Counterterms in Dimensionally Continued AdS Gravity

TL;DR

The work compares two regularization schemes for Lovelock gravity with AdS asymptotics—Dirichlet counterterms and Kounterterms—using Dimensionally Continued Gravity (CS-AdS in odd and BI-AdS in even dimensions) as a tractable test case. By extracting the Dirichlet counterterms from the divergent part of the Dirichlet variation, the authors show that in CS-AdS and BI-AdS the intrinsic counterterms can be reproduced as the difference between the Kounterterms series and the generalized Gibbons-Hawking term. This leads to the central conjecture that, for any Lovelock-AdS theory, the Dirichlet counterterms are generated by $c_d B_d-eta_d$ (up to finite terms), offering a unifying perspective on holographic renormalization in higher-curvature gravity. The findings have implications for constructing finite on-shell actions, stress tensors, and conserved charges in AdS/CFT for broader classes of gravitational theories.

Abstract

We revise two regularization mechanisms for Lovelock gravity with AdS asymptotics. The first one corresponds to the Dirichlet counterterm method, where local functionals of the boundary metric are added to the bulk action on top of a Gibbons-Hawking-Myers term that defines the Dirichlet problem in gravity. The generalized Gibbons-Hawking term can be found in any Lovelock theory following the Myers' procedure to achieve a well-posed action principle for a Dirichlet boundary condition on the metric, which is proved to be equivalent to the Hamiltonian formulation for a radial foliation of spacetime. In turn, a closed expression for the Dirichlet counterterms does not exist for a generic Lovelock gravity. The second method supplements the bulk action with boundary terms which depend on the extrinsic curvature (Kounterterms), and whose explicit form is independent of the particular theory considered. In this paper, we use Dimensionally Continued AdS Gravity (Chern-Simons-AdS in odd and Born-Infeld-AdS in even dimensions) as a toy model to perform the first explicit comparison between both regularization prescriptions. This can be done thanks to the fact that, in this theory, the Dirichlet counterterms can be readily integrated out from the divergent part of the Dirichlet variation of the action. The agreement between both procedures at the level of the boundary terms suggests the existence of a general property of any Lovelock-AdS gravity: intrinsic counterterms are generated as the difference between the Kounterterm series and the corresponding Gibbons-Hawking-Myers term.