Gauge fields on coherent sheaves

Abstract

Given a flat gauge field $\nabla$ on a vector bundle $F$ over a manifold $M$ we deduce a necessary and sufficient condition for the field $\nabla+ E$, with $E$ an ${\rm End}(F)$-valued $1$-form, to be a Yang-Mills field. For each curve of Yang-Mills fields on $F$ starting at $\nabla$, we define a cohomology class of $H^2(M,\,{\mathscr P})$, with ${\mathscr P}$ the sheaf of $\nabla$-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.