Perturbation theory of Dirac operators and homotopy groups of parameter spaces

05.02.2026, 16:15  –  Raum 1.22, Haus 9
Forschungsseminar Differentialgeometrie

Samuel Lockman (Regensburg)

I will begin by giving an overview of known results regarding Dirac minimal metrics and connections. Then, we will mainly be concerned with Dirac minimal connections, i.e. connections on a vector bundle for which the corresponding twisted Dirac operator has vanishing kernel or cokernel. Previous results by Anghel and Maier show that Dirac minimal connections on Hermitian twist bundles are generic for closed spin manifolds of dimension less than or equal to 4. We will see that in dimension 2 and 4, this space is k-connected, where k is the index of the corresponding Dirac operator. Motivated by an assumption which has appeared in Quantum Chromodynamics, we show that these results remain true when we furthermore restrict attention to trace-free connections.

Bei and Waterstraat show that for trivial twist bundles over odd-dimensional manifolds, with rank lying in a stable range, the space of Dirac-invertible connections has infinitely many connected components. I will show how to extend this beyond trivial twist bundles, and to other homotopy groups, including twist bundles over even-dimensional manifolds. If time permits, we will see some results within the stable range.