Abstract
The universal nature of black hole collapse in asymptotically AdS3 gravitational theories suggests that its holographic dual process, thermalization, should similarly be fixed by the universal features of 2d CFT with large central charge c. It is known that non-equilibrium states with scaling dimensions of order c can be sorted into states that eventually thermalize and those that fail to do so. By proving an equivalence between bounded Virasoro coadjoint orbits and certain (in)stability intervals of Hill’s equation it is shown that semi-classical CFTs possess a phase transition where a state that fails to thermalize can be promoted to a thermalizing state by preparing the system beforehand with an energy greater than an appropriate threshold energy. It is generally a difficult problem to ascertain whether a state will thermalize or not. As partial progress to this problem a set of lower bounds are presented for the threshold energy, which can alternatively be interpreted as criteria for thermalization.
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References
W. Magnus and S. Winkler, Hill’s equation, Interscience Publishers (1966).
S. Banerjee, J.-W. Brijan and G. Vos, On the universality of late-time correlators in semi-classical 2d CFTs, JHEP 08 (2018) 047 [arXiv:1805.06464] [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].
A.L. Fitzpatrick and J. Kaplan, Conformal Blocks Beyond the Semi-Classical Limit, JHEP 05 (2016) 075 [arXiv:1512.03052] [INSPIRE].
J. Balog, L. Feher and L. Palla, Coadjoint orbits of the Virasoro algebra and the global Liouville equation, Int. J. Mod. Phys. A 13 (1998) 315 [hep-th/9703045] [INSPIRE].
E. Witten, Coadjoint Orbits of the Virasoro Group, Commun. Math. Phys. 114 (1988) 1 [INSPIRE].
J. Cotler and K. Jensen, A theory of reparameterizations for AdS 3 gravity, arXiv:1808.03263 [INSPIRE].
J.L. Cardy, Operator Content of Two-Dimensional Conformally Invariant Theories, Nucl. Phys. B 270 (1986) 186 [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].
T. Hartman, Entanglement Entropy at Large Central Charge, arXiv:1303.6955 [INSPIRE].
B. Chen, J.-q. Wu and J.-j. Zhang, Holographic Description of 2D Conformal Block in Semi-classical Limit, JHEP 10 (2016) 110 [arXiv:1609.00801] [INSPIRE].
T. Anous, T. Hartman, A. Rovai and J. Sonner, Black Hole Collapse in the 1/c Expansion, JHEP 07 (2016) 123 [arXiv:1603.04856] [INSPIRE].
G. Segal, Unitarity Representations of Some Infinite Dimensional Groups, Commun. Math. Phys. 80 (1981) 301 [INSPIRE].
A. Garbarz and M. Leston, Classification of Boundary Gravitons in AdS 3 Gravity, JHEP 05 (2014) 141 [arXiv:1403.3367] [INSPIRE].
M. Bañados, Three-dimensional quantum geometry and black holes, AIP Conf. Proc. 484 (1999) 147 [hep-th/9901148] [INSPIRE].
E.J. Martinec, Conformal field theory, geometry and entropy, hep-th/9809021 [INSPIRE].
M.M. Sheikh-Jabbari and H. Yavartanoo, On quantization of AdS 3 gravity I: semi-classical analysis, JHEP 07 (2014) 104 [arXiv:1404.4472] [INSPIRE].
G. Compère, P. Mao, A. Seraj and M.M. Sheikh-Jabbari, Symplectic and Killing symmetries of AdS 3 gravity: holographic vs boundary gravitons, JHEP 01 (2016) 080 [arXiv:1511.06079] [INSPIRE].
M.M. Sheikh-Jabbari and H. Yavartanoo, On 3d bulk geometry of Virasoro coadjoint orbits: orbit invariant charges and Virasoro hair on locally AdS 3 geometries, Eur. Phys. J. C 76 (2016)493 [arXiv:1603.05272] [INSPIRE].
J. de Boer and D. Engelhardt, Remarks on thermalization in 2D CFT, Phys. Rev. D 94 (2016) 126019 [arXiv:1604.05327] [INSPIRE].
M. Srednicki, The approach to thermal equilibrium in quantized chaotic systems, J. Phys. A 32 (1999) 1163 [cond-mat/9809360].
O. Hulík, T. Procházka and J. Raeymaekers, Multi-centered AdS3 solutions from Virasoro conformal blocks, JHEP 03 (2017) 129 [arXiv:1612.03879] [INSPIRE].
L. Takhtajan and P. Zograf, Hyperbolic 2-spheres with conical singularities, accessory parameters and Kahler metrics on M(0, n), math/0112170.
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].
Al.B. Zamolodchikov, Conformal symmetry in two-dimensional space: Recursion representation of conformal block, Theor. Math. Phys. 73 (1987) 1088.
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].
J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].
P. Kraus, Lectures on black holes and the AdS 3 /CFT 2 correspondence, Lect. Notes Phys. 755 (2008) 193 [hep-th/0609074] [INSPIRE].
V. Balasubramanian and P. Kraus, A Stress tensor for Anti-de Sitter gravity, Commun. Math. Phys. 208 (1999) 413 [hep-th/9902121] [INSPIRE].
J.D. Brown and M. Henneaux, Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity, Commun. Math. Phys. 104 (1986) 207 [INSPIRE].
A.A. Kirillov, Elements of the Theory of Representations, Springer-Verlag (1976).
V.P. Lazutkin and T.F. Pankratova, Normal forms and versal deformations for Hill’s equation, Funct. Anal. Appl. 9 (1975) 306.
J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [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].
P. Kraus, A. Sivaramakrishnan and R. Snively, Late time Wilson lines, arXiv:1810.01439 [INSPIRE].
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Vos, G. Vacuum block thermalization in semi-classical 2d CFT. J. High Energ. Phys. 2019, 22 (2019). https://doi.org/10.1007/JHEP02(2019)022
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DOI: https://doi.org/10.1007/JHEP02(2019)022