Abstract
We compute four-point functions of two heavy and two “perturbatively heavy” operators in the semiclassical limit of Liouville theory on the sphere. We obtain these “Heavy-Heavy-Light-Light” (HHLL) correlators to leading order in the conformal weights of the light insertions in two ways: (a) via a path integral approach, combining different methods to evaluate correlation functions from complex solutions for the Liouville field, and (b) via the conformal block expansion. This latter approach identifies an integral over the continuum of normalizable states and a sum over an infinite tower of lighter discrete states, whose contribution we extract by analytically continuing standard results to our HHLL setting. The sum over this tower reproduces the sum over those complex saddlepoints of the path integral that contribute to the correlator. Our path integral computations reveal that when the two light operators are inserted at equal time in radial quantization, the leading-order HHLL correlator is independent of their separation, and more generally that at this order there is no short-distance singularity as the two light operators approach each other. The conformal block expansion likewise shows that in the discrete sum short-distance singularities are indeed absent for all intermediate states that contribute. In particular, the Virasoro vacuum block, which would have been singular at short distances, is not exchanged. The separation-independence of equal-time correlators is due to cancelations between the discrete contributions. These features lead to a Lorentzian singularity that, in conformal theories with anti-de Sitter (AdS) duals, would be associated to locality below the AdS scale.
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References
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].
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].
T. Hartman, Entanglement entropy at large central charge, arXiv:1303.6955 [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].
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].
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].
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].
A. Galliani, S. Giusto, E. Moscato and R. Russo, Correlators at large c without information loss, JHEP 09 (2016) 065 [arXiv:1606.01119] [INSPIRE].
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].
J. de Boer and D. Engelhardt, Remarks on thermalization in 2D CFT, Phys. Rev. D 94 (2016) 126019 [arXiv:1604.05327] [INSPIRE].
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].
J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].
E. Perlmutter, Bounding the space of holographic CFTs with chaos, JHEP 10 (2016) 069 [arXiv:1602.08272] [INSPIRE].
C.-M. Chang and Y.-H. Lin, Bootstrapping 2D CFTs in the semiclassical limit, JHEP 08 (2016) 056 [arXiv:1510.02464] [INSPIRE].
S. Collier, P. Kravchuk, Y.-H. Lin and X. Yin, Bootstrapping the spectral function: on the uniqueness of Liouville and the universality of BTZ, arXiv:1702.00423 [INSPIRE].
P. Menotti and G. Vajente, Semiclassical and quantum Liouville theory on the sphere, Nucl. Phys. B 709 (2005) 465 [hep-th/0411003] [INSPIRE].
P. Menotti and E. Tonni, Quantum Liouville theory on the pseudosphere with heavy charges, Phys. Lett. B 633 (2006) 404 [hep-th/0508240] [INSPIRE].
P. Menotti and E. Tonni, Liouville field theory with heavy charges. I. The pseudosphere, JHEP 06 (2006) 020 [hep-th/0602206] [INSPIRE].
P. Menotti and E. Tonni, Liouville field theory with heavy charges. II. The conformal boundary case, JHEP 06 (2006) 022 [hep-th/0602221] [INSPIRE].
D. Harlow, J. Maltz and E. Witten, Analytic continuation of Liouville theory, JHEP 12 (2011) 071 [arXiv:1108.4417] [INSPIRE].
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].
L. Hadasz and Z. Jaskolski, Liouville theory and uniformization of four-punctured sphere, J. Math. Phys. 47 (2006) 082304 [hep-th/0604187] [INSPIRE].
K. Krasnov, 3D gravity, point particles and Liouville theory, Class. Quant. Grav. 18 (2001) 1291 [hep-th/0008253] [INSPIRE].
M. Gary, S.B. Giddings and J. Penedones, Local bulk S-matrix elements and CFT singularities, Phys. Rev. D 80 (2009) 085005 [arXiv:0903.4437] [INSPIRE].
I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, Holography from conformal field theory, JHEP 10 (2009) 079 [arXiv:0907.0151] [INSPIRE].
A.B. Zamolodchikov and A.B. Zamolodchikov, Lectures on Liouville theory and matrix models, http://qft.itp.ac.ru/ZZ.pdf.
N. Seiberg, Notes on quantum Liouville theory and quantum gravity, Prog. Theor. Phys. Suppl. 102 (1990) 319 [INSPIRE].
J. Teschner, Liouville theory revisited, Class. Quant. Grav. 18 (2001) R153 [hep-th/0104158] [INSPIRE].
H. Dorn and H.J. Otto, Two and three point functions in Liouville theory, Nucl. Phys. B 429 (1994) 375 [hep-th/9403141] [INSPIRE].
M. Beccaria, A. Fachechi and G. Macorini, Virasoro vacuum block at next-to-leading order in the heavy-light limit, JHEP 02 (2016) 072 [arXiv:1511.05452] [INSPIRE].
A.L. Fitzpatrick and J. Kaplan, Conformal blocks beyond the semi-classical limit, JHEP 05 (2016) 075 [arXiv:1512.03052] [INSPIRE].
Ecyclopaedia Brittanica, https://www.britannica.com/biography/Apollonius-of-Perga.
Wikipedia, https://en.wikipedia.org/wiki/Circles_of_Apollonius.
M. Kulaxizi, A. Parnachev and G. Policastro, Conformal blocks and negativity at large central charge, JHEP 09 (2014) 010 [arXiv:1407.0324] [INSPIRE].
J.L. Cardy, O.A. Castro-Alvaredo and B. Doyon, Form factors of branch-point twist fields in quantum integrable models and entanglement entropy, J. Statist. Phys. 130 (2008) 129 [arXiv:0706.3384] [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].
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Balasubramanian, V., Bernamonti, A., Craps, B. et al. Heavy-heavy-light-light correlators in Liouville theory. J. High Energ. Phys. 2017, 45 (2017). https://doi.org/10.1007/JHEP08(2017)045
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DOI: https://doi.org/10.1007/JHEP08(2017)045