  To any differential system $d\Psi=\Phi\Psi$ where $\Psi$ belongs to a Lie
group (a fiber of a principal bundle) and $\Phi$ is a Lie algebra $\mathfrak g$
valued 1-form on a Riemann surface $\Sigma$, is associated an infinite sequence
of "correlators" $W_n$ that are symmetric $n$-forms on $\Sigma^n$. The goal of
this article is to prove that these correlators always satisfy "loop
equations", the same equations satisfied by correlation functions in random
matrix models, or the same equations as Virasoro or W-algebra constraints in
CFT.
