Phase transitions in 3D gravity and fractal dimension

  • Xi Dong
  • Shaun Maguire
  • Alexander Maloney
  • Henry Maxfield
Open Access
Regular Article - Theoretical Physics


We show that for three dimensional gravity with higher genus boundary conditions, if the theory possesses a sufficiently light scalar, there is a second order phase transition where the scalar field condenses. This three dimensional version of the holographic superconducting phase transition occurs even though the pure gravity solutions are locally AdS3. This is in addition to the first order Hawking-Page-like phase transitions between different locally AdS3 handlebodies. This implies that the Rényi entropies of holographic CFTs will undergo phase transitions as the Rényi parameter is varied, as long as the theory possesses a scalar operator which is lighter than a certain critical dimension. We show that this critical dimension has an elegant mathematical interpretation as the Hausdorff dimension of the limit set of a quotient group of AdS3, and use this to compute it, analytically near the boundary of moduli space and numerically in the interior of moduli space. We compare this to a CFT computation generalizing recent work of Belin, Keller and Zadeh, bounding the critical dimension using higher genus conformal blocks, and find a surprisingly good match.


AdS-CFT Correspondence Conformal and W Symmetry Conformal Field Theory 


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]
    J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    M. Bañados, C. Teitelboim and J. Zanelli, The black hole in three-dimensional space-time, Phys. Rev. Lett. 69 (1992) 1849 [hep-th/9204099] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    A. Strominger, Black hole entropy from near horizon microstates, JHEP 02 (1998) 009 [hep-th/9712251] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    J.D. Brown and M. Henneaux, Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity, Commun. Math. Phys. 104 (1986) 207 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    J.M. Maldacena and A. Strominger, AdS 3 black holes and a stringy exclusion principle, JHEP 12 (1998) 005 [hep-th/9804085] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  6. [6]
    R. Dijkgraaf, J.M. Maldacena, G.W. Moore and E.P. Verlinde, A black hole Farey tail, hep-th/0005003 [INSPIRE].
  7. [7]
    A. Maloney and E. Witten, Quantum Gravity Partition Functions in Three Dimensions, JHEP 02 (2010) 029 [arXiv:0712.0155] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    S.W. Hawking and D.N. Page, Thermodynamics of Black Holes in anti-de Sitter Space, Commun. Math. Phys. 87 (1983) 577 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  9. [9]
    D.R. Brill, Multi - black hole geometries in (2+1)-dimensional gravity, Phys. Rev. D 53 (1996) 4133 [gr-qc/9511022] [INSPIRE].
  10. [10]
    S. Aminneborg, I. Bengtsson, D. Brill, S. Holst and P. Peldan, Black holes and wormholes in (2+1)-dimensions, Class. Quant. Grav. 15 (1998) 627 [gr-qc/9707036] [INSPIRE].
  11. [11]
    K. Krasnov, Holography and Riemann surfaces, Adv. Theor. Math. Phys. 4 (2000) 929 [hep-th/0005106] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    M. Headrick, Entanglement Renyi entropies in holographic theories, Phys. Rev. D 82 (2010) 126010 [arXiv:1006.0047] [INSPIRE].ADSGoogle Scholar
  13. [13]
    J. Cardy, A. Maloney and H. Maxfield, A new handle on three-point coefficients: OPE asymptotics from genus two modular invariance, JHEP 10 (2017) 136 [arXiv:1705.05855] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    M. Cho, S. Collier and X. Yin, Genus Two Modular Bootstrap, arXiv:1705.05865 [INSPIRE].
  15. [15]
    C.A. Keller, G. Mathys and I.G. Zadeh, Bootstrapping Chiral CFTs at Genus Two, arXiv:1705.05862 [INSPIRE].
  16. [16]
    V. Balasubramanian, P. Hayden, A. Maloney, D. Marolf and S.F. Ross, Multiboundary Wormholes and Holographic Entanglement, Class. Quant. Grav. 31 (2014) 185015 [arXiv:1406.2663] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  17. [17]
    X. Yin, Partition Functions of Three-Dimensional Pure Gravity, Commun. Num. Theor. Phys. 2 (2008) 285 [arXiv:0710.2129] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    X. Yin, On Non-handlebody Instantons in 3D Gravity, JHEP 09 (2008) 120 [arXiv:0711.2803] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    H. Maxfield, S. Ross and B. Way, Holographic partition functions and phases for higher genus Riemann surfaces, Class. Quant. Grav. 33 (2016) 125018 [arXiv:1601.00980] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    S.S. Gubser, Breaking an Abelian gauge symmetry near a black hole horizon, Phys. Rev. D 78 (2008) 065034 [arXiv:0801.2977] [INSPIRE].ADSGoogle Scholar
  21. [21]
    S.A. Hartnoll, C.P. Herzog and G.T. Horowitz, Building a Holographic Superconductor, Phys. Rev. Lett. 101 (2008) 031601 [arXiv:0803.3295] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    A. Belin and A. Maloney, A New Instability of the Topological black hole, Class. Quant. Grav. 33 (2016) 215003 [arXiv:1412.0280] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 0406 (2004) P06002 [hep-th/0405152] [INSPIRE].zbMATHGoogle Scholar
  24. [24]
    A. Belin, A. Maloney and S. Matsuura, Holographic Phases of Renyi Entropies, JHEP 12 (2013) 050 [arXiv:1306.2640] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    A. Belin, L.-Y. Hung, A. Maloney and S. Matsuura, Charged Renyi entropies and holographic superconductors, JHEP 01 (2015) 059 [arXiv:1407.5630] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    M.A. Metlitski, C.A. Fuertes and S. Sachdev, Entanglement Entropy in the O(N) model, Phys. Rev. B 80 (2009) 115122 [arXiv:0904.4477] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    T. Faulkner, The Entanglement Renyi Entropies of Disjoint Intervals in AdS/CFT, arXiv:1303.7221 [INSPIRE].
  29. [29]
    T. Hartman, Entanglement Entropy at Large Central Charge, arXiv:1303.6955 [INSPIRE].
  30. [30]
    A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    T. Hartman, C.A. Keller and B. Stoica, Universal Spectrum of 2d Conformal Field Theory in the Large c Limit, JHEP 09 (2014) 118 [arXiv:1405.5137] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    H. Maxfield, Entanglement entropy in three dimensional gravity, JHEP 04 (2015) 031 [arXiv:1412.0687] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  33. [33]
    K. Skenderis and B.C. van Rees, Holography and wormholes in 2+1 dimensions, Commun. Math. Phys. 301 (2011) 583 [arXiv:0912.2090] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    A. Maloney, Geometric Microstates for the Three Dimensional Black Hole?, arXiv:1508.04079 [INSPIRE].
  35. [35]
    P. Bizoń and J. Jalmużna, Globally regular instability of AdS 3, Phys. Rev. Lett. 111 (2013) 041102 [arXiv:1306.0317] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    A. Belin, C.A. Keller and I.G. Zadeh, Genus two partition functions and Rényi entropies of large c conformal field theories, J. Phys. A 50 (2017) 435401 [arXiv:1704.08250] [INSPIRE].ADSzbMATHGoogle Scholar
  37. [37]
    M. Cho, S. Collier and X. Yin, Recursive Representations of Arbitrary Virasoro Conformal Blocks, arXiv:1703.09805 [INSPIRE].
  38. [38]
    T. Barrella, X. Dong, S.A. Hartnoll and V.L. Martin, Holographic entanglement beyond classical gravity, JHEP 09 (2013) 109 [arXiv:1306.4682] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  39. [39]
    M.R. Gaberdiel, C.A. Keller and R. Volpato, Genus Two Partition Functions of Chiral Conformal Field Theories, Commun. Num. Theor. Phys. 4 (2010) 295 [arXiv:1002.3371] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    A.B. Zamolodchikov, Conformal symmetry in two-dimensional space: recursion representation of conformal block, Theor. Math. Phys. 73 (1987) 1088.CrossRefGoogle Scholar
  41. [41]
    D. Harlow, J. Maltz and E. Witten, Analytic Continuation of Liouville Theory, JHEP 12 (2011) 071 [arXiv:1108.4417] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. [42]
    P.G. Zograf and L.A. Takhtadzhyan, On uniformization of Riemann surfaces and the Weil-Petersson metric on Teichmüller and Schottky spaces, Math. USSR Sb. 60 (1988) 297.MathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    M. Headrick, A. Maloney, E. Perlmutter and I.G. Zadeh, Rényi entropies, the analytic bootstrap and 3D quantum gravity at higher genus, JHEP 07 (2015) 059 [arXiv:1503.07111] [INSPIRE].ADSCrossRefGoogle Scholar
  44. [44]
    E. Witten, Three-Dimensional Gravity Revisited, arXiv:0706.3359 [INSPIRE].
  45. [45]
    P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J. Phys. A 42 (2009) 504005 [arXiv:0905.4013] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  46. [46]
    P. Kraus, A. Maloney, H. Maxfield, G.S. Ng and J.-q. Wu, Witten Diagrams for Torus Conformal Blocks, JHEP 09 (2017) 149 [arXiv:1706.00047] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  47. [47]
    A.B. Zamolodchikov, Conformal symmetry in two-dimensions: an explicit recurrence formula for the conformal partial wave amplitude, Commun. Math. Phys. 96 (1984) 419 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  48. [48]
    H. Maxfield, A view of the bulk from the worldline, arXiv:1712.00885 [INSPIRE].
  49. [49]
    C.-M. Chang and Y.-H. Lin, Bootstrap, universality and horizons, JHEP 10 (2016) 068 [arXiv:1604.01774] [INSPIRE].ADSCrossRefGoogle Scholar
  50. [50]
    H. Dorn and H.J. Otto, Two and three point functions in Liouville theory, Nucl. Phys. B 429 (1994) 375 [hep-th/9403141] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  51. [51]
    A.B. Zamolodchikov and A.B. Zamolodchikov, Structure constants and conformal bootstrap in Liouville field theory, Nucl. Phys. B 477 (1996) 577 [hep-th/9506136] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  52. [52]
    I.R. Klebanov and E. Witten, AdS/CFT correspondence and symmetry breaking, Nucl. Phys. B 556 (1999) 89 [hep-th/9905104] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  53. [53]
    D. Sullivan, The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math 50 (1979) 171.MathSciNetCrossRefzbMATHGoogle Scholar
  54. [54]
    S. Patterson, The Selberg zeta-function of a Kleinian group, in Number theory, trace formulas and discrete groups, Elsevier, (1989), pp. 409.Google Scholar
  55. [55]
    D. Mumford, C. Series and D. Wright, Indra’s pearls: the vision of Felix Klein, Cambridge University Press, (2002).Google Scholar
  56. [56]
    S.J. Patterson, The limit set of a Fuchsian group, Acta Math. 136 (1976) 241.MathSciNetCrossRefzbMATHGoogle Scholar
  57. [57]
    D. Sullivan et al., Related aspects of positivity in Riemannian geometry, J. Diff. Geom. 25 (1987) 327.MathSciNetCrossRefzbMATHGoogle Scholar
  58. [58]
    C.J. Bishop and P.W. Jones, Hausdorff dimension and Kleinian groups, Acta Math. 179 (1997) 1.MathSciNetCrossRefzbMATHGoogle Scholar
  59. [59]
    S. Giombi, A. Maloney and X. Yin, One-loop Partition Functions of 3D Gravity, JHEP 08 (2008) 007 [arXiv:0804.1773] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  60. [60]
    S.J. Patterson, P.A. Perry et al., The divisor of Selberg’s zeta function for Kleinian groups, Duke Math. J. 106 (2001) 321.MathSciNetCrossRefzbMATHGoogle Scholar
  61. [61]
    C.T. McMullen, Hausdorff dimension and conformal dynamics, III: Computation of dimension, Am. J. MAth. (1998) 691.Google Scholar
  62. [62]
    C.T. McMullen, Hausdorff dimension and conformal dynamics I: Strong convergence of Kleinian groups, J. Diff. Geom. 51 (1999) 471.MathSciNetCrossRefzbMATHGoogle Scholar
  63. [63]
    J. Dodziuk, T. Pignataro, B. Randol and D. Sullivan, Estimating small eigenvalues of Riemann surfaces, in The legacy of Sonya Kovalevskaya, volume 64, AMS (1987), pg. 93.Google Scholar
  64. [64]
    P. Calabrese, J. Cardy and E. Tonni, Entanglement negativity in quantum field theory, Phys. Rev. Lett. 109 (2012) 130502 [arXiv:1206.3092] [INSPIRE].ADSCrossRefGoogle Scholar
  65. [65]
    Y. Hou, On smooth moduli space of Riemann surfaces, arXiv:1610.03132.

Copyright information

© The Author(s) 2018

Authors and Affiliations

  • Xi Dong
    • 1
  • Shaun Maguire
    • 2
  • Alexander Maloney
    • 3
  • Henry Maxfield
    • 3
  1. 1.Department of PhysicsUniversity of CaliforniaSanta BarbaraU.S.A.
  2. 2.Institute for Quantum Information & MatterCalifornia Institute for TechnologyPasadenaU.S.A.
  3. 3.Department of PhysicsMcGill UniversityMontréalCanada

Personalised recommendations