Advertisement

Lifshitz spacetimes from AdS null and cosmological solutions

  • Koushik Balasubramanian
  • K. Narayan
Article

Abstract

We describe solutions of 10-dimensional supergravity comprising null deformations of AdS 5 × S 5 with a scalar field, which have z = 2 Lifshitz symmetries. The bulk Lifshitz geometry in 3 + 1-dimensions arises by dimensional reduction of these solutions. The dual field theory in this case is a deformation of the \( \mathcal{N} = 4 \) super Yang-Mills theory. We discuss the holographic 2-point function of operators dual to bulk scalars. We further describe time-dependent (cosmological) solutions which have anisotropic Lifshitz scaling symmetries. We also discuss deformations of AdS × X in 11-dimensional supergravity, which are somewhat similar to the solutions above. In some cases here, we expect the field theory duals to be deformations of the Chern-Simons theories on M2-branes stacked at singularities.

Keywords

Gauge-gravity correspondence Space-Time Symmetries 

References

  1. [1]
    D.T. Son, Toward an AdS/cold atoms correspondence: a geometric realization of the Schroedinger symmetry, Phys. Rev. D 78 (2008) 046003 [arXiv:0804.3972] [SPIRES].MathSciNetADSGoogle Scholar
  2. [2]
    K. Balasubramanian and J. McGreevy, Gravity duals for non-relativistic CFTs, Phys. Rev. Lett. 101 (2008) 061601 [arXiv:0804.4053] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  3. [3]
    J. Maldacena, D. Martelli and Y. Tachikawa, Comments on string theory backgrounds with non-relativistic conformal symmetry, JHEP 10 (2008) 072 [arXiv:0807.1100] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  4. [4]
    C.P. Herzog, M. Rangamani and S.F. Ross, Heating up Galilean holography, JHEP 11 (2008) 080 [arXiv:0807.1099] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  5. [5]
    A. Adams, K. Balasubramanian and J. McGreevy, Hot Spacetimes for Cold Atoms, JHEP 11 (2008) 059 [arXiv:0807.1111] [SPIRES].CrossRefADSGoogle Scholar
  6. [6]
    W.D. Goldberger, AdS/CFT duality for non-relativistic field theory, JHEP 03 (2009) 069 [arXiv:0806.2867] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  7. [7]
    J.L.F. Barbon and C.A. Fuertes, On the spectrum of nonrelativistic AdS/CFT, JHEP 09 (2008) 030 [arXiv:0806.3244] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  8. [8]
    S. Kachru, X. Liu and M. Mulligan, Gravity Duals of Lifshitz-like Fixed Points, Phys. Rev. D 78 (2008) 106005 [arXiv:0808.1725] [SPIRES].MathSciNetADSGoogle Scholar
  9. [9]
    M. Taylor, Non-relativistic holography, arXiv:0812.0530 [SPIRES].
  10. [10]
    S.A. Hartnoll and K. Yoshida, Families of IIB duals for nonrelativistic CFTs, JHEP 12 (2008) 071 [arXiv:0810.0298] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  11. [11]
    A. Donos and J.P. Gauntlett, Supersymmetric solutions for non-relativistic holography, JHEP 03 (2009) 138 [arXiv:0901.0818] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  12. [12]
    A. Donos and J.P. Gauntlett, Solutions of type IIB and D = 11 supergravity with Schrödinger (z) symmetry, JHEP 07 (2009) 042 [arXiv:0905.1098] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  13. [13]
    A. Donos and J.P. Gauntlett, Schrödinger invariant solutions of type IIB with enhanced supersymmetry, JHEP 10 (2009) 073 [arXiv:0907.1761] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  14. [14]
    N. Bobev, A. Kundu and K. Pilch, “Supersymmetric IIB Solutions with Schrodinger Symmetry”, JHEP 07 (2009) 107 [arXiv:0905.0673] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  15. [15]
    P. Hořava, Quantum Criticality and Yang-Mills Gauge Theory, arXiv:0811.2217 [SPIRES].
  16. [16]
    P. Hořava, Membranes at Quantum Criticality, JHEP 03 (2009) 020 [arXiv:0812.4287] [SPIRES]. ADSGoogle Scholar
  17. [17]
    U.H. Danielsson and L. Thorlacius, Black holes in asymptotically Lifshitz spacetime, JHEP 03 (2009) 070 [arXiv:0812.5088] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  18. [18]
    A. Adams, A. Maloney, A. Sinha and S.E. Vazquez, 1/N Effects in Non-Relativistic Gauge-Gravity Duality, JHEP 03 (2009) 097 [arXiv:0812.0166] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  19. [19]
    T. Azeyanagi, W. Li and T. Takayanagi, On String Theory Duals of Lifshitz-like Fixed Points, JHEP 06 (2009) 084 [arXiv:0905.0688] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  20. [20]
    A. Dhar, G. Mandal and S.R. Wadia, Asymptotically free four-fermi theory in 4 dimensions at the z=3 Lifshitz-like fixed point, Phys. Rev. D 80 (2009) 105018 [arXiv:0905.2928] [SPIRES]. ADSGoogle Scholar
  21. [21]
    S.R. Das and G. Murthy, CP N−1 Models at a Lifshitz Point, Phys. Rev. D 80 (2009) 065006 [arXiv:0906.3261] [SPIRES].ADSGoogle Scholar
  22. [22]
    W. Li, T. Nishioka and T. Takayanagi, Some No-go Theorems for String Duals of Non-relativistic Lifshitz-like Theories, JHEP 10 (2009) 015 [arXiv:0908.0363] [SPIRES].MathSciNetADSGoogle Scholar
  23. [23]
    K. Balasubramanian and J. McGreevy, An analytic Lifshitz black hole, Phys. Rev. D 80 (2009) 104039 [arXiv:0909.0263] [SPIRES].MathSciNetADSGoogle Scholar
  24. [24]
    K. Goldstein, S. Kachru, S. Prakash and S.P. Trivedi, Holography of Charged Dilaton Black Holes, arXiv:0911.3586 [SPIRES].
  25. [25]
    S.A. Hartnoll, J. Polchinski, E. Silverstein and D. Tong, Towards strange metallic holography, JHEP 04 (2010) 120 [arXiv:0912.1061] [SPIRES].CrossRefADSGoogle Scholar
  26. [26]
    J. Blaback, U.H. Danielsson and T. Van Riet, Lifshitz backgrounds from 10D supergravity, JHEP 02 (2010) 095 [arXiv:1001.4945] [SPIRES].CrossRefADSGoogle Scholar
  27. [27]
    T. Faulkner and J. Polchinski, Semi-HolographicFermi Liquids, arXiv:1001.5049 [SPIRES].
  28. [28]
    M. Mulligan, C. Nayak and S. Kachru, An Isotropic to Anisotropic Transition in a Fractional Quantum Hall State, arXiv:1004.3570 [SPIRES].
  29. [29]
    E.J. Brynjolfsson, U.H. Danielsson, L. Thorlacius and T. Zingg, Holographic models with anisotropic scaling, arXiv:1004.5566 [SPIRES].
  30. [30]
    P. Koroteev and M. Libanov, On Existence of Self-Tuning Solutions in Static Braneworlds without Singularities, JHEP 02 (2008) 104 [arXiv:0712.1136] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  31. [31]
    R.M. Hornreich, M. Luban and S. Shtrikman, Critical Behavior at the Onset of k-Space Instability on the λ Line, Phys. Rev. Lett. 35 (1975) 1678 [SPIRES].CrossRefADSGoogle Scholar
  32. [32]
    G. Grinstein, Anisotropic sine-Gordon model and infinite-order phase transitions in three dimensions, Phys. Rev. B 23 (1981) 4615. MathSciNetADSGoogle Scholar
  33. [33]
    E. Ardonne, P. Fendley and E. Fradkin, Topological order and conformal quantum critical points, Annals Phys. 310 (2004) 493 [cond-mat/0311466] [SPIRES].zbMATHCrossRefMathSciNetADSGoogle Scholar
  34. [34]
    C.L. Henley, Relaxation time for a dimer covering with height representation, J. Stat. Phys. 89 (1997) 483 [cond-mat/9607222].zbMATHCrossRefMathSciNetADSGoogle Scholar
  35. [35]
    C.L. Henley, From classical to quantum dynamics at Rokhsar-Kivelson points, in Proceedings of Highly Frustrated Magnetism 2003, Grenoble France, August 26–30 2003, [cond-mat/0311345].
  36. [36]
    P. Chaikin and T. Lubensky, Principles of condensed matter physics, Cambridge University Press, Cambridge U.K. (1995).Google Scholar
  37. [37]
    P. Ghaemi, A. Vishwanath and T. Senthil, Finite temperature properties of quantum Lifshitz transitions between valence bond solid phases: An example of ‘local’ quantum criticality, Phys. Rev. B 72 (2005) 024420 [cond-mat/0412409].ADSGoogle Scholar
  38. [38]
    S.R. Das, J. Michelson, K. Narayan and S.P. Trivedi, Time dependent cosmologies and their duals, Phys. Rev. D 74 (2006) 026002 [hep-th/0602107] [SPIRES].MathSciNetADSGoogle Scholar
  39. [39]
    S.R. Das, J. Michelson, K. Narayan and S.P. Trivedi, Cosmologies with Null Singularities and their Gauge Theory Duals, Phys. Rev. D 75 (2007) 026002 [hep-th/0610053] [SPIRES].MathSciNetADSGoogle Scholar
  40. [40]
    A. Awad, S.R. Das, K. Narayan and S.P. Trivedi, Gauge Theory Duals of Cosmological Backgrounds and their Energy Momentum Tensors, Phys. Rev. D 77 (2008) 046008 [arXiv:0711.2994] [SPIRES].MathSciNetADSGoogle Scholar
  41. [41]
    A. Awad, S.R. Das, S. Nampuri, K. Narayan and S.P. Trivedi, Gauge Theories with Time Dependent Couplings and their Cosmological Duals, Phys. Rev. D 79 (2009) 046004 [arXiv:0807.1517] [SPIRES].ADSGoogle Scholar
  42. [42]
    O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  43. [43]
    M. Benna, I. Klebanov, T. Klose and M. Smedback, Superconformal Chern-Simons Theories and AdS 4 /CFT 3 Correspondence, JHEP 09 (2008) 072 [arXiv:0806.1519] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  44. [44]
    S. Terashima and F. Yagi, Orbifolding the Membrane Action, JHEP 12 (2008) 041 [arXiv:0807.0368] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  45. [45]
    O. Aharony, O. Bergman and D.L. Jafferis, Fractional M2-branes, JHEP 11 (2008) 043 [arXiv:0807.4924] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  46. [46]
    D.L. Jafferis and A. Tomasiello, A simple class of N = 3 gauge/gravity duals, JHEP 10 (2008) 101 [arXiv:0808.0864] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  47. [47]
    Y. Imamura and K. Kimura, On the moduli space of elliptic Maxwell-Chern-Simons theories, Prog. Theor. Phys. 120 (2008) 509 [arXiv:0806.3727] [SPIRES].zbMATHCrossRefADSGoogle Scholar
  48. [48]
    Y. Imamura and K. Kimura, Quiver Chern-Simons theories and crystals, JHEP 10 (2008) 114 [arXiv:0808.4155] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  49. [49]
    K. Ueda and M. Yamazaki, Toric Calabi-Yau four-folds dual to Chern-Simons-matter theories, JHEP 12 (2008) 045 [arXiv:0808.3768] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  50. [50]
    S. Lee, Superconformal field theories from crystal lattices, Phys. Rev. D 75 (2007) 101901 [hep-th/0610204] [SPIRES].ADSGoogle Scholar
  51. [51]
    S. Lee, S. Lee and J. Park, Toric AdS 4 /CFT 3 duals and M-theory crystals, JHEP 05 (2007) 004 [hep-th/0702120] [SPIRES].CrossRefADSGoogle Scholar
  52. [52]
    S. Kim, S. Lee, S. Lee and J. Park, Abelian Gauge Theory on M2-brane and Toric Duality, Nucl. Phys. B 797 (2008) 340 [arXiv:0705.3540] [SPIRES].MathSciNetADSGoogle Scholar
  53. [53]
    D. Martelli and J. Sparks, Notes on toric Sasaki-Einstein seven-manifolds and AdS 4 /CFT 3, JHEP 11 (2008) 016 [arXiv:0808.0904] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  54. [54]
    D. Martelli and J. Sparks, Moduli spaces of Chern-Simons quiver gauge theories and AdS 4 /CFT 3, Phys. Rev. D 78 (2008) 126005 [arXiv:0808.0912] [SPIRES].MathSciNetADSGoogle Scholar
  55. [55]
    D. Martelli and J. Sparks, AdS 4 /CFT 3 duals from M2-branes at hypersurface singularities and their deformations, JHEP 12 (2009) 017 [arXiv:0909.2036] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  56. [56]
    S. Franco, I.R. Klebanov and D. Rodriguez-Gomez, M2-branes on Orbifolds of the Cone over Q 1,1,1, JHEP 08 (2009) 033 [arXiv:0903.3231] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  57. [57]
    A. Hanany and A. Zaffaroni, Tilings, Chern-Simons Theories and M2 Branes, JHEP 10 (2008) 111 [arXiv:0808.1244] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  58. [58]
    A. Hanany, D. Vegh and A. Zaffaroni, Brane Tilings and M2 Branes, JHEP 03 (2009) 012 [arXiv:0809.1440] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  59. [59]
    S. Franco, A. Hanany, J. Park and D. Rodriguez-Gomez, Towards M2-brane Theories for Generic Toric Singularities, JHEP 12 (2008) 110 [arXiv:0809.3237] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  60. [60]
    A. Hanany and Y.-H. He, M2-Branes and Quiver Chern-Simons: A Taxonomic Study, arXiv:0811.4044 [SPIRES].
  61. [61]
    A. Hanany and Y.-H. He, Chern-Simons: Fano and Calabi-Yau, arXiv:0904.1847 [SPIRES].
  62. [62]
    C.-S. Chu and P.-M. Ho, Time-dependent AdS/CFT duality and null singularity, JHEP 04 (2006) 013 [hep-th/0602054] [SPIRES].MathSciNetADSGoogle Scholar
  63. [63]
    C.-S. Chu and P.-M. Ho, Time-dependent AdS/CFT Duality II: Holographic Reconstruction of Bulk Metric and Possible Resolution of Singularity, JHEP 02 (2008) 058 [arXiv:0710.2640] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  64. [64]
    F.-L. Lin and W.-Y. Wen, Supersymmetric null-like holographic cosmologies, JHEP 05 (2006) 013 [hep-th/0602124] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  65. [65]
    F.-L. Lin and D. Tomino, One-loop effect of null-like cosmology’s holographic dual super-Yang-Mills, JHEP 03 (2007) 118 [hep-th/0611139] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  66. [66]
    A. Awad, S.R. Das, A. Ghosh, J.-H. Oh and S.P. Trivedi, Slowly Varying Dilaton Cosmologies and their Field Theory Duals, Phys. Rev. D 80 (2009) 126011 [arXiv:0906.3275] [SPIRES].ADSGoogle Scholar
  67. [67]
    K. Madhu and K. Narayan, String spectra near some null cosmological singularities, Phys. Rev. D 79 (2009) 126009 [arXiv:0904.4532] [SPIRES].MathSciNetADSGoogle Scholar
  68. [68]
    K. Narayan, Null cosmological singularities and free strings, Phys. Rev. D 81 (2010) 066005 [arXiv:0909.4731] [SPIRES].MathSciNetADSGoogle Scholar
  69. [69]
    R.M. Wald, General Relativity, The University of Chicago Press, Chicago U.S.A. (1984).zbMATHGoogle Scholar
  70. [70]
    M. Blau, J. Hartong and B. Rollier, Geometry of Schroedinger Space-Times, Global Coordinates and Harmonic Trapping, JHEP 07 (2009) 027 [arXiv:0904.3304] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  71. [71]
    M. Blau, J. Hartong and B. Rollier, Geometry of Schroedinger Space-Times II: Particle and Field Probes of the Causal Structure, arXiv:1005.0760 [SPIRES].
  72. [72]
    V.E. Hubeny and M. Rangamani, Causal structures of pp-waves, JHEP 12 (2002) 043 [hep-th/0211195] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  73. [73]
    V.E. Hubeny, M. Rangamani and S.F. Ross, Causal inheritance in plane wave quotients, Phys. Rev. D 69 (2004) 024007 [hep-th/0307257] [SPIRES].MathSciNetADSGoogle Scholar
  74. [74]
    Y. Nakayama, Anisotropic scale invariant cosmology, arXiv:0912.5118 [SPIRES].
  75. [75]
    Y. Nakayama, Universal time-dependent deformations of Schrödinger geometry, JHEP 04 (2010) 102 [arXiv:1002.0615] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  76. [76]
    L.D. Landau and E.M. Lifshitz, Classical Theory of Fields, Course of Theoretical Physics Vol. 2, Pergamon Press, New York U.S.A. (1987).Google Scholar
  77. [77]
    V.A. Belinsky, I.M. Khalatnikov and E.M. Lifshitz, Oscillatory approach to a singular point in the relativistic cosmology, Adv. Phys. 19 (1970) 525 [SPIRES]. CrossRefADSGoogle Scholar
  78. [78]
    V.A. Belinskii and I.M. Khalatnikov, Effect of scalar and vector fields on the nature of the cosmological singularity, Sov. Phys. JETP 36 (1973) 591.MathSciNetADSGoogle Scholar
  79. [79]
    C.W. Misner, Quantum cosmology. 1, Phys. Rev. 186 (1969) 1319 [SPIRES]. zbMATHCrossRefADSGoogle Scholar
  80. [80]
    T. Damour, M. Henneaux and H. Nicolai, Cosmological billiards, Class. Quant. Grav. 20 (2003) R145 [hep-th/0212256] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  81. [81]
    T. Damour and M. Henneaux, Chaos in superstring cosmology, Phys. Rev. Lett. 85 (2000) 920 [hep-th/0003139] [SPIRES].zbMATHCrossRefMathSciNetADSGoogle Scholar
  82. [82]
    J.P. Gauntlett, S. Kim, O. Varela and D. Waldram, Consistent supersymmetric Kaluza-Klein truncations with massive modes, JHEP 04 (2009) 102 [arXiv:0901.0676] [SPIRES].CrossRefADSGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2010

Authors and Affiliations

  1. 1.Center for Theoretical Physics, MITCambridgeU.S.A.
  2. 2.Chennai Mathematical Institute, SIPCOT IT ParkSiruseriIndia

Personalised recommendations