Uwe Semmelmann, University of Stuttgart
The Rarita-Schwinger operator is the Dirac operator twisted with the tangent bundle. It has several interesting applications in physics and differential geometry. In my talk I will consider this operator on Einstein manifolds where it has additional nice properties.
In contrast to the classical Dirac operator the Rarita-Schwinger operator can have a non-trivial kernel on compact manifolds with positive scalar curvature. I will discuss several examples for this. In particular, I will explain how for manifolds with special holonomy one can identify the kernel of the Rarita-Schwinger operator with subspaces of harmonic forms.
If time permits I will make a few comments on how to use the Rarita-Schwinger operator for proving stability of Einstein metrics. This works well for Ricci flat manifolds. However, in case of positive Ricci curvature it is so far only possible to use it to prove that metrics admitting Killing spinors are physically stable. My talk is based on an article with Yasushi Homma (Waseda University, Tokyo).