SIGMA 21 (2025), 088, 13 pages      arXiv:2408.16696      https://doi.org/10.3842/SIGMA.2025.088

Perturbations of APS Boundary Conditions for Lorentzian Dirac Operators

Lennart Ronge
Universität Potsdam, Haus 9, Karl-Liebknecht-Str. 24-25, 14476 Potsdam, Germany

Received May 20, 2025, in final form October 09, 2025; Published online October 20, 2025

Abstract
We study how far APS boundary conditions for a Lorentzian Dirac operator may be perturbed without destroying Fredholmness of the Dirac operator. This is done by developing criteria under which the perturbation of a compact pair of projections is a Fredholm pair.

Key words: index theory; Fredholm pairs; Lorentzian geometry; boundary value problems.

pdf (335 kb)   tex (17 kb)  

References

1. Avron J., Seiler R., Simon B., The index of a pair of projections, J. Funct. Anal. 120 (1994), 220-237.
2. Bär C., Hannes S., Boundary value problems for the Lorentzian Dirac operator, in Geometry and Physics. Vol. I, Oxford University Press, Oxford, 2018, 3-18, arXiv:1704.03224.
3. Bär C., Strohmaier A., An index theorem for Lorentzian manifolds with compact spacelike Cauchy boundary, Amer. J. Math. 141 (2019), 1421-1455, arXiv:1506.00959.
4. Bär C., Strohmaier A., Local index theory for Lorentzian manifolds, Ann. Sci. Éc. Norm. Supér. 57 (2024), 1693-1752, arXiv:2012.01364.
5. Doll N., Schulz-Baldes H., Waterstraat N., Spectral flow, De Gruyter Stud. Math., Vol. 94, De Gruyter, Berlin, 2023.
6. Lesch M., The uniqueness of the spectral flow on spaces of unbounded self-adjoint Fredholm operators, in Spectral Geometry of Manifolds with Boundary and Decomposition of Manifolds, Contemp. Math., Vol. 366, American Mathematical Society, Providence, RI, 2005, 193-224, arXiv:math.FA/0401411.