Supersymmetric solutions in six dimensions: a linear structure

  • Iosif Bena
  • Stefano Giusto
  • Masaki Shigemori
  • Nicholas P. Warner
Open Access


The equations underlying all supersymmetric solutions of six-dimensional minimal ungauged supergravity coupled to an anti-self-dual tensor multiplet have been known for quite a while, and their complicated non-linear form has hindered all attempts to systematically understand and construct supersymmetric solutions. In this paper we show that, by suitably re-parameterizing these equations, one can find a structure that allows one to construct supersymmetric solutions by solving a sequence of linear equations. We then illustrate this method by constructing a new class of geometries describing several parallel spirals carrying D1, D5 and P charge and parameterized by four arbitrary functions of one variable. A similar linear structure is known to exist in five dimensions, where it underlies the black hole, black ring and corresponding microstate geometries. The unexpected generalization of this to six dimensions will have important applications to the construction of new, more general such geometries.


Black Holes in String Theory Supergravity Models D-branes 


  1. [1]
    J.P. Gauntlett, J.B. Gutowski, C.M. Hull, S. Pakis and H.S. Reall, All supersymmetric solutions of minimal supergravity in five- dimensions, Class. Quant. Grav. 20 (2003) 4587 [hep-th/0209114] [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  2. [2]
    J.B. Gutowski and H.S. Reall, General supersymmetric AdS 5 black holes, JHEP 04 (2004) 048 [hep-th/0401129] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  3. [3]
    I. Bena and N.P. Warner, One ring to rule them all … and in the darkness bind them?, Adv. Theor. Math. Phys. 9 (2005) 667 [hep-th/0408106] [INSPIRE].MathSciNetMATHGoogle Scholar
  4. [4]
    I. Bena and N.P. Warner, Bubbling supertubes and foaming black holes, Phys. Rev. D 74 (2006) 066001 [hep-th/0505166] [INSPIRE].MathSciNetADSGoogle Scholar
  5. [5]
    P. Berglund, E.G. Gimon and T.S. Levi, Supergravity microstates for BPS black holes and black rings, JHEP 06 (2006) 007 [hep-th/0505167] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  6. [6]
    R. Güven, J.T. Liu, C. Pope and E. Sezgin, Fine tuning and six-dimensional gauged N=(1,0) supergravity vacua, Class. Quant. Grav. 21 (2004) 1001 [hep-th/0306201] [INSPIRE].ADSMATHCrossRefGoogle Scholar
  7. [7]
    J.B. Gutowski, D. Martelli and H.S. Reall, All Supersymmetric solutions of minimal supergravity in six- dimensions, Class. Quant. Grav. 20 (2003) 5049 [hep-th/0306235] [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  8. [8]
    A. Chamseddine, J.M. Figueroa-O’Farrill and W. Sabra, Supergravity vacua and Lorentzian Lie groups, hep-th/0306278 [INSPIRE].
  9. [9]
    M. Cariglia and O.A. Mac Conamhna, The General form of supersymmetric solutions of N=(1,0) U(1) and SU(2) gauged supergravities in six-dimensions, Class. Quant. Grav. 21 (2004) 3171 [hep-th/0402055] [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  10. [10]
    D. Jong, A. Kaya and E. Sezgin, 6D Dyonic String With Active Hyperscalars, JHEP 11 (2006) 047 [hep-th/0608034] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  11. [11]
    M. Akyol and G. Papadopoulos, Spinorial geometry and Killing spinor equations of 6 − D supergravity, Class. Quant. Grav. 28 (2011) 105001 [arXiv:1010.2632] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  12. [12]
    S.D. Mathur, The Fuzzball proposal for black holes: An Elementary review, Fortsch. Phys. 53 (2005) 793 [hep-th/0502050] [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  13. [13]
    I. Bena and N.P. Warner, Black holes, black rings and their microstates, Lect. Notes Phys. 755 (2008) 1 [hep-th/0701216] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  14. [14]
    S.D. Mathur, Fuzzballs and the information paradox: A Summary and conjectures, arXiv:0810.4525 [INSPIRE].
  15. [15]
    V. Balasubramanian, J. de Boer, S. El-Showk and I. Messamah, Black Holes as Effective Geometries, Class. Quant. Grav. 25 (2008) 214004 [arXiv:0811.0263] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    K. Skenderis and M. Taylor, The fuzzball proposal for black holes, Phys. Rept. 467 (2008) 117 [arXiv:0804.0552] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  17. [17]
    B.D. Chowdhury and A. Virmani, Modave Lectures on Fuzzballs and Emission from the D1 − D5 System, arXiv:1001.1444 [INSPIRE].
  18. [18]
    A. Saxena, G. Potvin, S. Giusto and A.W. Peet, Smooth geometries with four charges in four dimensions, JHEP 04 (2006) 010 [hep-th/0509214] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  19. [19]
    F. Denef, Supergravity flows and D-brane stability, JHEP 08 (2000) 050 [hep-th/0005049] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  20. [20]
    B. Bates and F. Denef, Exact solutions for supersymmetric stationary black hole composites, JHEP 11 (2011) 127 [hep-th/0304094] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  21. [21]
    I. Bena, C.-W. Wang and N.P. Warner, The Foaming three-charge black hole, Phys. Rev. D 75 (2007) 124026 [hep-th/0604110] [INSPIRE].MathSciNetADSGoogle Scholar
  22. [22]
    I. Bena, C.-W. Wang and N.P. Warner, Plumbing the Abyss: Black ring microstates, JHEP 07 (2008) 019 [arXiv:0706.3786] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  23. [23]
    J. de Boer, S. El-Showk, I. Messamah and D. Van den Bleeken, Quantizing N = 2 Multicenter Solutions, JHEP 05 (2009) 002 [arXiv:0807.4556] [INSPIRE].CrossRefGoogle Scholar
  24. [24]
    I. Bena, C.-W. Wang and N.P. Warner, Mergers and typical black hole microstates, JHEP 11 (2006) 042 [hep-th/0608217] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  25. [25]
    J. de Boer, S. El-Showk, I. Messamah and D. Van den Bleeken, A Bound on the entropy of supergravity?, JHEP 02 (2010) 062 [arXiv:0906.0011] [INSPIRE].CrossRefGoogle Scholar
  26. [26]
    I. Bena, N. Bobev, S. Giusto, C. Ruef and N.P. Warner, An Infinite-Dimensional Family of Black-Hole Microstate Geometries, JHEP 03 (2011) 022 [Erratum ibid. 1104 (2011) 059] [arXiv:1006.3497] [INSPIRE].
  27. [27]
    I. Bena, N. Bobev, C. Ruef and N.P. Warner, Entropy Enhancement and Black Hole Microstates, Phys. Rev. Lett. 105 (2010) 231301 [arXiv:0804.4487] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    I. Bena, N. Bobev, C. Ruef and N.P. Warner, Supertubes in Bubbling Backgrounds: Born-Infeld Meets Supergravity, JHEP 07 (2009) 106 [arXiv:0812.2942] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  29. [29]
    I. Bena, J. de Boer, M. Shigemori and N.P. Warner, Double, Double Supertube Bubble, JHEP 10 (2011) 116 [arXiv:1107.2650] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    J. de Boer and M. Shigemori, Exotic branes and non-geometric backgrounds, Phys. Rev. Lett. 104 (2010) 251603 [arXiv:1004.2521] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  31. [31]
    O. Lunin and S.D. Mathur, AdS/CFT duality and the black hole information paradox, Nucl. Phys. B 623 (2002) 342 [hep-th/0109154] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  32. [32]
    O. Lunin, J.M. Maldacena and L. Maoz, Gravity solutions for the D1 − D5 system with angular momentum, hep-th/0212210 [INSPIRE].
  33. [33]
    J. Ford, S. Giusto and A. Saxena, A Class of BPS time-dependent 3-charge microstates from spectral flow, Nucl. Phys. B 790 (2008) 258 [hep-th/0612227] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  34. [34]
    I. Bena, S. Giusto, C. Ruef and N.P. Warner, Supergravity Solutions from Floating Branes, JHEP 03 (2010) 047 [arXiv:0910.1860] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  35. [35]
    A. Dabholkar and J.A. Harvey, Nonrenormalization of the Superstring Tension, Phys. Rev. Lett. 63 (1989) 478 [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    A. Dabholkar, J.P. Gauntlett, J.A. Harvey and D. Waldram, Strings as solitons and black holes as strings, Nucl. Phys. B 474 (1996) 85 [hep-th/9511053] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  37. [37]
    O. Lunin and S.D. Mathur, Metric of the multiply wound rotating string, Nuclear Physics B 610 (2001) 49 [arXiv:hep-th/0105136].MathSciNetADSGoogle Scholar
  38. [38]
    G.T. Horowitz and D. Marolf, Counting states of black strings with traveling waves, Phys. Rev. D 55 (1997) 835 [arXiv:hep-th/9605224].MathSciNetADSGoogle Scholar
  39. [39]
    G.T. Horowitz and D. Marolf, Counting states of black strings with traveling waves. II, Phys. Rev. D 55 (1997) 846 [arXiv:hep-th/9606113].ADSGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2012

Authors and Affiliations

  • Iosif Bena
    • 1
  • Stefano Giusto
    • 2
    • 3
  • Masaki Shigemori
    • 4
  • Nicholas P. Warner
    • 5
  1. 1.Institut de Physique Théorique, CEA Saclay, CNRS-URA 2306Gif sur YvetteFrance
  2. 2.Dipartimento di Fisica “Galileo Galilei,”Università di PadovaPadovaItaly
  3. 3.INFN, Sezione di PadovaPadovaItaly
  4. 4.Kobayashi-Maskawa Institute for the Origin of Particles and the Universe, Nagoya UniversityNagoyaJapan
  5. 5.Department of Physics and AstronomyUniversity of Southern CaliforniaLos AngelesU.S.A.

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