We show that Holographic Renormalisation (HR) in eight dimensions is encoded in the unique conformal gravity theory that admits an Einstein sector with constant negative curvature. We explicitly relate HR to Topological Regularisation (TR), where the latter prescribes to add the Euler term to the Einstein-Hilbert Lagrangian density, with a precise coefficient so as to ensure that the resulting density is polynomial in the anti de Sitter (AdS) curvature. The polynomial is still asymptotically divergent. We find that the aforementioned, unique conformal gravity action in 8D, reproduces the polynomial, together with extra boundary terms that cancel the divergent terms in the action. We conjecture that, in arbitrary even dimension, HR is equivalent to conformally completing the Einstein-Hilbert action with negative cosmological constant, plus the Euler term with fixed coupling prescribed by TR.

The addition of Kounterterms to Einstein gravity leads to a finite action for asymptotically anti-de Sitter (AdS) spaces with a conformally flat boundary. In that sense, it provides a partial renormalization for AdS gravity when compared to standard holographic techniques, where the mismatch is given in terms of nontrivial conformal properties of the boundary. On the other hand, this method has the clear advantage that the variation of the action has a closed form in an arbitrary dimension. In this work, it is shown how to extract holographic information on conformal anomalies from the variation in -dimensional Einstein-AdS plus Kounterterms. Remarkably enough, a considerable part of the Weyl anomaly can be worked out for any odd dimension.

The asymptotic analysis for the metric of a generic solution of Einstein-Gauss-Bonnet AdS theory is provided by solving the field equations in the Fefferman-Graham frame. Using standard holographic renormalization, the counterterms that render the action finite are found up to seven spacetime dimensions. In the case of 6D, an equivalent formulation that permits a fully covariant determination of the counterterms is introduced, based on the finiteness of conformal invariants. It is shown that both schemes end up in the same holographic stress-energy tensor. Physical properties of six-dimensional topological Boulware-Deser black holes in Einstein-Gauss-Bonnet-AdS_6 gravity, whose boundary has nontrivial conformal features, are worked out in detail. Employing both renormalization prescriptions, finite asymptotic charges are found, and the correct black hole thermodynamics is recovered.

Abstract The vacuum entanglement entropy of a general conformal field theory (CFT) in d = 5 spacetime dimensions contains a universal term, F(A), which has a complicated and non-local dependence on the geometric details of the region A and the theory. Analogously to the previously known d = 3 case, we prove that for CFTs in d = 5 which are holographically dual to Einstein gravity, F(A) 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 A embedded in ℝ5. 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 d = 3 case, F(A) is not globally minimized by a round ball A = 𝔹4. Rather, F(A) can take arbitrarily positive and negative values as a function of A. Hence, we conclude that the round ball is not a global minimizer of F(A) for general five-dimensional CFTs.

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, LΣ, 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 LΣ 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.

We study the renormalization of a particular sector of Horndeski theory. In particular, we focus on the nonminimal coupling of a scalar field to the Gauss-Bonnet term and its kinetic coupling to the Einstein tensor. Adopting a power expansion on the scalar function that couples the Gauss-Bonnet term, we find specific conditions on their coefficients such that the action and charges are finite. To accomplish the latter, we add a finite set of intrinsic boundary terms. The contribution of the nonminimal coupling generates an effective scalar mass, allowing us to recover a modified Breitenlohner-Freedman bound. Furthermore, we compute the holographic 1-point functions and Ward identities associated with the scalar field and the metric. We constrain the parameter space of the theory by taking into account the preservation of scaling symmetry at the boundary.

We study the (anti-)self-duality conditions under which the electric and magnetic parts of the conserved charges of the dyonic Kerr-Newman-NUT-AdS solution become equivalent. Within a holographic framework, the stress tensor and the boundary Cotton tensor are computed from the electric/magnetic content of the Weyl tensor. The holographic stress tensor/Cotton tensor duality is recovered along the (anti-)self-dual curve in parameter space. We show that the latter not only implies a duality relation for the mass but also for the angular momentum. The partition function is computed to first order in the saddle-point approximation and a BPS bound is obtained. The ground state of the theory is enlarged to all the (anti-)self-dual configurations when the SO(4) and U(1) Pontryagin densities are introduced. We demonstrate this at the level of the action and variations thereof.

We explore the consistent truncation of conserved charges in quadratic curvature gravity (QCG) with anti--de Sitter asymptotics to the linear order in the Weyl tensor. The QCG action is given by the most general curvature-squared corrections to Einstein gravity, and it is suitably rendered finite by the addition of extrinsic counterterms (Kounterterms). The conserved charges derived from this action are, as a consequence, nonlinear in the spacetime Riemann tensor. A detailed analysis of the falloff of generic static solutions leads to a charge proportional to the electric part of the Weyl tensor, without loss of information on the energy of the system. The procedure followed provides firmer ground to the extension of the notion of Conformal Mass to higher-curvature gravity, as it appears as associated to a renormalized action. We observe that criticality condition in QCG poses an obstruction to the charge linearization, in contrast to previous results in Lovelock gravity, where degeneracy condition plays a key role.

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-AdS_6 theory.

In three-dimensional pseudo-Riemannian manifolds, the Cotton tensor arises as the variation of the gravitational Chern-Simons action with respect to the metric. It is Weyl-covariant, symmetric, traceless and covariantly conserved. Performing a reduction of the Cotton tensor with respect to Carrollian diffeomorphisms in a suitable frame, one discloses four sets of Cotton Carrollian relatives, which are conformal and obey Carrollian conservation equations. Each set of Carrollian Cotton tensors is alternatively obtained as the variation of a distinct Carroll-Chern-Simons action with respect to the degenerate metric and the clock form of a strong Carroll structure. The four Carroll-Chern-Simons actions emerge in the Carrollian reduction of the original Chern-Simons ascendant. They inherit its anomalous behaviour under diffeomorphisms and Weyl transformations. The extremums of these Carrollian actions are commented and illustrated.

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.

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.

We provide new formulas for the energy of black hole solutions in the anti--de Sitter (AdS) sector of a generic gravity theory with curvature-squared terms. This is achieved by the addition of counterterms of the extrinsic type (Kounterterms) which produces a finite variation of the total action. The procedure is compatible with the boundary conditions in asymptotically AdS spaces within the framework of quadratic curvature gravity (QCG) theory. As a consequence, without resorting to background-substraction, or linearized methods, the conserved charges correctly reproduce the energy of static black holes in QCG.

We study matching conditions for a spherically symmetric thin shell in Lovelock gravity which can be read off from the variation of the corresponding first-order action. In point of fact, the addition of Myers’ boundary terms to the gravitational action eliminates the dependence on the acceleration in this functional and such that the canonical momentum appears in the surface term in the variation of the total action. This procedure leads to junction conditions given by the discontinuity of the canonical momentum defined for an evolution normal to the boundary.In particular, we correct existing results in the literature for the thin shell collapse in generic Lovelock theories, which were mistakenly drawn from an inaccurate analysis of the total derivative terms in the system.

We study non-Einstein Bach-flat gravitational instanton solutions that can be regarded as the generalization of the Taub-NUT/Bolt and Eguchi-Hanson solutions of Einstein gravity to conformal gravity. These solutions include non-Einstein spaces which are either asymptotically locally flat spacetimes (ALF) or asymptotically locally Anti-de Sitter (AlAdS). Nevertheless, solutions with different asymptotic conditions exist: we find geometries that present a weakened AlAdS asymptotia, exhibiting the typical low decaying mode of conformal gravity. This permits to identify the simple Neumann boundary condition that, as it happens in the asymptotically AdS sector, selects the Einstein solution out of the solutions of conformal gravity. All the geometries present non-vanishing Hirzebruch signature and Euler characteristic, being single-centered instantons. We compute the topological charges as well as the Noether charges of the Taub-NUT/Bolt and Eguchi-Hanson spacetimes, which happen to be finite. This enables us to study the thermodynamic properties of these geometries.

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.

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 C-function candidate for CFTs of arbitrary dimension, and the type-B anomaly coefficient c for 4-dimensional CFTs.

The method of topological renormalization in anti--de Sitter (AdS) gravity consists in adding to the action a topological term which renders it finite, defining at the same time a well-posed variational problem. Here, we use this prescription to work out the thermodynamics of asymptotically locally anti--de Sitter (AlAdS) spacetimes, focusing on the physical properties of the Misner strings of both the Taub-NUT-AdS and Taub-Bolt-AdS solutions. We compute the contribution of the Misner string to the entropy by treating on the same footing the AdS and AlAdS sectors. As topological renormalization is found to correctly account for the physical quantities in the parity-preserving sector of the theory, we then investigate the holographic consequences of adding also the Chern-Pontryagin topological invariant to the bulk action; in particular, we discuss the emergence of the parity-odd contribution in the boundary stress tensor.

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.