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.

The Atiyah-Singer index theorem, the Euler number, and the Hirzebruch signature are derived via the supersymmetric path integral. Concisely, the supersymmetric path integral is a combination of a bosonic and a femionic path integral. The action in the supersymmetric path integral includes here bosonic, fermionic- and isospin fields (backgroundfields), where the cross terms in the Lagrangian are nicely eliminated due to scaling of the fields and using techniques from spontaneous breaking of supersymmetry (that give rise to a mechanism, analogous to the Higgs-mechanism, but here regarding the so called superparticles instead).  Thus, the supersymmetric path integral is a product of three pathintegrals over the three given fields, respectively, that can be evaluated exactly by means of Gaussian integrals. The closely related Witten index is a measure of the failure of spontaneous breaking of supersymmetry. In addition, the basic concepts of supersymmetry breaking are reviewed.

In this thesis, we investigate supersymmetric quantum mechanics (SUSYQM) and its relation to index theorems and equivariant cohomology. We define some basic constructions on super vector spaces in order to set the language for the rest of the thesis. The path integral in quantum mechanics is reviewed together with some related calculational methods and we give a path integral expression for the Witten index. Thereafter, we discuss the structure of SUSYQM in general. One shows that the Witten index can be taken to be the difference in dimension of the bosonic and fermionic zero energy eigenspaces. In the subsequent section, we derive index theorems. The models investigated are the supersymmetric non-linear sigma models with one or two supercharges. The former produces the index theorem for the spin-complex and the latter the Chern-Gauss-Bonnet Theorem. We then generalise to the case when a group action (by a compact connected Lie group) is included and want to consider the orbit space as the underlying space, in which case equivariant cohomology is introduced. In particular, the Weil and Cartan models are investigated and SUSYQM Lagrangians are derived using the obtained differentials. The goal was to relate this to gauge quantum mechanics, which was unfortunately not successful. However, what was shown was that the Euler characteristics of a closed oriented manifold and its homotopy quotient by U(1)n coincide.