Extrinsic Holographic Renormalization for a Scalar Field

TL;DR

This paper proposes an extrinsic holographic renormalization method for scalar fields in AdS space, focusing on boundary conditions and efficiency, especially for massless fields, with implications for the GKPW formula.

Contribution

It introduces a boundary-condition-based extrinsic renormalization scheme for scalar fields in AdS, differing from traditional methods by depending only on the field and its derivatives.

Findings

Extrinsic renormalization depends solely on the field and its radial derivatives.

The method is consistent with the Dirichlet principle when all non-normalizable modes are fixed.

Maximal efficiency observed for massless scalar fields, linking to the Kounterterm approach.

Abstract

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 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 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.