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High Energy Physics - Theory, hep-th,General Relativity and Quantum Cosmology, gr-qc
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 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
$d\bm{\Omega_{4n-1}}=tr(\bm{R}^{2n})$. 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
$\Omega_{4n-1}$. 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.
