Abstract
We consider the entanglement entropies in dSd sliced (A)dSd+1 in the presence of a hard radial cutoff for 2 ≤ d ≤ 6. By considering a one parameter family of analytical solutions, parametrized by their turning point in the bulk r*, we are able to compute the entanglement entropy for generic intervals on the cutoff slice. It has been proposed that the field theory dual of this scenario is a strongly coupled CFT, deformed by a certain irrelevant deformation — the so-called \( T\overline{T} \) deformation. Surprisingly, we find that we may write the entanglement entropies formally in the same way as the entanglement entropy for antipodal points on the sphere by introducing an effective radius Reff = R cos(βϵ), where R is the radius of the sphere and βϵ related to the length of the interval. Geometrically, this is equivalent to following the \( T\overline{T} \) trajectory until the generic interval corresponds to antipodal points on the sphere. Finally, we check our results by comparing the asymptotic behavior (no Dirichlet wall present) with the results of Casini, Huerta and Myers. We then switch on counterterms on the cutoff slice which are important with regards to the field theory calculation. We explicitly compute the contributions of the counterterms to the entanglement entropy by considering the Wald entropy. In the second part of this work, we extend the field theory calculation of the entanglement entropy for antipodal points for a d-dimensional field theory in context of DS/dS holography. We find excellent agreement with the results from holography and show, in particular, that the effects of the counterterms in the field theory calculation match the Wald entropy associated with the counterterms on the gravity side.
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F.A. Smirnov and A.B. Zamolodchikov, On space of integrable quantum field theories, Nucl. Phys. B 915 (2017) 363 [arXiv:1608.05499] [INSPIRE].
A. Cavaglià, S. Negro, I.M. Szécsényi and R. Tateo, \( T\overline{T} \)-deformed 2D quantum field theories, JHEP 10 (2016) 112 [arXiv:1608.05534] [INSPIRE].
A.B. Zamolodchikov, Expectation value of composite field \( T\overline{T} \) in two-dimensional quantum field theory, hep-th/0401146 [INSPIRE].
L. McGough, M. Mezei and H. Verlinde, Moving the CFT into the bulk with \( T\overline{T} \), JHEP 04 (2018) 010 [arXiv:1611.03470] [INSPIRE].
S. Dubovsky, V. Gorbenko and M. Mirbabayi, Asymptotic fragility, near AdS2 holography and \( T\overline{T} \), JHEP 09 (2017) 136 [arXiv:1706.06604] [INSPIRE].
H.-S. Jeong, K.-Y. Kim and M. Nishida, Entanglement and Rényi entropy of multiple intervals in \( T\overline{T} \)-deformed CFT and holography, arXiv:1906.03894 [INSPIRE].
J. Cardy, The \( T\overline{T} \) deformation of quantum field theory as random geometry, JHEP 10 (2018) 186 [arXiv:1801.06895] [INSPIRE].
L. Apolo and W. Song, Heating up holography for single-trace \( J\overline{T} \) deformations, arXiv:1907.03745 [INSPIRE].
W. Cottrell and A. Hashimoto, Comments on \( T\overline{T} \) double trace deformations and boundary conditions, Phys. Lett. B 789 (2019) 251 [arXiv:1801.09708] [INSPIRE].
M. Guica and R. Monten, \( T\overline{T} \) and the mirage of a bulk cutoff, arXiv:1906.11251 [INSPIRE].
P. Kraus, J. Liu and D. Marolf, Cutoff AdS3 versus the \( T\overline{T} \) deformation, JHEP 07 (2018) 027 [arXiv:1801.02714] [INSPIRE].
W. Donnelly and V. Shyam, Entanglement entropy and \( T\overline{T} \) deformation, Phys. Rev. Lett. 121 (2018) 131602 [arXiv:1806.07444] [INSPIRE].
T. Hartman, J. Kruthoff, E. Shaghoulian and A. Tajdini, Holography at finite cutoff with a T2 deformation, JHEP 03 (2019) 004 [arXiv:1807.11401] [INSPIRE].
M. Taylor, TT deformations in general dimensions, arXiv:1805.10287 [INSPIRE].
G. Bonelli, N. Doroud and M. Zhu, \( T\overline{T} \)-deformations in closed form, JHEP 06 (2018) 149 [arXiv:1804.10967] [INSPIRE].
B. Le Floch and M. Mezei, Solving a family of \( T\overline{T} \)-like theories, arXiv:1903.07606 [INSPIRE].
O. Aharony and T. Vaknin, The TT* deformation at large central charge, JHEP 05 (2018) 166 [arXiv:1803.00100] [INSPIRE].
S. Chakraborty, A. Giveon and D. Kutasov, \( T\overline{T},J\overline{T},T\overline{J} \) and string theory, J. Phys. A 52 (2019) 384003 [arXiv:1905.00051] [INSPIRE].
D.J. Gross, J. Kruthoff, A. Rolph and E. Shaghoulian, \( T\overline{T} \) in AdS2 and quantum mechanics, arXiv:1907.04873 [INSPIRE].
O. Aharony, S. Datta, A. Giveon, Y. Jiang and D. Kutasov, Modular invariance and uniqueness of \( T\overline{T} \) deformed CFT, JHEP 01 (2019) 086 [arXiv:1808.02492] [INSPIRE].
S. Datta and Y. Jiang, \( T\overline{T} \) deformed partition functions, JHEP 08 (2018) 106 [arXiv:1806.07426] [INSPIRE].
A. Giveon, N. Itzhaki and D. Kutasov, \( T\overline{T} \) and LST, JHEP 07 (2017) 122 [arXiv:1701.05576] [INSPIRE].
V. Shyam, Background independent holographic dual to \( T\overline{T} \) deformed CFT with large central charge in 2 dimensions, JHEP 10 (2017) 108 [arXiv:1707.08118] [INSPIRE].
C. Murdia, Y. Nomura, P. Rath and N. Salzetta, Comments on holographic entanglement entropy in TT deformed conformal field theories, Phys. Rev. D 100 (2019) 026011 [arXiv:1904.04408] [INSPIRE].
V. Shyam, Finite cutoff AdS5 holography and the generalized gradient flow, JHEP 12 (2018) 086 [arXiv:1808.07760] [INSPIRE].
Y. Jiang, Expectation value of \( T\overline{T} \) operator in curved spacetimes, arXiv:1903.07561 [INSPIRE].
C. Park, Holographic entanglement entropy in cutoff AdS, Int. J. Mod. Phys. A 33 (2019) 1850226 [arXiv:1812.00545] [INSPIRE].
M. Guica, An integrable Lorentz-breaking deformation of two-dimensional CFTs, SciPost Phys. 5 (2018) 048 [arXiv:1710.08415] [INSPIRE].
M. Baggio, A. Sfondrini, G. Tartaglino-Mazzucchelli and H. Walsh, On \( T\overline{T} \) deformations and supersymmetry, JHEP 06 (2019) 063 [arXiv:1811.00533] [INSPIRE].
C.-K. Chang, C. Ferko and S. Sethi, Supersymmetry and \( T\overline{T} \) deformations, JHEP 04 (2019) 131 [arXiv:1811.01895] [INSPIRE].
B. Chen, L. Chen and C.-Y. Zhang, Surface/State correspondence and \( T\overline{T} \) deformation, arXiv:1907.12110 [INSPIRE].
T. Ota, Comments on holographic entanglements in cutoff AdS, arXiv:1904.06930 [INSPIRE].
J. Cardy, \( T\overline{T} \) deformation of correlation functions, arXiv:1907.03394 [INSPIRE].
A. Banerjee, A. Bhattacharyya and S. Chakraborty, Entanglement Entropy for TT deformed CFT in general dimensions, Nucl. Phys. B 948 (2019) 114775 [arXiv:1904.00716] [INSPIRE].
P. Caputa, S. Datta and V. Shyam, Sphere partition functions & cut-off AdS, JHEP 05 (2019) 112 [arXiv:1902.10893] [INSPIRE].
V. Gorbenko, E. Silverstein and G. Torroba, dS/dS and \( T\overline{T} \), JHEP 03 (2019) 085 [arXiv:1811.07965] [INSPIRE].
A. Sfondrini and S.J. van Tongeren, \( T\overline{T} \) deformations as TsT transformations, arXiv:1908.09299 [INSPIRE].
S. Chakraborty, A. Giveon, N. Itzhaki and D. Kutasov, Entanglement beyond AdS, Nucl. Phys. B 935 (2018) 290 [arXiv:1805.06286] [INSPIRE].
Y. Jiang, Lectures on solvable irrelevant deformations of 2d quantum field theory, arXiv:1904.13376 [INSPIRE].
P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 0406 (2004) P06002 [hep-th/0405152] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
B. Chen, L. Chen and P.-X. Hao, Entanglement entropy in \( T\overline{T} \)-deformed CFT, Phys. Rev. D 98 (2018) 086025 [arXiv:1807.08293] [INSPIRE].
M. Alishahiha, A. Karch, E. Silverstein and D. Tong, The dS/dS correspondence, AIP Conf. Proc. 743 (2004) 393 [hep-th/0407125] [INSPIRE].
M. Alishahiha, A. Karch and E. Silverstein, Hologravity, JHEP 06 (2005) 028 [hep-th/0504056] [INSPIRE].
X. Dong, B. Horn, E. Silverstein and G. Torroba, Micromanaging de Sitter holography, Class. Quant. Grav. 27 (2010) 245020 [arXiv:1005.5403] [INSPIRE].
B. Freivogel, Y. Sekino, L. Susskind and C.-P. Yeh, A holographic framework for eternal inflation, Phys. Rev. D 74 (2006) 086003 [hep-th/0606204] [INSPIRE].
X. Dong et al., FRW solutions and holography from uplifted AdS/CFT, Phys. Rev. D 85 (2012) 104035 [arXiv:1108.5732] [INSPIRE].
A. Karch, Autolocalization in de Sitter space, JHEP 07 (2003) 050 [hep-th/0305192] [INSPIRE].
X. Dong, E. Silverstein and G. Torroba, De Sitter holography and entanglement entropy, JHEP 07 (2018) 050 [arXiv:1804.08623] [INSPIRE].
H. Geng, S. Grieninger and A. Karch, Entropy, entanglement and swampland bounds in DS/dS, JHEP 06 (2019) 105 [arXiv:1904.02170] [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. Holzhey, F. Larsen and F. Wilczek, Geometric and renormalized entropy in conformal field theory, Nucl. Phys. B 424 (1994) 443 [hep-th/9403108] [INSPIRE].
P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory: a non-technical introduction, Int. J. Quant. Inf. 4 (2006) 429 [quant-ph/0505193] [INSPIRE].
T. Nishioka, S. Ryu and T. Takayanagi, Holographic entanglement entropy: an overview, J. Phys. A 42 (2009) 504008 [arXiv:0905.0932] [INSPIRE].
H. Casini, Geometric entropy, area and strong subadditivity, Class. Quant. Grav. 21 (2004) 2351 [hep-th/0312238] [INSPIRE].
M.B. Plenio, J. Eisert, J. Dreissig and M. Cramer, Entropy, entanglement and area: analytical results for harmonic lattice systems, Phys. Rev. Lett. 94 (2005) 060503 [quant-ph/0405142] [INSPIRE].
M. Cramer, J. Eisert, M.B. Plenio and J. Dreissig, An entanglement-area law for general bosonic harmonic lattice systems, Phys. Rev. A 73 (2006) 012309 [quant-ph/0505092] [INSPIRE].
S. Das and S. Shankaranarayanan, How robust is the entanglement entropy: area relation?, Phys. Rev. D 73 (2006) 121701 [gr-qc/0511066] [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].
K. Skenderis, Lecture notes on holographic renormalization, Class. Quant. Grav. 19 (2002) 5849 [hep-th/0209067] [INSPIRE].
M. Taylor and W. Woodhead, Renormalized entanglement entropy, JHEP 08 (2016) 165 [arXiv:1604.06808] [INSPIRE].
J.H. Cooperman and M.A. Luty, Renormalization of entanglement entropy and the gravitational effective action, JHEP 12 (2014) 045 [arXiv:1302.1878] [INSPIRE].
R. Emparan, C.V. Johnson and R.C. Myers, Surface terms as counterterms in the AdS/CFT correspondence, Phys. Rev. D 60 (1999) 104001 [hep-th/9903238] [INSPIRE].
T. Faulkner, A. Lewkowycz and J. Maldacena, Quantum corrections to holographic entanglement entropy, JHEP 11 (2013) 074 [arXiv:1307.2892] [INSPIRE].
T. Faulkner et al., Gravitation from entanglement in holographic CFTs, JHEP 03 (2014) 051 [arXiv:1312.7856] [INSPIRE].
R.M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48 (1993) R3427 [gr-qc/9307038] [INSPIRE].
T. Jacobson, G. Kang and R.C. Myers, On black hole entropy, Phys. Rev. D 49 (1994) 6587 [gr-qc/9312023] [INSPIRE].
T. Jacobson, G. Kang and R.C. Myers, Black hole entropy in higher curvature gravity, talk given at the Heat Kernels and Quantum Gravity, August 2–6, Winnipeg, Canada (1994), gr-qc/9502009 [INSPIRE].
R. Brustein, D. Gorbonos and M. Hadad, Wald’s entropy is equal to a quarter of the horizon area in units of the effective gravitational coupling, Phys. Rev. D 79 (2009) 044025 [arXiv:0712.3206] [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].
H. Liu and M. Mezei, A refinement of entanglement entropy and the number of degrees of freedom, JHEP 04 (2013) 162 [arXiv:1202.2070] [INSPIRE].
H. Liu and M. Mezei, Probing renormalization group flows using entanglement entropy, JHEP 01 (2014) 098 [arXiv:1309.6935] [INSPIRE].
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Grieninger, S. Entanglement entropy and \( T\overline{T} \) deformations beyond antipodal points from holography. J. High Energ. Phys. 2019, 171 (2019). https://doi.org/10.1007/JHEP11(2019)171
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DOI: https://doi.org/10.1007/JHEP11(2019)171