Résume | A theorem due to Bär and Strohmaier (Amer. J. Math., 141 (5)) says that the Dirac operator on a Lorentzian manifold with compact Cauchy surface is Fredholm if Atiyah-Patodi-Singer boundary conditions are imposed at finite times. Furthermore, the index is given by a geometric formula that parallels as closely as possible the Atiyah-Patodi-Singer theorem in the Riemannian setting. In this talk I will report on joint work with Dawei Shen (Sorbonne Université) which extends this result to the infinite-time setting. Furthermore, we prove that in the infinite time situation, Fredholm inverses are Feynman parametrices in the sense of Duistermaat-Hörmander, a property which allows to show relationships with local aspects of the geometry. |