Non-Abelian Monopoles and Four-Manifold Topology - An Application of Virtual Morse-Bott Theory

This book explores Morse theory on the moduli space of non-Abelian monopoles over four-manifolds. The authors develop the foundations of gauge theory for anti-self-dual connections, Seiberg-Witten monopoles, and non-Abelian monopoles. They generalize Frankel's theorem for Morse theory of Hamiltonian functions for circle actions on symplectic manifolds. The authors prove that the Hitchin-Kobayashi correspondences between gauge-theoretic moduli spaces of Hermitian Yang-Mills connections or vortices and moduli spaces of polystable holomorphic vector bundles or pairs are circle-equivariant isomorphisms of real analytic spaces. Frankel's theorem and the preceding isomorphisms are used to compute the Morse indices of the singular strata of Seiberg-Witten moduli subspaces of moduli spaces of non-Abelian monopoles over Kahler surfaces.