Abstract
We consider the fist order, gradient-flow, description of the scalar fields coupled to spherically symmetric, asymptotically flat black holes in extended supergravities. Using the identification of the fake superpotential with Hamilton’s characteristic function we clarify some of its general properties, showing in particular (besides reviewing the issue of its duality invariance) that W has the properties of a Liapunov’s function, which implies that its extrema (associated with the horizon of extremal black holes) are asymptotically stable equilibrium points of the corresponding first order dynamical system (in the sense of Liapunov). Moreover, we show that the fake superpotential W has, along the entire radial flow, the same flat directions which exist at the attractor point. This allows to study properties of the ADM mass also for small black holes where in fact W has no critical points at finite distance in moduli space. In particular the W function for small non-BPS black holes can always be computed analytically, unlike for the large black-hole case.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Ceresole and G. Dall’Agata, Flow Equations for Non-BPS Extremal Black Holes, JHEP 03 (2007) 110 [hep-th/0702088] [SPIRES].
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] [SPIRES].
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] [SPIRES].
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] [SPIRES].
G. Bossard, Y. Michel and B. P ioline, Extremal black holes, nilpotent orbits and the true fake superpotential, JHEP 01 (2010) 038 [arXiv:0908.1742] [SPIRES].
A. Ceresole, G. Dall’Agata, S. Ferrara and A. Yeranyan, Universality of the superpotential for D = 4 extremal black holes, arXiv:0910.2697 [SPIRES].
D.Z. Freedman, C. Núñez, M. Schnabl and K. Skenderis, Fake Supergravity and Domain Wall Stability, Phys. Rev. D 69 (2004) 104027 [hep-th/0312055] [SPIRES].
S. Bellucci, S. Ferrara, M. Günaydin and A. Marrani, Charge orbits of symmetric special geometries and attractors, Int. J. Mod. Phys. A 21 (2006) 5043 [hep-th/0606209] [SPIRES].
S. Bellucci, S. Ferrara, R. Kallosh and A. Marrani, Extremal Black Hole and Flux Vacua Attractors, Lect. Notes Phys. 755 (2008) 115 [arXiv:0711.4547] [SPIRES].
V.I. Arnold, Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, Springer, (1997).
K. Meyer, G. Hall, Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, Springer-Verlag, New York U.S.A. (1992).
J. de Boer, E.P. Verlinde and H.L. Verlinde, On the holographic renormalization group, JHEP 08 (2000) 003 [hep-th/9912012] [SPIRES].
E.P. Verlinde and H.L. Verlinde, RG-flow, gravity and the cosmological constant, JHEP 05 (2000) 034 [hep-th/9912018] [SPIRES].
M. Fukuma, S. Matsuura and T. Sakai, Holographic renormalization group, Prog. Theor. Phys. 109 (2003) 489 [hep-th/0212314] [SPIRES].
K. Skenderis and P.K. Townsend, Hamilton-Jacobi method for Domain Walls and Cosmologies, Phys. Rev. D 74 (2006) 125008 [hep-th/0609056] [SPIRES].
P.K. Townsend, Hamilton-Jacobi Mechanics from Pseudo-Supersymmetry, Class. Quant. Grav. 25 (2008) 045017 [arXiv:0710.5178] [SPIRES].
K. Hotta, Holographic RG flow dual to attractor flow in extremal black holes, Phys. Rev. D 79 (2009) 104018 [arXiv:0902.3529] [SPIRES].
B. Janssen, P. Smyth, T. Van Riet and B. Vercnocke, A first-order formalism for timelike and spacelike brane solutions, JHEP 04 (2008) 007 [arXiv:0712.2808] [SPIRES].
W. Hahn, Stability of Motion, Springer-Verlag, Berlin Germany (1967).
N. Rouche and J. Mawhin, Ordinary Differential Equations. Stability and Periodic Solutions, Pitman, Boston/London/Melbourne, (1980).
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] [SPIRES].
M. Bianchi, S. Ferrara and R. Kallosh, Perturbative and Non-perturbative N = 8 Supergravity, Phys. Lett. B 690 (2010) 328 [arXiv:0910.3674] [SPIRES].
M. Bianchi, S. Ferrara and R. Kallosh, Observations on Arithmetic Invariants and U-duality Orbits in N = 8 Supergravity, JHEP 03 (2010) 081 [arXiv:0912.0057] [SPIRES].
B.L. Cerchiai, S. Ferrara, A. Marrani and B. Zumino, Duality, Entropy and ADM Mass in Supergravity, Phys. Rev. D 79 (2009) 125010 [arXiv:0902.3973] [SPIRES].
L. Andrianopoli, S. Ferrara, A. Marrani and M. Trigiante, Non-BPS Attractors in 5d and 6d Extended Supergravity, Nucl. Phys. B 795 (2008) 428 [arXiv:0709.3488] [SPIRES].
L. Andrianopoli, R. D’Auria, S. Ferrara and M. Trigiante, Extremal black holes in supergravity, Lect. Notes Phys. 737 (2008) 661 [hep-th/0611345] [SPIRES].
P. Aschieri, S. Ferrara and B. Zumino, Duality Rotations in Nonlinear Electrodynamics and in Extended Supergravity, Riv. Nuovo Cim. 31 (2008) 625 [arXiv:0807.4039] [SPIRES].
S. Bellucci, S. Ferrara, M. Günaydin and A. Marrani, SAM Lectures on Extremal Black Holes in D = 4 Extended Supergravity, arXiv:0905.3739 [SPIRES].
E.G. Gimon, F. Larsen and J. Simon, Black Holes in Supergravity: the non-BPS Branch, JHEP 01 (2008) 040 [arXiv:0710.4967] [SPIRES].
E. Cremmer and B. Julia, The SO(8) Supergravity, Nucl. Phys. B 159 (1979) 141 [SPIRES].
R. Kallosh and B. Kol, E7 Symmetric Area of the Black Hole Horizon, Phys. Rev. D 53 (1996) 5344 [hep-th/9602014] [SPIRES].
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Andrianopoli, L., D’Auria, R., Ferrara, S. et al. Fake superpotential for large and small extremal black holes. J. High Energ. Phys. 2010, 126 (2010). https://doi.org/10.1007/JHEP08(2010)126
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2010)126