Higgs bundles and four manifolds

Abstract

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 Kähler 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.