Existence of Energy Minimizers as Stable Knotted Solitons in the Faddeev Model

Abstract

In this paper, we study the existence of knot-like solitons realized as the energy-minimizing configurations in the Faddeev quantum field theory model. Topologically, these solitons are characterized by an Hopf invariant, Q, which is an integral class in the homotopy group π3(S 2 )= . We prove in the full space situation that there exists an infinite subset of such that for any m∈ , the Faddeev energy, E, has a minimizer among the topological class Q=m. Besides, we show that there always exists a least-positive-energy Faddeev soliton of non-zero Hopf invariant. In the bounded domain situation, we show that the existence of an energy minimizer holds for = . As a by-product, we obtain an important technical result which says that E and Q satisfy the sublinear inequality E≤C|Q|3/4, where C>0 is a universal constant. Such a fact explains why knotted (clustered soliton) configurations are preferred over widely separated unknotted (multisoliton) configurations when |Q| is sufficiently large.