Higgs Bundles and Four Manifolds

TL;DR

This paper introduces generalized gauge theories on Kähler manifolds using Higgs bundles, linking their partition functions to new invariants that could reveal more about smooth structures of four-manifolds beyond existing invariants.

Contribution

It proposes novel generalizations of Donaldson-Witten and Vafa-Witten theories based on Higgs bundles, connecting their partition functions to generalized invariants.

Findings

Partition function expressed as sum of generalized Donaldson-Witten and Seiberg-Witten invariants.

Generalized Seiberg-Witten invariants may encode additional smooth structure information.

Potential to distinguish smooth structures in non-simply-connected 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 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.