The Quantum Geometry of Polyhedral Surfaces: Non–linear σ Model and Ricci Flow

Abstract

In this chapter we discuss in great detail the connection between Non–Linear σ Model and Ricci Flow. In recent years, Ricci flow has been the point of departure and the motivating example for important developments in geometric analysis, most spectacularly for G. Perelman’s proof of Thurston’s geometrization program for three-manifolds and of the attendant Poincare conjecture. For one of those strange circumstances not unusual in the history of Science, the Ricci flow, introduced in the early 1980s by Richard Hamilton, independently appeared on the scene also in Physics. Indeed, Daniel Friedan, studying the weak coupling limit of the renormalization group flow for non-linear sigma models, introduced what later on came to be known as the Hamilton–DeTurck version of the Ricci flow. This QFT avatar of the Ricci flow was largely ignored in geometry until G. Perelman acknowledged that in his groundbreaking analysis he was somewhat inspired by the role that the effective action plays in non–linear σ–model theory. This soon called attention to the fact that in QFT the Ricci flow is naturally embedded into a more general geometric flow, the renormalization group flow for non–linear σ models, which, even if mathematically ill-defined, provides an interpretation for the Ricci flow which is open to generalizations. Here we discuss this connection in great detail, pinpointing both the many mathematical as well as physical subtle points of the perturbative embedding of the Ricci flow in the renormalization group flow. The geometry of the dilaton fields is discussed in depth using Riemannian metric measure spaces and their role in connecting NLσM effective action to Perelman’s \(\mathcal {F}\) energy. We do not know of any other published account of these matter which is so detailed and informative.