Supersymmetric Quantum Mechanics meets Topology

In the first part of this thesis, we will try to get familiar with some of the concepts of supersymmetry (SUSY), starting from the zero-dimensional Quantum Field Theory and apply these concepts to a toy model of (0+1) dimensional QFT otherwise known as Quantum Mechanics. Methods explored in this thesis allow for the computation of path integrals, mainly the partition function and its modification, the Witten Index, which "counts" the difference between bosonic and fermionic states of the SUSY model used. The second part will focus on how this is related to the topology, more precisely the topological invariants, of the manifold (target space). The model in question will be a particular supersymmetric sigma model (σ-model), where the Witten Index is related to the Euler Characteristic. These techniques can be further extended to higher dimensional QFTs or different topological invariants.