Quantum Numbers of Solitons and Theta-Terms in Non-Linear Sigma-Models

Solitons of a nonlinear field interacting with fermions often acquire a fermionic number or an electric charge if fermions carry a charge. We argue that the same mechanism gives solitons statistical and rotational properties of fermions. The latter are shown up through the geometrical phase of the nonlinear σ-model. We discuss the most interesting cases when the geometrical phase appears as a θ-term in nonlinear σ-models. This lecture summarizes the results of Ref.¹.