Moduli spaces of solutions of a noncommutative sigma model

Using a noncommutative version of the uniton theory, we study the space of those solutions of the noncommutative $U(1)$ sigma model that are representable as finite-dimensional perturbations of the identity operator. The basic integer-valued characteristics of such solutions are their normalized energy $e$, canonical rank $r$, and minimum uniton number $u$, which always satisfy $r\le e$ and $u\le e$. Starting with the so-called BPS solutions ($u=1$), we completely describe the sets of all solutions with $r=1,2,e-1,e$ (which forces $u\le2$) and all solutions of small energy ($e\le5$). The obtained results reveal a simple but nontrivial structure of the moduli spaces and lead to a series of conjectures.