Non-Abelian Sine-Gordon Solitons: Correspondence between $SU(N)$ Skyrmions and $\mathbb{C}{P}^{N-1}$ Lumps

Abstract

Topologically stable non-Abelian sine-Gordon solitons have been found recently in the $U(N)$ chiral Lagrangian and a $U(N)$ gauge theory with two $N$ by $N$ complex scalar fields coupled to each other. We construct the effective theory on a non-Abelian sine-Gordon soliton that is a nonlinear sigma model with the target space $\mathbb{R}\times \mathbb{C}{P}^{N-1}$. We then show that $\mathbb{C}{P}^{N-1}$ lumps on it represent $SU(N)$ Skyrmions in the bulk point of view, providing a physical realization of the rational map Ansatz for Skyrmions of the translational (Donaldson) type. We find therefore that Skyrmions can exist stably without the Skyrme term.

24 pages, 4 figures, published version