Abstract
We reconsider the one-loop correction to the holographic entanglement entropy in AdS 3 /CFT 2 by analysing the contributions due to a bulk higher spin s current or a scalar field with scaling dimension Δ. We consider the two-interval case and work perturbatively in their small cross ratio x. We provide various results for the entanglement entropy due to the so-called CDW elements of the associated Schottky uniformization group. In particular, in the higher spin current case, we obtain a closed formula for all the contributions of the form \( \mathcal{O} \)(x 2s+p) up to \( \mathcal{O} \)(x 4s), where 2-CDW elements are relevant. In the scalar field case, we calculate the similar contributions for generic values of Δ. The terms up to \( \mathcal{O} \)(x 2Δ+5) are compared with an explicit CFT calculation with full agreement. The analysis exploits various simplifications which are valid in the strict entanglement limit of the Rényi entropy. This allows to identify in a clean way the relevant operators that provide the gravity result. The 2-CDW contributions are also analysed and a closed formula for the leading \( \mathcal{O} \)(x 4s) coefficient is presented as a function of the generic spin s. As a specific application, we combine the CDW and 2-CDW calculations and present the complete \( \mathcal{O} \)(x 4s+2) entanglement entropy for a spin s = 2, 3, 4 higher spin currents.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
G. ’t Hooft, Dimensional reduction in quantum gravity, gr-qc/9310026 [INSPIRE].
L. Susskind, The World as a hologram, J. Math. Phys. 36 (1995) 6377 [hep-th/9409089] [INSPIRE].
J.D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333 [INSPIRE].
S. Hawking, Particle creation by black holes, Commun. Math. Phys. 43 (1975) 199.
J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [hep-th/9711200] [INSPIRE].
S. Ryu and T. Takayanagi, Aspects of holographic entanglement entropy, JHEP 08 (2006) 045 [hep-th/0605073] [INSPIRE].
T. Takayanagi, Entanglement entropy from a holographic viewpoint, Class. Quant. Grav. 29 (2012) 153001 [arXiv:1204.2450] [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].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
M. Headrick, Entanglement Renyi entropies in holographic theories, Phys. Rev. D 82 (2010) 126010 [arXiv:1006.0047] [INSPIRE].
T. Hartman, Entanglement entropy at large central charge, arXiv:1303.6955 [INSPIRE].
T. Faulkner, The entanglement Renyi entropies of disjoint intervals in AdS/CFT, arXiv:1303.7221 [INSPIRE].
L. Bombelli, R.K. Koul, J. Lee and R.D. Sorkin, A quantum source of entropy for black holes, Phys. Rev. D 34 (1986) 373 [INSPIRE].
M. Srednicki, Entropy and area, Phys. Rev. Lett. 71 (1993) 666 [hep-th/9303048] [INSPIRE].
P. Calabrese, J. Cardy and E. Tonni, Entanglement entropy of two disjoint intervals in conformal field theory, J. Stat. Mech. 0911 (2009) P11001 [arXiv:0905.2069] [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].
A. Maloney and E. Witten, Quantum gravity partition functions in three dimensions, JHEP 02 (2010) 029 [arXiv:0712.0155] [INSPIRE].
X. Yin, Partition functions of three-dimensional pure gravity, Commun. Num. Theor. Phys. 2 (2008) 285 [arXiv:0710.2129] [INSPIRE].
S. Giombi, A. Maloney and X. Yin, One-loop partition functions of 3D gravity, JHEP 08 (2008) 007 [arXiv:0804.1773] [INSPIRE].
S. Datta and J.R. David, Renyi entropies of free bosons on the torus and holography, arXiv:1311.1218 [INSPIRE].
B. Chen and J.-J. Zhang, On short interval expansion of Rényi entropy, JHEP 11 (2013) 164 [arXiv:1309.5453] [INSPIRE].
B. Chen, J. Long and J.-j. Zhang, Holographic Rényi entropy for CFT with W symmetry, arXiv:1312.5510 [INSPIRE].
B. Chen, F.-y. Song and J.-j. Zhang, Holographic Rényi entropy in AdS 3 /LCFT 2 correspondence, arXiv:1401.0261 [INSPIRE].
E. Perlmutter, Comments on Rényi entropy in AdS 3 /CFT 2, arXiv:1312.5740 [INSPIRE].
S. Datta, J.R. David, M. Ferlaino and S.P. Kumar, Higher spin entanglement entropy from CFT, arXiv:1402.0007 [INSPIRE].
M.A. Vasiliev, Higher spin gauge theories in four-dimensions, three-dimensions and two-dimensions, Int. J. Mod. Phys. D 5 (1996) 763 [hep-th/9611024] [INSPIRE].
M.A. Vasiliev, Higher spin matter interactions in (2 + 1)-dimensions, hep-th/9607135 [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, An AdS 3 dual for minimal model CFTs, Phys. Rev. D 83 (2011) 066007 [arXiv:1011.2986] [INSPIRE].
T. Nishioka, S. Ryu and T. Takayanagi, Holographic entanglement entropy: an overview, J. Phys. A 42 (2009) 504008 [arXiv:0905.0932] [INSPIRE].
P. Calabrese and J. Cardy, Entanglement entropy and quantum field theory: I, J. Stat. Mech. (2004) P06002 [hep-th/0405152] [INSPIRE].
P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J. Phys. A 42 (2009) 504005 [arXiv:0905.4013] [INSPIRE].
D. Harlow, J. Maltz and E. Witten, Analytic continuation of Liouville theory, JHEP 12 (2011) 071 [arXiv:1108.4417] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1402.0659
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Beccaria, M., Macorini, G. On the next-to-leading holographic entanglement entropy in AdS 3 /CFT 2 . J. High Energ. Phys. 2014, 45 (2014). https://doi.org/10.1007/JHEP04(2014)045
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2014)045