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.