Lecture Notes on Holographic Renormalization

TL;DR

The notes present a systematic holographic renormalization framework to compute renormalized QFT correlators from gravity in asymptotically AdS spacetimes. They develop a step-by-step procedure: solve bulk equations near the boundary, regulate and renormalize the on-shell action with covariant counterterms, and extract exact 1-point and higher-point functions, guaranteeing Ward identities and anomaly structures. The formalism is illustrated with a massive scalar in AdS, yielding explicit 2-point and 4-point functions, RG flow behavior, and the role of conformal anomalies. The work also discusses analytic continuation to de Sitter spaces and outlines extensions to RG flows, more general geometries, and higher-dimensional/defect setups, highlighting both the power and limitations of the approach.

Abstract

We review the formalism of holographic renormalization. We start by discussing mathematical results on asymptotically anti-de Sitter spacetimes. We then outline the general method of holographic renormalization. The method is illustrated by working all details in a simple example: a massive scalar field on anti-de Sitter spacetime. The discussion includes the derivation of the on-shell renormalized action, of holographic Ward identities, anomalies and RG equations, and the computation of renormalized one-, two- and four-point functions. We then discuss the application of the method to holographic RG flows. We also show that the results of the near-boundary analysis of asymptotically AdS spacetimes can be analytically continued to apply to asymptotically de Sitter spacetimes. In particular, it is shown that the Brown-York stress energy tensor of de Sitter spacetime is equal, up to a dimension dependent sign, to the Brown-York stress energy tensor of an associated AdS spacetime.