Geometry of [formula omitted] supersymmetry and the Atiyah-Singer index theorem

Abstract

We present an explicit realization of a conjecture by Atiyah and Witten on the geometrical interpretation of N= 1 2 supersymmetry. We apply our formalism to compute exactly (for all β) the path integral which yields the index of a Dirac operator for arbitrary gauge and gravitational background fields. Our approach is different from the original one by Alvarez-Gaumé, and by Friedan and Windey in the sense that we use the co-adjoint orbit method to realize the gauge symmetry. In this way we avoid the need to introduce a projection operator to the desired representation of the gauge group.