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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.17215 |
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Table of Contents:
- We establish 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 a geometric optimization process driven by a thermodynamic free energy. 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 -- defined as the continuous scale-dissipation rate of the free energy. We prove that the continuous evolution of the field distribution functional constitutes a functional JKO-Wasserstein gradient flow, which dynamically deforms the information metric on the theory space via the parametric Hessian of this entropic landscape. Crucially, an emergent information potential $Φ$ acts as a geometric gauge-fixing agent that generates the diffeomorphisms required to ensure the full tensorial consistency and general covariance of the flow. Our framework analytically demonstrates that the successive integration of high-energy degrees of freedom effectively smooths out the curvature of the information manifold, driving the system toward a steady Ricci soliton equilibrium at RG fixed points. These results provide a robust, first-principles foundation for characterizing the topological stability of quantum field theories and offer a novel synthesis connecting quantum field theory, optimal transport, and Perelman's theory of geometric evolution.