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Worldline approach to semi-classical conformal blocks

  • Eliot HijanoEmail author
  • Per Kraus
  • River Snively
Open Access
Regular Article - Theoretical Physics

Abstract

We extend recent results on semi-classical conformal blocks in 2d CFT and their relation to 3D gravity via the AdS/CFT correspondence. We consider four-point functions with two heavy and two light external operators, along with the exchange of a light operator. By explicit computation, we establish precise agreement between these CFT objects and a simple picture of particle worldlines joined by cubic vertices propagating in asymptotically AdS3 geometries (conical defects or BTZ black holes). We provide a simple argument that explains this agreement.

Keywords

AdS-CFT Correspondence Conformal and W Symmetry Black Holes 

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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    P. Di Francesco, P. Mathieu and D. Senechal, Conformal field theory, Springer, New York U.S.A. (1997).CrossRefzbMATHGoogle Scholar
  3. [3]
    A. Zamolodchikov, Conformal symmetry in two-dimensional space: recursion represenation of conformal block, Theor. Math. Phys. 73 (1987) 1088 [Teor. Mat. Fiz. 73 (1987) 103].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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, Holography from conformal field theory, JHEP 10 (2009) 079 [arXiv:0907.0151] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    I. Heemskerk and J. Sully, More holography from conformal field theory, JHEP 09 (2010) 099 [arXiv:1006.0976] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    S. El-Showk and K. Papadodimas, Emergent spacetime and holographic CFTs, JHEP 10 (2012) 106 [arXiv:1101.4163] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    A.L. Fitzpatrick and J. Kaplan, AdS field theory from conformal field theory, JHEP 02 (2013) 054 [arXiv:1208.0337] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    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
  10. [10]
    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
  11. [11]
    F.A. Dolan and H. Osborn, Conformal four point functions and the operator product expansion, Nucl. Phys. B 599 (2001) 459 [hep-th/0011040] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    F.A. Dolan and H. Osborn, Conformal partial waves and the operator product expansion, Nucl. Phys. B 678 (2004) 491 [hep-th/0309180] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    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].CrossRefzbMATHGoogle Scholar
  14. [14]
    P. Caputa, J. Simón, A. Štikonas and T. Takayanagi, Quantum entanglement of localized excited states at finite temperature, JHEP 01 (2015) 102 [arXiv:1410.2287] [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    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].ADSMathSciNetCrossRefGoogle Scholar
  16. [16]
    S. Jackson, L. McGough and H. Verlinde, Conformal bootstrap, universality and gravitational scattering, arXiv:1412.5205 [INSPIRE].
  17. [17]
    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
  18. [18]
    J. de Boer, A. Castro, E. Hijano, J.I. Jottar and P. Kraus, Higher spin entanglement and WN conformal blocks, arXiv:1412.7520 [INSPIRE].
  19. [19]
    D.A. Roberts and D. Stanford, Two-dimensional conformal field theory and the butterfly effect, arXiv:1412.5123 [INSPIRE].
  20. [20]
    A. Zamolodchikov, Two-dimensional conformal symmetry and critical four-spin correlation functions in the Ashkin-Teller model, Zh. Eksp. Teor. Fiz. 90 (1986) 1808 [JETP 63 (1986) 1061].Google Scholar
  21. [21]
    D. Harlow, J. Maltz and E. Witten, Analytic continuation of Liouville theory, JHEP 12 (2011) 071 [arXiv:1108.4417] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    T. Hartman, Entanglement entropy at large central charge, arXiv:1303.6955 [INSPIRE].
  23. [23]
    T. Faulkner, The entanglement Renyi entropies of disjoint intervals in AdS/CFT, arXiv:1303.7221 [INSPIRE].
  24. [24]
    A. Achucarro and P.K. Townsend, A Chern-Simons action for three-dimensional anti-de Sitter supergravity theories, Phys. Lett. B 180 (1986) 89 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  25. [25]
    E. Witten, (2 + 1)-dimensional gravity as an exactly soluble system, Nucl. Phys. B 311 (1988) 46 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  27. [27]
    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
  28. [28]
    L. Hadasz, Z. Jaskolski and M. Piatek, Classical geometry from the quantum Liouville theory, Nucl. Phys. B 724 (2005) 529 [hep-th/0504204] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    J. de Boer and J.I. Jottar, Entanglement entropy and higher spin holography in AdS3, JHEP 04 (2014) 089 [arXiv:1306.4347] [INSPIRE].CrossRefGoogle Scholar
  30. [30]
    M. Ammon, A. Castro and N. Iqbal, Wilson lines and entanglement entropy in higher spin gravity, JHEP 10 (2013) 110 [arXiv:1306.4338] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    A. Castro and E. Llabrés, Unravelling holographic entanglement entropy in higher spin theories, JHEP 03 (2015) 124 [arXiv:1410.2870] [INSPIRE].MathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  1. 1.Department of Physics and AstronomyUniversity of CaliforniaLos AngelesUnited States

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