arXiv 1506.083972015-06hep-thgr-qc

引力Chern-Simons项熵的压缩锥方法

Entropy for gravitational Chern-Simons terms by squashed cone method

Wu-Zhong Guo, Rong-Xin Miao

备注：: 19 page
DOI：: 10.1007/JHEP04(2016)006

摘要：

本文研究了具有非零外曲率视界的引力Chern-Simons项的熵和任意纠缠面的全息纠缠熵。在3D中，我们没有发现熵的异常出现。但直接使用压锥法不能得到正确的结果。对于高维数，熵的反常现象会出现，但不能直接使用挤压锥方法。这是因为Chern-Simons作用量不是规范不变的。为了得到合理的结果，我们提出了两种方法。一种是通过增加边界项来恢复规范不变性。这个边界项可以从Chern-Simons作用量的变分中导出。另一种是利用陈-西蒙斯关系式$d\bm{Ω_{4n-1}}=tr(\bm{R}^{2n})$。我们注意到$tr(\bm{R}^{2n})$的熵是局部全导数，即$S=d s_{CS}$。我们建议用引力Chern-Simons项$Ω_{4n-1}$的熵来识别$s_{CS}$。第一种方法可以得到任意维Wald熵的正确结果。在第二种方法中，除了Wald熵之外，我们还可以得到具有非零外曲率的熵的反常。我们的结果意味着拓扑不变量的熵，如庞特里亚金项$tr(\bm{R}^{2n})$和欧拉密度，是一个拓扑不变量的纠缠表面。

英文摘要：

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 we find no anomaly of entropy appears. But the 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 $d\bm{Ω_{4n-1}}=tr(\bm{R}^{2n})$. We notice that the entropy of $tr(\bm{R}^{2n})$ is a total derivative locally, i.e. $S=d s_{CS}$. We propose to identify $s_{CS}$ with the entropy of gravitational Chern-Simons terms $Ω_{4n-1}$. 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 $tr(\bm{R}^{2n})$ and the Euler density, is a topological invariant on the entangling surface.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文