Generic metrics, irreducible rank-one $\mathrm{PU}(2)$ monopoles, and transversality

Abstract

We prove that the moduli space of solutions to the PU(2) monopole equations is a smooth manifold of the expected dimension for simple, generic parameters such as (and including) the Riemannian metric on the given four-manifold. In a previous article, dg-ga/9710032, we proved transversality using an extension of the holonomy-perturbation methods of Donaldson, Floer, and Taubes, together with the existence of an Uhlenbeck compactification for the perturbed moduli space. However, it remained an important and interesting question to see whether there were simpler, more intrinsic alternatives to the holonomy perturbations and this is the issue we settle here. The idea that PU(2) monopoles might lead to a proof of Witten's conjecture (hep-th/9411102, hep-th/9709193) concerning the relation between the two types of four-manifold invariants was first proposed by Pidstrigach and Tyurin in 1994 (dg-ga/9507004): the space of PU(2) monopoles contains the moduli space of anti-self-dual connections together with copies of the various Seiberg-Witten moduli spaces, these forming singular loci in the higher-dimensional space of PU(2) monopoles. Results in this direction, due to the author and Leness, are surveyed in dg-ga/9709022, with a detailed account appearing in dg-ga/9712005. Our transversality theorem ensures that the anti-self-dual and Seiberg-Witten loci are the only singularities and that the PU(2) monopole moduli space forms a smooth - though non-compact, because of bubbling - cobordism between the links of the singularities.