  It is known that the Seiberg-Witten invariants, derived from supersymmetric
Yang-Mill theories in four-dimensions, do not distinguish smooth structure of
certain non-simply-connected four manifolds. We propose generalizations of
Donaldson-Witten and Vafa-Witten theories on a K\"{a}hler manifold based on
Higgs Bundles. We showed, in particular, that the partition function of our
generalized Vafa-Witten theory can be written as the sum of contributions our
generalized Donaldson-Witten invariants and generalized Seiberg-Witten
invariants. The resulting generalized Seiberg-Witten invariants might have,
conjecturally, information on smooth structure beyond the original
Seiberg-Witten invariants for non-simply-connected case.
