Discrete Approximation to the Faddeev Model

Hans Shmidheiser
Courant Institute of Mathematical Sciences
New York University

Abstract: We describe the Faddeev model for solitons, a topologically constrained variational problem for maps from the three sphere to the two sphere. The Faddeev solitons should be Faddeev energy minimizers in their Hopf class and solve a non-linear PDE. We investigate the question of how to pose the problem for maps from a discrete lattice (approximating the three sphere) to the two sphere. This is at first problematic, because one must rigorously address how to define a Hopf number for a map from a lattice. We describe how under appropriate discrete Faddeev energy bounds, maps are well-behaved enough to assign a Hopf number.

Moreover, we give a characterization of bounded energy maps, that essentially says, any singularities contain no topological information about the map. We then give an explicit summation formula, accurate to numerical precision for computing the Hopf number.

Time permitting we describe how this technique generalizes to other topologically constrained variational problems and how one may try to use the singularity characterization to prove a partial regularity result of the continuum minimizers.