Abstract
In this paper we investigate the entropy of gravitational Chern-Simons terms for the horizon with non-vanishing extrinsic curvatures, or the holographic entanglement entropy for arbitrary entangling surface. In 3D there is no anomaly of entropy. But the original squashed cone method can not be used directly to get the correct result. For higher dimensions the anomaly of entropy would appear, still, we can not use the squashed cone method directly. That is becasuse the Chern-Simons action is not gauge invariant. To get a reasonable result we suggest two methods. One is by adding a boundary term to recover the gauge invariance. This boundary term can be derived from the variation of the Chern-Simons action. The other one is by using the Chern-Simons relation dΩ4n−1 = tr(R 2n). We notice that the entropy of tr(R 2n) is a total derivative locally, i.e. S = ds CS . We propose to identify s CS with the entropy of gravitational Chern-Simons terms Ω4n − 1. In the first method we could get the correct result for Wald entropy in arbitrary dimension. In the second approach, in addition to Wald entropy, we can also obtain the anomaly of entropy with non-zero extrinsic curvatures. Our results imply that the entropy of a topological invariant, such as the Pontryagin term tr(R 2n) and the Euler density, is a topological invariant on the entangling surface.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J.D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333 [INSPIRE].
S.W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT,Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
R.M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48 (1993) 3427 [gr-qc/9307038] [INSPIRE].
A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [INSPIRE].
D.V. Fursaev, A. Patrushev and S.N. Solodukhin, Distributional geometry of squashed cones, Phys. Rev. D 88 (2013) 044054 [arXiv:1306.4000] [INSPIRE].
X. Dong and R.-X. Miao, Generalized gravitational entropy from total derivative action, JHEP 12 (2015) 100 [arXiv:1510.04273] [INSPIRE].
X. Dong, Holographic entanglement entropy for general higher derivative gravity, JHEP 01 (2014) 044 [arXiv:1310.5713] [INSPIRE].
J. Camps, Generalized entropy and higher derivative gravity, JHEP 03 (2014) 070 [arXiv:1310.6659] [INSPIRE].
R.-X. Miao and W.-Z. Guo, Holographic entanglement entropy for the most general higher derivative gravity, JHEP 08 (2015) 031 [arXiv:1411.5579] [INSPIRE].
R.-X. Miao, Universal terms of entanglement entropy for 6d CFTs, JHEP 10 (2015) 049 [arXiv:1503.05538] [INSPIRE].
Y. Huang and R.-X. Miao, A note on the resolution of the entropy discrepancy, Phys. Lett. B 749 (2015) 489 [arXiv:1504.02301] [INSPIRE].
A. Bhattacharyya, A. Kaviraj and A. Sinha, Entanglement entropy in higher derivative holography, JHEP 08 (2013) 012 [arXiv:1305.6694] [INSPIRE].
A. Bhattacharyya, M. Sharma and A. Sinha, On generalized gravitational entropy, squashed cones and holography, JHEP 01 (2014) 021 [arXiv:1308.5748] [INSPIRE].
A. Bhattacharyya and M. Sharma, On entanglement entropy functionals in higher derivative gravity theories, JHEP 10 (2014) 130 [arXiv:1405.3511] [INSPIRE].
S.L. Dubovsky and S.M. Sibiryakov, Spontaneous breaking of Lorentz invariance, black holes and perpetuum mobile of the 2nd kind, Phys. Lett. B 638 (2006) 509 [hep-th/0603158] [INSPIRE].
T. Jacobson and A.C. Wall, Black hole thermodynamics and Lorentz symmetry, Found. Phys. 40 (2010) 1076 [arXiv:0804.2720] [INSPIRE].
R.-X. Miao, M. Li and Y.-G. Miao, Violation of the first law of black hole thermodynamics in f(T) gravity, JCAP 11 (2011) 033 [arXiv:1107.0515] [INSPIRE].
S.N. Solodukhin, Holography with gravitational Chern-Simons, Phys. Rev. D 74 (2006) 024015 [hep-th/0509148] [INSPIRE].
R.F. Perez, Conserved current for the Cotton tensor, black hole entropy and equivariant Pontryagin forms, Class. Quant. Grav. 27 (2010) 135015 [arXiv:1004.3161] [INSPIRE].
P. Kraus and F. Larsen, Holographic gravitational anomalies, JHEP 01 (2006) 022 [hep-th/0508218] [INSPIRE].
M.-I. Park, BTZ black hole with gravitational Chern-Simons: thermodynamics and statistical entropy, Phys. Rev. D 77 (2008) 026011 [hep-th/0608165] [INSPIRE].
O. Mišković and R. Olea, Background-independent charges in topologically massive gravity, JHEP 12 (2009) 046 [arXiv:0909.2275] [INSPIRE].
P. Kraus and F. Larsen, Microscopic black hole entropy in theories with higher derivatives, JHEP 09 (2005) 034 [hep-th/0506176] [INSPIRE].
Y. Tachikawa, Black hole entropy in the presence of Chern-Simons terms, Class. Quant. Grav. 24 (2007) 737 [hep-th/0611141] [INSPIRE].
L. Bonora, M. Cvitan, P. Dominis Prester, S. Pallua and I. Smolic, Gravitational Chern-Simons Lagrangians and black hole entropy, JHEP 07 (2011) 085 [arXiv:1104.2523] [INSPIRE].
L. Bonora, M. Cvitan, P.D. Prester, S. Pallua and I. Smolic, Gravitational Chern-Simons terms and black hole entropy. Global aspects, JHEP 10 (2012) 077 [arXiv:1207.6969] [INSPIRE].
K. Jensen, R. Loganayagam and A. Yarom, Anomaly inflow and thermal equilibrium, JHEP 05 (2014) 134 [arXiv:1310.7024] [INSPIRE].
K. Jensen, R. Loganayagam and A. Yarom, Chern-Simons terms from thermal circles and anomalies, JHEP 05 (2014) 110 [arXiv:1311.2935] [INSPIRE].
T. Azeyanagi, R. Loganayagam, G.S. Ng and M.J. Rodriguez, Holographic thermal helicity, JHEP 08 (2014) 040 [arXiv:1311.2940] [INSPIRE].
T. Azeyanagi, R. Loganayagam, G.S. Ng and M.J. Rodriguez, Covariant Noether charge for higher dimensional Chern-Simons terms, JHEP 05 (2015) 041 [arXiv:1407.6364] [INSPIRE].
T. Azeyanagi, R. Loganayagam and G.S. Ng, Anomalies, Chern-Simons terms and black hole entropy, JHEP 09 (2015) 121 [arXiv:1505.02816] [INSPIRE].
S.N. Solodukhin, Holographic description of gravitational anomalies, JHEP 07 (2006) 003 [hep-th/0512216] [INSPIRE].
S. Deser, R. Jackiw and S. Templeton, Topologically massive gauge theories, Annals Phys. 140 (1982) 372 [Erratum ibid. 185 (1988) 406] [Erratum ibid. 281 (2000) 409] [INSPIRE].
A. Castro, S. Detournay, N. Iqbal and E. Perlmutter, Holographic entanglement entropy and gravitational anomalies, JHEP 07 (2014) 114 [arXiv:1405.2792] [INSPIRE].
J.-R. Sun, Note on Chern-Simons term correction to holographic entanglement entropy, JHEP 05 (2009) 061 [arXiv:0810.0967] [INSPIRE].
A. Schwimmer and S. Theisen, Entanglement entropy, trace anomalies and holography, Nucl. Phys. B 801 (2008) 1 [arXiv:0802.1017] [INSPIRE].
T. Azeyanagi, R. Loganayagam and G.S. Ng, Holographic entanglement for Chern-Simons terms, arXiv:1507.02298 [INSPIRE].
L.-Y. Hung, R.C. Myers and M. Smolkin, On holographic entanglement entropy and higher curvature gravity, JHEP 04 (2011) 025 [arXiv:1101.5813] [INSPIRE].
J. de Boer, M. Kulaxizi and A. Parnachev, Holographic entanglement entropy in Lovelock gravities, JHEP 07 (2011) 109 [arXiv:1101.5781] [INSPIRE].
W.-Z. Guo, S. He and J. Tao, Note on entanglement temperature for low thermal excited states in higher derivative gravity, JHEP 08 (2013) 050 [arXiv:1305.2682] [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: 1506.08397
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
Guo, WZ., Miao, RX. Entropy for gravitational Chern-Simons terms by squashed cone method. J. High Energ. Phys. 2016, 6 (2016). https://doi.org/10.1007/JHEP04(2016)006
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2016)006