On the Holographic Renormalization Group

TL;DR

This paper establishes a direct link between 5-dimensional supergravity equations and the renormalization group flow of 4-dimensional large N gauge theories, providing new insights into their holographic relationship.

Contribution

It derives first order flow equations for supergravity that mirror RG equations, supporting the identification of the supergravity action with the gauge theory's effective action.

Findings

Derived supergravity flow equations analogous to RG equations.

Identified relations between beta-functions and counterterms.

Supported the supergravity action as the gauge theory's effective action.

Abstract

We propose a direct correspondence between the classical evolution equations of 5-d supergravity and the renormalization group (RG) equations of the dual 4-d large $N$ gauge theory. Using standard Hamilton-Jacobi theory, we derive first order flow equations for the classical supergravity action $S$, that take the usual form of the Callan-Symanzik equations, including the corrections due to the conformal anomaly. This result gives direct support for the identification of $S$ with the quantum effective action of the gauge theory. In addition we find interesting new relations between the beta-functions and the counterterms that affect the 4-d cosmological and Newton constant.