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

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.