The Geometry of Supersymmetric Quantum Mechanics

TL;DR

This paper explores the geometric structures underlying one-dimensional supersymmetric quantum mechanics, detailing how target space geometry relates to the number of supersymmetries and their algebraic properties.

Contribution

It provides a geometric framework for N-extended supersymmetries in quantum mechanics, including explicit models and generalizations with central charges.

Findings

Complex structures satisfy a Clifford algebra for conventional supersymmetries.

Explicit models constructed for N=3 and N=4 supersymmetries.

Generalized models with non-quaternionic complex structures and central charges.

Abstract

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.