Journal of High Energy Physics

, 2018:42 | Cite as

Toward holographic reconstruction of bulk geometry from lattice simulations

  • Enrico RinaldiEmail author
  • Evan Berkowitz
  • Masanori Hanada
  • Jonathan Maltz
  • Pavlos Vranas
Open Access
Regular Article - Theoretical Physics


A black hole described in SU(N ) gauge theory consists of N D-branes. By separating one of the D-branes from others and studying the interaction between them, the black hole geometry can be probed. In order to obtain quantitative results, we employ the lattice Monte Carlo simulation. As a proof of the concept, we perform an explicit calculation in the matrix model dual to the black zero-brane in type IIA string theory. We demonstrate this method actually works in the high temperature region, where the stringy correction is large. We argue possible dual gravity interpretations.


Black Holes in String Theory Lattice Quantum Field Theory Gauge-gravity correspondence M(atrix) Theories 


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]
    T. Eguchi and H. Kawai, Reduction of Dynamical Degrees of Freedom in the Large-N Gauge Theory, Phys. Rev. Lett. 48 (1982) 1063 [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    G. Bhanot, U.M. Heller and H. Neuberger, The Quenched Eguchi-Kawai Model, Phys. Lett. B 113 (1982) 47 [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    D.J. Gross and Y. Kitazawa, A Quenched Momentum Prescription for Large-N Theories, Nucl. Phys. B 206 (1982) 440 [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    G. Parisi, A Simple Expression for Planar Field Theories, Phys. Lett. B 112 (1982) 463 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    A. Gonzalez-Arroyo and M. Okawa, The Twisted Eguchi-Kawai Model: A Reduced Model for Large-N Lattice Gauge Theory, Phys. Rev. D 27 (1983) 2397 [INSPIRE].ADSGoogle Scholar
  6. [6]
    E. Witten, Bound states of strings and p-branes, Nucl. Phys. B 460 (1996) 335 [hep-th/9510135] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    T. Banks, W. Fischler, S.H. Shenker and L. Susskind, M theory as a matrix model: A Conjecture, Phys. Rev. D 55 (1997) 5112 [hep-th/9610043] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  8. [8]
    R.C. Myers, Dielectric branes, JHEP 12 (1999) 022 [hep-th/9910053] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    G. ’t Hooft, Dimensional reduction in quantum gravity, Conf. Proc. C 930308 (1993) 284 [gr-qc/9310026] [INSPIRE].
  10. [10]
    L. Susskind, The World as a hologram, J. Math. Phys. 36 (1995) 6377 [hep-th/9409089] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    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].
  12. [12]
    T. Banks, M.R. Douglas, G.T. Horowitz and E.J. Martinec, AdS dynamics from conformal field theory, hep-th/9808016 [INSPIRE].
  13. [13]
    A. Hamilton, D.N. Kabat, G. Lifschytz and D.A. Lowe, Holographic representation of local bulk operators, Phys. Rev. D 74 (2006) 066009 [hep-th/0606141] [INSPIRE].ADSMathSciNetGoogle Scholar
  14. [14]
    I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, Holography from Conformal Field Theory, JHEP 10 (2009) 079 [arXiv:0907.0151] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  15. [15]
    I. Heemskerk, D. Marolf, J. Polchinski and J. Sully, Bulk and Transhorizon Measurements in AdS/CFT, JHEP 10 (2012) 165 [arXiv:1201.3664] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    B. de Wit, J. Hoppe and H. Nicolai, On the Quantum Mechanics of Supermembranes, Nucl. Phys. B 305 (1988) 545 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    J.M. Maldacena, Probing near extremal black holes with D-branes, Phys. Rev. D 57 (1998) 3736 [hep-th/9705053] [INSPIRE].ADSMathSciNetGoogle Scholar
  18. [18]
    A.A. Tseytlin and S. Yankielowicz, Free energy of N = 4 super Yang-Mills in Higgs phase and nonextremal D3-brane interactions, Nucl. Phys. B 541 (1999) 145 [hep-th/9809032] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  19. [19]
    N. Dorey, T.J. Hollowood, V.V. Khoze, M.P. Mattis and S. Vandoren, Multi-instanton calculus and the AdS/CFT correspondence in N = 4 superconformal field theory, Nucl. Phys. B 552 (1999) 88 [hep-th/9901128] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    I.L. Buchbinder, A. Yu. Petrov and A.A. Tseytlin, Two loop N = 4 super Yang-Mills effective action and interaction between D3-branes, Nucl. Phys. B 621 (2002) 179 [hep-th/0110173] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    N. Iizuka, D.N. Kabat, G. Lifschytz and D.A. Lowe, Probing black holes in nonperturbative gauge theory, Phys. Rev. D 65 (2002) 024012 [hep-th/0108006] [INSPIRE].ADSMathSciNetGoogle Scholar
  22. [22]
    S.M. Kuzenko, Self-dual effective action of N = 4 SYM revisited, JHEP 03 (2005) 008 [hep-th/0410128] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  23. [23]
    F. Ferrari, Emergent Space and the Example of AdS 5 XS 5, Nucl. Phys. B 869 (2013) 31 [arXiv:1207.0886] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  24. [24]
    F. Ferrari, D-Brane Probes in the Matrix Model, Nucl. Phys. B 880 (2014) 290 [arXiv:1311.4520] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    I.L. Buchbinder, E.A. Ivanov and I.B. Samsonov, The low-energy N = 4 SYM effective action in diverse harmonic superspaces, Phys. Part. Nucl. 48 (2017) 333 [arXiv:1603.02768] [INSPIRE].CrossRefGoogle Scholar
  26. [26]
    J.H. Schwarz, Gauge Theories on the Coulomb branch, Subnucl. Ser. 52 (2017) 167 [arXiv:1408.0852] [INSPIRE].Google Scholar
  27. [27]
    V. Sahakian, Y. Tawabutr and C. Yan, Emergent spacetime & quantum entanglement in matrix theory, JHEP 08 (2017) 140 [arXiv:1705.01128] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  28. [28]
    A. Mollabashi, N. Shiba and T. Takayanagi, Entanglement between Two Interacting CFTs and Generalized Holographic Entanglement Entropy, JHEP 04 (2014) 185 [arXiv:1403.1393] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    A. Karch and C.F. Uhlemann, Holographic entanglement entropy and the internal space, Phys. Rev. D 91 (2015) 086005 [arXiv:1501.00003] [INSPIRE].ADSGoogle Scholar
  30. [30]
    D. Berenstein and E. Dzienkowski, Numerical Evidence for Firewalls, arXiv:1311.1168 [INSPIRE].
  31. [31]
    C. Asplund, D. Berenstein and D. Trancanelli, Evidence for fast thermalization in the plane-wave matrix model, Phys. Rev. Lett. 107 (2011) 171602 [arXiv:1104.5469] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    E. Berkowitz, M. Hanada and J. Maltz, Chaos in Matrix Models and Black Hole Evaporation, Phys. Rev. D 94 (2016) 126009 [arXiv:1602.01473] [INSPIRE].ADSzbMATHGoogle Scholar
  33. [33]
    E. Berkowitz, M. Hanada and J. Maltz, A microscopic description of black hole evaporation via holography, Int. J. Mod. Phys. D 25 (2016) 1644002 [arXiv:1603.03055] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  34. [34]
    F. Ferrari, Black Hole Horizons and Bose-Einstein Condensation, arXiv:1601.08120 [INSPIRE].
  35. [35]
    L. Kofman, A.D. Linde, X. Liu, A. Maloney, L. McAllister and E. Silverstein, Beauty is attractive: Moduli trapping at enhanced symmetry points, JHEP 05 (2004) 030 [hep-th/0403001] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  36. [36]
    U.H. Danielsson, G. Ferretti and B. Sundborg, D particle dynamics and bound states, Int. J. Mod. Phys. A 11 (1996) 5463 [hep-th/9603081] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    D.N. Kabat and P. Pouliot, A Comment on zero-brane quantum mechanics, Phys. Rev. Lett. 77 (1996) 1004 [hep-th/9603127] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    M. Hanada, A simulation code prepared for the Monte Carlo String/M-theory Collaboration, downloadable at
  39. [39]
    T. Azeyanagi, M. Hanada, T. Hirata and H. Shimada, On the shape of a D-brane bound state and its topology change, JHEP 03 (2009) 121 [arXiv:0901.4073] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  40. [40]
    M. Berkooz and M.R. Douglas, Five-branes in M(atrix) theory, Phys. Lett. B 395 (1997) 196 [hep-th/9610236] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  41. [41]
    A. Karch and E. Katz, Adding flavor to AdS/CFT, JHEP 06 (2002) 043 [hep-th/0205236] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. [42]
    V.G. Filev and D. O’Connor, A Computer Test of Holographic Flavour Dynamics, JHEP 05 (2016) 122 [arXiv:1512.02536] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  43. [43]
    Y. Asano, V.G. Filev, S. Kováčik and D. O’Connor, The flavoured BFSS model at high temperature, JHEP 01 (2017) 113 [arXiv:1605.05597] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  44. [44]
    Y. Asano, V.G. Filev, S. Kováčik and D. O’Connor, A Computer Test of Holographic Flavour Dynamics II, arXiv:1612.09281 [INSPIRE].
  45. [45]
    J. Babington, J. Erdmenger, N.J. Evans, Z. Guralnik and I. Kirsch, Chiral symmetry breaking and pions in nonsupersymmetric gauge / gravity duals, Phys. Rev. D 69 (2004) 066007 [hep-th/0306018] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  46. [46]
    C. Hoyos-Badajoz, K. Landsteiner and S. Montero, Holographic meson melting, JHEP 04 (2007) 031 [hep-th/0612169] [INSPIRE].ADSCrossRefGoogle Scholar
  47. [47]
    D. Mateos, R.C. Myers and R.M. Thomson, Holographic phase transitions with fundamental matter, Phys. Rev. Lett. 97 (2006) 091601 [hep-th/0605046] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  48. [48]
    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
  49. [49]
    K.N. Anagnostopoulos, M. Hanada, J. Nishimura and S. Takeuchi, Monte Carlo studies of supersymmetric matrix quantum mechanics with sixteen supercharges at finite temperature, Phys. Rev. Lett. 100 (2008) 021601 [arXiv:0707.4454] [INSPIRE].ADSCrossRefGoogle Scholar
  50. [50]
    S. Catterall and T. Wiseman, Black hole thermodynamics from simulations of lattice Yang-Mills theory, Phys. Rev. D 78 (2008) 041502 [arXiv:0803.4273] [INSPIRE].ADSMathSciNetGoogle Scholar
  51. [51]
    M. Hanada, Y. Hyakutake, J. Nishimura and S. Takeuchi, Higher derivative corrections to black hole thermodynamics from supersymmetric matrix quantum mechanics, Phys. Rev. Lett. 102 (2009) 191602 [arXiv:0811.3102] [INSPIRE].ADSCrossRefGoogle Scholar
  52. [52]
    M. Hanada, Y. Hyakutake, G. Ishiki and J. Nishimura, Holographic description of quantum black hole on a computer, Science 344 (2014) 882 [arXiv:1311.5607] [INSPIRE].ADSCrossRefGoogle Scholar
  53. [53]
    D. Kadoh and S. Kamata, Gauge/gravity duality and lattice simulations of one dimensional SYM with sixteen supercharges, arXiv:1503.08499 [INSPIRE].
  54. [54]
    M. Hanada, Y. Hyakutake, G. Ishiki and J. Nishimura, Numerical tests of the gauge/gravity duality conjecture for D0-branes at finite temperature and finite N, Phys. Rev. D 94 (2016) 086010 [arXiv:1603.00538] [INSPIRE].ADSGoogle Scholar
  55. [55]
    E. Berkowitz, E. Rinaldi, M. Hanada, G. Ishiki, S. Shimasaki and P. Vranas, Precision lattice test of the gauge/gravity duality at large-N , Phys. Rev. D 94 (2016) 094501 [arXiv:1606.04951] [INSPIRE].ADSGoogle Scholar
  56. [56]
    Y. Sekino and L. Susskind, Fast Scramblers, JHEP 10 (2008) 065 [arXiv:0808.2096] [INSPIRE].ADSCrossRefGoogle Scholar
  57. [57]
    Y. Hyakutake, Quantum near-horizon geometry of a black 0-brane, PTEP 2014 (2014) 033B04 [arXiv:1311.7526] [INSPIRE].
  58. [58]
    S.D. Mathur, The Fuzzball proposal for black holes: An Elementary review, Fortsch. Phys. 53 (2005) 793 [hep-th/0502050] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  59. [59]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  60. [60]
    M. Hanada and J. Maltz, A proposal of the gauge theory description of the small Schwarzschild black hole in AdS 5×S 5, JHEP 02 (2017) 012 [arXiv:1608.03276] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  61. [61]
    T. Banks, W. Fischler, I.R. Klebanov and L. Susskind, Schwarzschild black holes from matrix theory, Phys. Rev. Lett. 80 (1998) 226 [hep-th/9709091] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Nuclear and Chemical Sciences DivisionLawrence Livermore National LaboratoryLivermoreU.S.A.
  2. 2.RIKEN-BNL Research CenterBrookhaven National LaboratoryUptonU.S.A.
  3. 3.Institut für Kernphysik and Institute for Advanced Simulation, Forschungszentrum JülichJülichGermany
  4. 4.Stanford Institute for Theoretical PhysicsStanford UniversityStanfordU.S.A.
  5. 5.Yukawa Institute for Theoretical PhysicsKyoto UniversityKyotoJapan
  6. 6.The Hakubi Center for Advanced ResearchKyoto UniversityKyotoJapan
  7. 7.Berkeley Center for Theoretical PhysicsUniversity of California at BerkeleyBerkeleyU.S.A.
  8. 8.Nuclear Science DivisionLawrence Berkeley National LaboratoryBerkeleyU.S.A.

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