The Principal chiral model as an integrable system
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 g
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Introduction
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 u-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 !Rl,l, 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) chira! 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. In Differential Geometry the interest in centred on harmonic maps from an Euclidean domain to some manifold, say a Lie group. The seminal paper [26] contains a brilliant analysis of these maps when the Lie group is U(N). The so called uniton
A. P. Fordy et al. (eds.), Harmonic Maps and Integrable Systems © Springer Fachmedien Wiesbaden 1994
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M. Manas
solutions can have finite energy, the flag factors [3, 29], and therefore be defined as maps from the two-sphere 8 2 into U(N), in fact one can characterizes these solutions by the minimal uniton number, which is less than N. This is an unexpected aspect of the theory, and a rather different situation to that of solitons, in that case there being no upper bound in the soliton number. Let us mention that the approach to the factorization problem of [26] and that of [1] is quite similar. In [29] a method is presented to obtain all the harmonic maps from 8 2 to U(N), in the cases N = 3,4 this overlap with the results of [20]. In [24] there is a Grassmannian model for this construction, and in [12] we find a clear exposition of the factorization problem, the natural action, and an infinite-dimensional Grassmannian description. Some of these results can be extended from the Riemann sphere to an arbitra
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