Abelian Hopfions of the CP^n model on R^{2n+1} and a fractionally powered topological lower bound

TL;DR

This paper generalizes the Skyrme-Faddeev model to higher dimensions using CP^n sigma models, establishing a fractional power topological energy bound and numerically constructing a Hopfion in R^5.

Contribution

It introduces CP^n sigma models on R^{2n+1} supporting finite energy Hopfions and proves a fractional power topological energy lower bound.

Findings

Existence of finite energy Hopfions in higher dimensions.

A fractional power topological energy bound depending on n.

Numerical construction of a ring-shaped Hopfion in R^5.

Abstract

Regarding the Skyrme-Faddeev model on $\mathbb R^3$ as a $\mathbb C \mathbb P^1$ sigma model, we propose $\mathbb C \mathbb P^n$ sigma models on $\mathbb R^{2n+1}$ 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 CP^n 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 R^3, is a fractional power of the topological charge, depending on $n$. The numerical construction of the simplest ring shaped un-knot Hopfion on R^5 is also discussed.