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An index theorem on asymptotically static spacetimes with compact Cauchy surface

Dawei Shen and Michał Wrochna

Vol. 4 (2022), No. 4, 727–766

We consider the Dirac operator on asymptotically static Lorentzian manifolds with an odd-dimensional compact Cauchy surface. We prove that if Atiyah–Patodi–Singer boundary conditions are imposed at infinite times then the Dirac operator is Fredholm. This generalizes a theorem due to Bär and Strohmaier (Amer. J. Math. 141:5 (2019), 1421–1455) in the case of finite times, and we also show that the corresponding index formula extends to the infinite setting. Furthermore, we demonstrate the existence of a Fredholm inverse which is at the same time a Feynman parametrix in the sense of Duistermaat and Hörmander. The proof combines methods from time-dependent scattering theory with a variant of Egorov’s theorem for pseudodifferential hyperbolic systems.

Dirac operator, index theory, microlocal analysis, hyperbolic partial differential equations
Mathematical Subject Classification
Primary: 35P25, 58J20, 58J40, 58J47
Secondary: 58J30, 58J45
Received: 1 June 2021
Revised: 9 December 2021
Accepted: 22 March 2022
Published: 21 January 2023
Dawei Shen
Laboratoire Jacques-Louis Lions
Sorbonne Université
Michał Wrochna
CY Cergy Paris Université