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

TL;DR

This paper investigates the statistical properties of solitons in the Skyrme-Faddeev model with target space P^N, focusing on the role of the Hopf term and the spectral flow of coupled fermions, revealing insights into their quantum nature.

Contribution

It analyzes how the Hopf term influences the statistics of solitons in P^N Skyrme-Faddeev models and explores the spectral flow of fermions to explain their quantum behavior.

Findings

The Hopf term's prefactor  is not quantized and varies with the physical system.

Spectral flow of fermions provides a mechanism for understanding soliton statistics.

The nature of soliton constituents, such as quarks, is linked to their statistical properties.

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.