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

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.

LaTeX 2e, 35 pages. Slightly revised version to appear in Topology and its Applications, (Proceedings of the Georgia Topology Conference, Atlanta, GA, June 1996). Physics reference and comment added

References in corpus (4)

• PU(2) Monopoles, I: Regularity, Uhlenbeck Compactness, and Transversality
• Moduli spaces of PU(2)-monopoles
• PU(2) monopoles. III: Existence of gluing and obstruction maps
• Uhlenbeck compactness and transversality for the moduli space of PU(2) monopoles

Cited by in corpus (12)

• An SO(3)-monopole cobordism formula relating Donaldson and Seiberg-Witten invariants
• SO(3) Monopoles, Level-One Seiberg-Witten Moduli Spaces, and Witten's Conjecture in Low Degrees
• PU(2) monopoles and links of top-level Seiberg-Witten moduli spaces
• PU(2) monopoles. II: Top-level Seiberg-Witten moduli spaces and Witten's conjecture in low degrees
• PU(2) monopoles and a conjecture of Marino, Moore, and Peradze
• Deformation theory of the blown-up Seiberg-Witten equation in dimension three
• What do Topologists want from Seiberg--Witten theory? (A review of four-dimensional topology for physicists)
• Generalised Seiberg-Witten equations and almost-Hermitian geometry
• Topological correlators of $SU(2)$, $\mathcal{N}=2^{\ast }$ SYM on four-manifolds
• A Kato-Yau inequality for harmonic spinors and decay estimate for eigenspinors
• Uhlenbeck compactness and transversality for the moduli space of PU(2) monopoles
• A Supersymmetric Quantum Field Theory Formulation of the Donaldson Polynomial Invariants