Harmonic maps into Lie groups (classical solutions of the chiral model). (English) Zbl 0677.58020

This paper deals with two aspects of the algebraic structure of \({\mathcal M}\), the space of harmonic maps from a simply-connected 2-dimensional domain (either Riemannian or Lorentzian) into a real Lie group \(G_{{\mathbb{R}}}\), the real form of a complex group G i.e. classical solutions of the chiral model are studied.
In the first part of the paper a representation of the loop group \({\mathcal A}(S^ 1,G_{{\mathbb{R}}})\) on \({\mathcal M}\) is constructed which corresponds to the Kac-Moody Lie algebra of infinitesimal deformations observed by Dolan. Here the main theorems are the description of the action on \({\mathcal M}\), and the description of the action of a subgroup on the space of harmonic mappings into Grassmannians.
The second part of the paper is restricted to a theory which applies only when \(\Omega\) is a 2-dimensional, simply-connected, Riemannian domain and the group \(G_{{\mathbb{R}}}=U(n)\). The author associates a non-negative integer called the uniton number to each solution and proves that the uniton number of every harmonic map from \(S^ 2\) into U(N) is finite. A construction is given for obtaining every harmonic map of uniton number n, which applies also to harmonic maps into Grassmannians. The moduli spaces of harmonic maps are described in terms of holomorphic sub-bundles of holomorphic bundles over \(\Omega\) satisfying certain additional conditions.
The last section contains questions which are left unanswered and might probably be investigated further.
Reviewer: B.Csikós


58E20 Harmonic maps, etc.
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