Abstract
We explain how to exploit systematically the structure of nilpotent orbits to obtain a solvable system of equations describing extremal solutions of (super-)gravity theories, i.e. systems that can be solved in a linear way. We present the procedure in the case of the STU model, where we show that all extremal solutions with a flat three-dimensional base are fully described with the help of three different nilpotent orbits: the BPS, the almost-BPS and the composite non-BPS. The latter describes a new class of solutions for which the orientation of half of the constituent branes have been inverted with respect to the BPS one, such that all the centres are intrinsically non-BPS, and interact with each others. We finally recover explicitly the ensemble of the almost-BPS solutions in our formalism and present an explicit two-centre solution of the new class.
Similar content being viewed by others
References
Strominger A., Vafa C.: Microscopic origin of the Bekenstein-Hawking entropy. Phys. Lett. B 379, 99 (1996)
Emparan R., Horowitz G.T.: Microstates of a neutral black hole in M theory. Phys. Rev. Lett. 97, 141601 (2006)
Dabholkar A., Sen A., Trivedi S.P.: Black hole microstates and attractor without supersymmetry. J. High Energy Phys. 0701, 096 (2007)
Astefanesei D., Goldstein K., Mahapatra S.: Moduli and (un)attractor black hole thermodynamics. Gen. Relativ. Gravit. 40, 2069 (2008)
Denef F.: Supergravity flows and D-brane stability. J. High Energy Phys. 0008, 050 (2000)
Ceresole A., Dall’Agata G.: Flow equations for non-BPS extremal black holes. JHEP 0703, 110 (2007)
Lopes Cardoso G., Ceresole A., Dall’Agata G. et al.: First-order flow equations for extremal black holes in very special geometry. J. High Energy Phys. 0710, 063 (2007)
Ceresole A., Dall’Agata G., Ferrara S., Yeranyan A.: First order flows for \({\mathcal{N} = 2}\) extremal black holes and duality invariants. Nucl. Phys. B 824, 239 (2010)
Goldstein K., Katmadas S.: Almost BPS black holes. J. High Energy Phys. 0905, 058 (2009)
Bena I., Dall’Agata G., Giusto S., Ruef C., Warner N.P.: Non-BPS black rings and black holes in Taub-NUT. J. High Energy Phys. 0906, 015 (2009)
Bena I., Giusto S., Ruef C., Warner N.P.: Multi-center non-BPS black holes: the solution. J. High Energy Phys. 0911, 032 (2009)
Bena I., Giusto S., Ruef C., Warner N.P.: Supergravity solutions from floating branes. J. High Energy Phys. 1003, 047 (2010)
Giusto, S., Dall’Agata, G., Ruef, C.: U-duality and non-BPS solutions. JHEP 1102, 074 (2011)
Bena, I., Giusto, S., Ruef, C.: A black ring with two angular momenta in Taub-NUT. JHEP 1106, 140 (2011)
Goldstein K., Iizuka N., Jena R.P., Trivedi S.P.: Non-supersymmetric attractors. Phys. Rev. D 72, 124021 (2005)
Hotta K., Kubota T.: Exact solutions and the attractor mechanism in non-BPS black holes. Prog. Theor. Phys. 118, 969 (2007)
Gimon E.G., Larsen F., Simon J.: Black holes in supergravity: the non-BPS branch. J. High Energy Phys. 0801, 040 (2008)
Gimon E.G., Larsen F., Simon J.: Constituent model of extremal non-BPS black holes. J. High Energy Phys. 0907, 052 (2009)
Gaiotto D., Li W.W., Padi M.: Non-supersymmetric attractor flow in symmetric spaces. J. High Energy Phys. 0712, 093 (2007)
Andrianopoli L., D’Auria R., Orazi E., Trigiante M.: First order description of black holes in moduli space. J. High Energy Phys. 0711, 032 (2007)
Andrianopoli L., D’Auria R., Orazi E., Trigiante M.: First order description of D = 4 static black holes and the Hamilton–Jacobi equation. Nucl. Phys. B 833, 1 (2010)
Bossard G., Michel Y., Pioline B.: Extremal black holes, nilpotent orbits and the true fake superpotential. J. High Energy Phys. 1001, 038 (2010)
Ceresole A., Dall’Agata G., Ferrara S., Yeranyan A.: Universality of the superpotential for d = 4 extremal black holes. Nucl. Phys. B 832, 358 (2010)
Perz J., Smyth P., Van Riet T., Vercnocke B.: First-order flow equations for extremal and non-extremal black holes. J. High Energy Phys. 0903, 150 (2009)
Kim S.S., Hornlund J.L., Palmkvist J., Virmani A.: Extremal solutions of the S 3 model and nilpotent orbits of G 2(2). J. High Energy Phys. 1008, 072 (2010)
Galli P., Goldstein K., Katmadas S., Perz J.: First-order flows and stabilisation equations for non-BPS extremal black holes. J. High Energy Phys. 1106, 070 (2011)
Galli, P., Ortin, T., Perz, J., Shahbazi, C.S.: Non-extremal black holes of \({\mathcal{N} = 2, d = 4}\) supergravity. JHEP 1107, 041 (2011)
Breitenlohner P., Maison D., Gibbons G.W.: Four-dimensional black holes from Kaluza–Klein theories. Commun. Math. Phys. 120, 295 (1988)
Hornlund, J.L.: On the symmetry orbits of black holes in non-linear sigma models (2011)
Bossard G., Nicolai H., Stelle K.S.: Universal BPS structure of stationary supergravity solutions. J. High Energy Phys. 0907, 003 (2009)
Bossard G., Nicolai H.: Multi-black holes from nilpotent Lie algebra orbits. Gen. Relativ. Gravit. 42, 509–537 (2010)
Bossard, G.: 1/8 BPS black hole composites (2010)
Bates, B., Denef, F.: Exact solutions for supersymmetric stationary black hole composites (2003)
Collingwood, D., McGovern, W.: Nilpotent orbits in semisimple Lie algebras. Van Nostrand Reinhold Mathematics Series, New York (1993)
Sekiguchi J.: Remarks on real nilpotent orbits of a symmetric pair. J. Math. Soc. Japan 39, 127–138 (1987)
Đoković D. Ž., Lemire N., Sekiguchi J.: The closure ordering of adjoint nilpotent orbits in \({\mathfrak{so}(p, q)}\) . Tohoku Math. J. 53, 395 (2001)
Denef, F., Moore, G.W.: Split states, entropy enigmas, holes and halos (2007)
Behrndt K., Kallosh R., Rahmfeld J. et al.: STU black holes and string triality. Phys. Rev. D 54, 6293 (1996)
Bena I., Warner N.P.: Bubbling supertubes and foaming black holes. Phys. Rev. D 74, 066001 (2006)
Berglund P., Gimon E.G., Levi T.S.: Supergravity microstates for BPS black holes and black rings. J. High Energy Phys. 0606, 007 (2006)
Saxena A., Potvin G., Giusto S., Peet A.W.: Smooth geometries with four charges in four dimensions. J. High Energy Phys. 0604, 010 (2006)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bossard, G., Ruef, C. Interacting non-BPS black holes. Gen Relativ Gravit 44, 21–66 (2012). https://doi.org/10.1007/s10714-011-1256-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10714-011-1256-9