Monopoles, Singularities and Hyperkähler Geometry

The main subjects of this thesis are monopoles. They are solutions to the Bogomolny equations on 3-manifolds, Calabi-Yau 3-folds and G2-manifolds. Monopoles, conjecturally, can be used to define invariants of manifolds. We prove the existence of non-trivial monopoles with Dirac singularities on rational homology 3-spheres, via a gluing construction. Furthermore, we will introduce some generalized Bogomolny equations in higher dimensions. The main difficulty in defining invariants of manifolds with special holonomy groups using these gauge-theoretic equations comes from the non-compactness of the moduli spaces of monopoles, which are governed by a first order differential operator, called the Fueter operator. The Fueter operator is a non-linear generalization of the Dirac operator over 3- and 4-manifolds, where the spinor bundle is replaced by a non-linear hyperkähler bundle. We prove partial compactness results by examining the different sources of non-compactness of the spaces of the Fueter sections and proving some of them, in fact, do not occur. Donaldson proposed the possibility of studying G2-manifolds from the viewpoint of coassociative fibrations and the adiabatic limit. This approach is expected to be helpful in understanding the non-compactness problems. The adiabatic picture led Donaldson and Scaduto to conjecture the existence of certain associative submanifolds in G2-manifolds with a coassociative K3-fibration near the adiabatic limit, which reduces to the question of the existence of certain asymptotically cylindrical special Lagrangians in certain Calabi-Yau 3-folds. We propose a strategy to prove this conjecture using the method of continuity and take several steps in that direction.