  A thin film superconductor shaped into a spherical shell at whose center lies
the end of long thin solenoid in which there is an integer flux $N\Phi_0$ has
been previously extensively studied numerically as a model of a two-dimensional
superconductor. The emergent flux from the solenoid produces a radial B-field
at the superconducting shell and $N$ vortices in the superconducting film. We
study here the effects of including a second solenoid (carrying a flux $f$)
which is inserted inside the first solenoid but passing right across the
sphere. This Aharonov-Bohm (AB) flux does not have to be quantized to make the
order parameter single valued. The Ginzburg-Landau (GL) free energy is
minimized at fixed $N$ as a function of $f$ and it is found that the minimum is
usually achieved when the AB flux $f$ is half a flux quantum, but depending on
$N$ the minimum may be at $f=0$ or values which are not obvious rational
fractions.
