S¹-invariant Monopoles on Hyperbolic Three Space

In this thesis we review various properties of the monopoles on three dimensional manifolds. In Chapter 2 we prove some basic property of the moduli space. We follow Braam's treatment in [B89] and define monopoles on three manifolds as instantons on four manifolds invariant under a rotation action. This correspondence allows us to study monopoles with similar methods developed for instantons. By constructing the Kuranishi map and applying implicit function theorem, we can obtain a local model of monopole moduli space. We prove that the moduli space is smooth and orientable. The proof uses same techniques as the instanton case. The monopole moduli space can be compactified by adding limit ideal connections. This is the method of Uhlenbeck compactification. The treatment is again similar to the four-manifold case with minor modifications regarding the invariance condition. To better understand the structure of the compactified space, we introduce a method of constructing monopoles. In Chapter 3 we review the ADHM construction and use it to describe the charge 1 instanton moduli space. In Chapter 4 we calculate two examples: monopoles on hyperbolic 3-space and monopoles on $S^2 \times \R$, in which we construct the $S^1$-equivariant $SU(2)$ bundles, define the charge and mass and describe the local model for the moduli space explicitly using the Kuranishi map. Moreover, we consider another $S^1$ action for both case and describe the $S^1$-invariant monopole moduli space. Lastly we finish by a brief introduction of the Sen's conjecture