Supersymmetric asymptotic AdS and Lifshitz solutions in Einstein-Weyl and conformal supergravities

  • H. Lü
  • Zhao-Long Wang


We show that there exist supersymmetric Lifshitz vacua in off-shell Einstein- Weyl supergravity, in addition to the BPS AdS4 vacuum. The Lifshitz exponents are determined by the product of the cosmological constant and the coupling of the Weyl- squared term. We then obtain the equations of the supersymmetric solutions that are asymptotic to the AdS or Lifshitz vacua. We obtain many examples of exact solutions as well as numerical ones. We find examples of extremal AdS black branes whose near-horizon geometry is AdS2 × T 2. We also find an extremal Lifshitz black hole with z = −2, whose horizon coincides with the curvature singularity. However the asymptotic Lifshitz solutions are in general smooth wormholes. In conformal supergravity, we find intriguing examples of non-extremal “BPS” AdS and Lifshitz black holes whose local Killing spinor is divergent on the horizon. We show that all the supersymmetric asymptotic AdS and Lifshitz solutions have the vanishing Noether charge associated with some scaling symmetry. We also study the integrability condition of the Killing spinor equation and the supersymmetric invariance of the action. Finally we show that the only spherically-symmetric BPS solution is the AdS vacuum.


Classical Theories of Gravity Supergravity Models 


  1. [1]
    J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1133 [Adv. Theor. Math. Phys. 2 (1998) 231] [hep-th/9711200] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  2. [2]
    S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].MathSciNetADSGoogle Scholar
  3. [3]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].MathSciNetADSMATHGoogle Scholar
  4. [4]
    S. Kachru, X. Liu and M. Mulligan, Gravity duals of Lifshitz-like fixed points, Phys. Rev. D 78 (2008) 106005 [arXiv:0808.1725] [INSPIRE].MathSciNetADSGoogle Scholar
  5. [5]
    P. Koroteev and M. Libanov, On existence of self-tuning solutions in static braneworlds without singularities, JHEP 02 (2008) 104 [arXiv:0712.1136] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  6. [6]
    M. Taylor, Non-relativistic holography, arXiv:0812.0530 [INSPIRE].
  7. [7]
    S.S. Gubser and A. Nellore, Ground states of holographic superconductors, Phys. Rev. D 80 (2009) 105007 [arXiv:0908.1972] [INSPIRE].ADSGoogle Scholar
  8. [8]
    S.A. Hartnoll and A. Tavanfar, Electron stars for holographic metallic criticality, Phys. Rev. D 83 (2011) 046003 [arXiv:1008.2828] [INSPIRE].ADSGoogle Scholar
  9. [9]
    U.H. Danielsson and L. Thorlacius, Black holes in asymptotically Lifshitz spacetime, JHEP 03 (2009) 070 [arXiv:0812.5088] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  10. [10]
    R.B. Mann, Lifshitz topological black holes, JHEP 06 (2009) 075 [arXiv:0905.1136] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    S.A. Hartnoll, J. Polchinski, E. Silverstein and D. Tong, Towards strange metallic holography, JHEP 04 (2010) 120 [arXiv:0912.1061] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    K. Balasubramanian and K. Narayan, Lifshitz spacetimes from AdS null and cosmological solutions, JHEP 08 (2010) 014 [arXiv:1005.3291] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  13. [13]
    A. Donos and J.P. Gauntlett, Lifshitz solutions of D = 10 and D = 11 supergravity, JHEP 12 (2010) 002 [arXiv:1008.2062] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  14. [14]
    R. Gregory, S.L. Parameswaran, G. Tasinato and I. Zavala, Lifshitz solutions in supergravity and string theory, JHEP 12 (2010) 047 [arXiv:1009.3445] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  15. [15]
    A. Donos, J.P. Gauntlett, N. Kim and O. Varela, Wrapped M 5-branes, consistent truncations and AdS/CMT, JHEP 12 (2010) 003 [arXiv:1009.3805] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  16. [16]
    D. Cassani and A.F. Faedo, Constructing Lifshitz solutions from AdS, JHEP 05 (2011) 013 [arXiv:1102.5344] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  17. [17]
    N. Halmagyi, M. Petrini and A. Zaffaroni, Non-relativistic solutions of N = 2 gauged supergravity, JHEP 08 (2011) 041 [arXiv:1102.5740] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  18. [18]
    I. Amado and A.F. Faedo, Lifshitz black holes in string theory, JHEP 07 (2011) 004 [arXiv:1105.4862] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  19. [19]
    H. Lü, Y. Pang, C. Pope and J. Vazquez-Poritz, AdS and Lifshitz black holes in conformal and Einstein-Weyl gravities, arXiv:1204.1062 [INSPIRE].
  20. [20]
    M. Alishahiha and R. Fareghbal, D-dimensional log gravity, Phys. Rev. D 83 (2011) 084052 [arXiv:1101.5891] [INSPIRE].ADSGoogle Scholar
  21. [21]
    T. Malek, Exact solutions of general relativity and quadratic gravity in arbitrary dimension, arXiv:1204.0291 [INSPIRE].
  22. [22]
    M. Gurses, T.C. Sisman and B. Tekin, New exact solutions of quadratic curvature gravity, arXiv:1204.2215 [INSPIRE].
  23. [23]
    P. Binetruy, G. Girardi and R. Grimm, Supergravity couplings: a geometric formulation, Phys. Rept. 343 (2001) 255 [hep-th/0005225] [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  24. [24]
    H. Lü, C. Pope, E. Sezgin and L. Wulff, Critical and non-critical Einstein-Weyl supergravity, JHEP 10 (2011) 131 [arXiv:1107.2480] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    H. Lü, C. Pope and P. Townsend, Domain walls from Anti-de Sitter space-time, Phys. Lett. B 391 (1997) 39 [hep-th/9607164] [INSPIRE].ADSGoogle Scholar
  26. [26]
    K. Stelle and P.C. West, Minimal auxiliary fields for supergravity, Phys. Lett. B 74 (1978) 330 [INSPIRE].ADSGoogle Scholar
  27. [27]
    S. Ferrara and P. van Nieuwenhuizen, The auxiliary fields of supergravity, Phys. Lett. B 74 (1978) 333 [INSPIRE].ADSGoogle Scholar
  28. [28]
    H. Lü and C. Pope, Critical gravity in four dimensions, Phys. Rev. Lett. 106 (2011) 181302 [arXiv:1101.1971] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    S. Deser et al., Critical points of D-dimensional extended gravities, Phys. Rev. D 83 (2011) 061502 [arXiv:1101.4009] [INSPIRE].ADSGoogle Scholar
  30. [30]
    E.A. Bergshoeff, O. Hohm, J. Rosseel and P.K. Townsend, Modes of log gravity, Phys. Rev. D 83 (2011) 104038 [arXiv:1102.4091] [INSPIRE].ADSGoogle Scholar
  31. [31]
    M. Porrati and M.M. Roberts, Ghosts of critical gravity, Phys. Rev. D 84 (2011) 024013 [arXiv:1104.0674] [INSPIRE].ADSGoogle Scholar
  32. [32]
    H. Liu, H. Lü and M. Luo, On black hole stability in critical gravities, Int. J. Mod. Phys. D 21 (2012) 1250020 [arXiv:1104.2623] [INSPIRE].ADSGoogle Scholar
  33. [33]
    G. Bertoldi, B.A. Burrington and A. Peet, Black holes in asymptotically Lifshitz spacetimes with arbitrary critical exponent, Phys. Rev. D 80 (2009) 126003 [arXiv:0905.3183] [INSPIRE].MathSciNetADSGoogle Scholar
  34. [34]
    H. Lü, C. Pope and Z.-L. Wang, Pseudo-supersymmetry, consistent sphere reduction and Killing spinors for the bosonic string, Phys. Lett. B 702 (2011) 442 [arXiv:1105.6114] [INSPIRE].ADSGoogle Scholar
  35. [35]
    H. Lü and Z.-L. Wang, Killing spinors for the bosonic string, Europhys. Lett. 97 (2012) 50010 [arXiv:1106.1664] [INSPIRE].CrossRefGoogle Scholar
  36. [36]
    H. Liu, H. Lü and Z.-L. Wang, Killing spinors for the bosonic string and the Kaluza-Klein theory with scalar potentials, Eur. Phys. J. C 72 (2012) 1853 [arXiv:1106.4566] [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    H. Lü, C. Pope and Z.-L. Wang, Pseudo-supergravity extension of the bosonic string, Nucl. Phys. B 854 (2012) 293 [arXiv:1106.5794] [INSPIRE].ADSCrossRefGoogle Scholar
  38. [38]
    H.-S. Liu, H. Lü, Z.-L. Wang, H. Lü and Z.-L. Wang, Gauged Kaluza-Klein AdS pseudo-supergravity, Phys. Lett. B 703 (2011) 524 [arXiv:1107.2659] [INSPIRE].ADSGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2012

Authors and Affiliations

  1. 1.Department of PhysicsBeijing Normal UniversityBeijingChina
  2. 2.School of PhysicsKorea Institute for Advanced StudySeoulKorea

Personalised recommendations