Actions of loop groups on harmonic maps

Abstract

We describe a general framework in which subgroups of the loop group Λ G l n C \Lambda G{l_n}\mathbb {C} act on the space of harmonic maps from S 2 {S^2} to G l n C G{l_n}\mathbb {C} . This represents a simplification of the action considered by Zakharov-Mikhailov-Shabat [ZM, ZS] in that we take the contour for the Riemann-Hilbert problem to be a union of circles; however, it reduces the basic ingredient to the well-known Birkhoff decomposition of Λ G l n C \Lambda G{l_n}\mathbb {C} , and this facilitates a rigorous treatment. We give various concrete examples of the action, and use these to investigate a suggestion of Uhlenbeck [Uh] that a limiting version of such an action ("completion") gives rise to her fundamental process of "adding a uniton". It turns out that this does not occur, because completion preserves the energy of harmonic maps. However, in the special case of harmonic maps from S 2 {S^2} to complex projective space, we describe a modification of this completion procedure which does indeed reproduce "adding a uniton".