  We study a simple quantum mechanical model of a spinning particle moving on a
sphere in the presence of a magnetic field. The system has two ground states.
As the magnetic field is varied, the ground states mix through a non-Abelian
Berry phase. We show that this Berry phase is the path ordered exponential of
the smooth SU(2) 't Hooft-Polyakov monopole. We further show that, by adjusting
a potential on the sphere, the monopole becomes BPS and obeys the Bogomolnyi
equations. For this choice of potential, it turns out that there is a hidden
supersymmetry underlying the system and the Bogomolnyi equations are analogous
to the tt* equations of Cecotti and Vafa. We conjecture that the Bogomolnyi
equations also govern the Berry phase of N=(2,2) supersymmetric sigma models
with other target spaces.
