Abstract
We construct and study stationary, asymptotically flat multicenter solutions describing regular black holes with non-Abelian hair (colored magnetic-monopole and dyon fields) in two models of \( \mathcal{N}=2 \), d = 4 Super-Einstein-Yang-Mills theories: the quadratic model \( {\overline{\mathrm{\mathbb{C}}\mathrm{\mathbb{P}}}}^3 \) and the cubic model ST[2, 6], which can be embedded in 10-dimensional Heterotic Supergravity. These solutions are based on the multicenter dyon recently discovered by one of us, which solves the SU(2) Bogomol’nyi and dyon equations on \( {\mathbb{E}}^3 \). In contrast to the well-known Abelian multicenter solutions, the relative positions of the non-Abelian black-hole centers are unconstrained.
We study necessary conditions on the parameters of the solutions that ensure the regularity of the metric. In the case of the \( {\overline{\mathrm{\mathbb{C}}\mathrm{\mathbb{P}}}}^3 \) model we show that it is enough to require the positivity of the “masses” of the individual black holes, the finiteness of each of their entropies and their superadditivity. In the case of the ST[2, 6] model we have not been able to show that analogous conditions are sufficient, but we give an explicit example of a regular solution describing thousands of non-Abelian dyonic black holes in equilibrium at arbitrary relative positions.
We also construct non-Abelian solutions that interpolate smoothly between just two aDS2×S2 vacua with different radii (dumbbell solutions).
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References
S.D. Majumdar, A class of exact solutions of Einstein’s field equations, Phys. Rev. 72 (1947) 390 [INSPIRE].
A. Papaetrou, A Static solution of the equations of the gravitational field for an arbitrary charge distribution, Proc. Roy. Irish Acad. A 51 (1947) 191 [INSPIRE].
A. Einstein and J. Grommer, Allgemeine Relativitätstheorie und Bewegungsgesetz, S. B. Preuss. Akad. Wiss. 1 (1927) 2.
A. Einstein, L. Infeld and B. Hoffmann, The Gravitational equations and the problem of motion, Annals Math. 39 (1938) 65 [INSPIRE].
V.A. Fock, Sur le mouvement de masses finies d’apres la théorie de gravitation einsteinienne, J. Phys. U.S.S.R. 1 (1939) 81.
A. Einstein and L. Infeld, The Gravitational equations and the problem of motion II, Annals Math. 41 (1940) 455 [INSPIRE].
A. Einstein and L. Infeld, On the motion of particles in general relativity theory, Can. J. Math. 1 (1949) 209.
A. Papapetrou, Equations of motion in General Relativity, Proc. Phys. Soc. A 64 (1951) 57.
W. Israel and K.A. Khan, Collinear Particles and Bondi Dipoles in General Relativity, Nuovo Cim. 33 (1964) 331.
R.C. Myers, Higher Dimensional Black Holes in Compactified Space-times, Phys. Rev. D 35 (1987) 455 [INSPIRE].
P.R. Wallace, On the relativistic equations of motion in electromagnetic theory, Ph.D. Thesis, University of Toronto, Toronto Canada (1940).
L. Infeld and P.R. Wallace, The Equations of Motion in Electrodynamics, Phys. Rev. 57 (1940) 797 [INSPIRE].
P.R. Wallace, Relativistic equations of motion in electromagnetic theory, Am. J. Math. 63 (1941) 729.
D.R. Brill and R.W. Lindquist, Interaction energy in geometrostatics, Phys. Rev. 131 (1963) 471 [INSPIRE].
T. Ortín, Time-symmetric initial data sets in four-dimensional dilaton gravity, Phys. Rev. D 52 (1995) 3392 [hep-th/9501094] [INSPIRE].
M. Cvetič, G.W. Gibbons and C.N. Pope, Super-Geometrodynamics, JHEP 03 (2015) 029 [arXiv:1411.1084] [INSPIRE].
J. Polchinski, Dirichlet Branes and Ramond-Ramond charges, Phys. Rev. Lett. 75 (1995) 4724 [hep-th/9510017] [INSPIRE].
Z. Perjés, Solutions of the coupled Einstein Maxwell equations representing the fields of spinning sources, Phys. Rev. Lett. 27 (1971) 1668 [INSPIRE].
W. Israel and G.A. Wilson, A class of stationary electromagnetic vacuum fields, J. Math. Phys. 13 (1972) 865 [INSPIRE].
J. Bellorın, P. Meessen and T. Ortín, Supersymmetry, attractors and cosmic censorship, Nucl. Phys. B 762 (2007) 229 [hep-th/0606201] [INSPIRE].
J.B. Hartle and S.W. Hawking, Solutions of the Einstein-Maxwell equations with many black holes, Commun. Math. Phys. 26 (1972) 87 [INSPIRE].
K.P. Tod, More on supercovariantly constant spinors, Class. Quant. Grav. 12 (1995) 1801 [INSPIRE].
E. Bergshoeff, R. Kallosh and T. Ortín, Stationary axion/dilaton solutions and supersymmetry, Nucl. Phys. B 478 (1996) 156 [hep-th/9605059] [INSPIRE].
J. Bellorín and T. Ortín, All the supersymmetric configurations of N = 4, d = 4 supergravity, Nucl. Phys. B 726 (2005) 171 [hep-th/0506056] [INSPIRE].
K. Behrndt, D. Lüst and W.A. Sabra, Stationary solutions of N = 2 supergravity, Nucl. Phys. B 510 (1998) 264 [hep-th/9705169] [INSPIRE].
G. Lopes Cardoso, B. de Wit, J. Kappeli and T. Mohaupt, Stationary BPS solutions in N = 2 supergravity with R 2 -interactions, JHEP 12 (2000) 019 [hep-th/0009234] [INSPIRE].
P. Meessen and T. Ortín, The Supersymmetric configurations of N = 2, d = 4 supergravity coupled to vector supermultiplets, Nucl. Phys. B 749 (2006) 291 [hep-th/0603099] [INSPIRE].
F. Denef, Supergravity flows and D-brane stability, JHEP 08 (2000) 050 [hep-th/0005049] [INSPIRE].
B. Bates and F. Denef, Exact solutions for supersymmetric stationary black hole composites, JHEP 11 (2011) 127 [hep-th/0304094] [INSPIRE].
C.W. Misner, The Flatter regions of Newman, Unti and Tamburino’s generalized Schwarzschild space, J. Math. Phys. 4 (1963) 924 [INSPIRE].
P. Bueno, P. Meessen, T. Ortín and P.F. Ramírez, \( \mathcal{N}=2 \) Einstein-Yang-Mills’ static two-center solutions, JHEP 12 (2014) 093 [arXiv:1410.4160] [INSPIRE].
S.A. Cherkis and B. Durcan, The ’t Hooft-Polyakov monopole in the presence of an ’t Hooft operator, Phys. Lett. B 671 (2009) 123 [arXiv:0711.2318] [INSPIRE].
S.A. Cherkis and B. Durcan, Singular monopoles via the Nahm transform, JHEP 04 (2008) 070 [arXiv:0712.0850] [INSPIRE].
P. Meessen, T. Ortın and P.F. Ramírez, Multicenter non-Abelian black hole and string solutions of N = 1, d = 5 supergravity, in preparation.
M.S. Volkov and D.V. Galtsov, Non-Abelian Einstein Yang-Mills black holes, JETP Lett. 50 (1989) 346 [Pisma Zh. Eksp. Teor. Fiz. 50 (1989) 312] [INSPIRE].
P. Bizon, Colored black holes, Phys. Rev. Lett. 64 (1990) 2844 [INSPIRE].
M.S. Volkov and D.V. Gal’tsov, Gravitating non-Abelian solitons and black holes with Yang-Mills fields, Phys. Rept. 319 (1999) 1 [hep-th/9810070] [INSPIRE].
D.V. Gal’tsov, Gravitating lumps, in proceedings of the 16th International Conference on General Relativity and Gravitation (GR16), Durban, South Africa, 15–21 July 2001, N.T. Bishop and S.D. Maharaj eds., World Scientific, Singapore (2002) [hep-th/0112038] [INSPIRE].
P.A. Cano, T. Ortın and P.F. Ramírez, A gravitating Yang-Mills instanton, JHEP 07 (2017) 011 [arXiv:1704.00504] [INSPIRE].
T. Ortín and C. Santoli, Supersymmetric solutions of SU(2)-Fayet-Iliopoulos-gauged \( \mathcal{N}=2 \) , d = 4 supergravity, Nucl. Phys. B 916 (2017) 37 [arXiv:1609.08694] [INSPIRE].
P. Meessen and T. Ortín, Supersymmetric solutions to gauged N = 2 d = 4 SUGRA: the full timelike shebang, Nucl. Phys. B 863 (2012) 65 [arXiv:1204.0493] [INSPIRE].
L. Andrianopoli et al., N = 2 supergravity and N = 2 super Yang-Mills theory on general scalar manifolds: Symplectic covariance, gaugings and the momentum map, J. Geom. Phys. 23 (1997) 111 [hep-th/9605032] [INSPIRE].
D.Z. Freedman and A.Van Proeyen, Supergravity, Cambridge University Press (2012) [INSPIRE].
T. Ortín, Gravity and Strings, 2nd edition, Cambridge University Press (2015) [INSPIRE].
M. Hübscher, P. Meessen, T. Ortín and S. Vaulà, N = 2 Einstein-Yang-Mills’s BPS solutions, JHEP 09 (2008) 099 [arXiv:0806.1477] [INSPIRE].
M. Hübscher, P. Meessen, T. Ortín and S. Vaulà, Supersymmetric N = 2 Einstein-Yang-Mills monopoles and covariant attractors, Phys. Rev. D 78 (2008) 065031 [arXiv:0712.1530] [INSPIRE].
P. Meessen, Supersymmetric coloured/hairy black holes, Phys. Lett. B 665 (2008) 388 [arXiv:0803.0684] [INSPIRE].
P. Meessen and T. Ortín, \( \mathcal{N}=2 \) super-EYM coloured black holes from defective Lax matrices, JHEP 04 (2015) 100 [arXiv:1501.02078] [INSPIRE].
E.B. Bogomolny, Stability of Classical Solutions, Sov. J. Nucl. Phys. 24 (1976) 449 [Yad. Fiz. 24 (1976) 861]. [INSPIRE].
G. ’t Hooft, Magnetic Monopoles in Unified Gauge Theories, Nucl. Phys. B 79 (1974) 276 [INSPIRE].
A.M. Polyakov, Particle Spectrum in the Quantum Field Theory, JETP Lett. 20 (1974) 194 [INSPIRE].
M.K. Prasad and C.M. Sommerfield, An Exact Classical Solution for the ’t Hooft Monopole and the Julia-Zee Dyon, Phys. Rev. Lett. 35 (1975) 760 [INSPIRE].
T.T. Wu and C.-N. Yang, Some Solutions Of The Classical Isotopic Gauge Field Equations, in Properties Of Matter Under Unusual Conditions, H. Mark and S. Fernbach eds., Wiley-Interscience, New York U.S.A. (1969), pp. 349-345 and in Selected Papers (1945-1980) of Chen Ning Yang, World Scientific (2005), pp. 400-405 [INSPIRE].
A.P. Protogenov, Exact Classical Solutions of Yang-Mills Sourceless Equations, Phys. Lett. B 67 (1977) 62 [INSPIRE].
P.F. Ramírez, Non-Abelian bubbles in microstate geometries, JHEP 11 (2016) 152 [arXiv:1608.01330] [INSPIRE].
G. Etesi and T. Hausel, New Yang-Mills instantons on multicentered gravitational instantons, Commun. Math. Phys. 235 (2003) 275 [hep-th/0207196] [INSPIRE].
P.A. Cano, P. Meessen, T. Ortín and P.F. Ramírez, Non-Abelian black holes in string theory, arXiv:1704.01134 [INSPIRE].
P.A. Cano, P. Meessen, T. Ortín and P.F. Ramírez, Four-dimensional non-Abelian black holes in string theory, work in progress.
P. Bueno, P. Meessen, T. Ortín and P.F. Ramírez, Resolution of SU(2) monopole singularities by oxidation, Phys. Lett. B 746 (2015) 109 [arXiv:1503.01044] [INSPIRE].
P.B. Kronheimer, Monopoles and Taub-NUT spaces, MSc Thesis, Oxford University, Oxford U.K. (1995).
A.A. Belavin, A.M. Polyakov, A.S. Schwartz and Y.S. Tyupkin, Pseudoparticle Solutions of the Yang-Mills Equations, Phys. Lett. B 59 (1975) 85 [INSPIRE].
P. Meessen, T. Ortín and P.F. Ramírez, Non-Abelian, supersymmetric black holes and strings in 5 dimensions, JHEP 03 (2016) 112 [arXiv:1512.07131] [INSPIRE].
T. Ortín and P.F. Ramírez, A non-Abelian Black Ring, Phys. Lett. B 760 (2016) 475 [arXiv:1605.00005] [INSPIRE].
P. Galli, P. Meessen and T. Ortín, The Freudenthal gauge symmetry of the black holes of \( \mathcal{N}=2 \) , d = 4 supergravity, JHEP 05 (2013) 011 [arXiv:1211.7296] [INSPIRE].
P. Dominic, T. Mandal and P.K. Tripathy, Multiple Single-Centered Attractors, JHEP 12 (2014) 158 [arXiv:1406.7147] [INSPIRE].
T. Mandal and P.K. Tripathy, On the Uniqueness of Supersymmetric Attractors, Phys. Lett. B 749 (2015) 221 [arXiv:1506.06276] [INSPIRE].
T. Mohaupt and O. Vaughan, The Hesse potential, the c-map and black hole solutions, JHEP 07 (2012) 163 [arXiv:1112.2876] [INSPIRE].
P.A. Cano, T. Ortín and C. Santoli, Non-Abelian black string solutions of \( \mathcal{N}=\left(2,0\right) \) , d = 6 supergravity, JHEP 12 (2016) 112 [arXiv:1607.02595] [INSPIRE].
P. Meessen, T. Ortín, J. Perz and C.S. Shahbazi, H-FGK formalism for black-hole solutions of N = 2, d = 4 and d = 5 supergravity, Phys. Lett. B 709 (2012) 260 [arXiv:1112.3332] [INSPIRE].
S. Chimento, A. Ruipérez and T. Ortín, work in progress.
M. Trigiante, Dual gauged supergravities, hep-th/0701218 [INSPIRE].
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Meessen, P., Ortín, T. & Ramírez, P.F. Dyonic black holes at arbitrary locations. J. High Energ. Phys. 2017, 66 (2017). https://doi.org/10.1007/JHEP10(2017)066
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DOI: https://doi.org/10.1007/JHEP10(2017)066