Singular defects with fractional topological charge\\ in $\sigma$ models and gauge theories

A novel class of self-dual solutions in $\sigma$ models and gauge theories is considered. The contribution of the corresponding fluctuations to the chiral condensate is calculated. We discuss the few tightly connected problems, such as the $U(1)$ problem, the $\theta$ dependence and the chiral symmetry breaking within a framework of this approach. Arguments in favour of significance of the configurations with fractional topological charge are given.