Right, but I think this is not as big a conceptual change as it first sounds like: eg the moduli space of bundles-with-connection over R^n is the same as the moduli space of connections on any fixed bundle over R^n , right? (since any two bundles are equivalent)
Perhaps it would be better to say that the fields are G-bundles with connection, rather than picking a particular one? Then the Poincare group does act on the space of fields by pullback.
Hitchin's eq are reduction of SDYM from 4d to 2d. Bogomolny eq are also reduction of SDYM, from 4d to 3d. In each case reduction from 4d gives connection A, adjoint field Phi. In 3d case Phi really is the physicist's Higgs field. Hitchin had studied 3d case, then carried the name to 2d case I think.
