Holographic Reconstruction of Spacetime and Renormalization in the AdS/CFT Correspondence

TL;DR

The paper develops a systematic, covariant holographic renormalization framework for AdS/CFT, enabling finite, well-defined correlators by regulating the bulk action, identifying divergences, and adding boundary counterterms before removing the regulator. It uses this procedure to reconstruct the bulk spacetime and bulk fields from conformal field theory data, showing that boundary sources plus the vacuum expectation value of the stress-energy tensor suffice to determine the asymptotic bulk geometry up to six dimensions, with explicit expressions for the holographic stress-energy tensor and anomalies. The work provides explicit formulas for the Brown–York-like stress tensor in various dimensions, clarifies conformal transformation properties (including anomaly contributions), and extends the analysis to scalar matter and backreaction, establishing a consistent framework for relating CFT data to bulk physics and suggesting avenues for further holographic explorations in RG flows and time-dependent settings.

Abstract

We develop a systematic method for renormalizing the AdS/CFT prescription for computing correlation functions. This involves regularizing the bulk on-shell supergravity action in a covariant way, computing all divergences, adding counterterms to cancel them and then removing the regulator. We explicitly work out the case of pure gravity up to six dimensions and of gravity coupled to scalars. The method can also be viewed as providing a holographic reconstruction of the bulk spacetime metric and of bulk fields on this spacetime, out of conformal field theory data. Knowing which sources are turned on is sufficient in order to obtain an asymptotic expansion of the bulk metric and of bulk fields near the boundary to high enough order so that all infrared divergences of the on-shell action are obtained. To continue the holographic reconstruction of the bulk fields one needs new CFT data: the expectation value of the dual operator. In particular, in order to obtain the bulk metric one needs to know the expectation value of stress-energy tensor of the boundary theory. We provide completely explicit formulae for the holographic stress-energy tensors up to six dimensions. We show that both the gravitational and matter conformal anomalies of the boundary theory are correctly reproduced. We also obtain the conformal transformation properties of the boundary stress-energy tensors.