Strongly coupled Skyrme-Faddeev-Niemi hopfions

The strongly coupled limit of the Skyrme-Faddeev-Niemi model (i.e. without quadratic kinetic term) with a potential is considered on the spacetime S(3) x R. For one-vacuum potentials two types of exact Hopf solitons are obtained. Depending on the value of the Hopf index, we find compact or non-compact hopfions. The compact hopfions saturate a Bogomolny bound and lead to a fractional energy-charge formula E similar to vertical bar Q vertical bar(1/2), whereas the non-compact solitons do not saturate the bound and give E similar to vertical bar Q vertical bar. In the case of potentials with two vacua compact shell-like hopfions are derived. Some remarks on the influence of the potential on topological solutions in the full Skyrme-Faddeev-Niemi model or in the (3 + 1) Minkowski space are also made.