  We show that if there are conserved flavor symmetries then some properties of
a monopole can depend on $\theta$ even when a fermion is massless. The
quantized nature of global symmetries and the fractional nature of the Witten
effect can lead to interesting structure. Seen from another point of view,
aside from possibly breaking baryon and lepton flavor symmetries (the
Callan-Rubakov effect), monopole boundary conditions can also break the axial
symmetry that otherwise could have been used to remove $\theta$ from the
Lagrangian. As an example, in a toy model, we calculate the $\theta$ dependence
of the mass of the monopole and properties of the non-zero charge density
surrounding the monopole.
