Quantization of the Nonlinear Sigma Model Revisited

TL;DR

This paper rigorously analyzes the quantization of the nonlinear sigma model in two dimensions, focusing on anomaly elimination related to symmetries, and connects the renormalization group flow to Ricci flow.

Contribution

It precisely formulates the cohomological problem of anomaly elimination and demonstrates the absence of anomalies for certain symmetries, linking renormalization to Ricci flow.

Findings

No anomalies for diffeomorphism covariance.

Anomaly-free under specific conditions for homogeneous target spaces.

Ricci flow as the one-loop renormalization group flow.

Abstract

We revisit the subject of perturbatively quantizing the nonlinear sigma model in two dimensions from a rigorous, mathematical point of view. Our main contribution is to make precise the cohomological problem of eliminating potential anomalies that may arise when trying to preserve symmetries under quantization. The symmetries we consider are twofold: (i) diffeomorphism covariance for a general target manifold; (ii) a transitive group of isometries when the target manifold is a homogeneous space. We show that there are no anomalies in case (i) and that (ii) is also anomaly-free under additional assumptions on the target homogeneous space, in agreement with the work of Friedan. We carry out some explicit computations for the $O(N)$-model. Finally, we show how a suitable notion of the renormalization group establishes the Ricci flow as the one loop renormalization group flow of the nonlinear sigma model.