Quantum Yang-Mills on a Riemann surface

Abstract

We obtain the quantum expectations of gauge-invariant functions of the connection on aG=SU(N) product bundle over a Riemann surface of genusg. We show that the spaceA/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 Path^2g G consists of 2g-tuples of paths inG subject to a relation on their endpoint values. Quantum expectations are iterated path integrals over first the fiber then over Path^2g G, each with respect to the push-forward toA/G _m of the measuree ^−S(A) D A. Here,S(A) denotes the Yang-Mills action onA. We exhibit a global section ofA/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 Path^2g G is the product of Wiener measures on the component paths, conditioned to preserve the endopoint relation. Conformal transformations of the metric onM 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 caseg=1.