Abstract
We explain when the first-order Hamilton-Jacobi equations for black holes (and domain walls) in (gauged) supergravity, reduce to the usual first-order equations derived from a fake superpotential. This turns out to be equivalent to the vanishing of a newly found constant of motion and we illustrate this with various examples. We show that fake supersymmetry is a necessary condition for having physically sensible extremal black hole solutions. We furthermore observe that small black holes become scaling solutions near the horizon. When combined with fake supersymmetry, this leads to a precise extension of the attractor mechanism to small black holes: the attractor solution is such that the scalars move on specific curves, determined by the black hole charges, that are purely geodesic, although there is a non-zero potential.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
D. Freedman, C. Núñez, M. Schnabl and K. Skenderis, Fake supergravity and domain wall stability, Phys. Rev. D 69 (2004) 104027 [hep-th/0312055] [INSPIRE].
A. Celi, A. Ceresole, G. Dall’Agata, A. Van Proeyen and M. Zagermann, On the fakeness of fake supergravity, Phys. Rev. D 71 (2005) 045009 [hep-th/0410126] [INSPIRE].
K. Skenderis and P.K. Townsend, Hidden supersymmetry of domain walls and cosmologies, Phys. Rev. Lett. 96 (2006) 191301 [hep-th/0602260] [INSPIRE].
A.R. Liddle and D.H. Lyth, Cosmological inflation and large-scale structure, Cambridge University Press, Cambridge U.K. (2000) [INSPIRE].
D. Bazeia, C.B. Gomes, L. Losano and R. Menezes, First-order formalism and dark energy, Phys. Lett. B 633 (2006) 415 [astro-ph/0512197] [INSPIRE].
V.I. Afonso, D. Bazeia and L. Losano, First-order formalism for bent brane, Phys. Lett. B 634 (2006) 526 [hep-th/0601069] [INSPIRE].
K. Skenderis and P.K. Townsend, Pseudo-supersymmetry and the domain-wall/cosmology correspondence, J. Phys. A 40 (2007) 6733 [hep-th/0610253] [INSPIRE].
K. Skenderis and P.K. Townsend, Hamilton-Jacobi method for curved domain walls and cosmologies, Phys. Rev. D 74 (2006) 125008 [hep-th/0609056] [INSPIRE].
P.K. Townsend, Hamilton-Jacobi mechanics from pseudo-supersymmetry, Class. Quant. Grav. 25 (2008) 045017 [arXiv:0710.5178] [INSPIRE].
P.K. Townsend, From wave geometry to fake supergravity, J. Phys. A 41 (2008) 304014 [arXiv:0710.5709] [INSPIRE].
J. Sonner and P.K. Townsend, Axion-dilaton domain walls and fake supergravity, Class. Quant. Grav. 24 (2007) 3479 [hep-th/0703276] [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].
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].
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].
G. Dall’Agata, Black holes in supergravity: flow equations and duality, arXiv:1106.2611 [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].
C.M. Miller, K. Schalm and E.J. Weinberg, Nonextremal black holes are BPS, Phys. Rev. D 76 (2007) 044001 [hep-th/0612308] [INSPIRE].
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] [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].
L. Andrianopoli, R. D’Auria, S. Ferrara and M. Trigiante, Fake superpotential for large and small extremal black holes, JHEP 08 (2010) 126 [arXiv:1002.4340] [INSPIRE].
S. Ferrara, R. Kallosh and A. Strominger, N = 2 extremal black holes, Phys. Rev. D 52 (1995) 5412 [hep-th/9508072] [INSPIRE].
S. Ferrara and R. Kallosh, Supersymmetry and attractors, Phys. Rev. D 54 (1996) 1514 [hep-th/9602136] [INSPIRE].
A.J. Tolley and D.H. Wesley, Scale-invariance in expanding and contracting universes from two-field models, JCAP 05 (2007) 006 [hep-th/0703101] [INSPIRE].
W. Chemissany, A. Ploegh and T. Van Riet, A note on scaling cosmologies, geodesic motion and pseudo-SUSY, Class. Quant. Grav. 24 (2007) 4679 [arXiv:0704.1653] [INSPIRE].
W. Chemissany, B. Janssen and T. Van Riet, Einstein branes, JHEP 10 (2011) 002 [arXiv:1107.1427] [INSPIRE].
I. Kanitscheider, K. Skenderis and M. Taylor, Precision holography for non-conformal branes, JHEP 09 (2008) 094 [arXiv:0807.3324] [INSPIRE].
J.L. Karthauser and P. Saffin, Scaling solutions and geodesics in moduli space, Class. Quant. Grav. 23 (2006) 4615 [hep-th/0604046] [INSPIRE].
P. Breitenlohner, D. Maison and G.W. Gibbons, Four-dimensional black holes from Kaluza-Klein theories, Commun. Math. Phys. 120 (1988) 295 [INSPIRE].
W. Chemissany et al., Black holes in supergravity and integrability, JHEP 09 (2010) 080 [arXiv:1007.3209] [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. de Antonio Martin, T. Ortín and C. Shahbazi, The FGK formalism for black p-branes in d dimensions, arXiv:1203.0260 [INSPIRE].
A. Butti, M. Graña, R. Minasian, M. Petrini and A. Zaffaroni, The baryonic branch of Klebanov-Strassler solution: a supersymmetric family of SU(3) structure backgrounds, JHEP 03 (2005) 069 [hep-th/0412187] [INSPIRE].
N. Halmagyi, J.T. Liu and P. Szepietowski, On N = 2 truncations of IIB on T 1,1, arXiv:1111.6567 [INSPIRE].
G. Giecold, Remark on the baryonic branch of the warped deformed conifold, arXiv:1112.1054 [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1203.3194
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Trigiante, M., Van Riet, T. & Vercnocke, B. Fake supersymmetry versus Hamilton-Jacobi. J. High Energ. Phys. 2012, 78 (2012). https://doi.org/10.1007/JHEP05(2012)078
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2012)078