Abstract
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.
Highlights
The Einstein-Hilbert action at high energies [1]
As it is standard in the saddle-point approximation of anti-de Sitter/conformal field theories (CFTs) (AdS/CFT) holography, the use of the on-shell classical gravity action as the generating functional for connected correlators of the dual CFT requires the finiteness of said action and requires the corresponding variational principle to be well-posed
In order to do so, we study the radial decomposition of the equations of motion (EOM) for the arbitrary HCG, expanded in the Poincare coordinate z considering the Fefferman-Graham (FG) expansion of the metric [27], which is standard for asymptotically locally AdS (AlAdS) manifolds
Summary
Where the Lagrangian includes any possible term constructed from arbitrary contractions of the Riemann tensor and the metric. The aim of this section is to obtain the form of the coefficient gi(j2) in terms of the conformal boundary metric gi(j0) for general theories of gravity While this could in principle be computed from the equations of motion Eνμ = 0 at order z2, here we will review the approach of [33], which can be applied directly to any HCG with an asymptotically AdS solution. The conditions that leave gi(j3) undetermined do not imply the degeneracy of the different AdS vacua It was suggested in [26], for higher-curvature theories with second-order linearized EOM around maximally symmetric backgrounds, that one can renormalize their actions using the same boundary term that is used in the Holographic Renormalization of Einstein-AdS gravity, i.e., the GHY term plus the HR counterterm, but with a coupling-dependent overall coefficient. Where χ(M2n) is the Euler characteristic of M2n, and E2n is the 2n−dimensional Euler density
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