Complex bundles with connection up to isomorphism form a semigroup under Whitney sum which is far from being a group. We define a new equivalence relation (structured equivalence) so that stable isomorphism classes up to structured equivalence form a group which is describable in terms of the Chern character form plus some finite type invariants from algebraic topology. The elements in this group also satisfy two further somewhat contradictory properties: a locality or gluing property and an integrality property. There is interest in using these objects as prequantum fields in gauge theory and M-theoiy.