Quantum Yang-Mills Theory on Compact Surfaces

We describe the construction of the Yang-Mills measure for the (Euclidean) quantum theory of gauge fields associated to a bundle over a compact surface. This measure is obtained from the product of a Gaussian measure and a Haar measure by means of a conditioning process which encodes the topology of the bundle. We then describe expectation values of important random variables (‘Wilson loops variables’) associated to systems of loops on the surface. We also discuss results relating to the limit of the Yang-Mills measure as an associated parameter goes to zero, and the symplectic nature, in certain situations, of the set of minima of the Yang-Mills functional.