Einstein-AdS action, renormalized volume/area and holographic Renyi 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 Renyi 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.

26 pages, a new appendix added, small clarifications made, accepted for publication in JHEP

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• Conformal renormalization of scalar-tensor theories
• Noether-Wald charges in six-dimensional Critical Gravity
• $\mathcal{N}=2$ AdS${}_4$ supergravity, holography and Ward identities
• Renormalized holographic entanglement entropy in Lovelock gravity
• Shape dependence of renormalized holographic entanglement entropy
• Renormalized AdS gravity and holographic entanglement entropy of even-dimensional CFTs
• Renormalized holographic entanglement entropy for Quadratic Curvature Gravity
• Conformal Renormalization of topological black holes in AdS${}_6$
• Renormalized entanglement entropy and curvature invariants
• Holographic Weyl anomaly for GJMS operators: one Laplacian to rule them all
• Topological invariants of the Ryu-Takayanagi ($RT$) surface used to observe holographic superconductor phase transition
• Non-equilibrium dynamics in Holography
• Conical defects and holography in topological AdS gravity
• Generalised proofs of the first law of entanglement entropy
• Disks globally maximize the entanglement entropy in $2+1$ dimensions
• Black Hole Energy in Lovelock Unique Vacuum theories