Counterterms, Kounterterms, and the variational problem in AdS gravity

TL;DR

This work analyzes when Kounterterms reproduce the holographic boundary counterterms in pure AdS gravity across dimensions. It develops a framework based on the dilatation/ Hamilton–Jacobi approach and the Chern–Gauss–Bonnet/topological origin of Kounterterms to compare the two boundary prescriptions. The authors prove that, in general, agreement requires the boundary Weyl tensor to vanish; in even boundary dimensions this condition is also sufficient, while in odd dimensions an additional Euler (conformal anomaly) constraint is needed. In the conformally flat sector, they derive explicit counterterms and Kounterterms, showing exact action-level matching in AdS$_4$ and clarifying the limits of the Kounterterm construction for higher dimensions and for odd dimensions with nonzero conformal anomaly.

Abstract

We show that the Kounterterms for pure AdS gravity in arbitrary even dimensions coincide with the boundary counterterms obtained through holographic renormalization if and only if the boundary Weyl tensor vanishes. In particular, the Kounterterms lead to a well posed variational problem for generic asymptotically locally AdS manifolds only in four dimensions. We determine the exact form of the counterterms for conformally flat boundaries and demonstrate that, in even dimensions, the Kounterterms take exactly the same form. This agreement can be understood as a consequence of Anderson's theorem for the renormalized volume of conformally compact Einstein 4-manifolds and its higher dimensional generalizations by Albin and Chang, Qing and Yang. For odd dimensional asymptotically locally AdS manifolds with a conformally flat boundary, the Kounterterms coincide with the boundary counterterms except for the logarithmic divergence associated with the holographic conformal anomaly, and finite local terms.