New Black Hole Solutions in = 2 and = 8 Gauged Supergravity †
Abstract
1. Introduction
Gauged Supergravities and Black Holes
2. The Model
2.1. Single-Dilaton Truncation
2.1.1. Vacua
2.1.2. Redefinitions
3. Results
3.1. Hairy Black Hole Solutions
3.1.1. Family 1 – Electric Solutions
- Boundary conditions, mass and thermodynamics for the electric solutions.
3.1.2. Family 2 – Magnetic Solutions
- Boundary conditions, mass and thermodynamics for the magnetic solutions.
3.1.3. Case or
4. Discussion
4.1. Duality Relation between the Two Families of Solutions
4.2. Supersymmetric Solutions
4.2.1. Family 1
4.2.2. Family 2
4.3. Old and New Solutions
4.3.1. Duff–Liu
4.3.2. Cacciatori–Klemm
4.4. New BPS Black Holes with Finite Area
4.4.1. Family 1: BPS Electric Black Holes
- k = 0:
- in the flat case the location of the horizon is very simple (see the above (91)) and it follows that and , so we conclude that only exists;
- k = −1:
- in the hyperbolic case, always exists while the solution exists provided ;
- k = +1:
- for spherical black hole only exists, provided .
4.4.2. Family 2: BPS Magnetic Black Holes
4.5. Hamilton–Jacobi Formulation
Flow equations
4.6. Truncations
4.6.1. Uncharged Case
4.6.2. Charged Case
- case.
Funding
Acknowledgments
Conflicts of Interest
Appendix A. Supersymmetric Black Hole Solutions
1 | The boundary theory is located at the UV critical point and is it possible to employ a UV/IR connection, which relates gravity degrees of freedom at large (small) radius with the corresponding counterparts in the dual field theory at high (low) energy regime. |
2 | Scalar self-interactions could be relevant for dynamic and thermodynamic stability of the configuration: naively, we can imagine the condition for the existence of hairy solutions as if the self-interaction of the scalar and the gravitational interaction were to combine such that the near-horizon hair did not collapse into the black hole, while the far-region hair did not escape to infinity. |
3 | |
4 | The STU model [62,63,64] is a supergravity coupled to vector multiplets and characterized, in a suitable symplectic frame, by the prepotential , together with symmetric scalar manifold of the form spanned by the three complex scalars (); this model is in turn a consistent truncation of the maximal theory in four dimensions with gauge group [65,66,67,68]. |
5 | The conditions come from the consistency of the axion field equations after the truncation. |
6 | The explicit form of the solution makes the uncharged limit well-defined, giving the hairy black hole configurations of [47]; the result should be not taken for granted, since in the standard literature the uncharged limit gives either Schwarzschild or Schwarzschild–AdS spacetime. |
7 | |
8 | |
9 | This particular class of solutions is noteworthy as it can be embedded in gauged supergravity [74]. |
10 | |
11 | The construction of these models was carried out by exploiting the freedom in the initial choice of the symplectic frame of the maximal theory, that is, gauging a group in different symplectic frames by rotating the original one [44,80] making use of a suitable symplectic matrix, thus obtaining a one-parameter class of inequivalent theories (-deformed models). |
12 | , label the 28 vectors of the maximal theory, while m, n are the symplectic indices of the 56 electric and magnetic charges. |
13 | We explicitly have , , [55]. |
14 | For the gamma-matrices we can use the conventions of App. A of [83]. |
References
- Maldacena, J.M. The Large N limit of superconformal field theories and supergravity. Int. J. Theor. Phys. 1999, 38, 1113–1133. [Google Scholar] [CrossRef]
- Susskind, L. The World as a hologram. J. Math. Phys. 1995, 36, 6377–6396. [Google Scholar] [CrossRef]
- Peet, A.W.; Polchinski, J. UV/IR relations in AdS dynamics. Phys. Rev. D 1999, 59, 065011. [Google Scholar] [CrossRef]
- Susskind, L.; Witten, E. The Holographic bound in anti-de Sitter space. arXiv 1998, arXiv:hep-th/9805114. [Google Scholar]
- de Boer, J.; Verlinde, E.P.; Verlinde, H.L. On the holographic renormalization group. JHEP 2000, 08, 003. [Google Scholar] [CrossRef]
- de Boer, J. The Holographic renormalization group. Fortsch. Phys. 2001, 49, 339–358. [Google Scholar] [CrossRef]
- Hawking, S.; Page, D.N. Thermodynamics of Black Holes in anti-De Sitter Space. Commun. Math. Phys. 1983, 87, 577. [Google Scholar] [CrossRef]
- Chamblin, A.; Emparan, R.; Johnson, C.V.; Myers, R.C. Charged AdS black holes and catastrophic holography. Phys. Rev. D 1999, 60, 064018. [Google Scholar] [CrossRef]
- Chamblin, A.; Emparan, R.; Johnson, C.V.; Myers, R.C. Holography, thermodynamics and fluctuations of charged AdS black holes. Phys. Rev. D 1999, 60, 104026. [Google Scholar] [CrossRef]
- Cvetic, M.; Gubser, S.S. Phases of R charged black holes, spinning branes and strongly coupled gauge theories. JHEP 1999, 04, 024. [Google Scholar] [CrossRef]
- Caldarelli, M.M.; Cognola, G.; Klemm, D. Thermodynamics of Kerr-Newman-AdS black holes and conformal field theories. Class. Quant. Grav. 2000, 17, 399–420. [Google Scholar] [CrossRef]
- Anabalón, A.; Astefanesei, D.; Gallerati, A.; Trigiante, M. Instability of supersymmetric black holes via quantum phase transitions. arXiv 2021, arXiv:hep-th/2105.08771. [Google Scholar]
- Strominger, A.; Vafa, C. Microscopic origin of the Bekenstein-Hawking entropy. Phys. Lett. 1996, B379, 99–104. [Google Scholar] [CrossRef]
- Cacciatori, S.L.; Klemm, D. Supersymmetric AdS(4) black holes and attractors. JHEP 2010, 01, 085. [Google Scholar] [CrossRef]
- Hristov, K.; Looyestijn, H.; Vandoren, S. BPS black holes in N = 2 D = 4 gauged supergravities. JHEP 2010, 08, 103. [Google Scholar] [CrossRef]
- Hristov, K.; Vandoren, S. Static supersymmetric black holes in AdS4 with spherical symmetry. JHEP 2011, 04, 047. [Google Scholar] [CrossRef][Green Version]
- Hristov, K.; Toldo, C.; Vandoren, S. On BPS bounds in D = 4 N = 2 gauged supergravity. JHEP 2011, 12, 014. [Google Scholar] [CrossRef]
- Toldo, C.; Vandoren, S. Static nonextremal AdS4 black hole solutions. JHEP 2012, 09, 048. [Google Scholar] [CrossRef]
- Chow, D.D.K.; Compère, G. Dyonic AdS black holes in maximal gauged supergravity. Phys. Rev. D 2014, 89, 065003. [Google Scholar] [CrossRef]
- Gnecchi, A.; Hristov, K.; Klemm, D.; Toldo, C.; Vaughan, O. Rotating black holes in 4d gauged supergravity. JHEP 2014, 01, 127. [Google Scholar] [CrossRef]
- Gnecchi, A.; Halmagyi, N. Supersymmetric black holes in AdS4 from very special geometry. JHEP 2014, 04, 173. [Google Scholar] [CrossRef]
- Lü, H.; Pang, Y.; Pope, C. An ω deformation of gauged STU supergravity. JHEP 2014, 04, 175. [Google Scholar] [CrossRef][Green Version]
- Faedo, F.; Klemm, D.; Nozawa, M. Hairy black holes in N=2 gauged supergravity. JHEP 2015, 11, 045. [Google Scholar] [CrossRef]
- Klemm, D.; Marrani, A.; Petri, N.; Santoli, C. BPS black holes in a non-homogeneous deformation of the stu model of N = 2, D = 4 gauged supergravity. JHEP 2015, 09, 205. [Google Scholar] [CrossRef]
- Chimento, S.; Klemm, D.; Petri, N. Supersymmetric black holes and attractors in gauged supergravity with hypermultiplets. JHEP 2015, 06, 150. [Google Scholar] [CrossRef][Green Version]
- Hristov, K.; Katmadas, S.; Toldo, C. Rotating attractors and BPS black holes in AdS4. JHEP 2019, 01, 199. [Google Scholar] [CrossRef]
- Daniele, N.; Faedo, F.; Klemm, D.; Ramírez, P.F. Rotating black holes in the FI-gauged N = 2, D = 4 ℂPn model. JHEP 2019, 03, 151. [Google Scholar] [CrossRef]
- Hertog, T. Towards a Novel no-hair Theorem for Black Holes. Phys. Rev. 2006, D74, 084008. [Google Scholar] [CrossRef]
- Hertog, T.; Maeda, K. Stability and thermodynamics of AdS black holes with scalar hair. Phys. Rev. 2005, D71, 024001. [Google Scholar] [CrossRef]
- Anabalon, A.; Astefanesei, D.; Martinez, C. Mass of asymptotically anti–de Sitter hairy spacetimes. Phys. Rev. 2015, D91, 041501. [Google Scholar] [CrossRef]
- Anabalon, A.; Astefanesei, D.; Choque, D.; Martinez, C. Trace Anomaly and Counterterms in Designer Gravity. JHEP 2016, 03, 117. [Google Scholar] [CrossRef]
- Lu, H.; Pope, C.N.; Wen, Q. Thermodynamics of AdS Black Holes in Einstein-Scalar Gravity. JHEP 2015, 03, 165. [Google Scholar] [CrossRef]
- Hertog, T.; Hollands, S. Stability in designer gravity. Class. Quant. Grav. 2005, 22, 5323–5342. [Google Scholar] [CrossRef]
- Amsel, A.J.; Hertog, T.; Hollands, S.; Marolf, D. A Tale of two superpotentials: Stability and instability in designer gravity. Phys. Rev. 2007, D75, 084008. [Google Scholar] [CrossRef]
- Faulkner, T.; Horowitz, G.T.; Roberts, M.M. New stability results for Einstein scalar gravity. Class. Quant. Grav. 2010, 27, 205007. [Google Scholar] [CrossRef][Green Version]
- Dall’Agata, G.; Inverso, G.; Trigiante, M. Evidence for a family of SO(8) gauged supergravity theories. Phys. Rev. Lett. 2012, 109, 201301. [Google Scholar] [CrossRef]
- Borghese, A.; Dibitetto, G.; Guarino, A.; Roest, D.; Varela, O. The SU(3)-invariant sector of new maximal supergravity. JHEP 2013, 1303, 082. [Google Scholar] [CrossRef]
- Tarrio, J.; Varela, O. Electric/magnetic duality and RG flows in AdS4/CFT3. JHEP 2014, 1401, 071. [Google Scholar] [CrossRef]
- Gallerati, A.; Samtleben, H.; Trigiante, M. The > 2 supersymmetric AdS vacua in maximal supergravity. JHEP 2014, 12, 174. [Google Scholar] [CrossRef][Green Version]
- Borghese, A.; Guarino, A.; Roest, D. All G2 invariant critical points of maximal supergravity. JHEP 2012, 1212, 108. [Google Scholar] [CrossRef]
- Kodama, H.; Nozawa, M. Classification and stability of vacua in maximal gauged supergravity. JHEP 2013, 01, 045. [Google Scholar] [CrossRef]
- Guarino, A. On new maximal supergravity and its BPS domain-walls. JHEP 2014, 02, 026. [Google Scholar] [CrossRef][Green Version]
- Guarino, A. CSOc superpotentials. Nucl. Phys. 2015, B900, 501–516. [Google Scholar] [CrossRef]
- De Wit, B.; Nicolai, H. N = 8 Supergravity. Nucl. Phys. 1982, B208, 323. [Google Scholar] [CrossRef]
- Dall’Agata, G.; Inverso, G. On the Vacua of N = 8 Gauged Supergravity in 4 Dimensions. Nucl. Phys. 2012, B859, 70–95. [Google Scholar] [CrossRef]
- Dall’Agata, G.; Inverso, G. De Sitter vacua in N = 8 supergravity and slow-roll conditions. Phys. Lett. B 2013, 718, 1132–1136. [Google Scholar] [CrossRef][Green Version]
- Anabalón, A.; Astefanesei, D.; Gallerati, A.; Trigiante, M. Hairy Black Holes and Duality in an Extended Supergravity Model. JHEP 2018, 04, 058. [Google Scholar] [CrossRef]
- Henneaux, M.; Martinez, C.; Troncoso, R.; Zanelli, J. Black holes and asymptotics of 2+1 gravity coupled to a scalar field. Phys. Rev. 2002, D65, 104007. [Google Scholar] [CrossRef]
- Martinez, C.; Troncoso, R.; Zanelli, J. Exact black hole solution with a minimally coupled scalar field. Phys. Rev. 2004, D70, 084035. [Google Scholar] [CrossRef]
- Hertog, T.; Maeda, K. Black holes with scalar hair and asymptotics in N = 8 supergravity. JHEP 2004, 07, 051. [Google Scholar] [CrossRef]
- Anabalon, A. Exact Black Holes and Universality in the Backreaction of non-linear Sigma Models with a potential in (A)dS4. JHEP 2012, 06, 127. [Google Scholar] [CrossRef]
- Feng, X.H.; Lu, H.; Wen, Q. Scalar Hairy Black Holes in General Dimensions. Phys. Rev. 2014, D89, 044014. [Google Scholar] [CrossRef]
- Dall’Agata, G.; Gnecchi, A. Flow equations and attractors for black holes in N = 2 U(1) gauged supergravity. JHEP 2011, 03, 037. [Google Scholar] [CrossRef]
- Gnecchi, A.; Toldo, C. On the non-BPS first order flow in N = 2 U(1)-gauged Supergravity. JHEP 2013, 03, 088. [Google Scholar] [CrossRef]
- Anabalon, A.; Astefanesei, D.; Gallerati, A.; Trigiante, M. New non-extremal and BPS hairy black holes in gauged N = 2 and N = 8 supergravity. JHEP 2021, 04, 047. [Google Scholar] [CrossRef]
- Anabalon, A.; Astefanesei, D. Black holes in ω-defomed gauged N = 8 supergravity. Phys. Lett. B 2014, 732, 137–141. [Google Scholar] [CrossRef]
- Anabalon, A.; Astefanesei, D.; Choque, D.; Gallerati, A.; Trigiante, M. Exact holographic RG flows in extended SUGRA. JHEP 2021, 04, 053. [Google Scholar] [CrossRef]
- Witten, E. Multitrace operators, boundary conditions, and AdS/CFT correspondence. arXiv 2001, arXiv:hep-th/0112258. [Google Scholar]
- Strominger, A. Special Geometry. Commun. Math. Phys. 1990, 133, 163–180. [Google Scholar] [CrossRef]
- Bagger, J.; Witten, E. Matter Couplings in N = 2 Supergravity. Nucl. Phys. B 1983, 222, 1–10. [Google Scholar] [CrossRef]
- Lauria, E.; Van Proeyen, A. = 2 Supergravity in D = 4, 5, 6 Dimensions; Springer: Berlin/Heidelberg, Germany, 2020; Volume 966. [Google Scholar] [CrossRef]
- Duff, M.J.; Liu, J.T.; Rahmfeld, J. Four-dimensional string-string-string triality. Nucl. Phys. B 1996, 459, 125–159. [Google Scholar] [CrossRef]
- Behrndt, K.; Kallosh, R.; Rahmfeld, J.; Shmakova, M.; Wong, W.K. STU black holes and string triality. Phys. Rev. D 1996, 54, 6293–6301. [Google Scholar] [CrossRef]
- Behrndt, K.; Lust, D.; Sabra, W.A. Stationary solutions of N = 2 supergravity. Nucl. Phys. B 1998, 510, 264–288. [Google Scholar] [CrossRef]
- Duff, M.; Liu, J.T. Anti-de Sitter black holes in gauged N = 8 supergravity. Nucl. Phys. B 1999, 554, 237–253. [Google Scholar] [CrossRef]
- Andrianopoli, L.; D’Auria, R.; Gallerati, A.; Trigiante, M. Extremal Limits of Rotating Black Holes. JHEP 2013, 1305, 071. [Google Scholar] [CrossRef][Green Version]
- Andrianopoli, L.; Gallerati, A.; Trigiante, M. On Extremal Limits and Duality Orbits of Stationary Black Holes. JHEP 2014, 01, 053. [Google Scholar] [CrossRef][Green Version]
- Andrianopoli, L.; D’Auria, R.; Gallerati, A.; Trigiante, M. On D = 4 Stationary Black Holes. J. Phys. Conf. Ser. 2013, 474, 012002. [Google Scholar] [CrossRef]
- Gallerati, A.; Trigiante, M. Introductory Lectures on Extended Supergravities and Gaugings. Springer Proc. Phys. 2016, 176, 41–109. [Google Scholar] [CrossRef]
- Trigiante, M. Gauged Supergravities. Phys. Rept. 2017, 680, 1–175. [Google Scholar] [CrossRef]
- Henneaux, M.; Martinez, C.; Troncoso, R.; Zanelli, J. Asymptotic behavior and Hamiltonian analysis of anti-de Sitter gravity coupled to scalar fields. Annals Phys. 2007, 322, 824–848. [Google Scholar] [CrossRef]
- Myers, R.C. Stress tensors and Casimir energies in the AdS / CFT correspondence. Phys. Rev. D 1999, 60, 046002. [Google Scholar] [CrossRef]
- Anabalon, A.; Andrade, T.; Astefanesei, D.; Mann, R. Universal Formula for the Holographic Speed of Sound. Phys. Lett. B 2018, 781, 547–552. [Google Scholar] [CrossRef]
- Luciani, J.F. Coupling of O(2) Supergravity with Several Vector Multiplets. Nucl. Phys. B 1978, 132, 325–332. [Google Scholar] [CrossRef]
- Gallerati, A. Constructing black hole solutions in supergravity theories. Int. J. Mod. Phys. 2020, A34, 1930017. [Google Scholar] [CrossRef]
- Ferrara, S.; Gibbons, G.W.; Kallosh, R. Black holes and critical points in moduli space. Nucl. Phys. 1997, B500, 75–93. [Google Scholar] [CrossRef]
- Andrianopoli, L.; D’Auria, R.; Ferrara, S.; Trigiante, M. Extremal black holes in supergravity. Lect. Notes Phys. 2008, 737, 661–727. [Google Scholar]
- Dall’Agata, G.; Inverso, G.; Marrani, A. Symplectic Deformations of Gauged Maximal Supergravity. JHEP 2014, 07, 133. [Google Scholar] [CrossRef]
- Inverso, G. Electric-magnetic deformations of D = 4 gauged supergravities. JHEP 2016, 03, 138. [Google Scholar] [CrossRef][Green Version]
- De Wit, B.; Nicolai, H. N=8 Supergravity with Local SO(8) x SU(8) Invariance. Phys. Lett. B 1982, 108, 285. [Google Scholar] [CrossRef]
- Cvetic, M.; Gubser, S.; Lu, H.; Pope, C. Symmetric potentials of gauged supergravities in diverse dimensions and Coulomb branch of gauge theories. Phys. Rev. D 2000, 62, 086003. [Google Scholar] [CrossRef]
- Cvetic, M.; Lu, H.; Pope, C.; Sadrzadeh, A. Consistency of Kaluza-Klein sphere reductions of symmetric potentials. Phys. Rev. D 2000, 62, 046005. [Google Scholar] [CrossRef]
- Andrianopoli, L.; Cerchiai, B.L.; D’Auria, R.; Gallerati, A.; Noris, R.; Trigiante, M.; Zanelli, J. N-extended D = 4 supergravity, unconventional SUSY and graphene. JHEP 2020, 01, 084. [Google Scholar] [CrossRef]
- Gaillard, M.K.; Zumino, B. Duality Rotations for Interacting Fields. Nucl. Phys. 1981, B193, 221. [Google Scholar] [CrossRef]
- Romans, L. Supersymmetric, cold and lukewarm black holes in cosmological Einstein-Maxwell theory. Nucl. Phys. B 1992, 383, 395–415. [Google Scholar] [CrossRef]
1 Anti-de Sitter 3 Anti-de Sitter | Figure 1a Figure 1b | |||
1 Anti-de Sitter | Figure 1c | |||
1 de Sitter 1 Anti-de Sitter | Figure 1d Figure 1c | |||
1 Anti-de Sitter | Figure 1c | |||
1 de Sitter 1 Anti-de Sitter | Figure 1d Figure 1c | |||
1 Anti-de Sitter | Figure 1c | |||
1 Anti-de Sitter 3 Anti-de Sitter | Figure 1e Figure 1f |
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Gallerati, A.
New Black Hole Solutions in
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New Black Hole Solutions in
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2021. "New Black Hole Solutions in
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