Duality covariant multi-centre black hole systems

  • Guillaume Bossard
  • Stefanos KatmadasEmail author
Open Access


We present a manifestly duality covariant formulation of the composite nonBPS and almost-BPS systems of multi-centre black hole solutions in four dimensions. The method of nilpotent orbits is used to define the two systems in terms of first order flow equations that transform covariantly under the duality group. Subsequently, we rewrite both systems of equations in terms of real, manifestly duality covariant, linear systems of Poisson equations. Somewhat unexpectedly, we find that the two systems are naturally described by the same equations involving space dependent abelian isometries that are conjugate to T-dualities by similarity transformations.


Black Holes Supergravity Models 


  1. [1]
    S. Ferrara, R. Kallosh and A. Strominger, \( \mathcal{N}=2 \) extremal black holes, Phys. Rev. D 52 (1995) 5412 [hep-th/9508072] [INSPIRE].MathSciNetADSGoogle Scholar
  2. [2]
    S. Ferrara and R. Kallosh, Supersymmetry and attractors, Phys. Rev. D 54 (1996) 1514 [hep-th/9602136] [INSPIRE].MathSciNetADSGoogle Scholar
  3. [3]
    A. Strominger, Macroscopic entropy of \( \mathcal{N}=2 \) extremal black holes, Phys. Lett. B 383 (1996) 39 [hep-th/9602111] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  4. [4]
    A. Strominger and C. Vafa, Microscopic origin of the Bekenstein-Hawking entropy, Phys. Lett. B 379 (1996) 99 [hep-th/9601029] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  5. [5]
    J.M. Maldacena, A. Strominger and E. Witten, Black hole entropy in M-theory, JHEP 12 (1997) 002 [hep-th/9711053] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  6. [6]
    F. Denef, Supergravity flows and D-brane stability, JHEP 08 (2000) 050 [hep-th/0005049] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  7. [7]
    K. Behrndt, D. Lüst and W.A. Sabra, Stationary solutions of \( \mathcal{N}=2 \) supergravity, Nucl. Phys. B 510 (1998) 264 [hep-th/9705169] [INSPIRE].ADSGoogle Scholar
  8. [8]
    B. Bates and F. Denef, Exact solutions for supersymmetric stationary black hole composites, JHEP 11 (2011) 127 [hep-th/0304094] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  9. [9]
    G. Lopes Cardoso, B. de Wit, J. Kappeli and T. Mohaupt, Stationary BPS solutions in \( \mathcal{N}=2 \) supergravity with R 2 interactions, JHEP 12(2000) 019 [hep-th/0009234] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    A. Dabholkar, F. Denef, G.W. Moore and B. Pioline, Precision counting of small black holes, JHEP 10 (2005) 096 [hep-th/0507014] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  11. [11]
    F. Denef and G.W. Moore, Split states, entropy enigmas, holes and halos, JHEP 11 (2011) 129 [hep-th/0702146] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  12. [12]
    G. Bossard and C. Ruef, Interacting non-BPS black holes, Gen. Rel. Grav. 44 (2012) 21 [arXiv:1106.5806] [INSPIRE].MathSciNetADSCrossRefzbMATHGoogle Scholar
  13. [13]
    G. Bossard, Octonionic black holes, JHEP 05 (2012) 113 [arXiv:1203.0530] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  14. [14]
    G. Bossard and S. Katmadas, Duality covariant non-BPS first order systems, JHEP 09 (2012) 100 [arXiv:1205.5461] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  15. [15]
    K. Goldstein and S. Katmadas, Almost BPS black holes, JHEP 05 (2009) 058 [arXiv:0812.4183] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  16. [16]
    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].MathSciNetADSCrossRefGoogle Scholar
  17. [17]
    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].MathSciNetADSCrossRefGoogle Scholar
  18. [18]
    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].ADSCrossRefGoogle Scholar
  19. [19]
    A. Dabholkar, A. Sen and S.P. Trivedi, Black hole microstates and attractor without supersymmetry, JHEP 01 (2007) 096 [hep-th/0611143] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  20. [20]
    I. Bena, N. Bobev and N.P. Warner, Spectral flow and the spectrum of multi-center solutions, Phys. Rev. D 77 (2008) 125025 [arXiv:0803.1203] [INSPIRE].MathSciNetADSGoogle Scholar
  21. [21]
    G. Dall’Agata, S. Giusto and C. Ruef, U-duality and non-BPS solutions, JHEP 02 (2011) 074 [arXiv:1012.4803] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  22. [22]
    S. Ferrara, A. Marrani, A. Shcherbakov and A. Yeranyan, Multi-centered first order formalism, JHEP 05 (2013) 127 [arXiv:1211.3262] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  23. [23]
    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].MathSciNetADSCrossRefGoogle Scholar
  24. [24]
    M. Cvetič and D. Youm, All the static spherically symmetric black holes of heterotic string on a six torus, Nucl. Phys. B 472 (1996) 249 [hep-th/9512127] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    T. Ortín, Extremality versus supersymmetry in stringy black holes, Phys. Lett. B 422 (1998) 93 [hep-th/9612142] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    R. Kallosh, N. Sivanandam and M. Soroush, Exact attractive non-BPS STU black holes, Phys. Rev. D 74 (2006) 065008 [hep-th/0606263] [INSPIRE].MathSciNetADSGoogle Scholar
  27. [27]
    G. Bossard and S. Katmadas, Multi-centre black hole solutions in four dimensions, in preparation.Google Scholar
  28. [28]
    B. de Wit and A. Van Proeyen, Isometries of special manifolds, hep-th/9505097 [INSPIRE].
  29. [29]
    M. Günaydin, G. Sierra and P. Townsend, The geometry of \( \mathcal{N}=2 \) Maxwell-Einstein supergravity and Jordan algebras, Nucl. Phys. B 242 (1984) 244 [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    K. Hristov, S. Katmadas and V. Pozzoli, Ungauging black holes and hidden supercharges, JHEP 01 (2013) 110 [arXiv:1211.0035] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  31. [31]
    J.R. Faulkner, A construction of Lie algebras from a class of ternary algebras, Trans. Amer. Math. Soc. 155 (1971) 397.MathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    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].ADSCrossRefGoogle Scholar
  33. [33]
    P. Breitenlohner, D. Maison and G.W. Gibbons, Four-dimensional black holes from Kaluza-Klein theories, Commun. Math. Phys. 120 (1988) 295 [INSPIRE].MathSciNetADSCrossRefzbMATHGoogle Scholar
  34. [34]
    G. Bossard, 1/8 BPS black hole composites, arXiv:1001.3157 [INSPIRE].
  35. [35]
    S. Ferrara and S. Sabharwal, Quaternionic manifolds for type II superstring vacua of Calabi-Yau spaces, Nucl. Phys. B 332 (1990) 317 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  36. [36]
    B. de Wit and A. Van Proeyen, Potentials and symmetries of general gauged \( \mathcal{N}=2 \) supergravity: Yang-Mills models, Nucl. Phys. B 245 (1984) 89 [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    B. de Wit, P. Lauwers and A. Van Proeyen, Lagrangians of \( \mathcal{N}=2 \) supergravity-matter systems, Nucl. Phys. B 255 (1985) 569 [INSPIRE].ADSCrossRefGoogle Scholar
  38. [38]
    A. Ceresole, R. D’Auria and S. Ferrara, The symplectic structure of \( \mathcal{N}=2 \) supergravity and its central extension, Nucl. Phys. Proc. Suppl. 46 (1996) 67 [hep-th/9509160] [INSPIRE].MathSciNetADSCrossRefzbMATHGoogle Scholar

Copyright information

© SISSA 2013

Authors and Affiliations

  1. 1.Centre de Physique Théorique, École Polytechnique, CNRSPalaiseauFrance
  2. 2.Institut de Physique Théorique, CEA Saclay, CNRS-URA 2306Gif sur YvetteFrance

Personalised recommendations