An index theorem on asymptotically static spacetimes with compact Cauchy surface

03.06.2021, 16:15  –  Online-Seminar
Forschungsseminar Differentialgeometrie

Michał Wrochna (Cergy Paris Université)

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 demonstrate 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.

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