Quark determinant in domain-like gluon fields

We address the computation of the determinant for the Dirac operator corresponding to a quark propagating in a background gluon field of the following type: the gauge field is covariantly constant and self-dual inside a hypersphere and with quark fields satisfying bag-like (chiral-violating) boundary conditions at the surface of the hypersphere. We find that the parity odd part of the logarithm of the determinant corresponds to the Abelian anomaly. However the parity even part also depends on the chiral angle associated with the boundary. This unexpected dependence is discussed.