  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.
