Supersymmetry. From quantum mechanics to lattice field theories

Supersymmetric quantum mechanical systems in arbitrary dimensions on curved spaces with nontrivial gauge fields are considered. The square of the Dirac operator serves as Hamiltonian. Supersymmetry can be used to solve for zero modes of the Dirac operator which is illustrated for the Kahler spaces CPn. Furthermore, a special supersymmetric quantum mechanical system is investigated, the supersymmetric hydrogen atom with its corresponding super-Laplace-Runge-Lenz vector. Finally, Wess-Zumino models in the continuum and on spatial lattices, which result in high dimensional supersymmetric quantum mechanical systems, are discussed in detail. The normalizable zero modes of the models with N=1 and N=2 supersymmetry are counted and constructed in the weak- and strong-coupling limits. Together with known methods from operator theory this gives us complete control of the zero mode sector of these theories for arbitrary coupling. (orig.)