Ricci Flow from the Renormalization of Nonlinear Sigma Models in the Framework of Euclidean Algebraic Quantum Field Theory

TL;DR

This paper demonstrates that, within Euclidean algebraic quantum field theory, the first-order renormalization group flow of nonlinear Sigma models corresponds to Ricci flow, linking quantum field theory and geometric analysis.

Contribution

It extends the Euclidean algebraic quantum field theory framework to nonlinear Sigma models and shows the Ricci flow emerges at first order in perturbation theory.

Findings

First-order renormalization group flow is Ricci flow.

Extended classification of Wick ordered powers in Euclidean setting.

Established connection between quantum field renormalization and geometric evolution.

Abstract

The perturbative approach to nonlinear Sigma models and the associated renormalization group flow are discussed within the framework of Euclidean algebraic quantum field theory and of the principle of general local covariance. In particular we show in an Euclidean setting how to define Wick ordered powers of the underlying quantum fields and we classify the freedom in such procedure by extending to this setting a recent construction of Khavkine, Melati and Moretti for vector valued free fields. As a by-product of such classification, we prove that, at first order in perturbation theory, the renormalization group flow of the nonlinear Sigma model is the Ricci flow.