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Klein bottle cosmology

Brian Greene^1,*, Daniel Kabat ^2,3,†, Janna Levin ^4,‡, and Massimo Porrati^5,§

¹Departments of Physics and Mathematics, Columbia University, 538 West 120th Street, New York, New York 10027, USA
²Department of Physics and Astronomy, Lehman College, City University of New York, 250 Bedford Park Boulevard W, Bronx, New York 10468, USA
³Graduate School and University Center, City University of New York, 365 Fifth Avenue, New York, New York 10016, USA
⁴Department of Physics and Astronomy, Barnard College of Columbia University, New York, New York 10027, USA
⁵Center for Cosmology and Particle Physics, Department of Physics, New York University, 726 Broadway, New York, New York 10003, USA

^*Contact author: brian.greene@columbia.edu
^†Contact author: daniel.kabat@lehman.cuny.edu
^‡Contact author: janna@astro.columbia.edu
^§Contact author: massimo.porrati@nyu.edu

Phys. Rev. D 113, 063576 – Published 31 March, 2026

DOI: https://doi.org/10.1103/dyym-ywsr

Abstract

We explore a higher-dimensional universe that is a product of Minkowski space and the nonorientable Klein bottle. The topology explicitly breaks important symmetries, such as translational invariance and ( $5 + 1$ )-dimensional $C P$ invariance. Somewhat surprisingly, the ( $3 + 1$ )-dimensional $c p$ of the Minkowski space can also be broken by the Klein bottle, both explicitly and in the presence of a brane. The topology enforces a background of fermion correlations that amounts to a condensate wall localized in the Klein bottle. The wall acts as an order parameter for the broken symmetries. If a brane passes through the wall, brane fermions that couple to the condensate are produced as quantified by the Bogoliubov coefficients for a time-dependent mass. The scenario meets the conditions, including $c p$ violation, to potentially generate the matter-antimatter asymmetry of the universe.

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References (14)

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Phys. Rev. D 113, 063576– Published 31 March, 2026

Vol. 113, Iss. 6 — 15 March 2026

Received 9 December 2025
Accepted 5 March 2026

DOI: https://doi.org/10.1103/dyym-ywsr

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

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Klein bottle cosmology

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