Hamiltonian analysis of gauged CP^1 model, with or without Hopf term, and fractional spin

TL;DR

This paper performs a Hamiltonian analysis of the gauged CP^1 model with and without the Hopf term, revealing that the Hopf term induces fractional spin depending on soliton number and nonabelian charge.

Contribution

It provides a detailed Hamiltonian analysis of the gauged CP^1 model and the effects of the Hopf term, highlighting the emergence of fractional spin.

Findings

Hamiltonian structures are similar with or without Hopf term.

The Hopf term leads to fractional spin.

Fractional spin depends on soliton number and nonabelian charge.

Abstract

Recently it has been shown by Cho and Kimm that the gauged $CP^1$ model, obtained by gauging the global SU(2) group of $CP^1$ model and adding a corresponding Chern-Simons term, has got its own soliton. These solitons are somewhat distinct from those of pure $CP^1$ model, as they cannot always be characterised by $\pi_2(CP^1)=Z$. In this paper, we first carry out the Hamiltonian analysis of this gauged $CP^1$ model. Then we couple the Hopf term, associated to these solitons and again carry out its Hamiltonian analysis. The symplectic structures, along with the structures of the constraints, of these two models (with or without Hopf term) are found to be essentially the same. The model with Hopf term, is then shown to have fractional spin, which however depends not only on the soliton number $N$ but also on the nonabelian charge.