Christian Bär (Potsdam)
An index theorem for Lorentzian manifolds with boundary
We show that the Dirac operator on a compact globally hyperbolic
Lorentzian spacetime with spacelike Cauchy boundary is a Fredholm
operator if appropriate boundary conditions are imposed. We prove that
the index of this operator is given by the same expression as in the
index formula of Atiyah-Patodi-Singer for Riemannian manifolds with
boundary. If time permits, an application to quantum field
theory (the computation of the chiral anomaly) will be sketched.
This is the first index theorem for Dirac operators on *Lorentzian*
manifolds and, from an analytic perspective, the methods to obtain it
are quite different from the classical Riemannian case. This is joint
work with Alexander Strohmaier.