Abstract
We derive the holographic entanglement entropy contribution from pure and mixed gravitational Chern-Simons(CS) terms in AdS2k+1. This is done through two different methods: first, by a direct evaluation of CS action in a holographic replica geometry and second by a descent of Dong’s derivation applied to the corresponding anomaly polynomial. In lower dimensions (k = 1, 2), the formula coincides with the Tachikawa formula for black hole entropy from gravitational CS terms. New extrinsic curvature corrections appear for k ≥ 3: we give explicit and concise expressions for the two pure gravitational CS terms in AdS7 and present various consistency checks, including agreements with the black hole entropy formula when evaluated at the bifurcation surface.
Highlights
AdS/CFT takes a state of CFT and recasts into geometry on the AdS side
New extrinsic curvature corrections appear for k ≥ 3: we give explicit and concise expressions for the two pure gravitational CS terms in AdS7 and present various consistency checks, including agreements with the black hole entropy formula when evaluated at the bifurcation surface
For reader’s convenience, we summarize here the main results obtained in this paper — the derivation of holographic formulae for Chern-Simons contribution to entanglement entropy
Summary
AdS/CFT takes a state of CFT and recasts into geometry on the AdS side. While we understand a lot about how this dictionary works, a clear cut algorithm on the field theory side to construct the dual geometry is missing. This, apart from resolving conceptual issues about covariance in previous works [33, 34], reproduces the correct odd parity Cardy type formula for higher dimensional black holes [40, 41] Another motivation for studying extrinsic corrections for Chern-Simons terms is to compare these two deformations of the pre-symplectic current. Corrections to the Tachikawa formula, the pure gravitational CS terms in AdS7 show an interesting dependence on extrinsic curvatures While this first method is a direct generalization of Dong’s original derivation and a simple abstract formula can be written down for any CS term (See eq (4.1)), the complicated Christoffel connection dependence in higher dimensional CS terms makes it more and more tedious to evaluate our formula explicitly as we move higher in dimensions.. Our answers by this method matches with a direct evaluation of CS action
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