Abstract
We exhibit the equivalence between the renormalized volume of asymptotically anti-de Sitter (AAdS) Einstein manifolds in four and six dimensions, and their renormalized Euclidean bulk gravity actions. The action is that of Einstein gravity, where the renormalization is achieved through the addition of a single topological term. We generalize this equivalence, proposing an explicit form for the renormalized volume of higher even-dimensional AAdS Einstein manifolds. We also show that evaluating the renormalized bulk gravity action on the conically singular manifold of the replica trick results in an action principle that corresponds to the renormalized volume of the regular part of the bulk, plus the renormalized area of a codimension-2 cosmic brane whose tension is related to the replica index. Renormalized Rényi entropy of odd-dimensional holographic CFTs can thus be obtained from the renormalized area of the brane with finite tension, including the effects of its backreaction on the bulk geometry. The area computation corresponds to an extremization problem for an enclosing surface that extends to the AdS boundary, where the newly defined renormalized volume is considered.
Highlights
To provide relevant information about the boundary geometry, in the form of conformal invariant quantities
We show that evaluating the renormalized bulk gravity action on the conically singular manifold of the replica trick results in an action principle that corresponds to the renormalized volume of the regular part of the bulk, plus the renormalized area of a codimension-2 cosmic brane whose tension is related to the replica index
We show that the relation between the EH part of the action and the bare volume given by eq (1.1), still holds between the renormalized action IEreHn [M2n] and the renormalized volume Volren [M2n], for the cases where a definite expression for the renormalized volume exists
Summary
The standard EH action, when evaluated on an AAdS Einstein manifold, is proportional to the volume of the manifold, which is divergent. We propose that for 2n−dimensional spacetimes, the renormalized Einstein-AdS action IEreHn is proportional to the corresponding renormalized volume of the bulk manifold. Considering the form of IEreHn [M2n] given in eq (2.4), we show that, as mentioned in the introduction, the renormalized volume of AAdS Einstein manifolds in four and six dimensions is proportional to the renormalized Einstein-AdS action, with the same proportionality factor considered in eq (1.1). These cases serve as examples for our conjectured relation in the general 2n−dimensional case, which we discuss afterwards
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