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

TL;DR

This paper investigates the quantum properties and spin statistics of solitons in the $ ext{CP}^N$ Skyrme-Faddeev model, including effects of the Hopf term and collective coordinate quantization, with specific results for $N=2$ and discussions for higher $N$.

Contribution

It introduces a collective coordinate quantization approach to analyze soliton spin statistics in the $ ext{CP}^N$ Skyrme-Faddeev model, incorporating the Abelian Chern-Simons (Hopf) term and exploring its implications.

Findings

For $N=1$, the Hopf term is an integer due to $ ext{Pi}_3( ext{CP}^1)= ext{Z}$.

For $N>1$, the Hopf term becomes perturbative as $ ext{Pi}_3( ext{CP}^N)$ is trivial.

Quantized mass spectra are calculated for $N=2$, showing soliton spin and statistics depend on the Hopf term angle $ heta$.

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 $N\geqq 3$.