Abstract
We study out-of-time ordered four-point functions in two dimensional conformal field theories by suitably analytically continuing the Euclidean correlator. For large central charge theories with a sparse spectrum, chaotic dynamics is revealed in an exponential decay; this is seen directly in the contribution of the vacuum block to the correlation function. However, contributions from individual non-vacuum blocks with large spin and small twist dominate over the vacuum block. We argue, based on holographic intuition, that suitable summations over such intermediate states in the block decomposition of the correlator should be sub-dominant, and attempt to use this criterion to constrain the OPE data with partial success. Along the way we also discuss the relation between the spinning Virasoro blocks and the on-shell worldline action of spinning particles in an asymptotically AdS spacetime.
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References
I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, Holography from Conformal Field Theory, JHEP 10 (2009) 079 [arXiv:0907.0151] [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].
M.R. Gaberdiel and R. Gopakumar, Tensionless string spectra on AdS 3, JHEP 05 (2018) 085 [arXiv:1803.04423] [INSPIRE].
G. Giribet, C. Hull, M. Kleban, M. Porrati and E. Rabinovici, Superstrings on AdS 3 at ∥ = 1, JHEP 08 (2018) 204 [arXiv:1803.04420] [INSPIRE].
J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].
A. Kitaev, A simple model of quantum holography, Talks at Kavli Institute for Theoretical Physics (KITP), Santa Barbara U.S.A. (2015).
J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].
A. Kitaev and S.J. Suh, The soft mode in the Sachdev-Ye-Kitaev model and its gravity dual, JHEP 05 (2018) 183 [arXiv:1711.08467] [INSPIRE].
S.H. Shenker and D. Stanford, Stringy effects in scrambling, JHEP 05 (2015) 132 [arXiv:1412.6087] [INSPIRE].
J. Murugan, D. Stanford and E. Witten, More on Supersymmetric and 2d Analogs of the SYK Model, JHEP 08 (2017) 146 [arXiv:1706.05362] [INSPIRE].
K. Bulycheva, \( \mathcal{N} \) = 2 SYK model in the superspace formalism, JHEP 04 (2018) 036 [arXiv:1801.09006] [INSPIRE].
C. Peng, \( \mathcal{N} \) = (0, 2) SYK, Chaos and Higher-Spins, JHEP 12 (2018) 065 [arXiv:1805.09325] [INSPIRE].
E. Perlmutter, Bounding the Space of Holographic CFTs with Chaos, JHEP 10 (2016) 069 [arXiv:1602.08272] [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].
A.L. Fitzpatrick and J. Kaplan, A Quantum Correction To Chaos, JHEP 05 (2016) 070 [arXiv:1601.06164] [INSPIRE].
C. Liu and D.A. Lowe, Notes on Scrambling in Conformal Field Theory, Phys. Rev. D 98 (2018) 126013 [arXiv:1808.09886] [INSPIRE].
H.R. Hampapura, A. Rolph and B. Stoica, Scrambling in Two-Dimensional Conformal Field Theories with Light and Smeared Operators, arXiv:1809.09651 [INSPIRE].
M. Blake, H. Lee and H. Liu, A quantum hydrodynamical description for scrambling and many-body chaos, JHEP 10 (2018) 127 [arXiv:1801.00010] [INSPIRE].
F.M. Haehl and M. Rozali, Effective Field Theory for Chaotic CFTs, JHEP 10 (2018) 118 [arXiv:1808.02898] [INSPIRE].
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].
A.B. Zamolodchikov, Conformal symmetry in two-dimensional space: Recursion representation of conformal block, Theor. Math. Phys. 73 (1987) 1088.
X.O. Camanho, J.D. Edelstein, J. Maldacena and A. Zhiboedov, Causality Constraints on Corrections to the Graviton Three-Point Coupling, JHEP 02 (2016) 020 [arXiv:1407.5597] [INSPIRE].
C.-M. Chang and Y.-H. Lin, Bootstrapping 2D CFTs in the Semiclassical Limit, JHEP 08 (2016) 056 [arXiv:1510.02464] [INSPIRE].
C.-M. Chang and Y.-H. Lin, Bootstrap, universality and horizons, JHEP 10 (2016) 068 [arXiv:1604.01774] [INSPIRE].
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].
Y. Kusuki, New Properties of Large-c Conformal Blocks from Recursion Relation, JHEP 07 (2018) 010 [arXiv:1804.06171] [INSPIRE].
Y. Kusuki, Large c Virasoro Blocks from Monodromy Method beyond Known Limits, JHEP 08 (2018) 161 [arXiv:1806.04352] [INSPIRE].
Y. Kusuki, Light Cone Bootstrap in General 2D CFTs and Entanglement from Light Cone Singularity, JHEP 01 (2019) 025 [arXiv:1810.01335] [INSPIRE].
R. Haag, Local quantum physics: Fields, particles, algebras, Springer-Verlag, Berlin Germany (1992).
T. Hartman, S. Jain and S. Kundu, Causality Constraints in Conformal Field Theory, JHEP 05 (2016) 099 [arXiv:1509.00014] [INSPIRE].
F.M. Haehl, R. Loganayagam, P. Narayan and M. Rangamani, Classification of out-of-time-order correlators, SciPost Phys. 6 (2019) 001 [arXiv:1701.02820] [INSPIRE].
J. Maldacena, D. Simmons-Duffin and A. Zhiboedov, Looking for a bulk point, JHEP 01 (2017) 013 [arXiv:1509.03612] [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].
E. Hijano, P. Kraus and R. Snively, Worldline approach to semi-classical conformal blocks, JHEP 07 (2015) 131 [arXiv:1501.02260] [INSPIRE].
S.H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP 03 (2014) 067 [arXiv:1306.0622] [INSPIRE].
S. Collier, Y.-H. Lin and X. Yin, Modular Bootstrap Revisited, JHEP 09 (2018) 061 [arXiv:1608.06241] [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].
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].
A. Belin, Permutation Orbifolds and Chaos, JHEP 11 (2017) 131 [arXiv:1705.08451] [INSPIRE].
Z. Komargodski and A. Zhiboedov, Convexity and Liberation at Large Spin, JHEP 11 (2013) 140 [arXiv:1212.4103] [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].
S. Collier, Y. Gobeil, H. Maxfield and E. Perlmutter, Quantum Regge Trajectories and the Virasoro Analytic Bootstrap, arXiv:1811.05710 [INSPIRE].
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].
J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].
M. Mathisson, Neue mechanik materieller systemes, Acta Phys. Polon. 6 (1937) 163 [INSPIRE].
A. Papapetrou, Spinning test particles in general relativity. 1., Proc. Roy. Soc. Lond. A 209 (1951) 248 [INSPIRE].
W.G. Dixon, Dynamics of extended bodies in general relativity. II. Moments of the charge-current vector, Proc. Roy. Soc. Lond. A 319 (1970) 509 [INSPIRE].
A. Castro, S. Detournay, N. Iqbal and E. Perlmutter, Holographic entanglement entropy and gravitational anomalies, JHEP 07 (2014) 114 [arXiv:1405.2792] [INSPIRE].
S. Caron-Huot, Analyticity in Spin in Conformal Theories, JHEP 09 (2017) 078 [arXiv:1703.00278] [INSPIRE].
D.A. Roberts, D. Stanford and L. Susskind, Localized shocks, JHEP 03 (2015) 051 [arXiv:1409.8180] [INSPIRE].
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Chang, CM., Ramirez, D.M. & Rangamani, M. Spinning constraints on chaotic large c CFTs. J. High Energ. Phys. 2019, 68 (2019). https://doi.org/10.1007/JHEP03(2019)068
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DOI: https://doi.org/10.1007/JHEP03(2019)068