Lifshitz holography: the whole shebang

  • Wissam Chemissany
  • Ioannis Papadimitriou
Open Access
Regular Article - Theoretical Physics


We provide a general algorithm for constructing the holographic dictionary for any asymptotically locally Lifshitz background, with or without hyperscaling violation, and for any values of the dynamical exponents z and θ, as well as the vector hyperscaling violating exponent [1, 2], that are compatible with the null energy condition. The analysis is carried out for a very general bottom up model of gravity coupled to a massive vector field and a dilaton with arbitrary scalar couplings. The solution of the radial Hamilton-Jacobi equation is obtained recursively in the form of a graded expansion in eigenfunctions of two commuting operators [3], which are the appropriate generalization of the dilatation operator for non scale invariant and Lorentz violating boundary conditions. The Fefferman-Graham expansions, the sources and 1-point functions of the dual operators, the Ward identities, as well as the local counterterms required for holographic renormalization all follow from this asymptotic solution of the radial Hamilton-Jacobi equation. We also find a family of exact backgrounds with z > 1 and θ > 0 corresponding to a marginal deformation shifting the vector hyperscaling violating parameter and we present an example where the conformal anomaly contains the only z = 2 conformal invariant in d = 2 with four spatial derivatives.


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


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


  1. [1]
    B. Gouteraux and E. Kiritsis, Quantum critical lines in holographic phases with (un)broken symmetry, JHEP 04 (2013) 053 [arXiv:1212.2625] [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    J. Gath, J. Hartong, R. Monteiro and N.A. Obers, Holographic Models for Theories with Hyperscaling Violation, JHEP 04 (2013) 159 [arXiv:1212.3263] [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    W. Chemissany and I. Papadimitriou, Generalized dilatation operator method for non-relativistic holography, Phys. Lett. B 737 (2014) 272 [arXiv:1405.3965] [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    P. Koroteev and M. Libanov, On Existence of Self-Tuning Solutions in Static Braneworlds without Singularities, JHEP 02 (2008) 104 [arXiv:0712.1136] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  5. [5]
    S. Kachru, X. Liu and M. Mulligan, Gravity duals of Lifshitz-like fixed points, Phys. Rev. D 78 (2008) 106005 [arXiv:0808.1725] [INSPIRE].ADSMathSciNetGoogle Scholar
  6. [6]
    M. Taylor, Non-relativistic holography, arXiv:0812.0530 [INSPIRE].
  7. [7]
    K. Balasubramanian and J. McGreevy, Gravity duals for non-relativistic CFTs, Phys. Rev. Lett. 101 (2008) 061601 [arXiv:0804.4053] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  8. [8]
    D.T. Son, Toward an AdS/cold atoms correspondence: a geometric realization of the Schrödinger symmetry, Phys. Rev. D 78 (2008) 046003 [arXiv:0804.3972] [INSPIRE].ADSMathSciNetGoogle Scholar
  9. [9]
    C. Charmousis, B. Gouteraux, B.S. 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
  10. [10]
    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].ADSCrossRefGoogle Scholar
  11. [11]
    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
  12. [12]
    N. Ogawa, T. Takayanagi and T. Ugajin, Holographic Fermi Surfaces and Entanglement Entropy, JHEP 01 (2012) 125 [arXiv:1111.1023] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  13. [13]
    E. Shaghoulian, Holographic Entanglement Entropy and Fermi Surfaces, JHEP 05 (2012) 065 [arXiv:1112.2702] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  14. [14]
    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
  15. [15]
    N. Iizuka et al., Bianchi Attractors: A Classification of Extremal Black Brane Geometries, JHEP 07 (2012) 193 [arXiv:1201.4861] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  16. [16]
    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].ADSMathSciNetGoogle Scholar
  17. [17]
    H.J. Boonstra, K. Skenderis and P.K. Townsend, The domain wall/QFT correspondence, JHEP 01 (1999) 003 [hep-th/9807137] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  18. [18]
    T. Gherghetta and Y. Oz, Supergravity, nonconformal field theories and brane worlds, Phys. Rev. D 65 (2002) 046001 [hep-th/0106255] [INSPIRE].ADSMathSciNetGoogle Scholar
  19. [19]
    I. Kanitscheider, K. Skenderis and M. Taylor, Precision holography for non-conformal branes, JHEP 09 (2008) 094 [arXiv:0807.3324] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  20. [20]
    T. Wiseman and B. Withers, Holographic renormalization for coincident Dp-branes, JHEP 10 (2008) 037 [arXiv:0807.0755] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  21. [21]
    A.W. Peet and J. Polchinski, UV/IR relations in AdS dynamics, Phys. Rev. D 59 (1999) 065011 [hep-th/9809022] [INSPIRE].ADSMathSciNetGoogle Scholar
  22. [22]
    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
  23. [23]
    M.J. Duff, G.W. Gibbons and P.K. Townsend, Macroscopic superstrings as interpolating solitons, Phys. Lett. B 332 (1994) 321 [hep-th/9405124] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  24. [24]
    S.S. Gubser, Curvature singularities: The Good, the bad and the naked, Adv. Theor. Math. Phys. 4 (2000) 679 [hep-th/0002160] [INSPIRE].zbMATHMathSciNetGoogle Scholar
  25. [25]
    K. Balasubramanian and K. Narayan, Lifshitz spacetimes from AdS null and cosmological solutions, JHEP 08 (2010) 014 [arXiv:1005.3291] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  26. [26]
    P. Bueno, W. Chemissany and C.S. Shahbazi, On hvLif -like solutions in gauged Supergravity, Eur. Phys. J. C 74 (2014) 2684 [arXiv:1212.4826] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    K. Narayan, On Lifshitz scaling and hyperscaling violation in string theory, Phys. Rev. D 85 (2012) 106006 [arXiv:1202.5935] [INSPIRE].ADSGoogle Scholar
  28. [28]
    K. Narayan, AdS null deformations with inhomogeneities, Phys. Rev. D 86 (2012) 126004 [arXiv:1209.4348] [INSPIRE].ADSGoogle Scholar
  29. [29]
    S.F. Ross and O. Saremi, Holographic stress tensor for non-relativistic theories, JHEP 09 (2009) 009 [arXiv:0907.1846] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  30. [30]
    P. Hořava and C.M. Melby-Thompson, Anisotropic Conformal Infinity, Gen. Rel. Grav. 43 (2011) 1391 [arXiv:0909.3841] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  31. [31]
    S.F. Ross, Holography for asymptotically locally Lifshitz spacetimes, Class. Quant. Grav. 28 (2011) 215019 [arXiv:1107.4451] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    R.B. Mann and R. McNees, Holographic Renormalization for Asymptotically Lifshitz Spacetimes, JHEP 10 (2011) 129 [arXiv:1107.5792] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  33. [33]
    M. Baggio, J. de Boer and K. Holsheimer, Hamilton-Jacobi Renormalization for Lifshitz Spacetime, JHEP 01 (2012) 058 [arXiv:1107.5562] [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    M. Baggio, J. de Boer and K. Holsheimer, Anomalous Breaking of Anisotropic Scaling Symmetry in the Quantum Lifshitz Model, JHEP 07 (2012) 099 [arXiv:1112.6416] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    T. Griffin, P. Hořava and C.M. Melby-Thompson, Conformal Lifshitz Gravity from Holography, JHEP 05 (2012) 010 [arXiv:1112.5660] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    T. Griffin, P. Hořava and C.M. Melby-Thompson, Lifshitz Gravity for Lifshitz Holography, Phys. Rev. Lett. 110 (2013) 081602 [arXiv:1211.4872] [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    D. Cassani and A.F. Faedo, Constructing Lifshitz solutions from AdS, JHEP 05 (2011) 013 [arXiv:1102.5344] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  38. [38]
    W. Chemissany and J. Hartong, From D3-branes to Lifshitz Space-Times, Class. Quant. Grav. 28 (2011) 195011 [arXiv:1105.0612] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  39. [39]
    W. Chemissany, D. Geissbuhler, J. Hartong and B. Rollier, Holographic Renormalization for z=2 Lifshitz Space-Times from AdS, Class. Quant. Grav. 29 (2012) 235017 [arXiv:1205.5777] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  40. [40]
    I. Papadimitriou, Holographic Renormalization of general dilaton-axion gravity, JHEP 08 (2011) 119 [arXiv:1106.4826] [INSPIRE].ADSCrossRefGoogle Scholar
  41. [41]
    M.H. Christensen, J. Hartong, N.A. Obers and B. Rollier, Torsional Newton-Cartan Geometry and Lifshitz Holography, Phys. Rev. D 89 (2014) 061901 [arXiv:1311.4794] [INSPIRE].ADSGoogle Scholar
  42. [42]
    M.H. Christensen, J. Hartong, N.A. Obers and B. Rollier, Boundary Stress-Energy Tensor and Newton-Cartan Geometry in Lifshitz Holography, JHEP 01 (2014) 057 [arXiv:1311.6471] [INSPIRE].CrossRefMathSciNetGoogle Scholar
  43. [43]
    Y. Korovin, K. Skenderis and M. Taylor, Lifshitz as a deformation of Anti-de Sitter, JHEP 08 (2013) 026 [arXiv:1304.7776] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  44. [44]
    Y. Korovin, K. Skenderis and M. Taylor, Lifshitz from AdS at finite temperature and top down models, JHEP 11 (2013) 127 [arXiv:1306.3344] [INSPIRE].ADSCrossRefGoogle Scholar
  45. [45]
    I. Kanitscheider and K. Skenderis, Universal hydrodynamics of non-conformal branes, JHEP 04 (2009) 062 [arXiv:0901.1487] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  46. [46]
    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].ADSCrossRefMathSciNetGoogle Scholar
  47. [47]
    J. de Boer, E.P. Verlinde and H.L. Verlinde, On the holographic renormalization group, JHEP 08 (2000) 003 [hep-th/9912012] [INSPIRE].CrossRefGoogle Scholar
  48. [48]
    D. Martelli and W. Mueck, Holographic renormalization and Ward identities with the Hamilton-Jacobi method, Nucl. Phys. B 654 (2003) 248 [hep-th/0205061] [INSPIRE].ADSCrossRefGoogle Scholar
  49. [49]
    I. Papadimitriou and K. Skenderis, AdS/CFT correspondence and geometry, hep-th/0404176 [INSPIRE].
  50. [50]
    I. Papadimitriou, Holographic renormalization as a canonical transformation, JHEP 11 (2010) 014 [arXiv:1007.4592] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  51. [51]
    I. Papadimitriou and K. Skenderis, Thermodynamics of asymptotically locally AdS spacetimes, JHEP 08 (2005) 004 [hep-th/0505190] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  52. [52]
    M. Henningson and K. Skenderis, The Holographic Weyl anomaly, JHEP 07 (1998) 023 [hep-th/9806087] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  53. [53]
    S. de Haro, S.N. Solodukhin and K. Skenderis, Holographic reconstruction of space-time and renormalization in the AdS/CFT correspondence, Commun. Math. Phys. 217 (2001) 595 [hep-th/0002230] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  54. [54]
    K. Skenderis, Lecture notes on holographic renormalization, Class. Quant. Grav. 19 (2002) 5849 [hep-th/0209067] [INSPIRE].CrossRefzbMATHMathSciNetGoogle Scholar
  55. [55]
    R. Arnowitt, S. Deser and C.W. Misner, Canonical Variables for General Relativity, Phys. Rev. 117 (1960) 1595.ADSCrossRefzbMATHMathSciNetGoogle Scholar
  56. [56]
    D.Z. Freedman, C. Núñez, M. Schnabl and K. Skenderis, Fake supergravity and domain wall stability, Phys. Rev. D 69 (2004) 104027 [hep-th/0312055] [INSPIRE].ADSGoogle Scholar
  57. [57]
    I. Papadimitriou and K. Skenderis, Correlation functions in holographic RG flows, JHEP 10 (2004) 075 [hep-th/0407071] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  58. [58]
    A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, CRC Press, Boca Raton U.S.A. (2002).CrossRefGoogle Scholar
  59. [59]
    P. Breitenlohner and D.Z. Freedman, Positive energy in anti-de Sitter backgrounds and gauged extended supergravity, Physics 115 (1982) 197.MathSciNetGoogle Scholar
  60. [60]
    P. Breitenlohner and D.Z. Freedman, Stability in gauged extended supergravity, Annals Phys. 144 (1982) 249.ADSCrossRefzbMATHMathSciNetGoogle Scholar
  61. [61]
    U. Gürsoy and E. Kiritsis, Exploring improved holographic theories for QCD: Part I, JHEP 02 (2008) 032 [arXiv:0707.1324] [INSPIRE].CrossRefGoogle Scholar
  62. [62]
    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
  63. [63]
    M.C.N. Cheng, S.A. Hartnoll and C.A. Keeler, Deformations of Lifshitz holography, JHEP 03 (2010) 062 [arXiv:0912.2784] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  64. [64]
    K. Holsheimer, On the Marginally Relevant Operator in ‡ = 2 Lifshitz Holography, JHEP 03 (2014) 084 [arXiv:1311.4539] [INSPIRE].ADSCrossRefGoogle Scholar
  65. [65]
    T. Nutma, xTras: A field-theory inspired xAct package for mathematica, Comput. Phys. Commun. 185 (2014) 1719 [arXiv:1308.3493] [INSPIRE].ADSCrossRefGoogle Scholar
  66. [66]
    P. McFadden and K. Skenderis, Holographic Non-Gaussianity, JCAP 05 (2011) 013 [arXiv:1011.0452] [INSPIRE].ADSCrossRefGoogle Scholar
  67. [67]
    I. Papadimitriou and A. Taliotis, Riccati equations for holographic 2-point functions, JHEP 04 (2014) 194 [arXiv:1312.7876] [INSPIRE].ADSCrossRefGoogle Scholar
  68. [68]
    C. Charmousis, B. Gouteraux, B.S. 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
  69. [69]
    M. Henneaux and C. Teitelboim, Quantization of Gauge Systems, Princeton University Press, Princeton U.S.A. (1992).zbMATHGoogle Scholar
  70. [70]
    S.-T. Hong and K.D. Rothe, Hamilton-Jacobi quantization of a free particle on a (n-1) sphere, hep-th/0302112 [INSPIRE].
  71. [71]
    K.D. Rothe and F.G. Scholtz, On the Hamilton-Jacobi equation for second class constrained systems, Annals Phys. 308 (2003) 639 [hep-th/0308142] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  72. [72]
    H.J. Rothe and K.D. Rother, Classical and Quantum Dynamics of Constrained Hamiltonian Systems, World Scientific, Singapore (2010).zbMATHGoogle Scholar

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© The Author(s) 2015

Authors and Affiliations

  1. 1.Department of Physics and SITPStanford UniversityStanfordU.S.A.
  2. 2.Instituto de Física Teórica UAM/CSICUniversidad Autónoma de MadridMadridSpain

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