Non-local boundary conditions for the Dirac operator on spacetimes with timelike boundary

19.05.2022, 16:00-17:30  –  2.09.0.14
Forschungsseminar Differentialgeometrie

Penelope Gehring

Non-local boundary conditions - for example the Atiyah-Patodi-Singer (APS) conditions -  for Dirac operators on Riemannian manifolds are rather well-understood, while not much is known for such operators on Lorentzian manifolds.   Recently, Bär-Strohmaier and Drago-Große-Murro introduced APS-like conditions for the spin Dirac operator on Lorentzian manifolds with spacelike and timelike boundary, respectively. While Bär--Strohmaier showed the Fredholmness of the Dirac operator with these boundary conditions, Drago-Große-Murro proved the well-posedness of the corresponding initial boundary value problem under certain geometric assumptions. 

In this talk, I will follow  in the footsteps of the latter authors and discuss  whether the APS-like conditions for Dirac operators on Lorentzian manifolds with timelike boundary can be replaced by more general conditions such that the associated initial boundary value problems are still well-posed.

After a  brief introduction to  globally hyperbolic manifolds with timelike boundary and the Dirac operator on these manifolds, I will discuss non-local boundary conditions and the related Cauchy problems in this setting. In the last part of my talk I will introduce important examples of non-local boundary conditions that lead to well-posedness of the corresponding Cauchy problem.