Conformal Gravity (CG) is a Weyl-invariant metric theory whose action is free from divergences for generic asymptotically anti-de Sitter spaces. For Neumann boundary conditions, it reduces to renormalized Einstein-AdS gravity at tree level. By evaluating CG’s action on a replica orbifold, one obtains a codimension-2 local conformal invariant functional, $\textit{L}$ , which reduces to the renormalized area, the reduced Hawking mass and the Willmore Energy, for a surface Σ. Although there is evidence supporting the idea that this functional should be finite, a detailed analysis of its asymptotic behavior near the conformal boundary is still lacking. In this work, the finiteness of the conformal surface functional $\textit{L}$ is shown for any boundary-anchored surface embedded in an arbitrary ambient spacetime which is a solution to CG. This conclusion is drawn regardless the fact the surface is minimal or not. This result implies that Conformal Renormalization method not only applies to the bulk action, but also codimension-2 functionals.

Chile

Giorgos Anastasiou

Martín Bravo

Rodrigo Olea

Asymptotic analysis of energy functionals in anti-de Sitter spacetimes

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.

Ignacio Araya

Spain

Pablo Bueno

Javier Moreno

Belgium

Alejandro Lopez

Higher-dimensional Willmore energy as holographic entanglement entropy

In the context of the holographic correspondence, we introduce a purely extrinsic renormalization prescription, exemplified with the case of a minimally-coupled scalar field in anti-de Sitter (AdS) space. The counterterms depend only on the field and its radial derivatives. This would seem to conflict with the Dirichlet variational principle, but we show that consistency follows from the fact that the asymptotic structure of asymptotically locally AdS spacetimes requires not only the leading, but also all of the subleading non-normalizable modes to be fixed as a boundary condition. Crucially, as seen from a path integral definition of the bulk partition function involved in the standard Gubser-Klevanov-Polyakov-Witten (GKPW) formula, this condition is valid away from the saddle. We find that the extrinsic renormalization prescription is maximally efficient when the scalar field is massless, which is suggestive of a connection with the Kounterterm method for renormalization of pure gravity.

Ignacio J. Araya

Mexico

Daniel Ávila

Alberto Güijosa

Sergio Patiño-López

Extrinsic holographic renormalization for a scalar field

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 study a conformally coupled scalar-tensor theory with a quartic potential possessing local conformal symmetry up to a boundary term. We show that requiring the restoration of the full local conformal symmetry fixes the counterterms that render the on-shell action finite. The building block of the resulting action is a conformally covariant tensor which is constructed out of the metric and the scalar field and it has the same conformal weight as the Weyl tensor. This allows us to obtain the counterterms for the scalar-tensor sector in a closed form. The finiteness of the conformally complete version of the action is suggestive on the validity of the conformal renormalization prescription. We extend this theory by adding the conformal gravity action and also the Einstein-AdS action written in MacDowell-Mansouri form. Even though the latter breaks the conformal symmetry, we find that the action is still renormalized provided a suitable falloff of the scalar field when considering asymptotically locally anti-de Sitter solutions. Black hole solutions in these theories are studied, for which the Hawking temperature and the partition function to first order in the saddle-point approximation are calculated, providing a concrete example of this renormalization scheme.

Mairym Busnego-Barrientos

Nelson Merino

Conformal renormalization of scalar-tensor theories

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.

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 derive a covariant expression for the renormalized holographic entanglement entropy for conformal field theories (CFTs) dual to quadratic curvature gravity in arbitrary dimensions. This expression is written as the sum of the bare entanglement entropy functional obtained using standard conical defect techniques, and a counterterm defined at the boundary of the extremal surface of the functional. The latter corresponds to the cod-2 self-replicating part of the extrinsic counterterms when evaluated on the replica orbifold. This renormalization method isolates the universal terms of the holographic entanglement entropy functional. We use it to compute the standard <math><mi>C</mi></math>-function candidate for CFTs of arbitrary dimension, and the type-B anomaly coefficient <math><mi>c</mi></math> for four-dimensional CFTs.

United States

France

David Rivera-Betancour

Renormalized holographic entanglement entropy for quadratic curvature gravity

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.