Renormalization group, secular term resummation and AdS (in)stability

TL;DR

This paper explores the application of renormalization group techniques to resummation of secular terms in non-linear AdS perturbations, providing insights into the stability and dynamics of AdS spacetime.

Contribution

It introduces a renormalization group approach to handle secular terms in non-linear gravitational perturbations in AdS, clarifying stability conditions.

Findings

Most secular terms predicted by linear spectrum do not appear in non-linear analysis.

Renormalization flow equations reveal the structure of non-linear interactions.

Resummation method improves understanding of AdS stability and dynamics.

Abstract

We revisit the issues of non-linear AdS stability, its relation to growing (secular) terms in naive perturbation theory around the AdS background, and the need and possible strategies for resumming such terms. To this end, we review a powerful and elegant resummation method, which is mathematically identical to the standard renormalization group treatment of ultraviolet divergences in perturbative quantum field theory. We apply this method to non-linear gravitational perturbation theory in the AdS background at first non-trivial order and display the detailed structure of the emerging renormalization flow equations. We prove, in particular, that a majority of secular terms (and the corresponding terms in the renormalization flow equations) that could be present on general grounds given the spectrum of frequencies of linear AdS perturbations, do not in fact arise.