Statistical Nature of Skyrme-Faddeev Models in 2+1 Dimensions and Normalizable Fermions

The Skyrme-Faddeev model has planar soliton solutions with the target space ℂPN. 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 П3(ℂP1) = ℤ, the term becomes an integer for N = 1. On the other hand, for N > 1, it becomes perturbative because П3(ℂPN) is trivial. The prefactor Θ of the Hopf term is not quantized, and its value depends on the physical system. We study the spectral flow of normalizable fermions coupled with the baby-Skyrme model (ℂPN Skyrme-Faddeev model). We discuss whether the statistical nature of solitons can be explained using their constituents, i.e., quarks.