QUANTUM YANG-MILLS ON A RIEMANN SURFACE

We obtain the quantum expectations of gauge-invariant functions of the connection on a G = SU(N) product bundle over a Riemann surface of genus g. We show that the space A/G(m) of connections modulo those gauge transformations which are the identity at one point is itself a principal bundle with affine linear fiber. The base space Path2g G consists of 2g-tuples of paths in G subject to a relation on their endpoint values. Quantum expectations are iterated path integrals over first the fiber then over Path2g G, each with respect to the push-forward to A/G(m) of the measure e-S(A) DA. Here, S(A) denotes the Yang-Mills action on A. We exhibit a global section of A/G(m) to define a choice of origin in each fiber, relative to which the measure on the fiber is Gaussian. The induced measure on Path2g G is the product of Wiener measures on the component paths, conditioned to preserve the endpoint relation. Conformal transformations of the metric on M act by reparametrizing these paths. We explicitly compute the partition function in the general case and the expectations of functions of certain products of Wilson loops in the case g = 1.