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.
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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].
P. Di Francesco, P. Mathieu and D. Senechal, Conformal field theory, Springer, New York U.S.A. (1997).
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].
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].
I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, Holography from conformal field theory, JHEP 10 (2009) 079 [arXiv:0907.0151] [INSPIRE].
I. Heemskerk and J. Sully, More holography from conformal field theory, JHEP 09 (2010) 099 [arXiv:1006.0976] [INSPIRE].
S. El-Showk and K. Papadodimas, Emergent spacetime and holographic CFTs, JHEP 10 (2012) 106 [arXiv:1101.4163] [INSPIRE].
A.L. Fitzpatrick and J. Kaplan, AdS field theory from conformal field theory, JHEP 02 (2013) 054 [arXiv:1208.0337] [INSPIRE].
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].
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].
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].
F.A. Dolan and H. Osborn, Conformal partial waves and the operator product expansion, Nucl. Phys. B 678 (2004) 491 [hep-th/0309180] [INSPIRE].
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].
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].
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].
S. Jackson, L. McGough and H. Verlinde, Conformal bootstrap, universality and gravitational scattering, arXiv:1412.5205 [INSPIRE].
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].
J. de Boer, A. Castro, E. Hijano, J.I. Jottar and P. Kraus, Higher spin entanglement and W N conformal blocks, arXiv:1412.7520 [INSPIRE].
D.A. Roberts and D. Stanford, Two-dimensional conformal field theory and the butterfly effect, arXiv:1412.5123 [INSPIRE].
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].
D. Harlow, J. Maltz and E. Witten, Analytic continuation of Liouville theory, JHEP 12 (2011) 071 [arXiv:1108.4417] [INSPIRE].
T. Hartman, Entanglement entropy at large central charge, arXiv:1303.6955 [INSPIRE].
T. Faulkner, The entanglement Renyi entropies of disjoint intervals in AdS/CFT, arXiv:1303.7221 [INSPIRE].
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].
E. Witten, (2 + 1)-dimensional gravity as an exactly soluble system, Nucl. Phys. B 311 (1988) 46 [INSPIRE].
J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].
T. Barrella, X. Dong, S.A. Hartnoll and V.L. Martin, Holographic entanglement beyond classical gravity, JHEP 09 (2013) 109 [arXiv:1306.4682] [INSPIRE].
L. Hadasz, Z. Jaskolski and M. Piatek, Classical geometry from the quantum Liouville theory, Nucl. Phys. B 724 (2005) 529 [hep-th/0504204] [INSPIRE].
J. de Boer and J.I. Jottar, Entanglement entropy and higher spin holography in AdS3, JHEP 04 (2014) 089 [arXiv:1306.4347] [INSPIRE].
M. Ammon, A. Castro and N. Iqbal, Wilson lines and entanglement entropy in higher spin gravity, JHEP 10 (2013) 110 [arXiv:1306.4338] [INSPIRE].
A. Castro and E. Llabrés, Unravelling holographic entanglement entropy in higher spin theories, JHEP 03 (2015) 124 [arXiv:1410.2870] [INSPIRE].
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ArXiv ePrint: 1501.02260
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Hijano, E., Kraus, P. & Snively, R. Worldline approach to semi-classical conformal blocks. J. High Energ. Phys. 2015, 131 (2015). https://doi.org/10.1007/JHEP07(2015)131
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DOI: https://doi.org/10.1007/JHEP07(2015)131