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Entanglement entropy, OTOC and bootstrap in 2D CFTs from Regge and light cone limits of multi-point conformal block

  • Yuya KusukiEmail author
  • Masamichi Miyaji
Open Access
Regular Article - Theoretical Physics
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Abstract

We explore the structures of light cone and Regge limit singularities of n-point Virasoro conformal blocks in c > 1 two-dimensional conformal field theories with no chiral primaries, using fusion matrix approach. These CFTs include not only holographic CFTs dual to classical gravity, but also their full quantum corrections, since this approach allows us to explore full 1/c corrections. As the important applications, we study time dependence of Renyi entropy after a local quench and out-of-time ordered correlator (OTOC) at late time.

We first show that, the n-th (n > 2) Renyi entropy after a local quench in our CFT grows logarithmically at late time, for any c and any conformal dimensions of excited primary. In particular, we find that this behavior is independent of c, contrary to the expectation that the finite c correction fixes the late time Renyi entropy to be constant. We also show that the constant part of the late time Renyi entropy is given by a monodromy matrix.

We also investigate OTOCs by using the monodromy matrix. We first rewrite the monodromy matrix in terms of fusion matrix explicitly. By this expression, we find that the OTOC decays exponentially in time, and the decay rates are divided into three patterns, depending on the dimensions of external operators. We note that our result is valid for any c > 1 and any external operator dimensions. Our monodromy matrix approach can be generalized to the Liouville theory and we show that the Liouville OTOC approaches constant in the late time regime.

We emphasize that, there is a number of other applications of the fusion and the monodromy matrix approaches, such as solving the conformal bootstrap equation. Therefore, it is tempting to believe that the fusion and monodromy matrix approaches provide a key to understanding the AdS/CFT correspondence.

Keywords

AdS-CFT Correspondence Conformal Field Theory 

Notes

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

References

  1. [1]
    S. Collier, Y.-H. Lin and X. Yin, Modular bootstrap revisited, JHEP 09 (2018) 061 [arXiv:1608.06241] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  2. [2]
    S. Collier, P. Kravchuk, Y.-H. Lin and X. Yin, Bootstrapping the spectral function: on the uniqueness of Liouville and the universality of BTZ, JHEP 09 (2018) 150 [arXiv:1702.00423] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  3. [3]
    A.L. Fitzpatrick, J. Kaplan and M.T. Walters, Virasoro conformal blocks and thermality from classical background fields, JHEP 11 (2015) 200 [arXiv:1501.05315] [INSPIRE].
  4. [4]
    K.B. Alkalaev and V.A. Belavin, Classical conformal blocks via AdS/CFT correspondence, JHEP 08 (2015) 049 [arXiv:1504.05943] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  5. [5]
    E. Hijano, P. Kraus, E. Perlmutter and R. Snively, Semiclassical Virasoro blocks from AdS 3 gravity, JHEP 12 (2015) 077 [arXiv:1508.04987] [INSPIRE].ADSzbMATHGoogle Scholar
  6. [6]
    C.T. Asplund, A. Bernamonti, F. Galli and T. Hartman, Holographic entanglement entropy from 2d CFT: heavy states and local quenches, JHEP 02 (2015) 171 [arXiv:1410.1392] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  7. [7]
    K.B. Alkalaev, Many-point classical conformal blocks and geodesic networks on the hyperbolic plane, JHEP 12 (2016) 070 [arXiv:1610.06717] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  8. [8]
    K.B. Alkalaev and V.A. Belavin, Holographic interpretation of 1-point toroidal block in the semiclassical limit, JHEP 06 (2016) 183 [arXiv:1603.08440] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  9. [9]
    K.B. Alkalaev and V.A. Belavin, Holographic duals of large-c torus conformal blocks, JHEP 10 (2017) 140 [arXiv:1707.09311] [INSPIRE].
  10. [10]
    H. Maxfield, A view of the bulk from the worldline, arXiv:1712.00885 [INSPIRE].
  11. [11]
    K. Alkalaev and V. Belavin, Large-c superconformal torus blocks, JHEP 08 (2018) 042 [arXiv:1805.12585] [INSPIRE].
  12. [12]
    Y. Hikida and T. Uetoko, Conformal blocks from Wilson lines with loop corrections, Phys. Rev. D 97 (2018) 086014 [arXiv:1801.08549] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  13. [13]
    Y. Hikida and T. Uetoko, Superconformal blocks from Wilson lines with loop corrections, JHEP 08 (2018) 101 [arXiv:1806.05836] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  14. [14]
    K. Alkalaev and M. Pavlov, Perturbative classical conformal blocks as Steiner trees on the hyperbolic disk, JHEP 02 (2019) 023 [arXiv:1810.07741] [INSPIRE].
  15. [15]
    H. Chen, A.L. Fitzpatrick, J. Kaplan and D. Li, The bulk-to-boundary propagator in black hole microstate backgrounds, JHEP 06 (2019) 107 [arXiv:1810.02436] [INSPIRE].zbMATHGoogle Scholar
  16. [16]
    Y. Kusuki, New properties of large-c conformal blocks from recursion relation, JHEP 07 (2018)010 [arXiv:1804.06171] [INSPIRE].
  17. [17]
    Y. Kusuki, Large-c Virasoro blocks from monodromy method beyond known limits, JHEP 08 (2018)161 [arXiv:1806.04352] [INSPIRE].
  18. [18]
    Y. Kusuki and T. Takayanagi, Renyi entropy for local quenches in 2D CFT from numerical conformal blocks, JHEP 01 (2018) 115 [arXiv:1711.09913] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  19. [19]
    Y. Kusuki, Light cone bootstrap in general 2D CFTs and entanglement from light cone singularity, JHEP 01 (2019) 025 [arXiv:1810.01335] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  20. [20]
    A.L. Fitzpatrick, J. Kaplan, D. Li and J. Wang, On information loss in AdS 3 /CFT 2, JHEP 05 (2016) 109 [arXiv:1603.08925] [INSPIRE].ADSGoogle Scholar
  21. [21]
    H. Chen, A.L. Fitzpatrick, J. Kaplan, D. Li and J. Wang, Degenerate operators and the 1/c expansion: Lorentzian resummations, high order computations and super-Virasoro blocks, JHEP 03 (2017) 167 [arXiv:1606.02659] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  22. [22]
    A.L. Fitzpatrick and J. Kaplan, On the late-time behavior of Virasoro blocks and a classification of semiclassical saddles, JHEP 04 (2017) 072 [arXiv:1609.07153] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  23. [23]
    H. Chen, C. Hussong, J. Kaplan and D. Li, A numerical approach to Virasoro blocks and the information paradox, JHEP 09 (2017) 102 [arXiv:1703.09727] [INSPIRE].ADSzbMATHGoogle Scholar
  24. [24]
    S. Collier, Y. Gobeil, H. Maxfield and E. Perlmutter, Quantum Regge trajectories and the Virasoro analytic bootstrap, JHEP 05 (2019) 212 [arXiv:1811.05710] [INSPIRE].ADSzbMATHGoogle Scholar
  25. [25]
    R. Rattazzi, V.S. Rychkov, E. Tonni and A. Vichi, Bounding scalar operator dimensions in 4D CFT, JHEP 12 (2008) 031 [arXiv:0807.0004] [INSPIRE].
  26. [26]
    V.S. Rychkov and A. Vichi, Universal constraints on conformal operator dimensions, Phys. Rev. D 80 (2009) 045006 [arXiv:0905.2211] [INSPIRE].ADSMathSciNetGoogle Scholar
  27. [27]
    F. Kos, D. Poland and D. Simmons-Duffin, Bootstrapping mixed correlators in the 3D Ising model, JHEP 11 (2014) 109 [arXiv:1406.4858] [INSPIRE].
  28. [28]
    D. Simmons-Duffin, A semidefinite program solver for the conformal bootstrap, JHEP 06 (2015) 174 [arXiv:1502.02033] [INSPIRE].
  29. [29]
    S. Hellerman, A universal inequality for CFT and quantum gravity, JHEP 08 (2011) 130 [arXiv:0902.2790] [INSPIRE].
  30. [30]
    D. Friedan and C.A. Keller, Constraints on 2d CFT partition functions, JHEP 10 (2013) 180 [arXiv:1307.6562] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  31. [31]
    J.L. Cardy, Operator content of two-dimensional conformally invariant theories, Nucl. Phys. B 270 (1986) 186 [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  32. [32]
    P. Kraus and A. Maloney, A Cardy formula for three-point coefficients or how the black hole got its spots, JHEP 05 (2017) 160 [arXiv:1608.03284] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  33. [33]
    D. Das, S. Datta and S. Pal, Universal asymptotics of three-point coefficients from elliptic representation of Virasoro blocks, Phys. Rev. D 98 (2018) 101901 [arXiv:1712.01842] [INSPIRE].ADSGoogle Scholar
  34. [34]
    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].ADSMathSciNetzbMATHGoogle Scholar
  35. [35]
    Y. Hikida, Y. Kusuki and T. Takayanagi, Eigenstate thermalization hypothesis and modular invariance of two-dimensional conformal field theories, Phys. Rev. D 98 (2018) 026003 [arXiv:1804.09658] [INSPIRE].ADSMathSciNetGoogle Scholar
  36. [36]
    A. Romero-Bermúdez, P. Sabella-Garnier and K. Schalm, A Cardy formula for off-diagonal three-point coefficients; or, how the geometry behind the horizon gets disentangled, JHEP 09 (2018) 005 [arXiv:1804.08899] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  37. [37]
    E.M. Brehm, D. Das and S. Datta, Probing thermality beyond the diagonal, Phys. Rev. D 98 (2018) 126015 [arXiv:1804.07924] [INSPIRE].ADSGoogle Scholar
  38. [38]
    B. Mukhametzhanov and A. Zhiboedov, Modular invariance, Tauberian theorems and microcanonical entropy, arXiv:1904.06359 [INSPIRE].
  39. [39]
    A.L. Fitzpatrick, J. Kaplan, D. Poland and D. Simmons-Duffin, The analytic bootstrap and AdS superhorizon locality, JHEP 12 (2013) 004 [arXiv:1212.3616] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  40. [40]
    Z. Komargodski and A. Zhiboedov, Convexity and liberation at large spin, JHEP 11 (2013) 140 [arXiv:1212.4103] [INSPIRE].ADSGoogle Scholar
  41. [41]
    L.F. Alday, A. Bissi and T. Lukowski, Large spin systematics in CFT, JHEP 11 (2015) 101 [arXiv:1502.07707] [INSPIRE].ADSzbMATHGoogle Scholar
  42. [42]
    A. Kaviraj, K. Sen and A. Sinha, Analytic bootstrap at large spin, JHEP 11 (2015) 083 [arXiv:1502.01437] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  43. [43]
    A. Kaviraj, K. Sen and A. Sinha, Universal anomalous dimensions at large spin and large twist, JHEP 07 (2015) 026 [arXiv:1504.00772] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  44. [44]
    L.F. Alday, Large spin perturbation theory for conformal field theories, Phys. Rev. Lett. 119 (2017) 111601 [arXiv:1611.01500] [INSPIRE].
  45. [45]
    D. Simmons-Duffin, The lightcone bootstrap and the spectrum of the 3d Ising CFT, JHEP 03 (2017) 086 [arXiv:1612.08471] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  46. [46]
    L.F. Alday and A. Zhiboedov, An algebraic approach to the analytic bootstrap, JHEP 04 (2017) 157 [arXiv:1510.08091] [INSPIRE].
  47. [47]
    C. Sleight and M. Taronna, Anomalous dimensions from crossing kernels, JHEP 11 (2018) 089 [arXiv:1807.05941] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  48. [48]
    A.L. Fitzpatrick, J. Kaplan and M.T. Walters, Universality of long-distance AdS physics from the CFT bootstrap, JHEP 08 (2014) 145 [arXiv:1403.6829] [INSPIRE].ADSGoogle Scholar
  49. [49]
    P. Banerjee, S. Datta and R. Sinha, Higher-point conformal blocks and entanglement entropy in heavy states, JHEP 05 (2016) 127 [arXiv:1601.06794] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  50. [50]
    V. Rosenhaus, Multipoint conformal blocks in the comb channel, JHEP 02 (2019) 142 [arXiv:1810.03244] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  51. [51]
    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].ADSMathSciNetzbMATHGoogle Scholar
  52. [52]
    P. Kraus and A. Sivaramakrishnan, Light-state dominance from the conformal bootstrap, arXiv:1812.02226 [INSPIRE].
  53. [53]
    S. He, T. Numasawa, T. Takayanagi and K. Watanabe, Quantum dimension as entanglement entropy in two dimensional conformal field theories, Phys. Rev. D 90 (2014) 041701 [arXiv:1403.0702] [INSPIRE].ADSGoogle Scholar
  54. [54]
    T. Numasawa, Scattering effect on entanglement propagation in RCFTs, JHEP 12 (2016) 061 [arXiv:1610.06181] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  55. [55]
    M. Nozaki, T. Numasawa and T. Takayanagi, Holographic local quenches and entanglement density, JHEP 05 (2013) 080 [arXiv:1302.5703] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  56. [56]
    P. Caputa, M. Nozaki and T. Takayanagi, Entanglement of local operators in large-N conformal field theories, PTEP 2014 (2014) 093B06 [arXiv:1405.5946] [INSPIRE].
  57. [57]
    J.R. David, S. Khetrapal and S.P. Kumar, Universal corrections to entanglement entropy of local quantum quenches, JHEP 08 (2016) 127 [arXiv:1605.05987] [INSPIRE].ADSGoogle Scholar
  58. [58]
    P. Caputa, Y. Kusuki, T. Takayanagi and K. Watanabe, Evolution of entanglement entropy in orbifold CFTs, J. Phys. A 50 (2017) 244001 [arXiv:1701.03110] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  59. [59]
    S. He, Conformal bootstrap to Renyi entropy in 2D Liouville and super-Liouville CFTs, Phys. Rev. D 99 (2019) 026005 [arXiv:1711.00624] [INSPIRE].ADSGoogle Scholar
  60. [60]
    W.-Z. Guo, S. He and Z.-X. Luo, Entanglement entropy in (1 + 1)D CFTs with multiple local excitations, JHEP 05 (2018) 154 [arXiv:1802.08815] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  61. [61]
    T. Shimaji, T. Takayanagi and Z. Wei, Holographic quantum circuits from splitting/joining local quenches, JHEP 03 (2019) 165 [arXiv:1812.01176] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  62. [62]
    L. Apolo, S. He, W. Song, J. Xu and J. Zheng, Entanglement and chaos in warped conformal field theories, JHEP 04 (2019) 009 [arXiv:1812.10456] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  63. [63]
    C.T. Asplund, A. Bernamonti, F. Galli and T. Hartman, Entanglement scrambling in 2d conformal field theory, JHEP 09 (2015) 110 [arXiv:1506.03772] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  64. [64]
    Y. Kusuki, T. Takayanagi and K. Umemoto, Holographic entanglement entropy on generic time slices, JHEP 06 (2017) 021 [arXiv:1703.00915] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  65. [65]
    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].ADSGoogle Scholar
  66. [66]
    A. Belin, A. Maloney and S. Matsuura, Holographic phases of Renyi entropies, JHEP 12 (2013) 050 [arXiv:1306.2640] [INSPIRE].ADSGoogle Scholar
  67. [67]
    A. Belin, L.-Y. Hung, A. Maloney and S. Matsuura, Charged Renyi entropies and holographic superconductors, JHEP 01 (2015) 059 [arXiv:1407.5630] [INSPIRE].ADSzbMATHGoogle Scholar
  68. [68]
    A. Belin, C.A. Keller and I.G. Zadeh, Genus two partition functions and Renyi entropies of large c conformal field theories, J. Phys. A 50 (2017) 435401 [arXiv:1704.08250] [INSPIRE].
  69. [69]
    X. Dong, S. Maguire, A. Maloney and H. Maxfield, Phase transitions in 3D gravity and fractal dimension, JHEP 05 (2018) 080 [arXiv:1802.07275] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  70. [70]
    S. Jackson, L. McGough and H. Verlinde, Conformal bootstrap, universality and gravitational scattering, Nucl. Phys. B 901 (2015) 382 [arXiv:1412.5205] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  71. [71]
    S.H. Shenker and D. Stanford, Stringy effects in scrambling, JHEP 05 (2015) 132 [arXiv:1412.6087] [INSPIRE].
  72. [72]
    D.A. Roberts and D. Stanford, Two-dimensional conformal field theory and the butterfly effect, Phys. Rev. Lett. 115 (2015) 131603 [arXiv:1412.5123] [INSPIRE].ADSGoogle Scholar
  73. [73]
    E. Perlmutter, Bounding the space of holographic CFTs with chaos, JHEP 10 (2016) 069 [arXiv:1602.08272] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  74. [74]
    P. Caputa, T. Numasawa and A. Veliz-Osorio, Out-of-time-ordered correlators and purity in rational conformal field theories, PTEP 2016 (2016) 113B06 [arXiv:1602.06542] [INSPIRE].
  75. [75]
    Y. Gu and X.-L. Qi, Fractional statistics and the butterfly effect, JHEP 08 (2016) 129 [arXiv:1602.06543] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  76. [76]
    R. Fan, Out-of-time-order correlation functions for unitary minimal models, arXiv:1809.07228 [INSPIRE].
  77. [77]
    P. Caputa, Y. Kusuki, T. Takayanagi and K. Watanabe, Out-of-time-ordered correlators in (T 2 ) n /Z n, Phys. Rev. D 96 (2017) 046020 [arXiv:1703.09939] [INSPIRE].ADSMathSciNetGoogle Scholar
  78. [78]
    J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  79. [79]
    A.L. Fitzpatrick and J. Kaplan, A quantum correction to chaos, JHEP 05 (2016) 070 [arXiv:1601.06164] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  80. [80]
    G.J. Turiaci, An inelastic bound on chaos, JHEP 07 (2019) 099 [arXiv:1901.04360] [INSPIRE].
  81. [81]
    J. Teschner, Liouville theory revisited, Class. Quant. Grav. 18 (2001) R153 [hep-th/0104158] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  82. [82]
    L. McGough and H. Verlinde, Bekenstein-Hawking entropy as topological entanglement entropy, JHEP 11 (2013) 208 [arXiv:1308.2342] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  83. [83]
    P. Caputa, N. Kundu, M. Miyaji, T. Takayanagi and K. Watanabe, Liouville action as path-integral complexity: from continuous tensor networks to AdS/CFT, JHEP 11 (2017) 097 [arXiv:1706.07056] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  84. [84]
    C. Liu and D.A. Lowe, Notes on scrambling in conformal field theory, Phys. Rev. D 98 (2018) 126013 [arXiv:1808.09886] [INSPIRE].
  85. [85]
    H.R. Hampapura, A. Rolph and B. Stoica, Scrambling in two-dimensional conformal field theories with light and smeared operators, Phys. Rev. D 99 (2019) 106010 [arXiv:1809.09651] [INSPIRE].ADSGoogle Scholar
  86. [86]
    C.-M. Chang, D.M. Ramirez and M. Rangamani, Spinning constraints on chaotic large c CFTs, JHEP 03 (2019) 068 [arXiv:1812.05585] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  87. [87]
    S. Ribault, Conformal field theory on the plane, arXiv:1406.4290 [INSPIRE].
  88. [88]
    O. Hulík, T. Procházka and J. Raeymaekers, Multi-centered AdS 3 solutions from Virasoro conformal blocks, JHEP 03 (2017) 129 [arXiv:1612.03879] [INSPIRE].
  89. [89]
    O. Hullík, J. Raeymaekers and O. Vasilakis, Multi-centered higher spin solutions from W N conformal blocks, JHEP 11 (2018) 101 [arXiv:1809.01387] [INSPIRE].
  90. [90]
    L. Cornalba, M.S. Costa, J. Penedones and R. Schiappa, Eikonal approximation in AdS/CFT: from shock waves to four-point functions, JHEP 08 (2007) 019 [hep-th/0611122] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  91. [91]
    L. Cornalba, M.S. Costa, J. Penedones and R. Schiappa, Eikonal approximation in AdS/CFT: conformal partial waves and finite N four-point functions, Nucl. Phys. B 767 (2007)327 [hep-th/0611123] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  92. [92]
    L. Cornalba, Eikonal methods in AdS/CFT: Regge theory and multi-reggeon exchange, arXiv:0710.5480 [INSPIRE].
  93. [93]
    M.S. Costa, V. Goncalves and J. Penedones, Conformal Regge theory, JHEP 12 (2012) 091 [arXiv:1209.4355] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  94. [94]
    G.W. Moore and N. Seiberg, Lectures on RCFT, in 1989 Banff NATO ASI: Physics, Geometry and Topology, Banff, Canada, 14–25 August 1989, pg. 1 [INSPIRE].
  95. [95]
    A.B. Zamolodchikov, Conformal symmetry in two-dimensional space: recursion representation of conformal block, Theor. Math. Phys. 73 (1987) 1088 [Teor. Mat. Fiz. 73 (1987)103].Google Scholar
  96. [96]
    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].ADSMathSciNetGoogle Scholar
  97. [97]
    M. Cho, S. Collier and X. Yin, Recursive representations of arbitrary Virasoro conformal blocks, JHEP 04 (2019) 018 [arXiv:1703.09805] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  98. [98]
    I. Esterlis, A.L. Fitzpatrick and D. Ramirez, Closure of the operator product expansion in the non-unitary bootstrap, JHEP 11 (2016) 030 [arXiv:1606.07458] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  99. [99]
    J.-B. Bae, K. Lee and S. Lee, Bootstrapping pure quantum gravity in AdS 3, arXiv:1610.05814 [INSPIRE].
  100. [100]
    Y.-H. Lin, S.-H. Shao, D. Simmons-Duffin, Y. Wang and X. Yin, N = 4 superconformal bootstrap of the K3 CFT, JHEP 05 (2017) 126 [arXiv:1511.04065] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  101. [101]
    P. Ruggiero, E. Tonni and P. Calabrese, Entanglement entropy of two disjoint intervals and the recursion formula for conformal blocks, J. Stat. Mech. 1811 (2018) 113101 [arXiv:1805.05975] [INSPIRE].MathSciNetGoogle Scholar
  102. [102]
    M. Nozaki, T. Numasawa and T. Takayanagi, Quantum entanglement of local operators in conformal field theories, Phys. Rev. Lett. 112 (2014) 111602 [arXiv:1401.0539] [INSPIRE].ADSGoogle Scholar
  103. [103]
    A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  104. [104]
    P. Bantay, Characters and modular properties of permutation orbifolds, Phys. Lett. B 419 (1998)175 [hep-th/9708120] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  105. [105]
    L. Hadasz, Z. Jaskolski and M. Piatek, Analytic continuation formulae for the BPZ conformal block, Acta Phys. Polon. B 36 (2005) 845 [hep-th/0409258] [INSPIRE].
  106. [106]
    G.W. Moore and N. Seiberg, Naturality in conformal field theory, Nucl. Phys. B 313 (1989) 16 [INSPIRE].ADSMathSciNetGoogle Scholar
  107. [107]
    A. Belin, Permutation orbifolds and chaos, JHEP 11 (2017) 131 [arXiv:1705.08451] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  108. [108]
    H. Dorn and H.J. Otto, Two and three point functions in Liouville theory, Nucl. Phys. B 429 (1994)375 [hep-th/9403141] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  109. [109]
    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].ADSzbMATHGoogle Scholar
  110. [110]
    S. Ribault and R. Santachiara, Liouville theory with a central charge less than one, JHEP 08 (2015) 109 [arXiv:1503.02067] [INSPIRE].
  111. [111]
    X. Cao, P. Le Doussal, A. Rosso and R. Santachiara, Liouville field theory and log-correlated random energy models, Phys. Rev. Lett. 118 (2017) 090601 [arXiv:1611.02193] [INSPIRE].
  112. [112]
    D. Bagrets, A. Altland and A. Kamenev, Sachdev-Ye-Kitaev model as Liouville quantum mechanics, Nucl. Phys. B 911 (2016) 191 [arXiv:1607.00694] [INSPIRE].ADSzbMATHGoogle Scholar
  113. [113]
    D. Bagrets, A. Altland and A. Kamenev, Power-law out of time order correlation functions in the SYK model, Nucl. Phys. B 921 (2017) 727 [arXiv:1702.08902] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  114. [114]
    T.G. Mertens, G.J. Turiaci and H.L. Verlinde, Solving the Schwarzian via the conformal bootstrap, JHEP 08 (2017) 136 [arXiv:1705.08408] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  115. [115]
    H.T. Lam, T.G. Mertens, G.J. Turiaci and H. Verlinde, Shockwave S-matrix from Schwarzian quantum mechanics, JHEP 11 (2018) 182 [arXiv:1804.09834] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  116. [116]
    M. Nozaki, Notes on quantum entanglement of local operators, JHEP 10 (2014) 147 [arXiv:1405.5875] [INSPIRE].
  117. [117]
    I. Ya. Aref’eva, M.A. Khramtsov and M.D. Tikhanovskaya, Thermalization after holographic bilocal quench, JHEP 09 (2017) 115 [arXiv:1706.07390] [INSPIRE].
  118. [118]
    D.S. Ageev and I. Ya. Aref’eva, Holographic instant conformal symmetry breaking by colliding conical defects, Theor. Math. Phys. 189 (2016) 1742 [Teor. Mat. Fiz. 189 (2016) 389] [arXiv:1512.03363] [INSPIRE].
  119. [119]
    B. Ponsot and J. Teschner, Liouville bootstrap via harmonic analysis on a noncompact quantum group, hep-th/9911110 [INSPIRE].
  120. [120]
    J. Teschner and G. Vartanov, 6j symbols for the modular double, quantum hyperbolic geometry and supersymmetric gauge theories, Lett. Math. Phys. 104 (2014) 527 [arXiv:1202.4698] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  121. [121]
    P. Di Francesco, P. Mathieu and D. Sénéchal, Conformal field theory, Springer Science & Business Media, (2012).Google Scholar

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© The Author(s) 2019

Authors and Affiliations

  1. 1.Center for Gravitational Physics, Yukawa Institute for Theoretical Physics (YITP)Kyoto UniversityKyotoJapan
  2. 2.Department of Physics, Faculty of ScienceThe University of TokyoTokyoJapan

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