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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 \sim |Q|^{1/2}, whereas the non-compact solitons do not saturate the bound and give E \sim |Q|. 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 (3+1) Minkowski space are also made.