Statistical nature of Skyrme-Faddeev models in $2+1$ dimensions and normalizable fermions

Abstract

The Skyrme-Faddeev model has planar soliton solutions with target space $\mathbb{C}P^N$. An Abelian Chern-Simons term (the Hopf term) in the Lagrangian of the model plays a crucial role for the statistical properties of the solutions. Because $\Pi_3(\mathbb{C}P^1)=\mathbb{Z}$, the term becomes an integer for $N=1$. On the other hand, for $N>1$, it becomes perturbative because $\Pi_3(\mathbb{C}P^N)$ is trivial. The prefactor $\Theta$ of the Hopf term is not quantized, and its value depends on the physical system. We study the spectral flow of the normalizable fermions coupled with the baby-Skyrme model ($\mathbb{C}P^N$ Skyrme-Faddeev model). We discuss whether the statistical nature of solitons can be explained using their constituents, i.e., the quarks.