An inquiry into the index of certain operators and extensions of supersymmetry

The basic properties of the one dimensional supersymmetric non-linear sigma-model are reviewed. A discussion of the relation between supersymmetry and the Atiyah-Singer index theorem is then given, followed by a look at how various examples of this theorem may be computed using the sigma-models. These ideas are applied to the concept of a horizontal supersymmetry: by considering a group manifold G as a principle H-bundle, the author restricts a N = 1/2 supersymmetry to lie entirely in the horizontal direction of the coset space G/H. This system is then employed for an index calculation to obtain a form of the Weyl character formula which is more general than the usual presentation. A review is then given of the extension of quantum mechanics and index theory to loop space. This serves as a motivation for attempting to compute topological invariants of the spaces M ap(X; M) by functional integral methods. He finds that there are no interesting results for dim X ≥ 2. To obtain a Dirac-type operator for higher dimensional spaces, the basic supersymmetry is extended to theories of differential forms. For a manifold of dimension 2k + 1, a supersymmetry between k-forms is demonstrated. The basis properties of the fermionic, even k version, particularly for k = 2, are explored in detail. The bosonic, odd k case is outlined