  The Atiyah-Singer index theorem is generalized to a two-dimensional SO(3)
Yang-Mills-Higgs (YMH) system. The generalized theorem is proven by using the
heat kernel method and a nonlinear realization of SU(2) gauge symmetry. This
theorem is applied to the problem of deriving a charge quantization condition
in the four-dimensional SO(3) YMH system with non-Abelian monopoles. The
resulting quantization condition, eg=n (n: integer), for an electric charge e
and a magnetic charge g is consistent with that found by Arafune, Freund and
Goebel. It is shown that the integer n is half of the index of a Dirac
operator.
