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

Abstract

We demonstrate an exact equivalence between the Functional Renormalization Group (FRG) and the Ricci flow modified by a potential-driven diffeomorphism. By reformulating the Polchinski exact renormalization group (RG) equation into an infinite-dimensional Fokker-Planck framework, we show that the evolution of the Fisher information metric on the coupling constant space is driven by a thermodynamic free energy. We construct a field-theoretic $\mathcal{F}$-entropy functional, defined as the continuous scale-dissipation rate of the free energy, which serves as an infinite-dimensional analogue of Perelman's $\mathcal{F}$-entropy. The evolution of the field distribution functional constitutes a functional JKO-Wasserstein gradient flow that deforms the information metric via the parametric Hessian of this entropic landscape. An emergent information potential $Φ$ generates the diffeomorphisms required to establish tensorial consistency and general covariance of the flow. This framework shows that the integration of high-energy degrees of freedom reduces the curvature of the information manifold, leading to a steady Ricci soliton equilibrium at RG fixed points. These results connect quantum field theory, optimal transport, and Perelman's theory of geometric evolution to characterize the topological stability of quantum field theories.