Lev Kapitanski: On the models of Skyrme and Faddeev

Both models describe heavy particles (baryons) as spatially localized solutions of PDEs (lumps). In the original Skyrme model, the fields are constant at |x| = ∞ maps from ℝ³ to the three-dimensional sphere, S³. All such maps are split into different sectors according to their topological charge — the degree of the map. One of the problems is to minimize the Skyrme energy in every sector. The Faddeev model adds internal structure — knottedness — into the lumps. The fields are maps from ℝ³ into the two-dimensional sphere, S². The homotopy classes of such maps are characterized by an integer — the Hopf number, which counts the linking of the pre-images of the regular values. In this talk I will give a (mathematical) review of the two models.