On the correspondence between fermionic number and statistics of solitons

Solitons of a nonlinear field interacting with fermions often acquire a fermionic number or an electric charge if fermions carry a charge. We establish a correspondence between charge and statistics (or spin) of solitons showing how the same mechanism (chiral anomaly) gives solitons statistical and rotational properties of fermions. These properties are encoded in a geometrical phase, i.e., an imaginary part of a euclidian action for a nonlinear σ-model. In the most interesting cases the geometrical phase is non-perturbative and has a form of an integer-valued theta-term.