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