Advertisement

Light cone bootstrap in general 2D CFTs and entanglement from light cone singularity

  • Yuya KusukiEmail author
Open Access
Regular Article - Theoretical Physics
  • 9 Downloads

Abstract

The light cone OPE limit provides a significant amount of information regarding the conformal field theory (CFT), like the high-low temperature limit of the partition function. We started with the light cone bootstrap in the general CFT 2 with c > 1. For this purpose, we needed an explicit asymptotic form of the Virasoro conformal blocks in the limit z → 1, which was unknown until now. In this study, we computed it in general by studying the pole structure of the fusion matrix (or the crossing kernel). Applying this result to the light cone bootstrap, we obtained the universal total twist (or equivalently, the universal binding energy) of two particles at a large angular momentum. In particular, we found that the total twist is saturated by the value \( \frac{c-1}{12} \) if the total Liouville momentum exceeds beyond the BTZ threshold. This might be interpreted as a black hole formation in AdS3.

As another application of our light cone singularity, we studied the dynamics of entanglement after a global quench and found a Renyi phase transition as the replica number was varied. We also investigated the dynamics of the 2nd Renyi entropy after a local quench.

We also provide a universal form of the Regge limit of the Virasoro conformal blocks from the analysis of the light cone singularity. This Regge limit is related to the general n-th Renyi entropy after a local quench and out of time ordered correlators.

Keywords

Conformal Field Theory AdS-CFT Correspondence Field Theories in Lower Dimensions 

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]
    A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory, Nucl. Phys. B 241 (1984) 333 [INSPIRE].
  2. [2]
    S. Ferrara, A.F. Grillo and R. Gatto, Tensor representations of conformal algebra and conformally covariant operator product expansion, Annals Phys. 76 (1973) 161 [INSPIRE].
  3. [3]
    A.M. Polyakov, Nonhamiltonian approach to conformal quantum field theory, Zh. Eksp. Teor. Fiz. 66 (1974) 23 [INSPIRE].Google Scholar
  4. [4]
    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].
  5. [5]
    J.L. Cardy, Operator Content of Two-Dimensional Conformally Invariant Theories, Nucl. Phys. B 270 (1986) 186 [INSPIRE].
  6. [6]
    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:1608.03284v1] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    J.L. 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
  8. [8]
    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].
  9. [9]
    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].ADSCrossRefzbMATHGoogle Scholar
  10. [10]
    E.M. Brehm, D. Das and S. Datta, Probing thermality beyond the diagonal, arXiv:1804.07924 [INSPIRE].
  11. [11]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Z. Komargodski and A. Zhiboedov, Convexity and Liberation at Large Spin, JHEP 11 (2013) 140 [arXiv:1212.4103] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    L.F. Alday, A. Bissi and T. Lukowski, Large spin systematics in CFT, JHEP 11 (2015) 101 [arXiv:1502.07707] [INSPIRE].ADSzbMATHGoogle Scholar
  14. [14]
    A. Kaviraj, K. Sen and A. Sinha, Analytic bootstrap at large spin, JHEP 11 (2015) 083 [arXiv:1502.01437] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    A. Kaviraj, K. Sen and A. Sinha, Universal anomalous dimensions at large spin and large twist, JHEP 07 (2015) 026 [arXiv:1504.00772] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    L.F. Alday, Large Spin Perturbation Theory for Conformal Field Theories, Phys. Rev. Lett. 119 (2017) 111601 [arXiv:1611.01500] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    D. Simmons-Duffin, The Lightcone Bootstrap and the Spectrum of the 3d Ising CFT, JHEP 03 (2017) 086 [arXiv:1612.08471] [INSPIRE].
  18. [18]
    L.F. Alday and A. Zhiboedov, An Algebraic Approach to the Analytic Bootstrap, JHEP 04 (2017) 157 [arXiv:1510.08091] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    C. Sleight and M. Taronna, Anomalous Dimensions from Crossing Kernels, JHEP 11 (2018) 089 [arXiv:1807.05941] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  20. [20]
    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].ADSCrossRefGoogle Scholar
  21. [21]
    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].
  22. [22]
    Y. Kusuki, New Properties of Large-c Conformal Blocks from Recursion Relation, JHEP 07 (2018) 010 [arXiv:1804.06171] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    Y. Kusuki, Large c Virasoro Blocks from Monodromy Method beyond Known Limits, JHEP 08 (2018) 161 [arXiv:1806.04352] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    A.L. Fitzpatrick and J. Kaplan, AdS Field Theory from Conformal Field Theory, JHEP 02 (2013) 054 [arXiv:1208.0337] [INSPIRE].
  25. [25]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    K.B. Alkalaev and V.A. Belavin, Classical conformal blocks via AdS/CFT correspondence, JHEP 08 (2015) 049 [arXiv:1504.05943] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    E. Hijano, P. Kraus, E. Perlmutter and R. Snively, Semiclassical Virasoro blocks from AdS 3 gravity, JHEP 12 (2015) 077 [arXiv:1508.04987] [INSPIRE].
  29. [29]
    E. Hijano, P. Kraus and R. Snively, Worldline approach to semi-classical conformal blocks, JHEP 07 (2015) 131 [arXiv:1501.02260] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    K.B. Alkalaev and V.A. Belavin, Monodromic vs. geodesic computation of Virasoro classical conformal blocks, Nucl. Phys. B 904 (2016) 367 [arXiv:1510.06685] [INSPIRE].
  31. [31]
    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].
  32. [32]
    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].ADSCrossRefGoogle Scholar
  33. [33]
    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].
  34. [34]
    B. Chen, J.-q. Wu and J.-j. Zhang, Holographic Description of 2D Conformal Block in Semi-classical Limit, JHEP 10 (2016) 110 [arXiv:1609.00801] [INSPIRE].
  35. [35]
    K.B. Alkalaev, Many-point classical conformal blocks and geodesic networks on the hyperbolic plane, JHEP 12 (2016) 070 [arXiv:1610.06717] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    O. Hulík, T. Procházka and J. Raeymaekers, Multi-centered AdS3 solutions from Virasoro conformal blocks, JHEP 03 (2017) 129 [arXiv:1612.03879] [INSPIRE].
  37. [37]
    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].ADSCrossRefzbMATHGoogle Scholar
  38. [38]
    K.B. Alkalaev and V.A. Belavin, Holographic duals of large-c torus conformal blocks, JHEP 10 (2017) 140 [arXiv:1707.09311] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    H. Maxfield, A view of the bulk from the worldline, arXiv:1712.00885 [INSPIRE].
  40. [40]
    K.B. Alkalaev and V.A. Belavin, Large-c superconformal torus blocks, JHEP 08 (2018) 042 [arXiv:1805.12585] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    Y. Hikida and T. Uetoko, Conformal blocks from Wilson lines with loop corrections, Phys. Rev. D 97 (2018) 086014 [arXiv:1801.08549] [INSPIRE].
  42. [42]
    S. Banerjee, J.-W. Brijan and G. Vos, On the universality of late-time correlators in semi-classical 2d CFTs, JHEP 08 (2018) 047 [arXiv:1805.06464] [INSPIRE].
  43. [43]
    Y. Hikida and T. Uetoko, Superconformal blocks from Wilson lines with loop corrections, JHEP 08 (2018) 101 [arXiv:1806.05836] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    V.A. Belavin and R. Geiko, c-Recursion for multi-point superconformal blocks. NS sector, JHEP 08 (2018) 112 [arXiv:1806.09563] [INSPIRE].
  45. [45]
    O. Hulík, J. Raeymaekers and O. Vasilakis, Multi-centered higher spin solutions from \( {\mathcal{W}}_N \) conformal blocks, JHEP 11 (2018) 101 [arXiv:1809.01387] [INSPIRE].
  46. [46]
    H. Chen, A.L. Fitzpatrick, J. Kaplan and D. Li, The Bulk-to-Boundary Propagator in Black Hole Microstate Backgrounds, arXiv:1810.02436 [INSPIRE].
  47. [47]
    A. Lewkowycz and J.M. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  48. [48]
    T. Hartman and J.M. Maldacena, Time Evolution of Entanglement Entropy from Black Hole Interiors, JHEP 05 (2013) 014 [arXiv:1303.1080] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  49. [49]
    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].
  50. [50]
    P. Caputa, M. Nozaki and T. Takayanagi, Entanglement of local operators in large-N conformal field theories, Prog. Theor. Exp. Phys. 2014 (2014) 093B06 [arXiv:1405.5946] [INSPIRE].
  51. [51]
    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].
  52. [52]
    T. Numasawa, Scattering effect on entanglement propagation in RCFTs, JHEP 12 (2016) 061 [arXiv:1610.06181v2] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  53. [53]
    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].
  54. [54]
    S. He, Conformal Bootstrap to Rényi Entropy in 2D Liouville and Super-Liouville CFTs, arXiv:1711.00624 [INSPIRE].
  55. [55]
    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].
  56. [56]
    P. Caputa, T. Numasawa and A. Veliz-Osorio, Out-of-time-ordered correlators and purity in rational conformal field theories, Prog. Theor. Exp. Phys. 2016 (2016) 113B06 [arXiv:1602.06542] [INSPIRE].
  57. [57]
    P. Caputa, Y. Kusuki, T. Takayanagi and K. Watanabe, Out-of-Time-Ordered Correlators in (T 2)n /n, Phys. Rev. D 96 (2017) 046020 [arXiv:1703.09939] [INSPIRE].
  58. [58]
    E. Perlmutter, Bounding the Space of Holographic CFTs with Chaos, JHEP 10 (2016) 069 [arXiv:1602.08272] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  59. [59]
    Y. Gu and X.-L. Qi, Fractional Statistics and the Butterfly Effect, JHEP 08 (2016) 129 [arXiv:1602.06543] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  60. [60]
    R. Fan, Out-of-Time-Order Correlation Functions for Unitary Minimal Models, arXiv:1809.07228 [INSPIRE].
  61. [61]
    L.F. Alday and J.M. Maldacena, Comments on operators with large spin, JHEP 11 (2007) 019 [arXiv:0708.0672] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  62. [62]
    S. Collier, Y.-H. Lin and X. Yin, Modular Bootstrap Revisited, JHEP 09 (2018) 061 [arXiv:1608.06241v1] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  63. [63]
    S. Collier, Y. Gobeil, H. Maxfield and E. Perlmutter, Quantum Regge Trajectories and the Virasoro Analytic Bootstrap, arXiv:1811.05710 [INSPIRE].
  64. [64]
    Y. Kusuki, T. Takayanagi and K. Umemoto, Holographic Entanglement Entropy on Generic Time Slices, JHEP 06 (2017) 021 [arXiv:1703.00915] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  65. [65]
    P. Calabrese and J.L. Cardy, Evolution of entanglement entropy in one-dimensional systems, J. Stat. Mech. 0504 (2005) P04010 [cond-mat/0503393] [INSPIRE].
  66. [66]
    P. Calabrese and J.L. Cardy, Entanglement entropy and conformal field theory, J. Phys. A 42 (2009) 504005 [arXiv:0905.4013] [INSPIRE].
  67. [67]
    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].
  68. [68]
    A. Belin, A. Maloney and S. Matsuura, Holographic Phases of Renyi Entropies, JHEP 12 (2013) 050 [arXiv:1306.2640] [INSPIRE].ADSCrossRefGoogle Scholar
  69. [69]
    A. Belin, L.-Y. Hung, A. Maloney and S. Matsuura, Charged Renyi entropies and holographic superconductors, JHEP 01 (2015) 059 [arXiv:1407.5630] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  70. [70]
    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].
  71. [71]
    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].
  72. [72]
    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].ADSCrossRefGoogle Scholar
  73. [73]
    P. Caputa, M. Nozaki and T. Takayanagi, Entanglement of local operators in large-N conformal field theories, Prog. Theor. Exp. Phys. 2014 (2014) 093B06 [arXiv:1405.5946v2] [INSPIRE].
  74. [74]
    K. Krasnov, Holography and Riemann surfaces, Adv. Theor. Math. Phys. 4 (2000) 929 [hep-th/0005106] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  75. [75]
    K. Krasnov, Three-dimensional gravity, point particles and Liouville theory, Class. Quant. Grav. 18 (2001) 1291 [hep-th/0008253] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  76. [76]
    S. Jackson, L. McGough and H. Verlinde, Conformal Bootstrap, Universality and Gravitational Scattering, Nucl. Phys. B 901 (2015) 382 [arXiv:1412.5205] [INSPIRE].
  77. [77]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  78. [78]
    J. Teschner, Liouville theory revisited, Class. Quant. Grav. 18 (2001) R153 [hep-th/0104158] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  79. [79]
    J. Teschner, From Liouville theory to the quantum geometry of Riemann surfaces, in proceedings of the 14th International Congress on Mathematical Physics (ICMP03), Lisbon, Portugal, 28 July-2 August 2003, hep-th/0308031 [INSPIRE].
  80. [80]
    L. Hadasz, Z. Jaskolski and P. Suchanek, Recursive representation of the torus 1-point conformal block, JHEP 01 (2010) 063 [arXiv:0911.2353] [INSPIRE].
  81. [81]
    N. Nemkov, Analytic properties of the Virasoro modular kernel, Eur. Phys. J. C 77 (2017) 368 [arXiv:1610.02000] [INSPIRE].
  82. [82]
    B. Ponsot and J. Teschner, Liouville bootstrap via harmonic analysis on a noncompact quantum group, hep-th/9911110 [INSPIRE].
  83. [83]
    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].
  84. [84]
    C.-M. Chang and Y.-H. Lin, Bootstrap, universality and horizons, JHEP 10 (2016) 068 [arXiv:1604.01774] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  85. [85]
    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].
  86. [86]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  87. [87]
    A.B. Zamolodchikov, Conformal symmetry in two-dimensional space: recursion representation of conformal block, Theor. Math. Phys. 73 (1987) 1088.CrossRefGoogle Scholar
  88. [88]
    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
  89. [89]
    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].CrossRefGoogle Scholar
  90. [90]
    J.-B. Bae, K. Lee and S. Lee, Bootstrapping Pure Quantum Gravity in AdS 3, arXiv:1610.05814 [INSPIRE].
  91. [91]
    Y.-H. Lin, S.-H. Shao, D. Simmons-Duffin, Y. Wang and X. Yin, \( \mathcal{N}=4 \) superconformal bootstrap of the K3 CFT, JHEP 05 (2017) 126 [arXiv:1511.04065] [INSPIRE].
  92. [92]
    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].ADSCrossRefzbMATHGoogle Scholar
  93. [93]
    E. Perlmutter, Virasoro conformal blocks in closed form, JHEP 08 (2015) 088 [arXiv:1502.07742v2] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  94. [94]
    M. Cho, S. Collier and X. Yin, Recursive Representations of Arbitrary Virasoro Conformal Blocks, arXiv:1703.09805 [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Center for Gravitational Physics, Yukawa Institute for Theoretical Physics (YITP)Kyoto UniversityKyotoJapan

Personalised recommendations