Analysis of the Faddeev model

In this paper we consider a generalization of the Faddeev model for the maps from a closed three-manifold into the two-sphere. We give a novel representation of smooth $S^2$-valued maps based on flat connections. This representation allows us to obtain an analytic description of the homotopy classes of $S^2$-valued maps that generalizes to Sobolev maps. It also leads to a new proof an old theorem of Pontrjagin. For the generalized Faddeev model, we prove the existence of minimizers in every homotopy class.