Counterterms in Dimensionally Continued AdS Gravity

TL;DR

This paper compares two regularization methods for Lovelock gravity with AdS asymptotics, showing their agreement in a specific model and suggesting a general relation between their boundary counterterms.

Contribution

It provides the first explicit comparison of Dirichlet counterterms and Kounterterms in dimensionally continued AdS gravity, revealing their intrinsic connection.

Findings

Both regularization methods agree in the studied model.

Intrinsic counterterms are the difference between Kounterterms and Gibbons-Hawking-Myers terms.

The comparison supports a general property of Lovelock-AdS gravity.

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.