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

TL;DR

This paper presents a systematic covariant method for renormalizing the AdS/CFT correspondence, reconstructing bulk spacetime from boundary data, and explicitly calculating stress-energy tensors up to six dimensions.

Contribution

It introduces a new holographic reconstruction technique and explicit formulas for stress-energy tensors, advancing the understanding of bulk-boundary relations in AdS/CFT.

Findings

Successfully renormalized the on-shell supergravity action in various dimensions.

Provided explicit holographic stress-energy tensor formulas up to six dimensions.

Confirmed the correct reproduction of conformal anomalies and transformation properties.

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.