Variational Methods in AdS/CFT

TL;DR

This paper introduces a variational approach to simplify the calculation and renormalization of 1-point functions in AdS/CFT, applicable to a broad class of theories and potentially beyond AdS/CFT.

Contribution

It provides a universal, simplified proof for renormalizing large radius divergences in 1-point functions using variational methods, applicable to any Lagrangian-derived theory.

Findings

Large radius divergences can be renormalized away in semiclassical approximation.

Renormalized 1-point functions are obtained from a simple variational problem involving finite quantities.

The approach is demonstrated with examples including scalar fields, gravity, and renormalization flows.

Abstract

We prove that the AdS/CFT calculation of 1-point functions can be drastically simplified by using variational arguments. We give a simple universal proof, valid for any theory that can be derived from a Lagrangian, that the large radius divergencies in 1-point functions can always be renormalized away (at least in the semiclassical approximation). The renormalized 1-point functions then follow by a simple variational problem involving only finite quantities. Several examples, a massive scalar, gravity, and renormalization flows, are discussed. Our results are general and can thus be used for dualities beyond AdS/CFT.