The quantum geometry of polyhedral surfaces: Non–linear σ model and ricci flow

we discuss in great detail the connection between Non–Linear
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 Poincaré 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 –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 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 NLM effective action to Perelman’s
F energy. We do not know of any other published account of these matter which is
so detailed and informative.