Revisiting the O(3) Non-linear Sigma Model and Its Pohlmeyer Reduction

TL;DR

This paper explores the relationship between solutions of the Euclidean O(3) non-linear sigma model and the Pohlmeyer reduced sinh-Gordon equation, revealing correspondences and differences among instantons, merons, and elliptic solutions.

Contribution

It provides a detailed mapping between known sigma model solutions and their counterparts in the Pohlmeyer reduced theory, clarifying the nature of these correspondences.

Findings

Instantons lack a Pohlmeyer counterpart

Merons correspond to the ground state in the reduced theory

Elliptic solutions show a two-to-one correspondence

Abstract

It is well known that sigma models in symmetric spaces accept equivalent descriptions in terms of integrable systems such as the sine-Gordon equation through Pohlmeyer reduction. In this paper, we study the mapping between known solutions of the Euclidean O(3) non-linear sigma model, such as instantons, merons and elliptic solutions that interpolate between the latter and solutions of the Pohlmeyer reduced theory, namely the sinh-Gordon equation. It turns out that instantons do not have a counterpart, merons correspond to the ground state, while the class of elliptic solutions is characterized by a two to one correspondence between solutions in the two descriptions.