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We construct one Yang-Mills measure on a compact surface for each isomorphismclass of principal bundles over this surface. For this, we define a newdiscrete gauge theory which is essentially a covering of the usual one. Weprove that the measures correponding to different isomorphism classes ofbundles or to different total areas of the surface are mutually singular. Wegive also a combinatorial computation of the partition functions based on theformalism of fat graphs.

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