PU(2) Monopoles, I: Regularity, Uhlenbeck Compactness, and Transversality

TL;DR

This paper establishes regularity, compactness, and transversality results for PU(2) monopole equations, facilitating the study of moduli spaces crucial for understanding the relationship between Donaldson and Seiberg-Witten invariants in four-manifold topology.

Contribution

It proves the existence of perturbations ensuring transversality and constructs an Uhlenbeck compactification for PU(2) monopole moduli spaces, advancing the analytical foundation for Witten's conjecture.

Findings

Existence of perturbations for transversality

Construction of Uhlenbeck compactification

Regularity results for PU(2) monopoles

Abstract

We prove the existence of perturbations for the PU(2) monopole equations, yielding transversality on the complement of the anti-self-dual or reducible solutions, and the existence of an Uhlenbeck compactification for the moduli space of solutions to these perturbed PU(2) monopole equations. In December 1994, V. Pidstrigach and A. Tyurin and then others proposed a method to prove Witten's conjecture concerning the relation between the Donaldson and Seiberg-Witten invariants of smooth four-manifolds. Their proposal uses a moduli space of solutions to the PU(2) monopole equations, which are a natural generalization of the U(1) monopole equations of Seiberg and Witten and the equation for anti-self-dual SO(3) connections, to construct a cobordism between links of compact moduli spaces of U(1) monopoles of Seiberg-Witten type and the moduli space of anti-self-dual connections, which appear as singularities in this larger moduli space. A basic requirement of this cobordism technique is the existence of an Uhlenbeck compactification for the moduli space of PU(2) monopoles and of generic-parameter transversality results for all the moduli spaces of PU(2) monopoles which appear in this compactification, on the complement of the anti-self-dual and U(1) solutions.