Topological quantum field theories, moduli spaces, and flat gauge connections.

: -We show how to construct a topological quantum field theory which corresponds to a given moduli space. We apply this method to the case of flat gauge connections defined over a Riemann surface and discuss its relations with the Chern-Simons theory and conformal field theory. The case of the SO(2,l) group is separately discussed. A topological field theory is linked to the moduli space of “self-dual” connections over Riemann surfaces. Another relation between the Chern-Simons theory and topological quantum field theory in three dimensions is established. We present the theory which corresponds to three dimensional gravity. Expressions for the Casson invariants are given. Possible generalizations are briefly discussed.