Journal of High Energy Physics

, 2012:61 | Cite as

Hyperscaling violation: a unified frame for effective holographic theories

  • Bom Soo KimEmail author


We investigate systematic classifications of low energy and lower dimensional effective holographic theories with Lifshitz and Schrödinger scaling symmetries only using metrics in terms of hyperscaling violation (θ) and dynamical (z) exponents. Their consistent parameter spaces are constrained by null energy and positive specific heat conditions, whose validity is explicitly checked against a previously known result. From dimensional reductions of many microscopic string solutions, we observe the classifications are tied with the number of scales in the original microscopic theories. Conformal theories do not generate a nontrivial θ for a simple sphere reduction. Theories with Lifshitz scaling with one scale are completely fixed by θ and z, and have a universal emblackening factor at finite temperature. Dimensional reduction of intersecting M2-M5 requires, we call, spatial anisotropic exponents (#), along with z = 1, θ = 0, because of another scale. Theories with Schrödinger scaling show similar simple classifications at zero temperature, while require more care due to an additional parameter being a thermodynamic variable at finite temperature.


Gauge-gravity correspondence AdS-CFT Correspondence Holography and condensed matter physics (AdS/CMT) D-branes 


  1. [1]
    S. Sachdev, The quantum phases of matter, arXiv:1203.4565 [INSPIRE].
  2. [2]
    D.S. Fisher, Scaling and critical slowing down in random-field Ising systems, Phys. Rev. Lett. 56 (1986) 416 [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    S. Sachdev, Quantum phase transitions, 2nd edition, Cambridge University Press, Cambridge U.K. (2011).zbMATHCrossRefGoogle Scholar
  4. [4]
    C. Charmousis, B. Gouteraux, B. Kim, E. Kiritsis and R. Meyer, Effective holographic theories for low-temperature condensed matter systems, JHEP 11 (2010) 151 [arXiv:1005.4690] [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    B. Gouteraux and E. Kiritsis, Generalized holographic quantum criticality at finite density, JHEP 12 (2011) 036 [arXiv:1107.2116] [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    N. Ogawa, T. Takayanagi and T. Ugajin, Holographic Fermi surfaces and entanglement entropy, JHEP 01 (2012) 125 [arXiv:1111.1023] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    L. Huijse, S. Sachdev and B. Swingle, Hidden Fermi surfaces in compressible states of gauge-gravity duality, Phys. Rev. B 85 (2012) 035121 [arXiv:1112.0573] [INSPIRE].ADSGoogle Scholar
  8. [8]
    X. Dong, S. Harrison, S. Kachru, G. Torroba and H. Wang, Aspects of holography for theories with hyperscaling violation, JHEP 06 (2012) 041 [arXiv:1201.1905] [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    E. Shaghoulian, Holographic entanglement entropy and Fermi surfaces, JHEP 05 (2012) 065 [arXiv:1112.2702] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    K. Narayan, On Lifshitz scaling and hyperscaling violation in string theory, Phys. Rev. D 85 (2012) 106006 [arXiv:1202.5935] [INSPIRE].ADSGoogle Scholar
  11. [11]
    B.S. Kim, Schrödinger Holography with and without hyperscaling violation, JHEP 06 (2012) 116 [arXiv:1202.6062] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    H. Singh, Lifshitz/Schrödinger Dp-branes and dynamical exponents, JHEP 07 (2012) 082 [arXiv:1202.6533] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    S.A. Hartnoll and E. Shaghoulian, Spectral weight in holographic scaling geometries, JHEP 07 (2012) 078 [arXiv:1203.4236] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    P. Dey and S. Roy, Lifshitz-like space-time from intersecting branes in string/M theory, JHEP 06 (2012) 129 [arXiv:1203.5381] [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    Y.S. Myung and T. Moon, Quasinormal frequencies and thermodynamic quantities for the Lifshitz black holes, Phys. Rev. D 86 (2012) 024006 [arXiv:1204.2116] [INSPIRE].ADSGoogle Scholar
  16. [16]
    P. Dey and S. Roy, Intersecting D-branes and Lifshitz-like space-time, Phys. Rev. D 86 (2012) 066009 [arXiv:1204.4858] [INSPIRE].ADSGoogle Scholar
  17. [17]
    E. Perlmutter, Hyperscaling violation from supergravity, JHEP 06 (2012) 165 [arXiv:1205.0242] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    M. Cadoni and S. Mignemi, Phase transition and hyperscaling violation for scalar black branes, JHEP 06 (2012) 056 [arXiv:1205.0412] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    C. Charmousis, B. Gouteraux and E. Kiritsis, Higher-derivative scalar-vector-tensor theories: black holes, Galileons, singularity cloaking and holography, JHEP 09 (2012) 011 [arXiv:1206.1499] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    M. Ammon, M. Kaminski and A. Karch, Hyperscaling-violation on probe D-branes, arXiv:1207.1726 [INSPIRE].
  21. [21]
    E. Kiritsis, Lorentz violation, gravity, dissipation and holography, arXiv:1207.2325 [INSPIRE].
  22. [22]
    J. Bhattacharya, S. Cremonini and A. Sinkovics, On the IR completion of geometries with hyperscaling violation, arXiv:1208.1752 [INSPIRE].
  23. [23]
    P. Dey and S. Roy, Holographic entanglement entropy of the near horizon 1/4 BPS F -Dp bound states, arXiv:1208.1820 [INSPIRE].
  24. [24]
    N. Kundu, P. Narayan, N. Sircar and S.P. Trivedi, Entangled dilaton dyons, arXiv:1208.2008 [INSPIRE].
  25. [25]
    M. Kulaxizi, A. Parnachev and K. Schalm, On holographic entanglement entropy of charged matter, JHEP 10 (2012) 098 [arXiv:1208.2937] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    M. Alishahiha and H. Yavartanoo, On holography with hyperscaling violation, arXiv:1208.6197 [INSPIRE].
  27. [27]
    C. Park, Membrane paradigm in the Einstein-dilaton theory, arXiv:1209.0842 [INSPIRE].
  28. [28]
    P. Dey and S. Roy, Lifshitz metric with hyperscaling violation from N S5-Dp states in string theory, arXiv:1209.1049 [INSPIRE].
  29. [29]
    J. Sadeghi, B. Pourhassan and A. Asadi, Thermodynamics of string black hole with hyperscaling violation, arXiv:1209.1235 [INSPIRE].
  30. [30]
    J. Sadeghi, B. Pourhasan and F. Pourasadollah, Schrödinger black holes with hyperscaling violation, arXiv:1209.1874 [INSPIRE].
  31. [31]
    S.S. Pal, Fermi-like liquid from Einstein-DBI-dilaton system, arXiv:1209.3559 [INSPIRE].
  32. [32]
    M. Alishahiha, E. O Colgain and H. Yavartanoo, Charged black branes with hyperscaling violating factor, arXiv:1209.3946 [INSPIRE].
  33. [33]
    P. Bueno, W. Chemissany, P. Meessen, T. Ortín and C. Shahbazi, Lifshitz-like solutions with hyperscaling violation in ungauged supergravity, arXiv:1209.4047 [INSPIRE].
  34. [34]
    K. Narayan, AdS null deformations with inhomogeneities, arXiv:1209.4348 [INSPIRE].
  35. [35]
    M. Cadoni and M. Serra, Hyperscaling violation for scalar black branes in arbitrary dimensions, arXiv:1209.4484 [INSPIRE].
  36. [36]
    J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [hep-th/9711200] [INSPIRE].MathSciNetADSzbMATHGoogle Scholar
  37. [37]
    O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri and Y. Oz, Large-N field theories, string theory and gravity, Phys. Rept. 323 (2000) 183 [hep-th/9905111] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  38. [38]
    L. Huijse and S. Sachdev, Fermi surfaces and gauge-gravity duality, Phys. Rev. D 84 (2011) 026001 [arXiv:1104.5022] [INSPIRE].ADSGoogle Scholar
  39. [39]
    S. Sachdev, Compressible quantum phases from conformal field theories in 2 + 1 dimensions, arXiv:1209.1637 [INSPIRE].
  40. [40]
    S.-S. Lee, A non-Fermi liquid from a charged black hole: a critical Fermi ball, Phys. Rev. D 79 (2009) 086006 [arXiv:0809.3402] [INSPIRE].ADSGoogle Scholar
  41. [41]
    H. Liu, J. McGreevy and D. Vegh, Non-Fermi liquids from holography, Phys. Rev. D 83 (2011) 065029 [arXiv:0903.2477] [INSPIRE].ADSGoogle Scholar
  42. [42]
    M. Cubrovic, J. Zaanen and K. Schalm, String theory, quantum phase transitions and the emergent Fermi-liquid, Science 325 (2009) 439 [arXiv:0904.1993] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  43. [43]
    T. Faulkner, H. Liu, J. McGreevy and D. Vegh, Emergent quantum criticality, Fermi surfaces and AdS 2, Phys. Rev. D 83 (2011) 125002 [arXiv:0907.2694] [INSPIRE].ADSGoogle Scholar
  44. [44]
    N. Doiron-Leyraud et al., Quantum oscillations and the Fermi surface in an underdoped high-T c superconductor, Nature 447 (2007) 565.ADSCrossRefGoogle Scholar
  45. [45]
    J. Eisert, M. Cramer and M. Plenio, Area laws for the entanglement entropyA review, Rev. Mod. Phys. 82 (2010) 277 [arXiv:0808.3773] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  46. [46]
    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  47. [47]
    S. Ryu and T. Takayanagi, Aspects of holographic entanglement entropy, JHEP 08 (2006) 045 [hep-th/0605073] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  48. [48]
    V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  49. [49]
    T. Takayanagi, Entanglement entropy from a holographic viewpoint, Class. Quant. Grav. 29 (2012) 153001 [arXiv:1204.2450] [INSPIRE].ADSCrossRefGoogle Scholar
  50. [50]
    K. Balasubramanian and J. McGreevy, An analytic Lifshitz black hole, Phys. Rev. D 80 (2009) 104039 [arXiv:0909.0263] [INSPIRE].MathSciNetADSGoogle Scholar
  51. [51]
    B. Gouteraux, B.S. Kim and R. Meyer, Charged dilatonic black holes and their transport properties, Fortsch. Phys. 59 (2011) 723 [arXiv:1102.4440] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  52. [52]
    R. Meyer, B. Gouteraux and B.S. Kim, Strange metallic behaviour and the thermodynamics of charged dilatonic black holes, Fortsch. Phys. 59 (2011) 741 [arXiv:1102.4433] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  53. [53]
    A. Hashimoto and N. Itzhaki, Noncommutative Yang-Mills and the AdS/CFT correspondence, Phys. Lett. B 465 (1999) 142 [hep-th/9907166] [INSPIRE].MathSciNetADSGoogle Scholar
  54. [54]
    J.M. Maldacena and J.G. Russo, Large-N limit of noncommutative gauge theories, JHEP 09 (1999) 025 [hep-th/9908134] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  55. [55]
    M. Alishahiha, Y. Oz and J.G. Russo, Supergravity and light-like noncommutativity, JHEP 09 (2000) 002 [hep-th/0007215] [INSPIRE].MathSciNetADSGoogle Scholar
  56. [56]
    V.E. Hubeny, M. Rangamani and S.F. Ross, Causal structures and holography, JHEP 07 (2005) 037 [hep-th/0504034] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  57. [57]
    A. Bergman, K. Dasgupta, O.J. Ganor, J.L. Karczmarek and G. Rajesh, Nonlocal field theories and their gravity duals, Phys. Rev. D 65 (2002) 066005 [hep-th/0103090] [INSPIRE].MathSciNetADSGoogle Scholar
  58. [58]
    A. Bergman and O.J. Ganor, Dipoles, twists and noncommutative gauge theory, JHEP 10 (2000) 018 [hep-th/0008030] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  59. [59]
    O.J. Ganor, A new Lorentz violating nonlocal field theory from string-theory, Phys. Rev. D 75 (2007) 025002 [hep-th/0609107] [INSPIRE].MathSciNetADSGoogle Scholar
  60. [60]
    O.J. Ganor, A. Hashimoto, S. Jue, B.S. Kim and A. Ndirango, Aspects of Puff field theory, JHEP 08 (2007) 035 [hep-th/0702030] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  61. [61]
    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
  62. [62]
    M. Taylor, Non-relativistic holography, arXiv:0812.0530 [INSPIRE].
  63. [63]
    C. Charmousis, B. Gouteraux and J. Soda, Einstein-Maxwell-dilaton theories with a Liouville potential, Phys. Rev. D 80 (2009) 024028 [arXiv:0905.3337] [INSPIRE].MathSciNetADSGoogle Scholar
  64. [64]
    S.S. Gubser and F.D. Rocha, Peculiar properties of a charged dilatonic black hole in AdS 5, Phys. Rev. D 81 (2010) 046001 [arXiv:0911.2898] [INSPIRE].MathSciNetADSGoogle Scholar
  65. [65]
    K. Goldstein, S. Kachru, S. Prakash and S.P. Trivedi, Holography of charged dilaton black holes, JHEP 08 (2010) 078 [arXiv:0911.3586] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  66. [66]
    M. Cadoni, G. D’Appollonio and P. Pani, Phase transitions between Reissner-Nordstrom and dilatonic black holes in 4D AdS spacetime, JHEP 03 (2010) 100 [arXiv:0912.3520] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  67. [67]
    M. Cadoni and P. Pani, Holography of charged dilatonic black branes at finite temperature, JHEP 04 (2011) 049 [arXiv:1102.3820] [INSPIRE].ADSCrossRefGoogle Scholar
  68. [68]
    G. Bertoldi, B.A. Burrington and A.W. Peet, Thermal behavior of charged dilatonic black branes in AdS and UV completions of Lifshitz-like geometries, Phys. Rev. D 82 (2010) 106013 [arXiv:1007.1464] [INSPIRE].ADSGoogle Scholar
  69. [69]
    G. Bertoldi, B.A. Burrington, A.W. Peet and I.G. Zadeh, Lifshitz-like black brane thermodynamics in higher dimensions, Phys. Rev. D 83 (2011) 126006 [arXiv:1101.1980] [INSPIRE].ADSGoogle Scholar
  70. [70]
    K. Goldstein et al., Holography of dyonic dilaton black branes, JHEP 10 (2010) 027 [arXiv:1007.2490] [INSPIRE].ADSCrossRefGoogle Scholar
  71. [71]
    N. Iizuka, N. Kundu, P. Narayan and S.P. Trivedi, Holographic Fermi and non-Fermi liquids with transitions in dilaton gravity, JHEP 01 (2012) 094 [arXiv:1105.1162] [INSPIRE].ADSCrossRefGoogle Scholar
  72. [72]
    B.-H. Lee, D.-W. Pang and C. Park, Strange metallic behavior in anisotropic background, JHEP 07 (2010) 057 [arXiv:1006.1719] [INSPIRE].ADSCrossRefGoogle Scholar
  73. [73]
    P. Berglund, J. Bhattacharyya and D. Mattingly, Charged dilatonic AdS black branes in arbitrary dimensions, JHEP 08 (2012) 042 [arXiv:1107.3096] [INSPIRE].ADSCrossRefGoogle Scholar
  74. [74]
    B. Gouteraux, J. Smolic, M. Smolic, K. Skenderis and M. Taylor, Holography for Einstein-Maxwell-dilaton theories from generalized dimensional reduction, JHEP 01 (2012) 089 [arXiv:1110.2320] [INSPIRE].ADSCrossRefGoogle Scholar
  75. [75]
    S.S. Gubser, Curvature singularities: the good, the bad and the naked, Adv. Theor. Math. Phys. 4 (2000) 679 [hep-th/0002160] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  76. [76]
    C. Hoyos and P. Koroteev, On the null energy condition and causality in Lifshitz holography, Phys. Rev. D 82 (2010) 084002 [Erratum ibid. D 82 (2010) 109905] [arXiv:1007.1428] [INSPIRE].ADSGoogle Scholar
  77. [77]
    B. Swingle and T. Senthil, Universal crossovers between entanglement entropy and thermal entropy, arXiv:1112.1069 [INSPIRE].
  78. [78]
    U. Gürsoy, E. Kiritsis and F. Nitti, Exploring improved holographic theories for QCD: part II, JHEP 02 (2008) 019 [arXiv:0707.1349] [INSPIRE].CrossRefGoogle Scholar
  79. [79]
    U. Gürsoy, E. Kiritsis, L. Mazzanti and F. Nitti, Holography and thermodynamics of 5D dilaton-gravity, JHEP 05 (2009) 033 [arXiv:0812.0792] [INSPIRE].CrossRefGoogle Scholar
  80. [80]
    C. Charmousis, R. Emparan and R. Gregory, Selfgravity of brane worlds: a new hierarchy twist, JHEP 05 (2001) 026 [hep-th/0101198] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  81. [81]
    D. Son, Toward an AdS/cold atoms correspondence: a geometric realization of the Schrödinger symmetry, Phys. Rev. D 78 (2008) 046003 [arXiv:0804.3972] [INSPIRE].MathSciNetADSGoogle Scholar
  82. [82]
    K. Balasubramanian and J. McGreevy, Gravity duals for non-relativistic CFTs, Phys. Rev. Lett. 101 (2008) 061601 [arXiv:0804.4053] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  83. [83]
    C.P. Herzog, M. Rangamani and S.F. Ross, Heating up galilean holography, JHEP 11 (2008) 080 [arXiv:0807.1099] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  84. [84]
    J. Maldacena, D. Martelli and Y. Tachikawa, Comments on string theory backgrounds with non-relativistic conformal symmetry, JHEP 10 (2008) 072 [arXiv:0807.1100] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  85. [85]
    A. Adams, K. Balasubramanian and J. McGreevy, Hot spacetimes for cold atoms, JHEP 11 (2008) 059 [arXiv:0807.1111] [INSPIRE].ADSCrossRefGoogle Scholar
  86. [86]
    D. Yamada, Thermodynamics of black holes in Schrödinger space, Class. Quant. Grav. 26 (2009) 075006 [arXiv:0809.4928] [INSPIRE].ADSCrossRefGoogle Scholar
  87. [87]
    M. Ammon, C. Hoyos, A. O’Bannon and J.M. Wu, Holographic flavor transport in Schrödinger spacetime, JHEP 06 (2010) 012 [arXiv:1003.5913] [INSPIRE].ADSCrossRefGoogle Scholar
  88. [88]
    M. Alishahiha and O.J. Ganor, Twisted backgrounds, PP waves and nonlocal field theories, JHEP 03 (2003) 006 [hep-th/0301080] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  89. [89]
    E.G. Gimon, A. Hashimoto, V.E. Hubeny, O. Lunin and M. Rangamani, Black strings in asymptotically plane wave geometries, JHEP 08 (2003) 035 [hep-th/0306131] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  90. [90]
    L. Mazzucato, Y. Oz and S. Theisen, Non-relativistic branes, JHEP 04 (2009) 073 [arXiv:0810.3673] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  91. [91]
    C. Duval, M. Hassaine and P. Horvathy, The geometry of Schrödinger symmetry in gravity background/non-relativistic CFT, Annals Phys. 324 (2009) 1158 [arXiv:0809.3128] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  92. [92]
    S.A. Hartnoll and K. Yoshida, Families of IIB duals for nonrelativistic CFTs, JHEP 12 (2008) 071 [arXiv:0810.0298] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  93. [93]
    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] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  94. [94]
    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] [INSPIRE].ADSCrossRefGoogle Scholar
  95. [95]
    A. Donos and J.P. Gauntlett, Supersymmetric solutions for non-relativistic holography, JHEP 03 (2009) 138 [arXiv:0901.0818] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  96. [96]
    S.S. Pal, Non-relativistic supersymmetric Dp-branes, Class. Quant. Grav. 26 (2009) 245014 [arXiv:0904.3620] [INSPIRE].ADSCrossRefGoogle Scholar
  97. [97]
    N. Bobev, A. Kundu and K. Pilch, Supersymmetric IIB solutions with Schrödinger symmetry, JHEP 07 (2009) 107 [arXiv:0905.0673] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  98. [98]
    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] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  99. [99]
    E. O Colgain, O. Varela and H. Yavartanoo, Non-relativistic M-theory solutions based on Kähler-Einstein spaces, JHEP 07 (2009) 081 [arXiv:0906.0261] [INSPIRE].ADSCrossRefGoogle Scholar
  100. [100]
    S. Cremonesi, D. Melnikov and Y. Oz, Stability of asymptotically Schrödinger RN Black hole and superconductivity, JHEP 04 (2010) 048 [arXiv:0911.3806] [INSPIRE].ADSCrossRefGoogle Scholar
  101. [101]
    J. Jeong, H.-C. Kim, S. Lee, E. O Colgain and H. Yavartanoo, Schrödinger invariant solutions of M-theory with enhanced supersymmetry, JHEP 03 (2010) 034 [arXiv:0911.5281] [INSPIRE].ADSCrossRefGoogle Scholar
  102. [102]
    N. Banerjee, S. Dutta and D.P. Jatkar, Geometry and phase structure of non-relativistic branes, Class. Quant. Grav. 28 (2011) 165002 [arXiv:1102.0298] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  103. [103]
    P. Kraus and E. Perlmutter, Universality and exactness of Schrödinger geometries in string and M-theory, JHEP 05 (2011) 045 [arXiv:1102.1727] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  104. [104]
    H.-C. Kim, S. Kim, K. Lee and J. Park, Emergent Schrödinger geometries from mass-deformed CFT, JHEP 08 (2011) 111 [arXiv:1106.4309] [INSPIRE].MathSciNetADSGoogle Scholar
  105. [105]
    C.M. Brown and O. DeWolfe, The Godel-Schrödinger spacetime and stringy chronology protection, JHEP 01 (2012) 032 [arXiv:1110.3840] [INSPIRE].ADSCrossRefGoogle Scholar
  106. [106]
    S. Hyun, J. Jeong and B.S. Kim, Finite temperature aging holography, JHEP 03 (2012) 010 [arXiv:1108.5549] [INSPIRE].ADSCrossRefGoogle Scholar
  107. [107]
    S. Hyun, J. Jeong and B.S. Kim, Aging logarithmic conformal field theory: a holographic view, arXiv:1209.2417 [INSPIRE].
  108. [108]
    M. Guica, K. Skenderis, M. Taylor and B.C. van Rees, Holography for Schrödinger backgrounds, JHEP 02 (2011) 056 [arXiv:1008.1991] [INSPIRE].ADSCrossRefGoogle Scholar
  109. [109]
    K. Balasubramanian and J. McGreevy, The particle number in galilean holography, JHEP 01 (2011) 137 [arXiv:1007.2184] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  110. [110]
    N. Itzhaki, J.M. Maldacena, J. Sonnenschein and S. Yankielowicz, Supergravity and the large-N limit of theories with sixteen supercharges, Phys. Rev. D 58 (1998) 046004 [hep-th/9802042] [INSPIRE].MathSciNetADSGoogle Scholar
  111. [111]
    H. Boonstra, K. Skenderis and P. Townsend, The domain wall/QFT correspondence, JHEP 01 (1999) 003 [hep-th/9807137] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  112. [112]
    B.S. Kim and D. Yamada, Properties of Schrödinger black holes from AdS space, JHEP 07 (2011) 120 [arXiv:1008.3286] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  113. [113]
    N. Bobev and B.C. van Rees, Schrödinger deformations of AdS 3 × S 3, JHEP 08 (2011) 062 [arXiv:1102.2877] [INSPIRE].ADSCrossRefGoogle Scholar
  114. [114]
    R. Caldeira Costa and M. Taylor, Holography for chiral scale-invariant models, JHEP 02 (2011) 082 [arXiv:1010.4800] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  115. [115]
    W.D. Goldberger, AdS/CFT duality for non-relativistic field theory, JHEP 03 (2009) 069 [arXiv:0806.2867] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  116. [116]
    J.L. Barbon and C.A. Fuertes, On the spectrum of nonrelativistic AdS/CFT, JHEP 09 (2008) 030 [arXiv:0806.3244] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  117. [117]
    B.S. Kim, E. Kiritsis and C. Panagopoulos, Holographic quantum criticality and strange metal transport, New J. Phys. 14 (2012) 043045 [arXiv:1012.3464] [INSPIRE].ADSCrossRefGoogle Scholar
  118. [118]
    K.-Y. Kim and D.-W. Pang, Holographic DC conductivities from the open string metric, JHEP 09 (2011) 051 [arXiv:1108.3791] [INSPIRE].ADSCrossRefGoogle Scholar
  119. [119]
    K.B. Fadafan, Strange metals at finitet Hooft coupling, arXiv:1208.1855 [INSPIRE].
  120. [120]
    G.T. Horowitz, J.M. Maldacena and A. Strominger, Nonextremal black hole microstates and U duality, Phys. Lett. B 383 (1996) 151 [hep-th/9603109] [INSPIRE].MathSciNetADSGoogle Scholar
  121. [121]
    A.A. Tseytlin, Composite BPS configurations of p-branes in ten-dimensions and eleven-dimensions, Class. Quant. Grav. 14 (1997) 2085 [hep-th/9702163] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  122. [122]
    J.M. Maldacena and H. Ooguri, Strings in AdS 3 and the SL(2, \( \mathbb{R} \)) WZW model. Part 3. Correlation functions, Phys. Rev. D 65 (2002) 106006 [hep-th/0111180] [INSPIRE].MathSciNetADSGoogle Scholar
  123. [123]
    H.J. Boonstra, B. Peeters and K. Skenderis, Duality and asymptotic geometries, Phys. Lett. B 411 (1997) 59 [hep-th/9706192] [INSPIRE].MathSciNetADSGoogle Scholar
  124. [124]
    M. Cvetič, H. Lü and C. Pope, Space-times of boosted p-branes and CFT in infinite momentum frame, Nucl. Phys. B 545 (1999) 309 [hep-th/9810123] [INSPIRE].ADSCrossRefGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2012

Authors and Affiliations

  1. 1.Raymond and Beverly Sackler School of Physics and AstronomyTel Aviv UniversityTel AvivIsrael

Personalised recommendations