Event

Talk
The kernel of the Dirac operator
Prof. Dr. Bernd Ammann
25 Oct 2011, 17:30 – 18:30

Title: The kernel of the Dirac operator.

We assume that $M$ is a compact connected oriented spin manifold equipped with a riemannian metric $g$. This setting allows to define spinors and a Dirac operator $D_g$ which is a first order elliptic differential operator acting on spinors. The spectrum of $D_g$ is real and discrete, and the eigenvalues can physically be interpreted as possible energy states
of a fermion.

In the talk we are mainly interested in the kernel of the Dirac operator.

The index theorem by Atiyah and Singer gives some information about the kernel, in particular it yields a lower bound on the dimension of this kernel. This lower bound is topological, i.e. it does not depend on $g$, but the kernel of $D_g$ does depend in general.

Two conjectures are natural:
(1) For generic metrics the dimension of the kernel of the Dirac operator is as small as allowed by the index theorem.
(2) For special metrics the dimension is arbitrarily large.

In a collaboration with Emmanuel Humbert and Mattias Dahl we proved (1), building on previous work by Christian Bär and Mattias Dahl. We will give a new proof which yields a ``local version'' (and thus stronger) version of Conjecture (1), and which uses the unique continuation property of the Dirac operator. We will also sketch the status of Conjecture (2) which has partially been verified.

Related to:

• Bernd Ammann