Einstein-AdS action, renormalized volume/area and holographic Rényi entropies

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.

Renormalized AdS gravity and holographic entanglement entropy of even-dimensional CFTs

“…An interesting geometric interpretation of (2.31) and (2.12) was established later in [82], by demonstrating explicitly that 4G N S ren is the renormalized area of the extremal surface Σ, and I ren [M] is proportional to the renormalized volume of the manifold M, in the context of conformal calculus [106][107][108]. These results are tied to the fact that, when examining the renormalized Einstein-AdS gravity action in the LM replica trick, the coupling of the cosmic brane which sources the conical singularity is through an action which is of Nambu-Goto form, but where instead of considering the area functional of the brane, one considers its renormalized area.…”

“…Another very attractive feature of the method is its deep connection with geometrical structures: topological invariants in the case of even D [76,78] (in which case the method is alternatively known as topological renormalization), and transgression forms in the case of odd D [79]. Also of interest is the fact that the Kounterterms lead to a renormalized action that makes direct contact with the definition of renormalized volume for AAdS previously studied in the mathematical literature [82], and is intriguingly linked on-shell with critical gravity [83,84] and conformal gravity [85,86].…”

Entanglement entropy for TT deformed CFT in general dimensions

“…For d = 2, this is related to the Hagedorn spectrum of LST [5,6,7,8,9].3 In 2d, an exact match between the result from field theory and holography is demonstrated in [20]. 4 For related connection between REE and renormalized volume, one can see [54] and related references.…”

Shape dependence of renormalized holographic entanglement entropy

“…In section 2, we review Kounterterms renormalization scheme in connection to the renormalized EE for a spherical entangling region developed in refs. [49,58,59]. This quantity is found to be proportional to the renormalized volume of the RT surface.…”