Gauge fields on coherent sheaves

Given a flat gauge field ∇ on a vector bundle F over a manifold M we deduce a necessary and sufficient condition for the field ∇ + E, with E an End(F )-valued 1-form, to be a YangMills field. For each curve of Yang-Mills fields on F starting at ∇, we define a cohomology class of H(M, P), with P the sheaf of ∇-parallel sections of F . This cohomology class vanishes when the curve consists of flat fields. We prove the existence of a curve of Yang-Mills fields on a bundle over the torus T 2 connecting two vacuum states. We define holomorphic and meromorphic gauge fields on a coherent sheaf and the corresponding Yang-Mills functional. In this setting, we analyze the Aharonov-Bohm effect and the Wong equation. MSC 2010: 53C05, 14F06, 58E15, 32C35