Lecture Notes on Holographic Renormalization

TL;DR

This paper reviews holographic renormalization, detailing the mathematical framework, methods, and applications to AdS and dS spacetimes, including derivations of key quantities and their physical implications.

Contribution

It provides a comprehensive overview of holographic renormalization, including explicit calculations and extensions to de Sitter spacetimes, enhancing understanding of holographic dualities.

Findings

Derivation of on-shell renormalized action for scalar fields.

Establishment of holographic Ward identities and anomalies.

Analytic continuation between AdS and dS spacetimes for stress tensors.

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.