Verification & discretization of the index theorem for a two dimensional dirac operator - Lim, Kim Song (2015)
The famous Atiyah-Singer Index Theorem states that for an elliptic partial differential operator D on a compact manifold, the analytical index (related to the solution space of the partial differential equation Df = 0) is equal to the topological index (defined in terms of some topological data of D). This project consists of two different parts. The first part of the thesis will verify the theorem for a simple two dimensional Dirac operator with certain boundary conditions where the topological data enters. The second part of the thesis will then describe how the analytic index can be defined in the discretized setting of lattice gauge theory. This is a subtle issue because the usual definition of the index automatically vanishes in the discretized setting. Therefore another, indirect approach is needed. Finally, numerical results for the index are presented in some examples which verify that the index theorem holds in the discrete setting when the definition of the analytic index that we described is used.