Global Color Is Not Always Defined

Far away from a fundamental magnetic monopole of the SU(5) grand unified theory, the full theory reduces to one whose gauge group is unbroken SU(3) ⊗ U(1). The algebra of generators of smooth gauge transformations in this unbroken theory, however, will not in general contain any subalgebra isomorphic to su(3). This means that global color rotations are not always defined. The resulting inability to classify semiclassical dyon states by their color climinates an apparent paradox related to the vacuum angle $\theta$.