Theory and Dirac Operators
Dirac operators play an important role in modern physics and
mathematics. Originally, this operator was introduced by P. Dirac
(1928), when looking for an equation to describe
spin 1/2 particles (fermions, e.g. electrons) that would fit into the framework of both special relativity and quantum mechanics.
In course of the development of the index theory of elliptic operators it was observed that it is possible to define an analogeous operator on special Riemannian manifolds, the spin manifolds, and that its analytical properties are strongly related to the geometry of the Riemannina manifold. In the lecture we define the Dirac operator of a Riemannian spin manifold and study its analytic properties in relation to the underlying geometry.
The definition and the study of spinors and Dirac operators on
Riemannian spin manifolds require knowledge from differential geometry
on fibre bundles. Depending on the prior knowledge of the participants
we will introduce or recall the techniques and facts from gauge theory
in the first part of the lecture.