Supersymmetry in Quantum Mechanics

Abstract: The introduction of supersymmetry has led to great progress in the study of quantum field theories. Notably, with supersymmetry, properties of a quantum field theory can be computed with higher precision than what would otherwise be possible. In this project, we investigate supersymmetry in the context of quantum mechanics. In particular, we show how the Witten index is insensitive to the details of the supersymmetric quantum mechanical system, making it a robust quantity when considering variations in the system’s parameters. Explicit calculations of the supersymmetric ground states are carried out to identify what determines the Witten index. The concept of superpotential is introduced and we relate Morse theory to the Witten index by identifying the superpotential as a Morse function. Moreover, we consider supersymmetric quantum mechanics on compact orientable Riemann manifolds. We show how the structure of supersymmetric quantum mechanics has a close connection to topological properties of the target manifolds. Specifically, the Witten index is shown to be the Euler characteristic, a topological invariant.