Flat connections, geometric invariants and energy of harmonic functions on compact Riemann surfaces

This work grew out of an attempt to generalize the construction of Chern-Simons invariants. In this paper, we associate a geometric invariant to the space of flat connection on a SU(2)-bundle on a compact Riemann surface and relate it to the energy of harmonic functions on the surface. Our set up is as follows. Let G = SU(2) and M be a compact Riemann surface and E ~ M be the trivial G-bundle. (Any SU(2)-bundle over M is topologically trivial). Let ~g be the space of all connections and ~the subspace of all flat connections on this G-bundle. We endow on ~g the Frechet topology and the subspace topology on o~-. Given a loop ~r:S 1 ~ , we can extend tr to the closed unit disc t~:D2 ~cg since is contractible. On the trivial G-bundle E x D 2 ~ M x D 2 we define a tautological connection form 8 ~ as follows