Abstract
Using algebraic tools inspired by the study of nilpotent orbits in simple Lie algebras, we obtain a large class of solutions describing interacting non-BPS black holes in \( \mathcal{N} = 8 \) supergravity, which depend on 44 harmonic functions. For this purpose, we consider a truncation \( {E_{{{6}({6})}}}/S{p_{\text{c}}}\left( {8,\mathbb{R}} \right) \subset {E_{{{8}({8})}}}/{\text{Spin}}_{\text{c}}^{ * }\left( {16} \right) \) of the non-linear sigma model describing stationary solutions of the theory, which permits a reduction of algebraic computations to the multiplication of 27 by 27 matrices. The lift to \( \mathcal{N} = 8 \) supergravity is then carried out without loss of information by using a pertinent representation of the moduli parametrizing E7(7)/SUc (8) in terms of complex valued Hermitian matrices over the split octonions, which generalise the projective coordinates of exceptional special K¨ahler manifolds. We extract the electromagnetic charges, mass and angular momenta of the solutions, and exhibit the duality invariance of the black holes distance separations. We discuss in particular a new type of interaction which appears when interacting non-BPS black holes are not aligned. Finally we will explain the possible generalisations toward the description of the most general stationary black hole solutions of \( \mathcal{N} = 8 \) supergravity.
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References
A. Strominger and C. Vafa, Microscopic origin of the Bekenstein-Hawking entropy, Phys. Lett. B 379 (1996) 99 [hep-th/9601029] [INSPIRE].
R. Emparan and G.T. Horowitz, Microstates of a Neutral Black Hole in M-theory, Phys. Rev. Lett. 97 (2006) 141601 [hep-th/0607023] [INSPIRE].
A. Dabholkar, A. Sen and S.P. Trivedi, Black hole microstates and attractor without supersymmetry, JHEP 01 (2007) 096 [hep-th/0611143] [INSPIRE].
D. Astefanesei, K. Goldstein and S. Mahapatra, Moduli and (un)attractor black hole thermodynamics, Gen. Rel. Grav. 40 (2008) 2069 [hep-th/0611140] [INSPIRE].
F. Denef, Supergravity flows and D-brane stability, JHEP 08 (2000) 050 [hep-th/0005049] [INSPIRE].
F. Denef and G.W. Moore, Split states, entropy enigmas, holes and halos, JHEP 11 (2011) 129 [hep-th/0702146] [INSPIRE].
J. Manschot, B. Pioline and A. Sen, A Fixed point formula for the index of multi-centered N = 2 black holes, JHEP 05 (2011) 057 [arXiv:1103.1887] [INSPIRE].
A. Sen, Equivalence of Three Wall Crossing Formulae, arXiv:1112.2515 [INSPIRE].
A. Ceresole and G. Dall’Agata, Flow Equations for Non-BPS Extremal Black Holes, JHEP 03 (2007) 110 [hep-th/0702088] [INSPIRE].
G. Lopes Cardoso, A. Ceresole, G. Dall’Agata, J.M. Oberreuter and J. Perz, First-order flow equations for extremal black holes in very special geometry, JHEP 10 (2007) 063 [arXiv:0706.3373] [INSPIRE].
A. Ceresole, G. Dall’Agata, S. Ferrara and A. Yeranyan, First order flows for N = 2 extremal black holes and duality invariants, Nucl. Phys. B 824 (2010) 239 [arXiv:0908.1110] [INSPIRE].
K. Goldstein and S. Katmadas, Almost BPS black holes, JHEP 05 (2009) 058 [arXiv:0812.4183] [INSPIRE].
I. Bena, G. Dall’Agata, S. Giusto, C. Ruef and N.P. Warner, Non-BPS Black Rings and Black Holes in Taub-NUT, JHEP 06 (2009) 015 [arXiv:0902.4526] [INSPIRE].
I. Bena, S. Giusto, C. Ruef and N.P. Warner, Multi-Center non-BPS Black Holes: the Solution, JHEP 11 (2009) 032 [arXiv:0908.2121] [INSPIRE].
I. Bena, S. Giusto, C. Ruef and N.P. Warner, Supergravity Solutions from Floating Branes, JHEP 03 (2010) 047 [arXiv:0910.1860] [INSPIRE].
G. Dall’Agata, S. Giusto and C. Ruef, U-duality and non-BPS solutions, JHEP 02 (2011) 074 [arXiv:1012.4803] [INSPIRE].
I. Bena, S. Giusto and C. Ruef, A Black Ring with two Angular Momenta in Taub-NUT, JHEP 06 (2011) 140 [arXiv:1104.0016] [INSPIRE].
K. Goldstein, N. Iizuka, R.P. Jena and S.P. Trivedi, Non-supersymmetric attractors, Phys. Rev. D 72 (2005) 124021 [hep-th/0507096] [INSPIRE].
K. Hotta and T. Kubota, Exact Solutions and the Attractor Mechanism in Non-BPS Black Holes, Prog. Theor. Phys. 118 (2007) 969 [arXiv:0707.4554] [INSPIRE].
E.G. Gimon, F. Larsen and J. Simon, Black holes in Supergravity: The Non-BPS branch, JHEP 01 (2008) 040 [arXiv:0710.4967] [INSPIRE].
E.G. Gimon, F. Larsen and J. Simon, Constituent Model of Extremal non-BPS Black Holes, JHEP 07 (2009) 052 [arXiv:0903.0719] [INSPIRE].
D. Gaiotto, W. Li and M. Padi, Non-Supersymmetric Attractor Flow in Symmetric Spaces, JHEP 12 (2007) 093 [arXiv:0710.1638] [INSPIRE].
L. Andrianopoli, R. D’Auria, E. Orazi and M. Trigiante, First order description of black holes in moduli space, JHEP 11 (2007) 032 [arXiv:0706.0712] [INSPIRE].
S. Bellucci, S. Ferrara, A. Marrani and A. Yeranyan, stu Black Holes Unveiled, Entropy 10 (2008) 507 [arXiv:0807.3503] [INSPIRE].
L. Andrianopoli, R. D’Auria, E. Orazi and M. Trigiante, First Order Description of D = 4 static Black Holes and the Hamilton-Jacobi equation, Nucl. Phys. B 833 (2010) 1 [arXiv:0905.3938] [INSPIRE].
G. Bossard, Y. Michel and B. Pioline, Extremal black holes, nilpotent orbits and the true fake superpotential, JHEP 01 (2010) 038 [arXiv:0908.1742] [INSPIRE].
A. Ceresole, G. Dall’Agata, S. Ferrara and A. Yeranyan, Universality of the superpotential for D = 4 extremal black holes, Nucl. Phys. B 832 (2010) 358 [arXiv:0910.2697] [INSPIRE].
S. Ferrara, A. Marrani and E. Orazi, Maurer-Cartan Equations and Black Hole Superpotentials in N = 8 Supergravity, Phys. Rev. D 81 (2010) 085013 [arXiv:0911.0135] [INSPIRE].
J. Perz, P. Smyth, T. Van Riet and B. Vercnocke, First-order flow equations for extremal and non-extremal black holes, JHEP 03 (2009) 150 [arXiv:0810.1528] [INSPIRE].
S.-S. Kim, J. Lindman Hornlund, J. Palmkvist and A. Virmani, Extremal Solutions of the S3 Model and Nilpotent Orbits of G2(2) , JHEP 08 (2010) 072 [arXiv:1004.5242] [INSPIRE].
P. Galli, K. Goldstein, S. Katmadas and J. Perz, First-order flows and stabilisation equations for non-BPS extremal black holes, JHEP 06 (2011) 070 [arXiv:1012.4020] [INSPIRE].
D. Astefanesei, K. Goldstein, R.P. Jena, A. Sen and S.P. Trivedi, Rotating attractors, JHEP 10 (2006) 058 [hep-th/0606244] [INSPIRE].
P. Galli, T. Ortín, J. Perz and C.S. Shahbazi, Non-extremal black holes of N = 2, D = 4 supergravity, JHEP 07 (2011) 041 [arXiv:1105.3311] [INSPIRE].
G. Bossard and C. Ruef, Interacting non-BPS black holes, Gen. Rel. Grav. 44 (2012) 21 [arXiv:1106.5806] [INSPIRE].
P. Breitenlohner, D. Maison and G.W. Gibbons, Four-Dimensional Black Holes from Kaluza-Klein Theories, Commun. Math. Phys. 120 (1988) 295 [INSPIRE].
J.L. Hornlund, On the symmetry orbits of black holes in non-linear σ-models, JHEP 08 (2011) 090 [arXiv:1104.4949] [INSPIRE].
G. Bossard, H. Nicolai and K. Stelle, Universal BPS structure of stationary supergravity solutions, JHEP 07 (2009) 003 [arXiv:0902.4438] [INSPIRE].
G. Bossard and H. Nicolai, Multi-black holes from nilpotent Lie algebra orbits, Gen. Rel. Grav. 42 (2010) 509 [arXiv:0906.1987] [INSPIRE].
G. Bossard, 1/8 BPS Black Hole Composites, arXiv:1001.3157 [INSPIRE].
D. Rasheed, The Rotating dyonic black holes of Kaluza-Klein theory, Nucl. Phys. B 454 (1995) 379 [hep-th/9505038] [INSPIRE].
M. Günaydin, G. Sierra and P. Townsend, Exceptional Supergravity Theories and the MAGIC Square, Phys. Lett. B 133 (1983) 72 [INSPIRE].
M. Günaydin, G. Sierra and P. Townsend, The Geometry of N = 2 Maxwell-Einstein Supergravity and Jordan Algebras, Nucl. Phys. B 242 (1984) 244 [INSPIRE].
E. Cremmer and B. Julia, The SO(8) Supergravity, Nucl. Phys. B 159 (1979) 141 [INSPIRE].
M. Günaydin, K. Koepsell and H. Nicolai, Conformal and quasiconformal realizations of exceptional Lie groups, Commun. Math. Phys. 221 (2001) 57 [hep-th/0008063] [INSPIRE].
M. Günaydin and O. Pavlyk, Quasiconformal realizations of E6(6) , E7(7) , E8(8) and SO(n + 3, m + 3), N ≥ 4 supergravity and spherical vectors, [INSPIRE].
T.A. Springer and F.D. Veldkamp, Octonions, Jordan algebras, and exceptional groups, Springer Monographs in Mathematics (2000).
S. Ferrara, E.G. Gimon and R. Kallosh, Magic supergravities, N = 8 and black hole composites, Phys. Rev. D 74 (2006) 125018 [hep-th/0606211] [INSPIRE].
D. Collingwood and W. McGovern, Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold Mathematics Series, New York (1993).
D. Ž. Đoković, The closure diagrams for nilpotent orbits of real form of E6 , J. Lie Theory 11 (2001) 381.
F. Larsen, Rotating Kaluza-Klein black holes, Nucl. Phys. B 575 (2000) 211 [hep-th/9909102] [INSPIRE].
S. Ferrara and R. Kallosh, On N = 8 attractors, Phys. Rev. D 73 (2006) 125005 [hep-th/0603247] [INSPIRE].
S. Ferrara and A. Marrani, On the Moduli Space of non-BPS Attractors for N = 2 Symmetric Manifolds, Phys. Lett. B 652 (2007) 111 [arXiv:0706.1667] [INSPIRE].
B. de Wit and H. Nicolai, N = 8 Supergravity, Nucl. Phys. B 208 (1982) 323 [INSPIRE].
S. Ferrara and M. Günaydin, Orbits of exceptional groups, duality and BPS states in string theory, Int. J. Mod. Phys. A 13 (1998) 2075 [hep-th/9708025] [INSPIRE].
B. Bates and F. Denef, Exact solutions for supersymmetric stationary black hole composites, JHEP 11 (2011) 127 [hep-th/0304094] [INSPIRE].
S. Ferrara and J.M. Maldacena, Branes, central charges and U duality invariant BPS conditions, Class. Quant. Grav. 15 (1998) 749 [hep-th/9706097] [INSPIRE].
L. Andrianopoli, R. D’Auria, S. Ferrara, A. Marrani and M. Trigiante, Two-Centered Magical Charge Orbits, JHEP 04 (2011) 041 [arXiv:1101.3496] [INSPIRE].
S. Ferrara, A. Marrani and A. Yeranyan, On Invariant Structures of Black Hole Charges, JHEP 02 (2012) 071 [arXiv:1110.4004] [INSPIRE].
G. Bossard, The Extremal black holes of N = 4 supergravity from so(8, 2 + n) nilpotent orbits, Gen. Rel. Grav. 42 (2010) 539 [arXiv:0906.1988] [INSPIRE].
D. Ž. Đoković, Classification of nilpotent elements in simple exceptional real Lie algebras of inner type and description of their centralizers, J. Algebra 112 (1988) 503.
D. Ž. Đoković, The closure diagram for nilpotent orbits of the split real form of E8, Cent. Eur. J. Math. 4 (2003) 573
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Bossard, G. Octonionic black holes. J. High Energ. Phys. 2012, 113 (2012). https://doi.org/10.1007/JHEP05(2012)113
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DOI: https://doi.org/10.1007/JHEP05(2012)113