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

TL;DR

This paper demonstrates the existence of knot solitons (Hopfions) in an $SU(3)$-based Skyrme-Faddeev-Niemi model, deriving solutions via two ansatz types and analyzing their quantum properties.

Contribution

It introduces new $SU(3)$ knot soliton solutions using embedding and non-embedding ansatzes, linking them to $CP^1$ models, and explores their quantum features.

Findings

Existence of $SU(3)$ Hopfions confirmed.

Euler-Lagrange equations reduce to $CP^1$ model.

Quantum analysis via zero-mode quantization performed.

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 B\"{a}cklund transformation. The resulting Euler-Lagrange equations for both ansatz reduce exactly to those of the $CP^1$ Skyrme-Faddeev-Niemi model. We also examine some quantum aspects of the solutions using the collective coordinate zero-mode quantization method.