The Hesse potential, the c-map and black hole solutions. (English) Zbl 1397.83078
Summary: We present a new formulation of the local \(c\)-map, which makes use of a symplectically covariant real formulation of special Kähler geometry. We obtain an explicit and simple expression for the resulting quaternionic, or, in the case of reduction over time, para-quaternionic Kähler metric in terms of the Hesse potential, which is similar to the expressions for the metrics obtained from the rigid \(r\)- and \(c\)-map, and from the local \(r\)-map.As an application we use the temporal version of the \(c\)-map to derive the black hole attractor equations from geometric properties of the scalar manifold, without imposing supersymmetry or spherical symmetry. We observe that for general (non-symmetric) \(c\)-map spaces static BPS solutions are related to a canonical family of totally isotropic, totally geodesic submanifolds. Static non-BPS solutions can be obtained by applying a field rotation matrix which is subject to a non-trivial compatibility condition. We show that for a class of prepotentials, which includes the very special (’cubic’) prepotentials as a subclass, axion-free solutions always admit a non-trivial field rotation matrix.
MSC:
| 83C57 | Black holes |
| 83E30 | String and superstring theories in gravitational theory |
| 83E50 | Supergravity |
References:
| [1] | Wit, B.; Proeyen, A., Potentials and symmetries of general gauged \(N\) = 2 supergravity: Yang-Mills models, Nucl. Phys., B 245, 89, (1984) · doi:10.1016/0550-3213(84)90425-5 |
| [2] | Seiberg, N.; Witten, E., Electric-magnetic duality, monopole condensation and confinement in \(N\) = 2 supersymmetric Yang-Mills theory, Nucl. Phys., B 426, 19, (1994) · Zbl 0996.81510 · doi:10.1016/0550-3213(94)90124-4 |
| [3] | Seiberg, N.; Witten, E., Monopoles, duality and chiral symmetry breaking in \(N\) = 2 supersymmetric QCD, Nucl. Phys., B 431, 484, (1994) · Zbl 1020.81911 · doi:10.1016/0550-3213(94)90214-3 |
| [4] | Kachru, S.; Vafa, C., Exact results for \(N\) = 2 compactifications of heterotic strings, Nucl. Phys., B 450, 69, (1995) · Zbl 0957.14509 · doi:10.1016/0550-3213(95)00307-E |
| [5] | Ferrara, S.; Harvey, JA; Strominger, A.; Vafa, C., Second quantized mirror symmetry, Phys. Lett., B 361, 59, (1995) · Zbl 0899.32012 · doi:10.1016/0370-2693(95)01074-Z |
| [6] | Kachru, S.; Klemm, A.; Lerche, W.; Mayr, P.; Vafa, C., Nonperturbative results on the point particle limit of \(N\) = 2 heterotic string compactifications, Nucl. Phys., B 459, 537, (1996) · Zbl 1003.81524 · doi:10.1016/0550-3213(95)00574-9 |
| [7] | Ferrara, S.; Kallosh, R.; Strominger, A., \(N\) = 2 extremal black holes, Phys. Rev., D 52, 5412, (1995) |
| [8] | Strominger, A.; Vafa, C., Microscopic origin of the bekenstein-Hawking entropy, Phys. Lett., B 379, 99, (1996) · Zbl 1376.83026 · doi:10.1016/0370-2693(96)00345-0 |
| [9] | Maldacena, JM; Strominger, A.; Witten, E., Black hole entropy in \(M\)-theory, JHEP, 12, 002, (1997) · Zbl 0951.83034 · doi:10.1088/1126-6708/1997/12/002 |
| [10] | Lopes Cardoso, G.; Wit, B.; Mohaupt, T., Corrections to macroscopic supersymmetric black hole entropy, Phys. Lett., B 451, 309, (1999) · Zbl 1058.83516 · doi:10.1016/S0370-2693(99)00227-0 |
| [11] | Lopes Cardoso, G.; Wit, B.; Kappeli, J.; Mohaupt, T., Stationary BPS solutions in \(N\) = 2 supergravity with \(R\)\^{}{2} interactions, JHEP, 12, 019, (2000) · Zbl 0990.83567 · doi:10.1088/1126-6708/2000/12/019 |
| [12] | Ooguri, H.; Strominger, A.; Vafa, C., Black hole attractors and the topological string, Phys. Rev., D 70, 106007, (2004) |
| [13] | Lopes Cardoso, G.; Wit, B.; Kappeli, J.; Mohaupt, T., Asymptotic degeneracy of dyonic \(N\) = 4 string states and black hole entropy, JHEP, 12, 075, (2004) |
| [14] | Lopes Cardoso, G.; Wit, B.; Kappeli, J.; Mohaupt, T., Black hole partition functions and duality, JHEP, 03, 074, (2006) · Zbl 1226.83028 · doi:10.1088/1126-6708/2006/03/074 |
| [15] | Dixon, LJ; Kaplunovsky, V.; Louis, J., Moduli dependence of string loop corrections to gauge coupling constants, Nucl. Phys., B 355, 649, (1991) · doi:10.1016/0550-3213(91)90490-O |
| [16] | Lopes Cardoso, G.; Wit, B.; Mohaupt, T., Deviations from the area law for supersymmetric black holes, Fortsch. Phys., 48, 49, (2000) · Zbl 0952.83034 · doi:10.1002/(SICI)1521-3978(20001)48:1/3<49::AID-PROP49>3.0.CO;2-O |
| [17] | Lopes Cardoso, G.; Wit, B.; Mohaupt, T., Macroscopic entropy formulae and nonholomorphic corrections for supersymmetric black holes, Nucl. Phys., B 567, 87, (2000) · Zbl 0951.81039 · doi:10.1016/S0550-3213(99)00560-X |
| [18] | Cardoso, G.; Wit, B.; Mahapatra, S., Subleading and non-holomorphic corrections to \(N\) = 2 BPS black hole entropy, JHEP, 02, 006, (2009) · Zbl 1245.81152 · doi:10.1088/1126-6708/2009/02/006 |
| [19] | Freed, DS, Special Kähler manifolds, Commun. Math. Phys., 203, 31, (1999) · Zbl 0940.53040 · doi:10.1007/s002200050604 |
| [20] | Cardoso, G.; Wit, B.; Mahapatra, S., Black hole entropy functions and attractor equations, JHEP, 03, 085, (2007) · doi:10.1088/1126-6708/2007/03/085 |
| [21] | Cardoso, G.; Wit, B.; Mahapatra, S., BPS black holes, the Hesse potential and the topological string, JHEP, 06, 052, (2010) · Zbl 1290.81107 · doi:10.1007/JHEP06(2010)052 |
| [22] | V. Cortés, in Proceedings of the international conference ”PDEs, Submanifolds and Affine Differential Geometry”. Vol. 57, Warsaw Poland (2000), B. Opozda, U. Simon and M. Wiehe eds., Banach Center Publications (Polish Academy of Sciences, Institute of Mathematics), Warsaw Poland (2000), pg. 11-16. · Zbl 1029.53017 |
| [23] | Alekseevsky, D.; Cortés, V.; Devchand, C., Special complex manifolds, J. Geom. Phys., 42, 85, (2002) · Zbl 1004.53038 · doi:10.1016/S0393-0440(01)00078-X |
| [24] | Cortés, V.; Mohaupt, T., Special geometry of Euclidean supersymmetry III: the local \(r\)-map, instantons and black holes, JHEP, 07, 066, (2009) · doi:10.1088/1126-6708/2009/07/066 |
| [25] | Ferrara, S.; Macia, O., Real symplectic formulation of local special geometry, Phys. Lett., B 637, 102, (2006) · Zbl 1247.53090 · doi:10.1016/j.physletb.2006.04.010 |
| [26] | Mohaupt, T.; Vaughan, O., Developments in special geometry, J. Phys. Conf. Ser., 343, 012078, (2012) · doi:10.1088/1742-6596/343/1/012078 |
| [27] | Günaydin, M.; Sierra, G.; Townsend, P., The geometry of \(N\) = 2 Maxwell-Einstein supergravity and Jordan algebras, Nucl. Phys., B 242, 244, (1984) · doi:10.1016/0550-3213(84)90142-1 |
| [28] | Bagger, J.; Witten, E., Matter couplings in \(N\) = 2 supergravity, Nucl. Phys., B 222, 1, (1983) · doi:10.1016/0550-3213(83)90605-3 |
| [29] | Cecotti, S.; Ferrara, S.; Girardello, L., Geometry of type II superstrings and the moduli of superconformal field theories, Int. J. Mod. Phys., A 4, 2475, (1989) · Zbl 0681.58044 · doi:10.1142/S0217751X89000972 |
| [30] | Ferrara, S.; Sabharwal, S., Quaternionic manifolds for type II superstring vacua of Calabi-Yau spaces, Nucl. Phys., B 332, 317, (1990) · doi:10.1016/0550-3213(90)90097-W |
| [31] | Wit, B.; Proeyen, A., Special geometry, cubic polynomials and homogeneous quaternionic spaces, Commun. Math. Phys., 149, 307, (1992) · Zbl 0824.53043 · doi:10.1007/BF02097627 |
| [32] | Wit, B.; Vanderseypen, F.; Proeyen, A., Symmetry structure of special geometries, Nucl. Phys., B 400, 463, (1993) · Zbl 0941.83529 · doi:10.1016/0550-3213(93)90413-J |
| [33] | Günaydin, M.; Neitzke, A.; Pioline, B.; Waldron, A., BPS black holes, quantum attractor flows and automorphic forms, Phys. Rev., D 73, 084019, (2006) |
| [34] | Neitzke, A.; Pioline, B.; Vandoren, S., Twistors and black holes, JHEP, 04, 038, (2007) · doi:10.1088/1126-6708/2007/04/038 |
| [35] | Günaydin, M.; Neitzke, A.; Pioline, B.; Waldron, A., Quantum attractor flows, JHEP, 09, 056, (2007) · doi:10.1088/1126-6708/2007/09/056 |
| [36] | Gaiotto, D.; Li, W.; Padi, M., Non-supersymmetric attractor flow in symmetric spaces, JHEP, 12, 093, (2007) · Zbl 1246.81244 · doi:10.1088/1126-6708/2007/12/093 |
| [37] | Bossard, G.; Michel, Y.; Pioline, B., Extremal black holes, nilpotent orbits and the true fake superpotential, JHEP, 01, 038, (2010) · Zbl 1269.81108 · doi:10.1007/JHEP01(2010)038 |
| [38] | Roček, M.; Vafa, C.; Vandoren, S., Hypermultiplets and topological strings, JHEP, 02, 062, (2006) |
| [39] | M. Roček, C. Vafa and S. Vandoren, Quaternion-Kahler spaces, hyperkahler cones, and the c-map, math/0603048. |
| [40] | Robles-Llana, D.; Saueressig, F.; Vandoren, S., String loop corrected hypermultiplet moduli spaces, JHEP, 03, 081, (2006) · Zbl 1226.81211 · doi:10.1088/1126-6708/2006/03/081 |
| [41] | Robles-Llana, D.; Roček, M.; Saueressig, F.; Theis, U.; Vandoren, S., Nonperturbative corrections to 4\(D\) string theory effective actions from SL(2, \(\mathbb{Z}\)) duality and supersymmetry, Phys. Rev. Lett., 98, 211602, (2007) · Zbl 1228.81241 · doi:10.1103/PhysRevLett.98.211602 |
| [42] | Alexandrov, S.; Pioline, B.; Saueressig, F.; Vandoren, S., \(D\)-instantons and twistors, JHEP, 03, 044, (2009) · doi:10.1088/1126-6708/2009/03/044 |
| [43] | Gaiotto, D.; Moore, GW; Neitzke, A., Four-dimensional wall-crossing via three-dimensional field theory, Commun. Math. Phys., 299, 163, (2010) · Zbl 1225.81135 · doi:10.1007/s00220-010-1071-2 |
| [44] | D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin Systems and the WKB Approximation, arXiv:0907.3987 [INSPIRE]. · Zbl 1358.81150 |
| [45] | Cortes, V.; Mayer, C.; Mohaupt, T.; Saueressig, F., Special geometry of Euclidean supersymmetry. II. hypermultiplets and the \(c\)-map, JHEP, 06, 025, (2005) · doi:10.1088/1126-6708/2005/06/025 |
| [46] | V. Cortes, T. Mohaupt and H. Xu, Completeness in supergravity constructions, arXiv:1101.5103 [INSPIRE]. · Zbl 1247.83224 |
| [47] | Mohaupt, T.; Waite, K., Instantons, black holes and harmonic functions, JHEP, 10, 058, (2009) · doi:10.1088/1126-6708/2009/10/058 |
| [48] | Mohaupt, T.; Waite, K., Euclidean actions, instantons, solitons and supersymmetry, J. Phys., A 44, 175403, (2011) · Zbl 1214.81275 |
| [49] | Mohaupt, T.; Vaughan, O., Non-extremal black holes, harmonic functions and attractor equations, Class. Quant. Grav., 27, 235008, (2010) · Zbl 1207.83034 · doi:10.1088/0264-9381/27/23/235008 |
| [50] | Breitenlohner, P.; Maison, D.; Gibbons, GW, Four-dimensional black holes from Kaluza-Klein theories, Commun. Math. Phys., 120, 295, (1988) · Zbl 0661.53064 · doi:10.1007/BF01217967 |
| [51] | Cortes, V.; Mayer, C.; Mohaupt, T.; Saueressig, F., Special geometry of Euclidean supersymmetry. 1. vector multiplets, JHEP, 03, 028, (2004) · doi:10.1088/1126-6708/2004/03/028 |
| [52] | Behrndt, K.; etal., Classical and quantum \(N\) = 2 supersymmetric black holes, Nucl. Phys., B 488, 236, (1997) · Zbl 0925.83086 |
| [53] | Bergshoeff, E.; Kallosh, R.; Ortín, T., Stationary axion/Dilaton solutions and supersymmetry, Nucl. Phys., B 478, 156, (1996) · Zbl 0925.83105 · doi:10.1016/0550-3213(96)00408-7 |
| [54] | Behrndt, K.; Lüst, D.; Sabra, WA, Stationary solutions of \(N\) = 2 supergravity, Nucl. Phys., B 510, 264, (1998) · Zbl 0953.83049 |
| [55] | A. Ceresole, R. D’Auria, S. Ferrara and A. Van Proeyen, On electromagnetic duality in locally supersymmetric N = 2 Yang-Mills theory, hep-th/9412200 [INSPIRE]. |
| [56] | Wit, B., \(N\) = 2 electric-magnetic duality in a chiral background, Nucl. Phys. Proc. Suppl., 49, 191, (1996) · Zbl 0957.81693 · doi:10.1016/0920-5632(96)00335-0 |
| [57] | Alvarez-Gaume, L.; Freedman, DZ, Kahler geometry and the renormalization of supersymmetric \(σ\)-models, Phys. Rev., D 22, 846, (1980) |
| [58] | Mohaupt, T.; Cortés, V. (ed.), Special geometry, black holes and Euclidean supersymmetry, (2010), Zürich Switzerland · Zbl 1208.83002 |
| [59] | Mohaupt, T., Black hole entropy, special geometry and strings, Fortsch. Phys., 49, 3, (2001) · Zbl 0985.83001 · doi:10.1002/1521-3978(200102)49:1/3<3::AID-PROP3>3.0.CO;2-# |
| [60] | B. de Wit, Supergravity, hep-th/0212245 [INSPIRE]. |
| [61] | Cremmer, E.; etal., Vector multiplets coupled to \(N\) = 2 supergravity: superhiggs effect, flat potentials and geometric structure, Nucl. Phys., B 250, 385, (1985) · doi:10.1016/0550-3213(85)90488-2 |
| [62] | Denef, F., Supergravity flows and \(D\)-brane stability, JHEP, 08, 050, (2000) · Zbl 0990.83553 · doi:10.1088/1126-6708/2000/08/050 |
| [63] | Ceresole, A.; Dall’Agata, G., Flow equations for non-BPS extremal black holes, JHEP, 03, 110, (2007) · doi:10.1088/1126-6708/2007/03/110 |
| [64] | Lopes Cardoso, G.; Ceresole, A.; Dall’Agata, G.; Oberreuter, JM; Perz, J., First-order flow equations for extremal black holes in very special geometry, JHEP, 10, 063, (2007) · doi:10.1088/1126-6708/2007/10/063 |
| [65] | Perz, J.; Smyth, P.; Riet, T.; Vercnocke, B., First-order flow equations for extremal and non-extremal black holes, JHEP, 03, 150, (2009) · doi:10.1088/1126-6708/2009/03/150 |
| [66] | Bleeken, D., BPS dyons and Hesse flow, JHEP, 02, 067, (2012) · Zbl 1309.81152 · doi:10.1007/JHEP02(2012)067 |
| [67] | Bellorın, J.; Meessen, P.; Ortín, T., Supersymmetry, attractors and cosmic censorship, Nucl. Phys., B 762, 229, (2007) · Zbl 1116.83020 · doi:10.1016/j.nuclphysb.2006.11.004 |
| [68] | Bossard, G.; Nicolai, H.; Stelle, K., Gravitational multi-NUT solitons, komar masses and charges, Gen. Rel. Grav., 41, 1367, (2009) · Zbl 1177.83064 · doi:10.1007/s10714-008-0720-7 |
| [69] | Khuri, RR; Ortín, T., A nonsupersymmetric dyonic extreme Reissner-Nordstrom black hole, Phys. Lett., B 373, 56, (1996) · doi:10.1016/0370-2693(96)00139-6 |
| [70] | Ortín, T., Extremality versus supersymmetry in stringy black holes, Phys. Lett., B 422, 93, (1998) · doi:10.1016/S0370-2693(98)00040-9 |
| [71] | T. Ortín, Nonsupersymmetric (but) extreme black holes, scalar hair and other open problems, hep-th/9705095 [INSPIRE]. |
| [72] | Goldstein, K.; Iizuka, N.; Jena, RP; Trivedi, SP, Non-supersymmetric attractors, Phys. Rev., D 72, 124021, (2005) |
| [73] | Tripathy, PK; Trivedi, SP, Non-supersymmetric attractors in string theory, JHEP, 03, 022, (2006) · Zbl 1226.81234 · doi:10.1088/1126-6708/2006/03/022 |
| [74] | Kallosh, R., New attractors, JHEP, 12, 022, (2005) · doi:10.1088/1126-6708/2005/12/022 |
| [75] | Galli, P.; Ortín, T.; Perz, J.; Shahbazi, CS, Non-extremal black holes of \(N\) = 2, \(D\) = 4 supergravity, JHEP, 07, 041, (2011) · Zbl 1298.81284 · doi:10.1007/JHEP07(2011)041 |
| [76] | Meessen, P.; Ortín, T., Non-extremal black holes of \(N\) = 2, \(D\) = 5 supergravity, Phys. Lett., B 707, 178, (2012) · doi:10.1016/j.physletb.2011.12.006 |
| [77] | Lozano-Tellechea, E.; Ortín, T., The general, duality invariant family of nonbps black hole solutions of \(N\) = 4, \(D\) = 4 supergravity, Nucl. Phys., B 569, 435, (2000) · Zbl 0953.83064 · doi:10.1016/S0550-3213(99)00762-2 |
| [78] | Behrndt, K.; Kallosh, R.; Rahmfeld, J.; Shmakova, M.; Wong, WK, STU black holes and string triality, Phys. Rev., D 54, 6293, (1996) |
| [79] | Louis, J.; Sonnenschein, J.; Theisen, S.; Yankielowicz, S., Nonperturbative properties of heterotic string vacua compactified on \(K\)3 × \(T\)\^{}{2}, Nucl. Phys., B 480, 185, (1996) · Zbl 0925.81208 · doi:10.1016/S0550-3213(96)00429-4 |
| [80] | Morrison, DR; Vafa, C., Compactifications of \(F\)-theory on Calabi-Yau threefolds. 2, Nucl. Phys., B 476, 437, (1996) · Zbl 0925.14007 · doi:10.1016/0550-3213(96)00369-0 |
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