Abstract
Recently, two groups have made distinct proposals for a de Sitter space that is emergent from conformal field theory (CFT). The first proposal is that, for two-dimensional holographic CFTs, the kinematic space of geodesics on a space-like slice of the asymptotically anti-de Sitter bulk is two-dimensional de Sitter space (dS2), with a metric that can be derived from the entanglement entropy of intervals in the CFT. In the second proposal, de Sitter dynamics emerges naturally from the first law of entanglement entropy for perturbations around the vacuum state of CFTs. We provide support for the equivalence of these two emergent spacetimes in the vacuum case and beyond. In particular, we study the kinematic spaces of nontrivial solutions of 3d gravity, including the BTZ black string, BTZ black hole, and conical singularities. We argue that the resulting spaces are generically globally hyperbolic spacetimes that support dynamics given boundary conditions at future infinity. For the BTZ black string, corresponding to a thermal state of the CFT, we show that both prescriptions lead to an emergent hyperbolic patch of dS2. We offer a general method for relating kinematic space and the auxiliary de Sitter space that is valid in the vacuum and thermal cases.
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S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
S. Ryu and T. Takayanagi, Aspects of Holographic Entanglement Entropy, JHEP 08 (2006) 045 [hep-th/0605073] [INSPIRE].
V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].
B. Freivogel, R.A. Jefferson, L. Kabir, B. Mosk and I.-S. Yang, Casting Shadows on Holographic Reconstruction, Phys. Rev. D 91 (2015) 086013 [arXiv:1412.5175] [INSPIRE].
N. Engelhardt and S. Fischetti, Covariant Constraints on Hole-ography, Class. Quant. Grav. 32 (2015) 195021 [arXiv:1507.00354] [INSPIRE].
J. Lin, Bulk Locality from Entanglement in Gauge/Gravity Duality, arXiv:1510.02367 [INSPIRE].
J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].
M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav. 42 (2010) 2323 [arXiv:1005.3035] [INSPIRE].
N. Lashkari, M.B. McDermott and M. Van Raamsdonk, Gravitational dynamics from entanglement ‘thermodynamics’, JHEP 04 (2014) 195 [arXiv:1308.3716] [INSPIRE].
T. Faulkner, M. Guica, T. Hartman, R.C. Myers and M. Van Raamsdonk, Gravitation from Entanglement in Holographic CFTs, JHEP 03 (2014) 051 [arXiv:1312.7856] [INSPIRE].
M. Freedman and M. Headrick, Bit threads and holographic entanglement, arXiv:1604.00354 [INSPIRE].
B. Swingle, Entanglement Renormalization and Holography, Phys. Rev. D 86 (2012) 065007 [arXiv:0905.1317] [INSPIRE].
M. Nozaki, S. Ryu and T. Takayanagi, Holographic Geometry of Entanglement Renormalization in Quantum Field Theories, JHEP 10 (2012) 193 [arXiv:1208.3469] [INSPIRE].
N. Bao et al., Consistency conditions for an AdS multiscale entanglement renormalization ansatz correspondence, Phys. Rev. D 91 (2015) 125036 [arXiv:1504.06632] [INSPIRE].
B. Czech, L. Lamprou, S. McCandlish and J. Sully, Integral Geometry and Holography, JHEP 10 (2015) 175 [arXiv:1505.05515] [INSPIRE].
B. Czech, L. Lamprou, S. McCandlish and J. Sully, Tensor Networks from Kinematic Space, JHEP 07 (2016) 100 [arXiv:1512.01548] [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].
R.C. Myers, J. Rao and S. Sugishita, Holographic Holes in Higher Dimensions, JHEP 06 (2014) 044 [arXiv:1403.3416] [INSPIRE].
M. Headrick, R.C. Myers and J. Wien, Holographic Holes and Differential Entropy, JHEP 10 (2014) 149 [arXiv:1408.4770] [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 et al., Tensor network quotient takes the vacuum to the thermal state, Phys. Rev. B 94 (2016) 085101 [arXiv:1510.07637] [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].
M. Headrick and T. Takayanagi, A holographic proof of the strong subadditivity of entanglement entropy, Phys. Rev. D 76 (2007) 106013 [arXiv:0704.3719] [INSPIRE].
P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 0406 (2004) P06002 [hep-th/0405152] [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].
M. Bañados, C. Teitelboim and J. Zanelli, The Black hole in three-dimensional space-time, Phys. Rev. Lett. 69 (1992) 1849 [hep-th/9204099] [INSPIRE].
M. Bañados, M. Henneaux, C. Teitelboim and J. Zanelli, Geometry of the (2+1) black hole, Phys. Rev. D 48 (1993) 1506 [Erratum ibid. D 88 (2013) 069902] [gr-qc/9302012] [INSPIRE].
D. Brill, Black holes and wormholes in (2+1)-dimensions, gr-qc/9904083 [INSPIRE].
V.E. Hubeny, H. Maxfield, M. Rangamani and E. Tonni, Holographic entanglement plateaux, JHEP 08 (2013) 092 [arXiv:1306.4004] [INSPIRE].
B. Chen and J.-q. Wu, Large interval limit of Rényi entropy at high temperature, Phys. Rev. D 92 (2015) 126002 [arXiv:1412.0763] [INSPIRE].
B. Chen and J.-q. Wu, Holographic calculation for large interval Rényi entropy at high temperature, Phys. Rev. D 92 (2015) 106001 [arXiv:1506.03206] [INSPIRE].
S.W. Hawking and D.N. Page, Thermodynamics of Black Holes in anti-de Sitter Space, Commun. Math. Phys. 87 (1983) 577 [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].
V. Balasubramanian, J. de Boer, E. Keski-Vakkuri and S.F. Ross, Supersymmetric conical defects: Towards a string theoretic description of black hole formation, Phys. Rev. D 64 (2001) 064011 [hep-th/0011217] [INSPIRE].
V. Balasubramanian, P. Kraus and M. Shigemori, Massless black holes and black rings as effective geometries of the D1-D5 system, Class. Quant. Grav. 22 (2005) 4803 [hep-th/0508110] [INSPIRE].
P. Caputa, M. Nozaki and T. Takayanagi, Entanglement of local operators in large-N conformal field theories, PTEP 2014 (2014) 093B06 [arXiv:1405.5946] [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].
J.L. Cardy, Operator Content of Two-Dimensional Conformally Invariant Theories, Nucl. Phys. B 270 (1986) 186 [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].
M. Casals, A. Fabbri, C. Martínez and J. Zanelli, Quantum dress for a naked singularity, Phys. Lett. B 760 (2016) 244 [arXiv:1605.06078] [INSPIRE].
R. Emparan, A. Fabbri and N. Kaloper, Quantum black holes as holograms in AdS brane worlds, JHEP 08 (2002) 043 [hep-th/0206155] [INSPIRE].
S. Aminneborg, I. Bengtsson, D. Brill, S. Holst and P. Peldan, Black holes and wormholes in (2+1)-dimensions, Class. Quant. Grav. 15 (1998) 627 [gr-qc/9707036] [INSPIRE].
F. Dal’Bo, Geodesic and horocyclic trajectories, Springer Science & Business Media (2010).
V. Balasubramanian, P. Hayden, A. Maloney, D. Marolf and S.F. Ross, Multiboundary Wormholes and Holographic Entanglement, Class. Quant. Grav. 31 (2014) 185015 [arXiv:1406.2663] [INSPIRE].
R.M. Wald, General Relativity, Chicago University Press (1984).
H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].
D.D. Blanco, H. Casini, L.-Y. Hung and R.C. Myers, Relative Entropy and Holography, JHEP 08 (2013) 060 [arXiv:1305.3182] [INSPIRE].
P. Di Francesco, P. Mathieu and D. Sénéchal, Conformal Field Theory, Springer, New York, U.S.A. (1997).
T. Hartman and N. Afkhami-Jeddi, Speed Limits for Entanglement, arXiv:1512.02695 [INSPIRE].
P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J. Phys. A 42 (2009) 504005 [arXiv:0905.4013] [INSPIRE].
P. Calabrese and A. Lefevre, Entanglement spectrum in one-dimensional systems, Phys. Rev. A 78 (2008) 032329 [arXiv:0806.3059].
L.-Y. Hung, R.C. Myers, M. Smolkin and A. Yale, Holographic Calculations of Renyi Entropy, JHEP 12 (2011) 047 [arXiv:1110.1084] [INSPIRE].
T. Barrella, X. Dong, S.A. Hartnoll and V.L. Martin, Holographic entanglement beyond classical gravity, JHEP 09 (2013) 109 [arXiv:1306.4682] [INSPIRE].
S. Datta and J.R. David, Rényi entropies of free bosons on the torus and holography, JHEP 04 (2014) 081 [arXiv:1311.1218] [INSPIRE].
E. Perlmutter, Comments on Renyi entropy in AdS 3 /CFT 2, JHEP 05 (2014) 052 [arXiv:1312.5740] [INSPIRE].
B. Chen, J.-B. Wu and J.-j. Zhang, Short interval expansion of Rényi entropy on torus, JHEP 08 (2016) 130 [arXiv:1606.05444] [INSPIRE].
D.L. Jafferis, A. Lewkowycz, J. Maldacena and S.J. Suh, Relative entropy equals bulk relative entropy, JHEP 06 (2016) 004 [arXiv:1512.06431] [INSPIRE].
D. Anninos, De Sitter Musings, Int. J. Mod. Phys. A 27 (2012) 1230013 [arXiv:1205.3855] [INSPIRE].
V. Mukhanov, Physical Foundations of Cosmology, Cambridge University Press, Cambridge (2005).
W. Israel, Thermo field dynamics of black holes, Phys. Lett. A 57 (1976) 107 [INSPIRE].
S.G. Avery and B.D. Chowdhury, No Holography for Eternal AdS Black Holes, arXiv:1312.3346 [INSPIRE].
S.D. Mathur, What is the dual of two entangled CFTs?, arXiv:1402.6378 [INSPIRE].
B.D. Chowdhury, Limitations of holography, Int. J. Mod. Phys. D 24 (2014) 1550008 [arXiv:1405.4292] [INSPIRE].
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Asplund, C.T., Callebaut, N. & Zukowski, C. Equivalence of emergent de Sitter spaces from conformal field theory. J. High Energ. Phys. 2016, 154 (2016). https://doi.org/10.1007/JHEP09(2016)154
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DOI: https://doi.org/10.1007/JHEP09(2016)154