Higher dimensional formulation of counterterms

TL;DR

This paper explores the renormalization of on-shell actions in asymptotically AdS solutions, revealing that for certain geometries the counterterm action becomes non-covariant due to boundary degeneracy.

Contribution

It demonstrates that while counterterms can renormalize the action for asymptotically AdS x S solutions, the resulting counterterm action lacks covariance because of boundary degeneracy.

Findings

Counterterm actions are coordinate and gauge dependent.

Renormalization is possible despite boundary degeneracy.

Counterterm actions are not covariant in degenerate boundary cases.

Abstract

It is by now well established that divergences of the on-shell action for asymptotically AdS solutions can be cancelled by adding covariant local boundary counterterms to the action. Here we show that although one can still renormalise the action for asymptotically $AdS_p \times S^q$ solutions using local boundary counterterms the counterterm action is not covariant since the conformal boundary is degenerate. Any given counterterm action is defined with respect to specific coordinate frame and gauge choices.