Functional Renormalization Group as a Ricci Flow: An $\mathcal{F}$-Entropy Perspective on Information Metric Dynamics

TL;DR

This paper reveals a deep connection between the Functional Renormalization Group and Ricci flow, using an $ extit{F}$-entropy functional to interpret RG evolution as a geometric optimization on the information metric.

Contribution

It introduces an $ extit{F}$-entropy functional as a Lyapunov potential, linking RG flow to Ricci flow and geometric analysis in an infinite-dimensional setting.

Findings

The RG evolution of the Fisher information metric is a gradient flow of the $ extit{F}$-entropy.

The effective action's log acts as a scalar potential generating necessary diffeomorphisms.

High-energy mode integration smooths the information manifold's curvature, leading to Ricci soliton equilibrium.

Abstract

We establish an equivalence between the Functional Renormalization Group (FRG) and the Ricci flow modified by a diffeomorphism. By reformulating the Polchinski exact renormalization group equation into an infinite-dimensional Fokker-Planck framework, we show that the evolution of the Fisher information metric on the coupling constant space is a geometric optimization process. Central to this mapping is our construction of a field-theoretic $\mathcal{F}$-entropy functional - an infinite-dimensional analogue of Perelman's $\mathcal{F}$-entropy functional - which acts as a Lyapunov potential for the theory. We prove that the continuous scale evolution of the field distribution constitutes a Riemannian gradient flow of this $\mathcal{F}$-entropy, which in turn deforms the information metric on the coupling constant space via the parametric Hessian of the entropic landscape. Crucially, the log of the effective action serves as a scalar potential $\Phi$ that generates the diffeomorphisms required to ensure the tensorial consistency of the flow. Our framework demonstrates that successive integration of high-energy degrees of freedom effectively smooths out the curvature of the information manifold, driving the system toward a Ricci soliton equilibrium. These results provide a novel foundation for characterizing the stability of RG fixed points and offer a first-principles bridge connecting quantum field theory, information geometry, and Perelman's theory of geometric evolution.