Samuel Lockman: Perturbation theory of Dirac operators and homotopy groups of parameter spaces

By a theorem of Ammann and Dahl, the space of Dirac-minimal Riemannian metrics on a closed spin manifold of dimension 2 or 4 is connected. In the first part of the talk, I will show that for any surface of genus at least 3, equipped with any bounding spin structure, the space of Dirac-minimal (i.e. Dirac-invertible) Riemannian metrics is not simply connected.

In the second part of the talk, I will discuss Dirac minimal connections for twisted Dirac operators. Previous results by Anghel and Maier show that Dirac minimal connections on Hermitian twist bundles are dense for closed spin manifolds of dimension less than or equal to 4. I will show how to extend their results to dimensions 7 and 8. We will also see that in dimensions 2 and 4, the space of Dirac minimal connections are in fact k-connected, where k is the index of the corresponding Dirac operator. Further variations such as trace-free connections and connections on real twist bundles will be discussed. Lastly, I will show how to construct non-trivial elements in some homotopy groups of Dirac invertible connections.