$SU(3)$ Knot Solitons: Hopfions in the $F_2$ Skyrme-Faddeev-Niemi model

Abstract

We discuss the existence of knot solitons (Hopfions) in a Skryme-Faddeev-Niemi-type model on the target space $SU(3)/U(1{)}^2$, which can be viewed as an effective theory of both the $SU(3)$ Yang-Mills theory and the $SU(3)$ anti-ferromagnetic Heisenberg model. We derive the knot solitons with two different types of ansatz: the first is a trivial embedding configuration of $SU(2)$ into $SU(3)$, and the second is a non-embedding configuration that can be generated through the Bäcklund transformation. The resulting Euler-Lagrange equations for both ansatz reduce exactly to those of the $C{P}^1$ Skyrme-Faddeev-Niemi model. We also examine some quantum aspects of the solutions using the collective coordinate zero-mode quantization method.

arXiv comment8 pages

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• Flag manifold sigma models: spin chains and integrable theories
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• Fractional Skyrmion molecules in a $\mathbb{C}{P}^{N-1}$ model
• Topological terms of (2+1)d flag-manifold sigma models
• Sigma-model analysis of $SU(3)$ antiferromagnetic spins on the triangular lattice
• Oscillator Calculus on Coadjoint Orbits and Index Theorems