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

We establish a geometric correspondence between the Functional Renormalization Group (FRG) and a Ricci flow modified by a potential-driven diffeomorphism. By rewriting the Polchinski exact RG equation as an infinite-dimensional Fokker–Planck equation for field-distribution functionals, we show how a probability flow driven by a "thermodynamic" free-energy functional induces the evolution of the Fisher information metric on the coupling-constant manifold. Using the continuous scale-dissipation rate of this free-energy functional, we construct an RG-flow entropy functional that provides an infinite-dimensional counterpart of Perelman's $\mathcal{F}$-entropy. The parametric Hessian of this RG-flow entropy then encodes the scale deformation of the Fisher information metric, thereby linking the JKO–Wasserstein flow in field-configuration space to the geometry of the coupling-constant manifold. An emergent scalar information potential $\Phi$ encodes the potential-driven diffeomorphism component, restoring the tensorial form of the flow under reparametrizations of the coupling coordinates. In this representation, the successive integration of high-energy degrees of freedom effectively smooths out the curvature of the information manifold, so that RG fixed points are realized as steady Ricci soliton equilibria. These results connect quantum field theory, optimal transport, and Perelman's theory of geometric evolution, providing a geometric framework for characterizing the stability, universality, and topological structure of quantum field theories.