Homotopy RG Flow and the Non-Linear $\sigma$-model

TL;DR

This paper provides a mathematical formulation of the low energy effective theory of the 2D sigma model, linking its beta function to the Ricci curvature of the target manifold, thus connecting physics and geometry.

Contribution

It offers a rigorous mathematical treatment of the sigma model's effective theory and relates the beta function to Ricci curvature, extending Friedan's physical results.

Findings

The effective theory encodes the topology and geometry of the target manifold.

The beta function is related to the Ricci curvature of the target.

Mathematical formulation of the low energy limit of the sigma model.

Abstract

The purpose of this note is two give a mathematical treatment to the low energy effective theory of the two-dimensional sigma model. Perhaps surprisingly, our low energy effective theory encodes much of the topology and geometry of the target manifold. In particular, we relate the $\beta$-function of our theory to the Ricci curvature of the target, recovering the physical result of Friedan.