Abstract
Renormalized entanglement entropy can be defined using the replica trick for any choice of renormalization scheme; renormalized entanglement entropy in holographic settings is expressed in terms of renormalized areas of extremal surfaces. In this paper we show how holographic renormalized entanglement entropy can be expressed in terms of the Euler invariant of the surface and renormalized curvature invariants. For a spherical entangling region in an odd-dimensional CFT, the renormalized entanglement entropy is proportional to the Euler invariant of the holographic entangling surface, with the coefficient of proportionality capturing the (renormalized) F quantity. Variations of the entanglement entropy can be expressed elegantly in terms of renormalized curvature invariants, facilitating general proofs of the first law of entanglement.
Highlights
The focus in this paper will be on the holographic definition of renormalized entanglement entropy in terms of the renormalized area of entangling surfaces, as shown in (2.1) and (3.2)
Renormalized entanglement entropy can be defined using the replica trick for any choice of renormalization scheme; renormalized entanglement entropy in holographic settings is expressed in terms of renormalized areas of extremal surfaces
In this paper we show how holographic renormalized entanglement entropy can be expressed in terms of the Euler invariant of the surface and renormalized curvature invariants
Summary
Consider a codimension two static minimal surface Σ with boundary ∂Σ in an asymptotically locally AdS4 spacetime. G is the metric on the minimal surface and h is the metric at the boundary of the minimal surface It was shown in [17] that the renormalized area can be expressed in terms of the Euler characteristic of the surface and an integral of local invariants. The analysis of [17] was for two dimensional minimal surfaces in (d + 1)-dimensional asymptotically locally hyperbolic Einstein spaces i.e. Euclidean signature. In the case of a static Ryu-Takayanagi entangling surface, the extrinsic curvature in the time direction is zero and by tracelessness of the Weyl curvature the renormalized area reduces to. Weyl tensor in this way is to match with our higher dimensional result shown in the later section
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