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

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.

23 pages, 1 figure

References in corpus (5)

• Stationary ring solitons in field theory - knots and vortons
• Black holes with only one Killing field
• Rotating Boson Stars in 5 Dimensions
• Notes on Yang-Mills--Higgs monopoles and dyons on R^D, and Chern-Simons--Higgs solitons on \R^{D-2}: Dimensional reduction of Chern-Pontryagin densities
• Massive Hopfions

Cited by in corpus (3)

• Fractional Skyrmion molecules in a $\mathbb{C}{P}^{N-1}$ model
• $SU(3)$ Knot Solitons: Hopfions in the $F_2$ Skyrme-Faddeev-Niemi model
• Self-Dual Skyrmions on the Spheres $S^{2N+1}$