The Principal chiral model as an integrable system

Abstract

The study of harmonic maps is an important subject of research not only in Differential Geometry, but also in Theoretical Physics and Mathematical Physics under the name of chiral fields [9]. These are maps with values in nonlinear manifolds such as Lie groups, Grassmannians, projective spaces, spheres, Stiefel manifolds, etc; therefore the equations defining these maps are nonlinear. The two-dimensional case can be solved exactly (with the exception of the Stiefel manifold case, up to now). In Mathematical Physics this was known for the non-linear σ-model since [21], but it was the Russian school in integrable systems who made an exhaustive study of the principal chiral model (chiral fields with values in a Lie group), see [11, 18, 10]. The Zakharov-Shabat dressing method was applied in [31, 32] to construct solutions of this model in a systematic way. The analysis performed in those papers concerns chiral fields from a Minkowski space-time ℝ1,1, hence one is dealing with a hyperbolic evolution equation, and the solutions found there were of soliton type, see [18] for a detailed exposition of the dynamics of these solitons. The integrable character of the principal chiral model is also reflected in the existence of a doubly infinite family of local conservations laws, see [8, 19, 4, 5, 7]. Both aspects can be deduced from the zero-curvature formulation [21, 31, 32] of the model and the consequent Birkhoff factorization technique for constructing solutions. In [1] one can find a beautiful analysis of some particular soliton type solution of the U(N) chiral model, in particular Morse theory is used to describe the dynamics of the one-soliton, and also there is a description of the factorization problem in terms of an infinite-dimensional Grassmannian.