XXV Workshop on Geometric Methods in Physics 2-8.07.2006

Andrei Domrin

Noncommutative unitons

A solution of the noncommutative $U(n)$ sigma-model is said to be finite-type if it may be presented as a finite-dimensional perturbation of a solution of energy~0. We show that every finite-type solution admits a factorization into unitons, which is a noncommutative analogue of Uhlenbeck's description of harmonic maps from $S^2$ to $U(n)$. As a corollary, we establish the phenomenon of quantization of energy for such solutions and obtain an explicit description of the moduli space of finite-type solutions for small values of energy. We also use the uniton factorization to construct large families of previously unknown ``genuinely non-Grassmannian'' solutions.