Non-supersymmetric attractors in R**2 gravities. JHEP 0608, 004 (2006) [arXiv:hep-th/0602022]
[187]	Parvizi, S., Tavanfar, A.: Partition function of non-supersymmetric black holes in the supergravity limit. arXiv:hep-th/0602292 · Zbl 1159.83339
[188]	Kallosh, R.: From BPS to non-BPS black holes canonically. arXiv:hep-th/0603003
[189]	Exirifard, G.: The alpha’ stretched horizon in heterotic string. JHEP 0610, 070 (2006) [arXiv:hep-th/0604021]
[190]	Chandrasekhar, B.: Born–Infeld corrections to the entropy function of heterotic black holes. arXiv:hep-th/0604028
[191]	Ghodsi, A.: R**4 corrections to D1D5p black hole entropy from entropy function formalism. Phys. Rev. D 74, 124026 (2006) [arXiv:hep-th/0604106]
[192]	Alishahiha, M., Ebrahim, H.: New attractor, entropy function, black hole partition function. JHEP 0611, 017 (2006) [arXiv:hep-th/0605279] · Zbl 1226.83055
[193]	Kar, S., Majumdar, S.: Noncommutative D(3)-brane, black holes, attractor mechanism. Phys. Rev. D 74, 066003 (2006) [arXiv:hep-th/0606026]
[194]	Sarkar, T., Sengupta, G., Nath Tiwari, B.: On the thermodynamic geometry of BTZ black holes. JHEP 0611, 015 (2006) [arXiv:hep-th/0606084]
[195]	Cai, R.G., Pang, D.W.: Entropy function for 4-charge extremal black holes in type IIA superstring theory. Phys. Rev. D 74, 064031 (2006) [arXiv:hep-th/0606098]
[196]	Sinha, A., Suryanarayana, N.V.: Two-charge small black hole entropy: string-loops, multi-strings. JHEP 0610, 034 (2006) [arXiv:hep-th/0606218]
[197]	Kallosh, R., Sivanandam, N., Soroush, M.: Exact attractive non-BPS STU black holes. Phys. Rev. D 74, 065008 (2006) [arXiv:hep-th/0606263] · Zbl 1226.83032
[198]	Kaura, P., Misra, A.: On the existence of non-supersymmetric black hole attractors for two-parameter Calabi–Yau’s, attractor equations. arXiv:hep-th/0607132 · Zbl 1116.83017
[199]	Cardoso, G.L., Grass, V., Lust, D., Perz, J.: Extremal non-BPS black holes, entropy extremization. JHEP 0609, 078 (2006) [arXiv:hep-th/0607202]
[200]	Pioline, B.: Lectures on on black holes, topological strings, quantum attractors. Class. Quant. Grav. 23, S981 (2006) [arXiv:hep-th/0607227] · Zbl 1155.83305
[201]	Bellucci, S., Ferrara, S., Marrani, A., Yeranyan, A.: Mirror Fermat Calabi–Yau threefolds, Landau-Ginzburg black hole attractors. arXiv:hep-th/0608091 · Zbl 1179.83049
[202]	Arfaei, H., Fareghbal, R.: Double-horizon limit, decoupling of the dynamics at the horizon. arXiv:hep-th/0608222 · Zbl 1190.83090
[203]	Astefanesei, D., Goldstein, K., Mahapatra, S.: Moduli, (un)attractor black hole thermodynamics. arXiv:hep-th/0611140 · Zbl 1152.83370
[204]	Chandrasekhar, B., Yavartanoo, H., Yun, S.: Non-supersymmetric attractors in BI black holes. arXiv:hep-th/0611240
[205]	Andrianopoli, L., D’Auria, R., Ferrara, S., Trigiante, M.: Extremal black holes in supergravity. arXiv:hep-th/0611345 · Zbl 1274.83063
[206]	Mosaffa, A.E., Randjbar-Daemi, S., Sheikh-Jabbari, M.M.: Non-Abelian magnetized blackholes, unstable attractors. arXiv:hep-th/0612181 · Zbl 1151.83024
[207]	Cardoso, G.L., de Wit, B., Mahapatra, S.: Black hole entropy functions and attractor equations. arXiv:hep-th/0612225 · Zbl 1245.81152
[208]	Miller, C.M., Schalm, K., Weinberg, E.J.: Nonextremal black holes are BPS. arXiv:hep-th/0612308
[209]	Chen, C.M., Gal’tsov, D.V., Orlov, D.G.: Extremal black holes in D = 4 Gauss–Bonnet gravity. arXiv:hep-th/0701004
[210]	D’Auria, R., Ferrara, S., Trigiante, M.: Critical points of the black-hole potential for homogeneous special geometries. arXiv:hep-th/0701090
[211]	Cai, R.G., Pang, D.W.: Entropy function for non-extremal black holes in string theory. arXiv:hep-th/0701158
[212]	Bena, I., Warner, N.P.: Black holes, black rings, their microstates. arXiv:hep-th/0701216 · Zbl 1155.83301
[213]	Goldstein, K., Jena, R.P.: One entropy function to rule them all. arXiv:hep-th/0701221 · Zbl 1245.83031
[214]	Bellucci, S., Ferrara, S., Marrani, A.: Attractor horizon geometries of extremal black holes. arXiv:hep-th/0702019 · Zbl 1100.83001
[215]	Cai, R.G., Pang, D.W.: On entropy function for supersymmetric black rings. JHEP 0704, 027 (2007) [arXiv:hep-th/0702040]
[216]	Ceresole, A., Dall’Agata, G.: Flow equations for non-BPS extremal black holes. JHEP 0703, 110 (2007) [arXiv:hep-th/0702088]
[217]	Raeymaekers, J.: Open string attractors. JHEP 0704, 075 (2007) [arXiv:hep-th/0702142]
[218]	Hyun, S., Kim, W., Oh, J.J., Son, E.J.: Entropy function, universal entropy of two-dimensional extremal black holes. JHEP 0704, 057 (2007) [arXiv:hep-th/0702170]
[219]	Mohaupt, T.: Supersymmetric black holes in string theory. arXiv:hep-th/0703035 · Zbl 1117.83070
[220]	Mohaupt, T.: Duality, black hole partition functions. arXiv:hep-th/0703036 · Zbl 1226.83028
[221]	Mohaupt, T.: Special geometry, black holes, Euclidean supersymmetry. arXiv:hep-th/0703037 · Zbl 1208.83002
[222]	Andrianopoli, L., D’Auria, R., Ferrara, S., Trigiante, M.: Black-hole attractors in N = 1 supergravity. arXiv:hep-th/0703178
[223]	Saraikin, K., Vafa, C.: Non-supersymmetric black holes, topological strings. arXiv:hep-th/0703214 · Zbl 1140.83382
[224]	D’Errico, L., Muck, W., Pettorino, R.: Stretched horizon, entropy of superstars. arXiv:hep-th/0703223
[225]	Garousi, M.R., Ghodsi, A.: On attractor mechanism, entropy function for non-extremal black holes/branes. arXiv:hep-th/0703260
[226]	Suryanarayana, N.V., Wapler, M.C.: Charges from Attractors. arXiv:0704.0955 [hep-th] · Zbl 1206.83107
[227]	Cai, R.G., Cao, L.M.: On the entropy function, the attractor mechanism for spherically symmetric extremal black holes. arXiv:0704.1239 [hep-th]
[228]	Dias, O.J.C., Silva, P.J.: Attractors, the quantum statistical relation for extreme (BPS or not) black holes. arXiv:0704.1405 [hep-th]
[229]	de Wit, B.: BPS black holes. arXiv:0704.1452 [hep-th] · Zbl 1006.83027
[230]	Hanaki, K., Ohashi, K., Tachikawa, Y.: Comments on charges, near-horizon data of black rings. arXiv:0704.1819 [hep-th] · Zbl 1246.83204
[231]	Guica, M., Strominger, A.: Cargese lectures on string theory with eight supercharges. arXiv:0704.3295 [hep-th]
[232]	Castro, A., Davis, J.L., Kraus, P., Larsen, F.: Precision entropy of spinning black holes. arXiv:0705.1847 [hep-th]
[233]	Garousi, M.R., Ghodsi, A.: Entropy function for non-extremal D1D5, D2D6NS5-branes. arXiv:0705.2149 [hep-th]
[234]	Cho, J.H., Nam, S.: Non-supersymmetric attractor with the cosmological constant. JHEP 0707, 011 (2007) [arXiv:0705.2892 [hep-th]]
[235]	Ferrara, S., Marrani, A.: N = 8 non-BPS attractors, fixed scalars, magic supergravities. arXiv:0705.3866 [hep-th] · Zbl 1225.83080
[236]	Nampuri, S., Tripathy, P.K., Trivedi, S.P.: On the stability of non-supersymmetric attractors in string theory. arXiv:0705.4554 [hep-th]
[237]	Cvitan, M., Prester, P.D., Pallua, S., Smolic, I.: Extremal black holes in D = 5: SUSY vs. Gauss–Bonnet corrections. arXiv:0706.1167 [hep-th] · Zbl 1245.83027
[238]	Ferrara, S., Marrani, A.: On the moduli space of non-BPS attractors for N = 2 symmetric manifolds. arXiv:0706.1667 [hep-th] · Zbl 1248.83138
[239]	Astefanesei, D., Yavartanoo, H.: Stationary black holes, attractor mechanism. arXiv:0706.1847 [hep-th] · Zbl 1273.83087
[240]	Yun, S.: Non-supersymmetric unattractors in Born–Infeld black holes. arXiv:0706.2046 [hep-th]
[241]	Misra, A., Shukla, P.: Area codes, large volume (non-)perturbative alpha’- Instanton–corrected non-supersymmetric (A)dS minimum, the inverse problem, fake superpotentials for multiple-singular-loci-two-parameter Calabi–Yau’s. arXiv:0707.0105 [hep-th]
[242]	Ceresole, A., Ferrara, S., Marrani, A.: 4d/5d correspondence for the black hole potential, its critical points. arXiv:0707.0964 [hep-th] · Zbl 1148.83010
[243]	Anber, M.M., Kastor, D.: The attractor mechanism in Gauss–Bonnet gravity. arXiv:0707.1464 [hep-th]
[244]	Bellucci, S., Marrani, A., Orazi, E., Shcherbakov, A.: Attractors with vanishing central charge. arXiv:0707.2730 [hep-th] · Zbl 1246.83097
[245]	Hotta, K., Kubota, T.: Exact solutions, the attractor mechanism in non-BPS black holes. arXiv:0707.4554 [hep-th]
[246]	Arfaei, H., Fareghbal, R.: Double horizon limit, AdS geometry, entropy function. arXiv:0708.0240 [hep-th] · Zbl 1190.83090
[247]	Kawai, T., Yamada, Y., Yang, S.K.: Elliptic genera, N = 2 superconformal field theory. Nucl. Phys. B 414, 191 (1994) [arXiv:hep-th/9306096] · Zbl 0980.58500
[248]	Witten E. (1982). Constraints on supersymmetry breaking. Nucl. Phys. B 202: 253 · Zbl 0499.53056
[249]	Lerche W., Vafa C. and Warner N.P. (1989). Chiral rings in N = 2 superconformal theories. Nucl. Phys. B 324: 427
[250]	Dixon L.J., Kaplunovsky V. and Louis J. (1991). Moduli dependence of string loop corrections to gauge coupling constants. Nucl. Phys. B 355: 649
[251]	Harvey, J.A., Moore, G.W.: Algebras, BPS States, Strings. Nucl. Phys. B 463, 315 (1996) [arXiv:hep-th/9510182] · Zbl 0912.53056
[252]	Kawai, T.: N = 2 heterotic string threshold correction, K3 surface and generalized Kac–Moody superalgebra. Phys. Lett. B 372, 59 (1996) [arXiv:hep-th/9512046] · Zbl 1055.81606
[253]	Neumann, C.D.D.: The elliptic genus of Calabi–Yau 3-folds, 4-folds, product formulae, generalized Kac-Moody algebras. J. Geom. Phys. 29, 5 (1999) [arXiv:hep-th/9607029] · Zbl 0944.32024
[254]	Igusa J. (1962). On siegel modular varieties of genus two. Amer. J. Math. 84: 175200 · Zbl 0133.33301
[255]	Igusa J. (1962). On siegel modular varieties of genus two (ii). Amer. J. Math. 86: 392412
[256]	Borcherds R. (1995). Automorphic forms on O(s+2,2;R), infinite products. Invent. Math. 120: 161 · Zbl 0932.11028
[257]	Gritsenko, V.A., Nikulin, V.V.: Siegel automorphic form corrections of some Lorentzian Kac–Moody Lie algebras. Amer. J. Math. 119, 181 (1997) [arXiv:alg-geom/9504006] · Zbl 0914.11020
[258]	Ibukiyama T. (1991). On siegel modular varieties of level 3. Int. J. Math 2(1): 1735 · Zbl 0721.11018
[259]	Hayashida, S., Ibukiyama, T.: Siegel modular forms of half integral weights, a lifting conjecture. Journal of Kyoto Univ 45 (2005) · Zbl 1122.11028
[260]	Aoki H. and Ibukiyama T. (2005). Simple graed rings of siegel modular forms, differential operators, borcherds products. Int. J. Math. 16: 249279 · Zbl 1068.11030
[261]	Eichler M. and Zagier D. (1985). The Theory of Jacobi Forms. Birkhauser, Basel · Zbl 0554.10018
[262]	Skoruppa N.-P. (1992). Math. Comput. 58: 381
[263]	Manickam M., Ramakrishnan B. and Vasudevan T.C. (1993). Manuscripta. Math. 81: 161 · Zbl 0820.11029
[264]	Vafa, C., Nucl. Phys. B 463, 435 (1996) [arXiv:hep-th/9512078]