  We consider supersymmetric quantum mechanical systems in arbitrary dimensions
on curved spaces with nontrivial gauge fields. The square of the Dirac operator
serves as Hamiltonian. We derive a relation between the number of supercharges
that exist and restrictions on the geometry of the underlying spaces as well as
the admissible gauge field configurations. From the superalgebra with two or
more real supercharges we infer the existence of integrability conditions and
obtain a corresponding superpotential. This potential can be used to deform the
supercharges and to determine zero modes of the Dirac operator. The general
results are applied to the Kahler spaces CP^n.

High Energy Physics - Theory