Simple counterterms for asymptotically AdS spacetimes in Lovelock gravity

TL;DR

This work develops a practical, foliation-based method to construct boundary counterterms that renormalize gravitational actions in asymptotically AdS spacetimes, restricted to conformally flat boundaries of the form $S^m \times H^{d-m}$. The author derives explicit counterterms up to sixth order in the Riemann tensor for the Einstein-Hilbert action, extends the approach to Gauss-Bonnet and third-order Lovelock theories, and provides Lovelock counterterms up to $m=3$ so that actions are finite in up to fourteen boundary dimensions. A key application demonstrates how these counterterms yield finite holographic entanglement entropy via the Casini construction, yielding $s = \frac{2}{\pi} a$ with $a$ a central charge (e.g., $a = \pi^2 (L/l_P)^3 (1 - 6 \lambda f_\infty)$ in Einstein-Gauss-Bonnet). Overall, the results broaden the holographic toolkit for dual CFTs with differing central charges and enable robust thermodynamic and entanglement calculations in higher dimensions and Lovelock gravity.

Abstract

Although gravitational actions diverge in asymptotically AdS spacetimes, boundary counterterms can be added in order to cancel out those divergences; such counterterms are known in general to third order in the Riemann tensor for the Einstein-Hilbert action. Considering foliations of AdS with an $S^m \times H^{d-m}$ boundary, we discuss a simple algorithm which we use to generate counterterms up to sixth order in the Riemann tensor, for the Einstein-Hilbert, Gauss-Bonnet and third-order-Lovelock Lagrangians. We also comment on other theories such as $F(R)$ gravity.