Notes on Fluctuations and Correlation Functions in Holographic Renormalization Group Flows

TL;DR

This work analyzes fluctuations of scalars and the metric around holographic RG-flow backgrounds in five-dimensional N=8 gauged supergravity to compute boundary two-point functions. It develops a framework for the coupled graviton–active-scalar system, reduces the single-active-scalar case to a third-order equation and casts fluctuations in Schrödinger form to apply SUSY QM insights. The authors test the approach on the N=1 SYM and Coulomb-branch flows, obtaining inert-scalar correlators with discrete spectra and solving active-scalar fluctuations via hypergeometric methods, while identifying significant obstacles in extracting physical correlators due to coupling and interior singularities. They conclude that standard holographic renormalization in the coupled sector is insufficient, pointing to the need for refined prescriptions and noting related insights for domain-wall stability and holographic Hamilton–Jacobi RG.

Abstract

We study the coupled equations describing fluctuations of scalars and the metric about background solutions of N=8 gauged supergravity which are dual to boundary field theories with renormalization group flow. For the case of a kink solution with a single varying scalar, we develop a procedure to decouple the equations, and we solve them in particular examples. However, difficulties occur in the calculation of correlation functions from the fluctuations, presumably because the AdS/CFT correspondence has not yet been properly implemented in the coupled scalar-gravity sector. Some new examples of correlators of operators dual to simpler uncoupled bulk scalars are given and are satisfactory. As byproducts of our study we make some observations relevant to the stability of domain walls in the brane-world scenario and to the Hamilton-Jacobi formulation of holographic RG flows.