Hamiltonian Analysis of Gauged CP1CP^1 Model, the Hopf term, and fractional spin

Recently it was shown by Cho and Kimm that the gauged $CP^1$ model, obtained by gauging the global $SU(2)$ group 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 a detailed Hamiltonian analysis of this gauged $CP^1$ model. This reveals that the model has only $SU(2)$ as the gauge invariance, rather than $SU(2) \times U(1)$. The $U(1)$ gauge invariance of the original (ungauged) $CP^1$ model is actually contained in the $SU(2)$ group itself. 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 a Hopf term is shown to have fractional spin which, when computed in the radiation gauge, is found to depend not only on the soliton number $N$, but also on the nonabelian charge. We then carry out a reduced (partially) phase space analysis in a different physical sector of the model where the degrees of freedom associated with the $CP^1$ fields are transformed away. The model now reduces to a $U(1)$ gauge theory with two Chern-Simons gauge fields getting mass-like terms and one remaining massless. In this case the fractional spin is computed in terms of the dynamical degrees of freedom and shown to depend purely on the charge of the surviving abelian symmetry. Although this reduced model is shown to have its own solitonic configuration, it turns out to be trivial.Comment: Latex, 26 pages, accepted for publication in Phys. Rev.

Hamiltonian Formulation of Quantum Hall Skyrmions with Hopf Term

We study the nonrelativistic nonlinear sigma model with Hopf term in this paper. This is an important issue beacuse of its relation to the currently interesting studies in skyrmions in quantum Hall systems. We perform the Hamiltonian analysis of this system in $CP^1$ variables. When the coefficient of the Hopf term becomes zero we get the Landau-Lifshitz description of the ferromagnets. The addition of Hopf term dramatically alters the Hamiltonian analysis. The spin algebra is modified giving a new structure and interpretation to the system. We point out momentum and angular momentum generators and new features they bring in to the system.Comment: 16pages, Latex file, typos correcte