Quantum Yang-Mills on the two-sphere

Abstract

We obtain the quantum expectations of gauge-invariant functions of the connection on a principalG=SU(N) bundle overS ². We show that the spaceA/g _m of connections modulo gauge transformations which are the identity at one point is itself a principal bundle over ΩG, based loops in the symmetry group. The fiber inA/g _m is an affine linear space. Quantum expectations are iterated path integrals first over this fiber then over ΩG, each with respect to the push-forward toA/g _m of the measure s^-S(A) DA.S(A) denotes the Yang-Mills action onA. There is a global section ofA/g _m on which the first integral is a Gaussian. The resulting measure on ΩG is the conditional Wiener measure. We explicitly compute the expectations of a special class of Wilson loops.