Yang–Mills Moduli Spaces and Donaldson’s Diagonalization Theorem

Abstract

Donaldson’s introduction of gauge-theoretic methods into 4-manifold topology in the early 1980s revealed a sharp divergence between the topological and smooth classifications in dimension four. This thesis develops the analytic and topological foundations of four-dimensional gauge theory, culminating in a proof of Donaldson’s diagonalization theorem on definite intersection forms of smooth 4-manifolds. On the topological side, we introduce smooth 4-manifolds and their intersection forms, classify unimodular symmetric bilinear forms, and develop the homotopy classification of vector bundles via classifying spaces. On the analytic side, we review the differential geometry of connections and curvature, develop Chern–Weil theory, and introduce the Sobolev spaces and infinite-dimensional manifold framework needed for the analysis of the space of connections. We then study anti-self-dual connections and the topology of their moduli space, which provides the cobordism underlying Donaldson’s theorem. Throughout, we aim for a self-contained and accessible exposition of the machinery that four-dimensional gauge theory requires.