Abstract
We propose a renormalization scheme for entanglement entropy of three-dimensional CFTs with a four-dimensional asymptotically AdS gravity dual in the context of the gauge/gravity correspondence. The procedure consists in adding the Chern form as a boundary term to the area functional of the Ryu-Takayanagi minimal surface. We provide an explicit prescription for the renormalized entanglement entropy, which is derived via the replica trick. This is achieved by considering a Euclidean gravitational action renormalized by the addition of the Chern form at the spacetime boundary, evaluated in the conically-singular replica manifold. We show that the addition of this boundary term cancels the divergent part of the entanglement entropy, recovering the results obtained by Taylor and Woodhead. We comment on how this prescription for renormalizing the entanglement entropy is in line with the general program of topological renormalization in asymptotically AdS gravity.
Highlights
In the context of the AdS=CFT correspondence [1,2,3], the entanglement entropy (EE) of an entangling region A in a CFT with an asymptotically AdS (AAdS) Einstein gravity dual, can be computed as the volume of a codimension-2 minimal surface
This is done by evaluating the usual counterterms for Einstein gravity at the conically singular spacetime boundary, which is conformal to the manifold of the Replica CFT
This boundary term corresponds to the Chern form evaluated at the boundary of the RT minimal surface, which is conformal to the entangling surface that bounds the entangling region in the CFT
Summary
In the context of the AdS=CFT correspondence [1,2,3], the entanglement entropy (EE) of an entangling region A in a CFT with an asymptotically AdS (AAdS) Einstein gravity dual, can be computed as the volume of a codimension-2 minimal surface. As it was shown by Taylor and Woodhead [8], it is possible to renormalize the EE by adding counterterms constructed through the Replica Trick [8,9,10] from the standard Holographic Renormalization procedure [11,12,13,14,15,16,17] This is done by evaluating the usual counterterms for Einstein gravity at the conically singular spacetime boundary, which is conformal to the manifold of the Replica CFT. We propose an alternative regularization prescription that has the advantage of giving the countertem for the EE as a single boundary term, which can be written in closed form for CFTs of arbitrary (odd) dimensions that have an (evendimensional) AAdS E-H gravity dual This boundary term corresponds to the Chern form evaluated at the boundary of the RT minimal surface, which is conformal to the entangling surface that bounds the entangling region in the CFT. IV, we give a general outlook of the method and comment on possible generalizations thereof
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