Path Integration in Curved Space and Extended Self-Dual Supergravity.

This dissertation is devoted to the study of two separate topics. First we discuss the relation between the operator formalism and path integration for quantum mechanical systems in curved space. Starting with a time -discretized approach to the transition element, we obtain the action to be used in the path integral, and also give a proper derivation of the Feynman rules needed for actual loop computations. We then apply our findings to the computation of anomalies in quantum field theories. We then discuss N = 8 self-dual supergravity in 2 + 2 dimensions. For the case of ungauged supergravity, we obtain the equations of motion for the component fields from projections of the Bianchi identities in chiral superspace. These equations are then derived from a manifestly Lorentz invariant action.