On the geometry of spontaneous symmetry breaking

Given a principal bundle P→X with a structure group G and an associated Higgs bundle Σ with a standard fiber G/H, the case of a matter bundle E whose standard fiber admits action only of an exact symmetry subgroup H of G is examined. In the presence of a fixed Higgs field σ: X→Σ, matter fields are represented by sections of a matter bundle Eh associated with the corresponding reduced subbundle Ph of P. The totality of matter fields and Higgs fields is described by sections of the bundle Ẽ which is the composite bundle EH→Σ→X where EH→Σ is the bundle associated with the principal H-bundle P→Σ. The bundle Ẽ fails to be associated with a principal bundle. To construct a connection Γ: Ẽ→J1Ẽ on Ẽ, the canonical jet bundle morphism J1EH×J1Σ→J1Ẽ is used.