The Geometry of Supersymmetric Quantum Mechanics

One-dimensional sigma-models with N supersymmetries are considered. For conventional supersymmetries there must be N − 1 complex structures satisfying a Clifford algebra and the constraints on the target space geometry can be formulated in terms of these. In the cases in which the complex structures are simultaneously integrable, a conventional extended superspace formulation is given, with the geometry determined by a 2-form potential for N = 2, by a 1-form potential for N = 3 and a scalar potential for N = 4; for N > 4 it is given by a scalar potential satisfying differential constraints. This gives explicit constructions of models with N = 3 but not N = 4 supersymmetry and of N = 4 models in which the complex structures do not satisfy a quaternionic algebra. Generalisations with central terms in the superalgebra are also considered.