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Entropy for gravitational Chern-Simons terms by squashed cone method

  • Wu-Zhong GuoEmail author
  • Rong-Xin Miao
Open Access
Regular Article - Theoretical Physics

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.

Keywords

AdS-CFT Correspondence Classical Theories of Gravity 

Notes

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.

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Copyright information

© The Author(s) 2016

Authors and Affiliations

  1. 1.State Key Laboratory of Theoretical Physics, Institute of Theoretical PhysicsChinese Academy of ScienceBeijingP.R. China
  2. 2.Yukawa Institute for Theoretical Physics (YITP)Kyoto UniversityKyotoJapan
  3. 3.Max Planck Institute for Gravitational Physics (Albert Einstein Institute)GolmGermany

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