Abelian Hopfions of the CPn model on R2n+1 and a fractionally powered topological lower bound

Regarding the Skyrme-Faddeev model on R3 as a CP1 sigma model, we propose CPn sigma models on R2n+1 as generalisations which may support finite energy Hopfion solutions in these dimensions. The topological charge stabilising these field configurations is the Chern-Simons charge, namely the volume integral of the Chern-Simons density which has a local expression in terms of the composite connection and curvature of the CPn field. It turns out that subject to the sigma model constraint, this density is a total divergence. We prove the existence of a topological lower bound on the energy, which, as in the Vakulenko-Kapitansky case in R3, is a fractional power of the topological charge, depending on n. The numerical construction of the simplest ring shaped un-knot Hopfion on R5 is also discussed.