Eugen Radu, D. H. Tchrakian, Yisong Yang
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.
View original:
http://arxiv.org/abs/1305.4784
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