Spontaneous symmetry breaking, topological defects and fractional quantum numbers
Topological defects or texture solitons appear as a consequence of spontaneous symmetry breaking in physical systems of various space dimension d: in particular, a domain wall (kink) at d = 1, a vortex at d = 2, and a monopole at d = 3. If a fermion field is quantized in the background of a topological defect, then, generally, this gives rize to states with fractional values of quantum numbers. Charge fractionalization in d = 1 systems resolves puzzle of the spin-charge anomaly in molecular polymer chains of the polyacetylene type. Induced vacuum charge is related in general to spectral asymmetry of the Dirac Hamiltonian operator. The method of a self-adjoint extension of a symmetric operator is used to impose a boundary condition for the quantized fermion field at the edge of the defect. We review cases of charge fractionalization in the background of a magnetic monopole, as well as of a Skyrmeon and a chiral bag. In addition to charge, other quantum numbers fractionalize also, and appropriate examples for d = 2 systems in the magnetic vortex background are considered.