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

The Skyrme–Faddeev model has planar soliton solutions with the target space $\mathcal{P}^N$. 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 $\Pi_3(\mathcal{P}^1)=\mathbb{Z}$, the term becomes an integer for $N=1$. On the other hand, for $N>1$, it becomes perturbative because $\Pi_3(\mathcal{P}^N)$ 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 normalizable fermions coupled with the baby-Skyrme model $(\mathcal{P}^N$ Skyrme–Faddeev model$)$. We discuss whether the statistical nature of solitons can be explained using their constituents, i.e., quarks.