The vacuum entanglement entropy of a general conformal field theory (CFT) in = 5 spacetime dimensions contains a universal term, ( ), which has a complicated and non-local dependence on the geometric details of the region and the theory. Analogously to the previously known = 3 case, we prove that for CFTs in = 5 which are holographically dual to Einstein gravity, ( ) is equal to a four-dimensional version of the “Willmore energy” associated to a doubled and closed version of the Ryu-Takayanagi (RT) surface of embedded in ℝ . This generalized Willmore energy is shown to arise from a conformal-invariant codimension-two functional obtained by evaluating six-dimensional Conformal Gravity on the conically-singular orbifold of the replica trick. The new functional involves an integral over the doubled RT surface of a linear combination of three quartic terms in extrinsic curvatures and is free from ultraviolet divergences by construction. We verify explicitly the validity of our new formula for various entangling regions and argue that, as opposed to the = 3 case, ( ) is not globally minimized by a round ball = 𝔹 . Rather, ( ) can take arbitrarily positive and negative values as a function of . Hence, we conclude that the round ball is not a global minimizer of ( ) for general five-dimensional CFTs.

Chile

Giorgos Anastasiou

Ignacio Araya

Spain

Pablo Bueno

Javier Moreno

Rodrigo Olea

Belgium

Alejandro Lopez

Higher-dimensional Willmore energy as holographic entanglement entropy

We describe a method to generate scalar–tensor theories with Weyl symmetry, starting from arbitrary purely metric higher derivative gravity theories. The method consists in the definition of a conformally-invariant metric  $$\hat{g}_{\mu \nu }$$  <math> <msub> <mover> <mi>g</mi> <mo>^</mo> </mover> <mrow> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </math>  , that is a rank (0,2)-tensor constructed out of the metric tensor and the scalar field. This new object has zero conformal weight and is given by  $$\phi ^{2/\Delta }g_{\mu \nu }$$  <math> <mrow> <msup> <mi>ϕ</mi> <mrow> <mn>2</mn> <mo>/</mo> <mi>Δ</mi> </mrow> </msup> <msub> <mi>g</mi> <mrow> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mrow> </math>  , where  $$(-\Delta )$$  <math> <mrow> <mo>(</mo> <mo>-</mo> <mi>Δ</mi> <mo>)</mo> </mrow> </math>   is the conformal dimension of the scalar. As  $$g_{\mu \nu }$$  <math> <msub> <mi>g</mi> <mrow> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </math>   has conformal dimension of 2, the resulting tensor is trivially a conformal invariant. Then, the generated scalar–tensor theory, which we call the Weyl uplift of the original purely metric theory, is obtained by replacing the metric by  $$\hat{g}_{\mu \nu }$$  <math> <msub> <mover> <mi>g</mi> <mo>^</mo> </mover> <mrow> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </math>   in the action that defines the original theory. This prescription allowed us to define the Weyl uplift of theories with terms of higher order in the Riemannian curvature. Furthermore, the prescription for scalar–tensor theories coming from terms that have explicit covariant derivatives in the Lagrangian is discussed. The same mechanism can also be used for the derivation of the equations of motion of the scalar–tensor theory from the original field equations in the Einstein frame. Applying this method of Weyl uplift allowed us to reproduce the known result for the conformal scalar coupling to Lovelock gravity and to derive that of Einsteinian cubic gravity. Finally, we show that the cancellation of the volume divergences in the theory given by the conformal scalar coupling to Einstein–Anti-de Sitter gravity is achieved by the Weyl uplift of the original theory augmented by counterterms, which is relevant in the framework of conformal renormalization.

Ireland

Avik Chakraborty

Weyl–invariant scalar–tensor gravities from purely metric theories

We present a streamlined proof that any Einstein-AdS space is a solution of the Lu, Pang and Pope conformal gravity theory in six dimensions. The reduction of conformal gravity into Einstein theory manifestly shows that the action of the latter can be written as the Einstein-Hilbert term plus the Euler topological density and an additional contribution that depends on the Laplacian of the bulk Weyl tensor squared. The prescription for obtaining this form of the action by embedding the Einstein theory into a Weyl-invariant purely metric theory, was dubbed Conformal Renormalization and its resulting action was shown to be equivalent to the one obtained by holographic renormalization. As a non-trivial application of the method, we compute the Noether-Wald charges and thermodynamic quantities for topological black hole solutions with generic transverse section in Einstein-AdS6 theory.

Cristóbal Corral

Conformal Renormalization of topological black holes in AdS6

We find a covariant expression for the universal part of the holographic entanglement entropy which is valid for CFTs dual to generic higher curvature gravities in up to five bulk dimensions. We use this functional to compute universal coefficients of stress-tensor correlators in three-dimensional CFTs dual to Cubic Curvature Gravity. Using gauge/gravity duality, we work out an expression for the entanglement entropy of deformed entangling regions and read the coefficients from the power expansion of the entropy in the deformation parameter. In particular, we obtain the t 4 coefficient of the 3-point function and exhibit a difference between the results obtained using the entanglement entropy functional for minimal and non-minimal splittings. We compare the obtained expressions for t 4 derived considering both splittings with results obtained through other holographic methods which are splitting-independent. We find agreement with the result obtained from the non-minimal splitting, whereas the result derived from the minimal splitting is inconsistent and it is therefore ruled out.

Mexico

Andrés Argandoña

CFT correlators from shape deformations in Cubic Curvature Gravity

We provide a new derivation of the Hawking mass and Willmore energy functionals for asymptotically AdS spacetimes, by embedding Einstein-AdS gravity in Conformal Gravity. By construction, the evaluation of the four-dimensional Conformal Gravity action in a manifold with a conical defect produces a codimension-2 conformal invariant functional L Σ. The energy functionals are then particular cases of L Σ for Einstein-AdS and pure AdS ambient spaces, respectively. The bulk action is finite for AdS asymptotics and both Hawking mass and Willmore energy are finite as well. The result suggests a generic relation between conformal invariance and renormalization, where the codimension-2 properties are inherited from the bulk gravity action.

Energy functionals from Conformal Gravity

We implement a universal method for renormalizing AdS gravity actions applicable to arbitrary higher curvature theories in up to five dimensions. The renormalization procedure considers the extrinsic counterterm for Einstein-AdS gravity given by the Kounterterms scheme, but with a theory-dependent coupling constant that is fixed by the requirement of renormalization for the vacuum solution. This method is shown to work for a generic higher curvature gravity with arbitrary couplings except for a zero measure subset, which includes well-known examples where the asymptotic behavior is modified and the AdS vacua are degenerate, such as Chern-Simons gravity in 5D, Conformal Gravity in 4D and New Massive Gravity in 3D. In order to show the universality of the scheme, we perform a decomposition of the equations of motion into their normal and tangential components with respect to the Poincare coordinate and study the Fefferman-Graham expansion of the metric. We verify the cancellation of divergences of the on-shell action and the well-posedness of the variational principle.

José Edelstein

Alberto Sánchez

David Rodríguez

Alejandro López

Universal renormalization procedure for higher curvature gravities in D ≤ 5

It has been recently shown that there is a particular combination of conformal invariants in six dimensions which accepts a generic Einstein space as a solution. The Lagrangian of this Conformal Gravity theory — originally found by Lu, Pang and Pope (LPP) — can be conveniently rewritten in terms of products and covariant derivatives of the Weyl tensor. This allows one to derive the corresponding Noether prepotential and Noether-Wald charges in a compact form. Based on this expression, we calculate the Noether-Wald charges of six-dimensional Critical Gravity at the bicritical point, which is defined by the difference of the actions for Einstein-AdS gravity and the LPP Conformal Gravity. When considering Einstein manifolds, we show the vanishing of the Noether prepotential of Critical Gravity explicitly, which implies the triviality of the Noether-Wald charges. This result shows the equivalence between Einstein-AdS gravity and Conformal Gravity within its Einstein sector not only at the level of the action but also at the level of the charges.

Noether-Wald charges in six-dimensional Critical Gravity

We study the renormalization of Entanglement Entropy in holographic CFTs dual to Lovelock gravity. It is known that the holographic EE in Lovelock gravity is given by the Jacobson-Myers (JM) functional. As usual, due to the divergent Weyl factor in the Fefferman-Graham expansion of the boundary metric for Asymptotically AdS spaces, this entropy functional is infinite. By considering the Kounterterm renormalization procedure, which utilizes extrinsic boundary counterterms in order to renormalize the on-shell Lovelock gravity action for AAdS spacetimes, we propose a new renormalization prescription for the Jacobson-Myers functional. We then explicitly show the cancellation of divergences in the EE up to next-to-leading order in the holographic radial coordinate, for the case of spherical entangling surfaces. Using this new renormalization prescription, we directly find the C−function candidates for odd and even dimensional CFTs dual to Lovelock gravity. Our results illustrate the notable improvement that the Kounterterm method affords over other approaches, as it is non-perturbative and does not require that the Lovelock theory has limiting Einstein behavior.

Canada

Robert Mann

Renormalized holographic entanglement entropy in Lovelock gravity

We extend Maldacena’s argument, namely, obtaining Einstein gravity from Conformal Gravity, to six dimensional manifolds. The proof relies on a particular combination of conformal (and topological) invariants, which makes manifest the fact that 6D Conformal Gravity admits an Einstein sector. Then, by taking generalized Neumann boundary conditions, the Conformal Gravity action reduces to the renormalized Einstein-AdS action. These restrictions are implied by the vanishing of the traceless Ricci tensor, which is the defining property of any Einstein spacetime. The equivalence between Conformal and Einstein gravity renders trivial the Einstein solutions of 6D Critical Gravity at the bicritical point.

Einstein gravity from Conformal Gravity in 6D

We derive a general formula for renormalized entanglement entropy in even- dimensional CFTs holographically dual to Einstein gravity in one dimension higher. In order to renormalize, we adapt the Kounterterm method to asymptotically locally AdS manifolds with conical singularities. On the gravity side, the computation considers extrin- sic counterterms and the use of the replica trick à la Lewkowycz-Maldacena. The boundary counterterm B  d  is shown to satisfy a key property, in direct analogy to the Euler density: when evaluated on a conically singular manifold, it decomposes into a regular part plus a codimension-2 version of itself located at the conical singularity. The renormalized entropy thus obtained is shown to correspond to the universal part of the holographic entangle- ment entropy, which for spherical entangling surfaces is proportional to the central charge a that is the subject of the a-theorem. We also review and elucidate various aspects of the Kounterterm approach, including in particular its full compatibility with the Dirichlet condition for the metric at the conformal boundary, that is of standard use in holography.