Discrete and continuous Yang-Mills measure for non-trivial bundles over compact surfaces

In this paper, we construct one Yang-Mills measure on an orientable compact surface for each isomorphism class of principal bundles with compact connected structure group over this surface. For this, we refine the discretization procedure used in a previous construction [9] and define a discrete theory on a new configuration space which is essentially a covering of the usual one. We prove that the measures corresponding to different isomorphism classes of bundles or to different total areas of the base space are mutually singular. We give also a combinatorial computation of the partition functions which relies on the formalism of fat graphs.