The Wasserstein geometry of non-linear sigma models and the Hamilton-Perelman Ricci flow

TL;DR

This paper explores the connection between non-linear sigma models and Ricci flow using Wasserstein geometry, providing a rigorous framework for their relationship and generalizations.

Contribution

It introduces a Wasserstein geometric framework to relate non-linear sigma models with Ricci flow, extending Hamilton-Perelman Ricci flow through renormalization group analysis.

Findings

Wasserstein geometry models the sigma models-Ricci flow relation.

Provides a rigorous embedding of Ricci flow into sigma models.

Characterizes a generalized Ricci flow with monotonicity and gradient flow properties.

Abstract

Non linear sigma models are quantum field theories describing, in the large deviations sense, random fluctuations of harmonic maps between a Riemann surface and a Riemannian manifold. Via their formal renormalization group analysis, they provide a framework for possible generalizations of the Hamilton-Perelman Ricci flow. By exploiting the heat kernel embedding introduced by N. Gigli and C. Mantegazza, we show that the Wasserstein geometry of the space of probability measures over Riemannian metric measure spaces provides a natural setting for discussing the relation between non-linear sigma models and Ricci flow theory. This approach provides a rigorous model for the embedding of Ricci flow into the renormalization group flow for non linear sigma models, and characterizes a non-trivial generalization of the Hamilton-Perelman version of the Ricci flow. We discuss in detail the monotonicity and gradient flow properties of this extended flow.