  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.
