Abstract
We show that the dynamics of the kinematic space of a 2-dimensional CFT is gravitational and described by Jackiw-Teitelboim theory. We discuss the first law of this 2-dimensional dilaton gravity theory to support the relation between modular Hamiltonian and dilaton that underlies the kinematic space construction. It is further argued that Jackiw-Teitelboim gravity can be derived from a 2-dimensional version of Jacobson’s maximal vacuum entanglement hypothesis. Applied to the kinematic space context, this leads us to the statement that the kinematic space of a 2-dimensional boundary CFT can be obtained from coupling the boundary CFT to JT gravity through a maximal vacuum entanglement principle.
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References
B. Czech, L. Lamprou, S. McCandlish and J. Sully, Integral Geometry and Holography, JHEP 10 (2015) 175 [arXiv:1505.05515] [INSPIRE].
J. de Boer, M.P. Heller, R.C. Myers and Y. Neiman, Holographic de Sitter Geometry from Entanglement in Conformal Field Theory, Phys. Rev. Lett. 116 (2016) 061602 [arXiv:1509.00113] [INSPIRE].
B. Czech, L. Lamprou, S. McCandlish, B. Mosk and J. Sully, A Stereoscopic Look into the Bulk, JHEP 07 (2016) 129 [arXiv:1604.03110] [INSPIRE].
J. de Boer, F.M. Haehl, M.P. Heller and R.C. Myers, Entanglement, holography and causal diamonds, JHEP 08 (2016) 162 [arXiv:1606.03307] [INSPIRE].
B. Czech, L. Lamprou, S. McCandlish, B. Mosk and J. Sully, Equivalent Equations of Motion for Gravity and Entropy, JHEP 02 (2017) 004 [arXiv:1608.06282] [INSPIRE].
N. Callebaut and H. Verlinde, Entanglement Dynamics in 2D CFT with Boundary: Entropic origin of JT gravity and Schwarzian QM, arXiv:1808.05583 [INSPIRE].
A. Karch, J. Sully, C.F. Uhlemann and D.G.E. Walker, Boundary Kinematic Space, JHEP 08 (2017) 039 [arXiv:1703.02990] [INSPIRE].
T. Jacobson, Entanglement Equilibrium and the Einstein Equation, Phys. Rev. Lett. 116 (2016) 201101 [arXiv:1505.04753] [INSPIRE].
C. Holzhey, F. Larsen and F. Wilczek, Geometric and renormalized entropy in conformal field theory, Nucl. Phys. B 424 (1994) 443 [hep-th/9403108] [INSPIRE].
N. Seiberg, Notes on quantum Liouville theory and quantum gravity, Prog. Theor. Phys. Suppl. 102 (1990) 319 [INSPIRE].
A.C. Wall, Testing the Generalized Second Law in 1+1 dimensional Conformal Vacua: An Argument for the Causal Horizon, Phys. Rev. D 85 (2012) 024015 [arXiv:1105.3520] [INSPIRE].
S. de Haro, S.N. Solodukhin and K. Skenderis, Holographic reconstruction of space-time and renormalization in the AdS/CFT correspondence, Commun. Math. Phys. 217 (2001) 595 [hep-th/0002230] [INSPIRE].
M.M. Roberts, Time evolution of entanglement entropy from a pulse, JHEP 12 (2012) 027 [arXiv:1204.1982] [INSPIRE].
O. Coussaert, M. Henneaux and P. van Driel, The Asymptotic dynamics of three-dimensional Einstein gravity with a negative cosmological constant, Class. Quant. Grav. 12 (1995) 2961 [gr-qc/9506019] [INSPIRE].
K. Bautier, F. Englert, M. Rooman and P. Spindel, The Fefferman-Graham ambiguity and AdS black holes, Phys. Lett. B 479 (2000) 291 [hep-th/0002156] [INSPIRE].
S. Carlip, Dynamics of asymptotic diffeomorphisms in (2 + 1)-dimensional gravity, Class. Quant. Grav. 22 (2005) 3055 [gr-qc/0501033] [INSPIRE].
H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].
C.T. Asplund, N. Callebaut and C. Zukowski, Equivalence of Emergent de Sitter Spaces from Conformal Field Theory, JHEP 09 (2016) 154 [arXiv:1604.02687] [INSPIRE].
J. Cardy and E. Tonni, Entanglement hamiltonians in two-dimensional conformal field theory, J. Stat. Mech. 1612 (2016) 123103 [arXiv:1608.01283] [INSPIRE].
S. Lloyd, The quantum geometric limit, arXiv:1206.6559 [INSPIRE].
P. Bueno, V.S. Min, A.J. Speranza and M.R. Visser, Entanglement equilibrium for higher order gravity, Phys. Rev. D 95 (2017) 046003 [arXiv:1612.04374] [INSPIRE].
S. Gao and R.M. Wald, The ’Physical process’ version of the first law and the generalized second law for charged and rotating black holes, Phys. Rev. D 64 (2001) 084020 [gr-qc/0106071] [INSPIRE].
M. Spradlin and A. Strominger, Vacuum states for AdS 2 black holes, JHEP 11 (1999) 021 [hep-th/9904143] [INSPIRE].
J. Maldacena, D. Stanford and Z. Yang, Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space, PTEP 2016 (2016) 12C104 [arXiv:1606.01857] [INSPIRE].
V. Iyer and R.M. Wald, Some properties of Noether charge and a proposal for dynamical black hole entropy, Phys. Rev. D 50 (1994) 846 [gr-qc/9403028] [INSPIRE].
P.D. Hislop and R. Longo, Modular Structure of the Local Algebras Associated With the Free Massless Scalar Field Theory, Commun. Math. Phys. 84 (1982) 71 [INSPIRE].
C.R. Nappi and A. Pasquinucci, Thermodynamics of two-dimensional black holes, Mod. Phys. Lett. A 7 (1992) 3337 [gr-qc/9208002] [INSPIRE].
D. Grumiller and R. McNees, Thermodynamics of black holes in two (and higher) dimensions, JHEP 04 (2007) 074 [hep-th/0703230] [INSPIRE].
R.B. Mann and S.F. Ross, The D → 2 limit of general relativity, Class. Quant. Grav. 10 (1993) 1405 [gr-qc/9208004] [INSPIRE].
M. Cadoni, Conformal equivalence of 2 − D dilaton gravity models, Phys. Lett. B 395 (1997) 10 [hep-th/9610201] [INSPIRE].
D. Grumiller and R. Jackiw, Liouville gravity from Einstein gravity, in Recent developments in theoretical physics, S. Gosh and G. Kar eds., World Scientific, Singapore (2010), pg. 331 [arXiv:0712.3775] [INSPIRE].
J. Engelsöy, T.G. Mertens and H. Verlinde, An investigation of AdS 2 backreaction and holography, JHEP 07 (2016) 139 [arXiv:1606.03438] [INSPIRE].
K. Jensen, Chaos in AdS 2 Holography, Phys. Rev. Lett. 117 (2016) 111601 [arXiv:1605.06098] [INSPIRE].
T.M. Fiola, J. Preskill, A. Strominger and S.P. Trivedi, Black hole thermodynamics and information loss in two-dimensions, Phys. Rev. D 50 (1994) 3987 [hep-th/9403137] [INSPIRE].
J. Gegenberg, G. Kunstatter and D. Louis-Martinez, Observables for two-dimensional black holes, Phys. Rev. D 51 (1995) 1781 [gr-qc/9408015] [INSPIRE].
R.B. Mann, Conservation laws and 2 − D black holes in dilaton gravity, Phys. Rev. D 47 (1993) 4438 [hep-th/9206044] [INSPIRE].
M.D. McGuigan, C.R. Nappi and S.A. Yost, Charged black holes in two-dimensional string theory, Nucl. Phys. B 375 (1992) 421 [hep-th/9111038] [INSPIRE].
A. Achucarro and M.E. Ortiz, Relating black holes in two-dimensions and three-dimensions, Phys. Rev. D 48 (1993) 3600 [hep-th/9304068] [INSPIRE].
D. Grumiller, W. Kummer and D.V. Vassilevich, Dilaton gravity in two-dimensions, Phys. Rept. 369 (2002) 327 [hep-th/0204253] [INSPIRE].
V. Balasubramanian, B.D. Chowdhury, B. Czech, J. de Boer and M.P. Heller, Bulk curves from boundary data in holography, Phys. Rev. D 89 (2014) 086004 [arXiv:1310.4204] [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. Strominger, Black hole entropy from near horizon microstates, JHEP 02 (1998) 009 [hep-th/9712251] [INSPIRE].
B. Czech, P. Hayden, N. Lashkari and B. Swingle, The Information Theoretic Interpretation of the Length of a Curve, JHEP 06 (2015) 157 [arXiv:1410.1540] [INSPIRE].
V. Balasubramanian, B.D. Chowdhury, B. Czech and J. de Boer, Entwinement and the emergence of spacetime, JHEP 01 (2015) 048 [arXiv:1406.5859] [INSPIRE].
T.G. Mertens, G.J. Turiaci and H.L. Verlinde, Solving the Schwarzian via the Conformal Bootstrap, JHEP 08 (2017) 136 [arXiv:1705.08408] [INSPIRE].
T.G. Mertens, The Schwarzian theory — origins, JHEP 05 (2018) 036 [arXiv:1801.09605] [INSPIRE].
J. Haegeman, T.J. Osborne, H. Verschelde and F. Verstraete, Entanglement Renormalization for Quantum Fields in Real Space, Phys. Rev. Lett. 110 (2013) 100402 [arXiv:1102.5524] [INSPIRE].
G. Vidal, Entanglement Renormalization, Phys. Rev. Lett. 99 (2007) 220405 [cond-mat/0512165] [INSPIRE].
F.A. Smirnov and A.B. Zamolodchikov, On space of integrable quantum field theories, Nucl. Phys. B 915 (2017) 363 [arXiv:1608.05499] [INSPIRE].
S. Dubovsky, V. Gorbenko and M. Mirbabayi, Asymptotic fragility, near AdS 2 holography and \( T\overline{T} \), JHEP 09 (2017) 136 [arXiv:1706.06604] [INSPIRE].
S. Dubovsky, V. Gorbenko and G. Hernández-Chifflet, T T partition function from topological gravity, JHEP 09 (2018) 158 [arXiv:1805.07386] [INSPIRE].
P. Caputa, N. Kundu, M. Miyaji, T. Takayanagi and K. Watanabe, Liouville Action as Path-Integral Complexity: From Continuous Tensor Networks to AdS/CFT, JHEP 11 (2017) 097 [arXiv:1706.07056] [INSPIRE].
R. Jackiw, Lower Dimensional Gravity, Nucl. Phys. B 252 (1985) 343 [INSPIRE].
C. Teitelboim, Gravitation and Hamiltonian Structure in Two Space-Time Dimensions, Phys. Lett. B 126 (1983) 41 [INSPIRE].
A. Almheiri and J. Polchinski, Models of AdS 2 backreaction and holography, JHEP 11 (2015) 014 [arXiv:1402.6334] [INSPIRE].
V. Iyer, Lagrangian perfect fluids and black hole mechanics, Phys. Rev. D 55 (1997) 3411 [gr-qc/9610025] [INSPIRE].
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Callebaut, N. The gravitational dynamics of kinematic space. J. High Energ. Phys. 2019, 153 (2019). https://doi.org/10.1007/JHEP02(2019)153
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DOI: https://doi.org/10.1007/JHEP02(2019)153