Extrinsic holographic renormalization for a scalar field

In the context of the holographic correspondence, we introduce a purely extrinsic renormalization prescription, exemplified with the case of a minimally-coupled scalar field in anti-de Sitter (AdS) space. The counterterms depend only on the field and its radial derivatives. This would seem to conflict with the Dirichlet variational principle, but we show that consistency follows from the fact that the asymptotic structure of asymptotically locally AdS spacetimes requires not only the leading, but also all of the subleading non-normalizable modes to be fixed as a boundary condition. Crucially, as seen from a path integral definition of the bulk partition function involved in the standard Gubser-Klevanov-Polyakov-Witten (GKPW) formula, this condition is valid away from the saddle. We find that the extrinsic renormalization prescription is maximally efficient when the scalar field is massless, which is suggestive of a connection with the Kounterterm method for renormalization of pure gravity.