Finite gravitational action for higher derivative and stringy gravities

TL;DR

This paper extends the holographic renormalization program by deriving a local surface counterterm for higher-derivative gravity, including Weyl (stringy) gravity, in bulk dimensions $d+1=4$ and $5$. Adding these counterterms renders the action finite for asymptotically AdS spacetimes and yields a well-defined energy-momentum tensor, enabling the holographic trace anomaly for $d=2$ and $d=4$ boundary theories to be computed consistently with prior results. The authors also compute the AdS black hole mass within this scheme, showing agreement with the standard Einstein gravity results and showing how parameter choices in the counterterms can reconcile differences in the Weyl case, highlighting a form of regularization dependence. The work suggests universality of the local counterterm approach and points to extensions to higher dimensions and asymptotically flat spaces, with implications for string-inspired HD gravity models and early-universe applications.

Abstract

We generalize the local surface counterterm prescription suggested in Einstein gravity for higher derivative (HD) and Weyl gravities. Explicitly, the surface counterterm is found for three- and five-dimensional HD gravities. As a result, the gravitational action for asymptotically AdS spaces is finite and gravitational energy-momentum tensor is well-defined. The holographic trace anomaly for d2 and d4 boundary (gauge) QFT dual to above HD gravity is calculated from gravitational energy-momentum tensor. The calculation of AdS black hole mass in HD gravity is presented within above prescrition. The comparison with the standard prescription (using reference spacetime) is done.