Collective coordinate quantization and spin statistics of the solitons in the $\mathbb{C}P^N$ Skyrme-Faddeev model

Abstract

The $\mathbb{C}P^N$ extended Skyrme-Faddeev model possesses planar soliton solutions. We consider quantum aspects of the solutions applying collective coordinate quantization in regime of rigid body approximation. In order to discuss statistical properties of the solutions we include an Abelian Chern-Simons term (the Hopf term) in the Lagrangian. Since $\Pi_3(\mathbb{C}P^1)=\mathbb{Z}$ then for $N=1$ the term becomes an integer. On the other hand for $N>1$ it became perturbative because $\Pi_3(\mathbb{C}P^N)$ is trivial. The prefactor of the Hopf term (anyon angle) $\Theta$ is not quantized and its value depends on the physical system. The corresponding fermionic models can fix value of the angle $\Theta$ for all $N$ in a way that the soliton with $N=1$ is not an anyon type whereas for $N>1$ it is always an anyon even for $\Theta=n\pi, n\in \mathbb{Z}$. We quantize the solutions and calculate several mass spectra for $N=2$. Finally we discuss generalization for $N\geqq 3$.