The CpN-1 model: the fractional topological charge and theorem about the index for bounded manifolds

A novel class of self-dual solutions defined on a manifold with a boundary and possessing a topological charge Q=1/N is considered. The contribution of the corresponding fluctuations to the fermion condensate is calculated for the supersymmetric variant of the CPN-1 model. The result turns out to be finite, and this signifies spontaneous breaking of discrete chiral symmetry. The Atiaah-Patodi-Singer theorem of the index for manifolds with a boundary is discussed in detail. The necessity of imposing global boundary conditions in the case under consideration is explained