ACTIONS OF LOOP-GROUPS ON HARMONIC MAPS

We describe a general framework in which subgroups of the loop group LAMBDA-Gl(n)C act on the space of harmonic maps from S2 to Gl(n)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 LAMBDA-Gl(n)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 S2 to complex projective space, we describe a modification of this completion procedure which does indeed reproduce "adding a uniton".