PU(2) monopoles and relations between four-manifold invariants

TL;DR

This paper reviews the program to relate Donaldson and Seiberg-Witten invariants of four-manifolds via PU(2) monopoles, highlighting analytical challenges and key results in the moduli space approach.

Contribution

It outlines the PU(2) monopole framework as a cobordism between different invariants and discusses transversality, compactness, and gluing issues in this context.

Findings

Established transversality and compactness results for PU(2) monopoles

Calculated Donaldson invariants in terms of Seiberg-Witten invariants

Outlined analytical difficulties in PU(2) monopole gluing theory

Abstract

Using quantum field-theoretic arguments, Witten has established a relation between the Donaldson and Seiberg-Witten invariants of smooth four-manifolds. In this survey article, we describe the program to prove this relation using a moduli space of PU(2) = SO(3) monopoles as a cobordism between the Donaldson moduli space of anti-self-dual SO(3) connections and moduli spaces of U(1) monopoles. We provide an overview of some of our transversality and Uhlenbeck compactness results for PU(2) monopoles, along with some of our calculations of Donaldson invariants in terms of Seiberg-Witten invariants. We give a brief outline of issues concerning the gluing theory, focussing on some of the analytical difficulties that are particular to PU(2) monopoles, and its application to the PU(2) monopole program to prove the relation between Donaldson and Seiberg-Witten invariants.