GEOMETRY OF N = 1/2 SUPERSYMMETRY AND THE ATIYAH-SINGER INDEX THEOREM

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 beta) 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-Gaume, 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.