  In this paper, as a step towards a unified mathematical treatment of the
gauge functionals from quantum field theory that have found profound
applications in mathematics, we generalize the Seiberg-Witten functional that
in particular includes the Kapustin-Witten functional as a special case. We
first demonstrate the smoothness of weak solutions to this generalized
functional. We then establish the existence of weak solutions under the
assumption that the structure group of the bundle is abelian, by verifying the
Palais-Smale compactness.
