Dirac operators over the flat 3-torus

We determine spectrum and eigenspaces of some families of SpinC Dirac operators over the flat 3-torus. Our method relies on projections onto appropriate 2-tori on which we use complex geometry. Furthermore we investigate those families by means of spectral sections (in the sense of Melrose/Piazza). Our aim is to give a hands-on approach to this concept. First we calculate the relevant indices with the help of spectral flows. Then we define the concept of a system of infinitesimal spectral sections which allows us to classify spectral sections for small parameters R up to equivalence in K-theory. We undertake these classifications for the families of operators mentioned above. Our aim is therefore twofold: On the one hand we want to understand the behavior of SpinC Dirac operators over a 3-torus, especially for situations which are induced from a 4-manifold with boundary T . This has prospective applications in generalized Seiberg-Witten theory. On the other hand we want to make the term “spectral section”, for which one normally only knows existence results, more concrete by giving a detailed description in a special situation. M.S.C. 2010: 47A10, 58C40, 58J30.