Electromagnetism, metric deformations, ellipticity and gauge operators on conformal 4-manifolds

On Riemannian signature conformal 4-manifolds we give a conformally invariant extension of the Maxwell operator on 1-forms. We show the extension is in an appropriate sense injectively elliptic, and recovers the invariant gauge operator of Eastwood and Singer. The extension has a natural compatibility with the de Rham complex and we prove that, given a certain restriction, its conformally invariant null space is isomorphic to the first de Rham cohomology. General machinery for extending this construction is developed and as a second application we describe an elliptic extension of a natural operator on perturbations of conformal structure. This operator is closely linked to a natural sequence of invariant operators that we construct explictly. In the conformally flat setting this yields a complex known as the conformal deformation complex and for this we describe a conformally invariant Hodge theory which parallels the de Rham result.