Abstract
We consider two dimensional conformal field theory (CFT) with large central charge c in an excited state obtained by the insertion of an operator Φ with large dimension ΔΦ ∼ O(c) at spatial infinities in the thermal state. We argue that correlation functions of light operators in such a state can be viewed as thermal correlators with a rescaled effective temperature. The effective temperature controls the growth of out-of-time order (OTO) correlators and results in a violation of the universal upper bound on the associated Lyapunov exponent when ΔΦ < 0 and the CFT is nonunitary. We present a specific realization of this situation in the holographic Chern-Simons formulation of a CFT with \( {\mathrm{W}}_3^{(2)} \) symmetry also known as the Bershadsky-Polyakov algebra. We examine the precise correspondence between the semiclassical (large-c) representations of this algebra and the Chern-Simons formulation, and infer that the holographic CFT possesses a discretuum of degenerate ground states with negative conformal dimension \( {\Delta}_{\Phi}=-\frac{c}{8} \). Using the Wilson line prescription to compute entanglement entropy and OTO correlators in the holographic CFT undergoing a local quench, we find the Lyapunov exponent \( {\uplambda}_L=\frac{4\pi }{\beta } \), violating the universal chaos bound.
References
P.H. Ginsparg, Applied Conformal Field Theory, in proceedings of the Les Houches Summer School in Theoretical Physics: Fields, Strings, Critical Phenomena, Les Houches, France, 28 June–5 August 1988, pp. 1–168 [hep-th/9108028] [INSPIRE].
D.M. Hofman and J. Maldacena, Conformal collider physics: Energy and charge correlations, JHEP05 (2008) 012 [arXiv:0803.1467] [INSPIRE].
D.M. Hofman, D. Li, D. Meltzer, D. Poland and F. Rejon-Barrera, A Proof of the Conformal Collider Bounds, JHEP06 (2016) 111 [arXiv:1603.03771] [INSPIRE].
T. Hartman, S. Jain and S. Kundu, Causality Constraints in Conformal Field Theory, JHEP05 (2016) 099 [arXiv:1509.00014] [INSPIRE].
T. Hartman, S. Kundu and A. Tajdini, Averaged Null Energy Condition from Causality, JHEP07 (2017) 066 [arXiv:1610.05308] [INSPIRE].
J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP08 (2016) 106 [arXiv:1503.01409] [INSPIRE].
N. Afkhami-Jeddi, K. Colville, T. Hartman, A. Maloney and E. Perlmutter, Constraints on higher spin CFT2 , JHEP05 (2018) 092 [arXiv:1707.07717] [INSPIRE].
J.R. David, S. Khetrapal and S.P. Kumar, Local quenches and quantum chaos from higher spin perturbations, JHEP10 (2017) 156 [arXiv:1707.07166] [INSPIRE].
P. Narayan and J. Yoon, Chaos in Three-dimensional Higher Spin Gravity, JHEP07 (2019) 046 [arXiv:1903.08761] [INSPIRE].
A.L. Fitzpatrick, J. Kaplan and M.T. Walters, Virasoro Conformal Blocks and Thermality from Classical Background Fields, JHEP11 (2015) 200 [arXiv:1501.05315] [INSPIRE].
C.T. Asplund, A. Bernamonti, F. Galli and T. Hartman, Holographic Entanglement Entropy from 2d CFT: Heavy States and Local Quenches, JHEP02 (2015) 171 [arXiv:1410.1392] [INSPIRE].
A.L. Fitzpatrick, J. Kaplan and M.T. Walters, Universality of Long-Distance AdS Physics from the CFT Bootstrap, JHEP08 (2014) 145 [arXiv:1403.6829] [INSPIRE].
D. Bianchini, O.A. Castro-Alvaredo, B. Doyon, E. Levi and F. Ravanini, Entanglement Entropy of Non Unitary Conformal Field Theory, J. Phys.A 48 (2015) 04FT01 [arXiv:1405.2804] [INSPIRE].
A.M. Polyakov, Gauge Transformations and Diffeomorphisms, Int. J. Mod. Phys.A 5 (1990) 833 [INSPIRE].
M. Bershadsky, Conformal field theories via Hamiltonian reduction, Commun. Math. Phys.139 (1991) 71 [INSPIRE].
M. Ammon, M. Gutperle, P. Kraus and E. Perlmutter, Spacetime Geometry in Higher Spin Gravity, JHEP10 (2011) 053 [arXiv:1106.4788] [INSPIRE].
J.R. David, M. Ferlaino and S.P. Kumar, Thermodynamics of higher spin black holes in 3D, JHEP11 (2012) 135 [arXiv:1210.0284] [INSPIRE].
J. de Boer and J.I. Jottar, Entanglement Entropy and Higher Spin Holography in AdS3 , JHEP04 (2014) 089 [arXiv:1306.4347] [INSPIRE].
M. Ammon, A. Castro and N. Iqbal, Wilson Lines and Entanglement Entropy in Higher Spin Gravity, JHEP10 (2013) 110 [arXiv:1306.4338] [INSPIRE].
E. Perlmutter, Bounding the Space of Holographic CFTs with Chaos, JHEP10 (2016) 069 [arXiv:1602.08272] [INSPIRE].
J.L. Cardy, Operator Content of Two-Dimensional Conformally Invariant Theories, Nucl. Phys.B 270 (1986) 186 [INSPIRE].
J. de Boer, A. Castro, E. Hijano, J.I. Jottar and P. Kraus, Higher spin entanglement and WN conformal blocks, JHEP07 (2015) 168 [arXiv:1412.7520] [INSPIRE].
P. Banerjee, S. Datta and R. Sinha, Higher-point conformal blocks and entanglement entropy in heavy states, JHEP05 (2016) 127 [arXiv:1601.06794] [INSPIRE].
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].
T. Anous and J. Sonner, Phases of scrambling in eigenstates, SciPost Phys.7 (2019) 003 [arXiv:1903.03143] [INSPIRE].
P. Goddard, A. Kent and D.I. Olive, Virasoro Algebras and Coset Space Models, Phys. Lett.B 152 (1985) 88 [INSPIRE].
P. Goddard, A. Kent and D.I. Olive, Unitary Representations of the Virasoro and Supervirasoro Algebras, Commun. Math. Phys.103 (1986) 105 [INSPIRE].
V.G. Drinfeld and V.V. Sokolov, Lie algebras and equations of Korteweg- de Vries type, J. Sov. Math.30 (1984) 1975 [INSPIRE].
N. Wyllard, W-algebras and surface operators in N = 2 gauge theories, J. Phys.A 44 (2011) 155401 [arXiv:1011.0289] [INSPIRE].
T. Arakawa, Rationality of Bershadsky-Polyakov vertex algebras, Commun. Math. Phys.323 (2013) 627 [arXiv:1005.0185] [INSPIRE].
M. Gutperle and P. Kraus, Higher Spin Black Holes, JHEP05 (2011) 022 [arXiv:1103.4304] [INSPIRE].
P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech.0406 (2004) P06002 [hep-th/0405152] [INSPIRE].
P. Caputa, J. Simón, A. Ŝtikonas and T. Takayanagi, Quantum Entanglement of Localized Excited States at Finite Temperature, JHEP01 (2015) 102 [arXiv:1410.2287] [INSPIRE].
P. Caputa, J. Simón, A. Ŝtikonas, T. Takayanagi and K. Watanabe, Scrambling time from local perturbations of the eternal BTZ black hole, JHEP08 (2015) 011 [arXiv:1503.08161] [INSPIRE].
J.R. David, S. Khetrapal and S.P. Kumar, Universal corrections to entanglement entropy of local quantum quenches, JHEP08 (2016) 127 [arXiv:1605.05987] [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].
M. Henneaux and S.-J. Rey, Nonlinear Winfinity as Asymptotic Symmetry of Three-Dimensional Higher Spin Anti-de Sitter Gravity, JHEP12 (2010) 007 [arXiv:1008.4579] [INSPIRE].
A. Campoleoni, S. Fredenhagen, S. Pfenninger and S. Theisen, Asymptotic symmetries of three-dimensional gravity coupled to higher-spin fields, JHEP11 (2010) 007 [arXiv:1008.4744] [INSPIRE].
M. Ammon, M. Gutperle, P. Kraus and E. Perlmutter, Black holes in three dimensional higher spin gravity: A review, J. Phys.A 46 (2013) 214001 [arXiv:1208.5182] [INSPIRE].
P. Calabrese, J. Cardy and E. Tonni, Entanglement negativity in quantum field theory, Phys. Rev. Lett.109 (2012) 130502 [arXiv:1206.3092] [INSPIRE].
P. Calabrese, J. Cardy and E. Tonni, Entanglement negativity in extended systems: A field theoretical approach, J. Stat. Mech.1302 (2013) P02008 [arXiv:1210.5359] [INSPIRE].
S. Datta, J.R. David, M. Ferlaino and S.P. Kumar, Higher spin entanglement entropy from CFT, JHEP06 (2014) 096 [arXiv:1402.0007] [INSPIRE].
S. Datta, J.R. David, M. Ferlaino and S.P. Kumar, Universal correction to higher spin entanglement entropy, Phys. Rev.D 90 (2014) 041903 [arXiv:1405.0015] [INSPIRE].
R. de Mello Koch, W. LiMing, H.J.R. Van Zyl and J.P. Rodrigues, Chaos in the Fishnet, Phys. Lett. B 793 (2019) 169 [arXiv:1902.06409] [INSPIRE].
V. Jahnke, K.-Y. Kim and J. Yoon, On the Chaos Bound in Rotating Black Holes, JHEP05 (2019) 037 [arXiv:1903.09086] [INSPIRE].
R.R. Poojary, BTZ dynamics and chaos, arXiv:1812.10073 [INSPIRE].
A. Ŝtikonas, Scrambling time from local perturbations of the rotating BTZ black hole, JHEP02 (2019) 054 [arXiv:1810.06110] [INSPIRE].
M. Ferlaino, T. Hollowood and S.P. Kumar, Asymptotic symmetries and thermodynamics of higher spin black holes in AdS3 , Phys. Rev.D 88 (2013) 066010 [arXiv:1305.2011] [INSPIRE].
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David, J.R., Hollowood, T.J., Khetrapal, S. et al. Chaos bound in Bershadsky-Polyakov theory. J. High Energ. Phys. 2019, 77 (2019). https://doi.org/10.1007/JHEP10(2019)077
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DOI: https://doi.org/10.1007/JHEP10(2019)077
Keywords
- AdS-CFT Correspondence
- Conformal and W Symmetry
- Conformal Field Theory
- Higher Spin Gravity