AdS/CFT correspondence and Geometry

TL;DR

The paper addresses extracting QFT data (correlators, Ward identities, anomalies) from AdS/CFT by reformulating holographic renormalization in a Hamiltonian framework. By treating the radial coordinate as time and organizing the near-boundary solution covariantly in dilatation weight, the authors derive universal, dimension-spanning counterterms and renormalized one-point functions directly from canonical momenta. They extend the method from pure gravity to gravity coupled to scalars, providing recursion relations and, in explicit examples, a superpotential construction that streamline the computation of holographic observables. The approach offers a more efficient, conceptually transparent alternative to standard holographic renormalization with broad applicability to higher dimensions and complex matter content.

Abstract

In the first part of this paper we provide a short introduction to the AdS/CFT correspondence and to holographic renormalization. We discuss how QFT correlation functions, Ward identities and anomalies are encoded in the bulk geometry. In the second part we develop a Hamiltonian approach to the method of holographic renormalization, with the radial coordinate playing the role of time. In this approach regularized correlation functions are related to canonical momenta and the near-boundary expansions of the standard approach are replaced by covariant expansions where the various terms are organized according to their dilatation weight. This leads to universal expressions for counterterms and one-point functions (in the presence of sources) that are valid in all dimensions. The new approach combines optimally elements from all previous methods and supersedes them in efficiency.