Time-Independent Solutions to the Two-Dimensional Non-Linear O(3) Sigma Model and Surfaces of Constant Mean Curvature

TL;DR

This paper establishes a correspondence between time-independent solutions of the 2D non-linear O(3) sigma model and constant mean curvature surfaces, providing geometric insights into known solutions like solitons and antisolitons.

Contribution

It introduces a novel geometric framework linking sigma model solutions to surfaces of constant mean curvature, including explicit constructions for classical solutions.

Findings

Belavin-Polyakov solutions relate to spherical and minimal surfaces

Purkait-Ray solutions correspond to constant-mean-curvature helicoids

The geometric approach offers new perspectives on the role of the Hopf invariant

Abstract

It is shown that time-independent solutions to the (2+1)-dimensional non- linear O(3) sigma model may be placed in correspondence with surfaces of constant mean curvature in three-dimensional Euclidean space. The tools required to establish this correspondence are provided by the classical differential geometry of surfaces. A constant-mean-curvature surface induces a solution to the O(3) model through the identification of the Gauss map, or normal vector, of the surface with the field vector of the sigma model. Some explicit solutions, including the solitons and antisolitons discovered by Belavin and Polyakov, and a more general solution due to Purkait and Ray, are considered and the surfaces giving rise to them are found explicitly. It is seen, for example, that the Belavin-Polyakov solutions are induced by the Gauss maps of surfaces which are conformal to their spherical images, i.e. spheres and minimal surfaces, and that the Purkait-Ray solution corresponds to the family of constant-mean-curvature helicoids first studied by do Carmo and Dajczer in 1982. A generalisation of this method to include time-dependence may shed new light on the role of the Hopf invariant in this model.