Solving formally the Auxiliary System of $O(N)$ Non Linear Sigma Model

TL;DR

This paper demonstrates that the integrability of the $SO(N)/SO(N-1)$ Principal Chiral Model stems from the Pohlmeyer reduction of the $O(N)$ Non Linear Sigma Model, providing a formal solution to its auxiliary system.

Contribution

It establishes a formal connection between the Lax pair of the PCM and the zero curvature condition derived from Pohlmeyer reduction, revealing the origin of integrability.

Findings

Lax pair of PCM related to zero curvature condition

Solution of auxiliary system from Pohlmeyer reduction

Integrability rooted in flatness of the enhanced space

Abstract

We show that the integrability of the $SO(N)/SO(N-1)$ Principal Chiral Model (PCM) originates from the Pohlmeyer reduction of the $O(N)$ Non Linear Sigma Model (NLSM). In particular, we show that the Lax pair of the PCM is related upon redefinitions and identification of parameters to the zero curvature condition, which is a consequence of the flatness of the enhanced space used in the Pohlmeyer reduction. This identification provides the solution of the auxiliary system that corresponds to an arbitrary NLSM/PCM solution.