Abstract
We extend our topological renormalization scheme for Entanglement Entropy to holographic CFTs of arbitrary odd dimensions in the context of the AdS/CFT correspondence. The procedure consists in adding the Chern form as a boundary term to the area functional of the Ryu-Takayanagi minimal surface. The renormalized Entanglement Entropy thus obtained can be rewritten in terms of the Euler characteristic and the AdS curvature of the minimal surface. This prescription considers the use of the Replica Trick to express the renormalized Entanglement Entropy in terms of the renormalized gravitational action evaluated on the conically-singular replica manifold extended to the bulk. This renormalized action is obtained in turn by adding the Chern form as the counterterm at the boundary of the 2n-dimensional asymptotically AdS bulk manifold. We explicitly show that, up to next-to-leading order in the holographic radial coordinate, the addition of this boundary term cancels the divergent part of the Entanglement Entropy. We discuss possible applications of the method for studying CFT parameters like central charges.
Highlights
In Ref. [1], we presented an alternative renormalization scheme for the entanglement entropy (EE) of 3D conformal field theories (CFTs) with 4D asymptotically anti–de Sitter (AAdS) gravity duals, in the context of the gauge/gravity duality [2,3,4]
We generalize this method to holographic CFTs of arbitrary odd dimensions by considering the properties of squashed cones [11,14] and the renormalized gravitational action given in Ref. [5]
We begin by reviewing the usual RT proposal [19,20], which considers that the EE of an entangling region A in a holographic CFT is given by the volume of a codimension-2 extremal surface Σ in the AAdS bulk
Summary
In Ref. [1], we presented an alternative renormalization scheme for the entanglement entropy (EE) of 3D conformal field theories (CFTs) with 4D asymptotically anti–de Sitter (AAdS) gravity duals, in the context of the gauge/gravity duality [2,3,4]. The renormalized EE obtained, corresponds to a modification of the Ryu-Takayanagi (RT) area functional [19,20], which includes the addition of the Chern form with a fixed coefficient We generalize this method to holographic CFTs of arbitrary odd dimensions by considering the properties of squashed cones [11,14] and the renormalized gravitational action given in Ref. ; ð4Þ is the AdS curvature of Σ, which depends on its intrinsic Riemann curvature tensor, and χ1⁄2Σ is the Euler characteristic of Σ This topological form of SrEeEn is useful for computing the renormalized EE of certain entangling regions, which give rise to constant curvature minimal surfaces in the bulk.
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