- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Spatially flat cosmological quantum spacetimes

Christian Gaß* and Harold C. Steinacker†

  • *Contact author: christian.gass@univie.ac.at
  • †Contact author: harold.steinacker@univie.ac.at
PDF

Phys. Rev. D 113, 024050 – Published 23 January, 2026

DOI: https://doi.org/10.1103/6l9d-njcr

Abstract

We recently described a cosmological quantum spacetime of vanishing spatial curvature, which can be considered as background for the Ishibashi-Kitazawa-Kawai-Tsuchiya (IKKT) matrix model, assuming that the resulting gauge theory couples weakly. Building on this example, we construct a large class of spatially flat cosmological quantum spacetimes. We also elaborate on various details of their algebraic and semiclassical structure as well as the higher spin modes present in these models. In particular, we introduce the notion of approximate diffeomorphisms on the cosmological quantum spacetimes that stem from gauge transformations of the underlying matrix model, and investigate how different gauges are related in the semiclassical regime by approximate diffeomorphisms. Finally, we briefly outline how the described quantum spacetimes could be incorporated into the full IKKT model.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (37)

  1. S. Fernando and M. Gunaydin, Minimal unitary representation of SO*(8)=SO(6,2) and its SU(2) deformations as massless 6D conformal fields and their supersymmetric extensions, Nucl. Phys. B841, 339 (2010).
  2. H. C. Steinacker, Quantum Geometry, Matrix Theory, and Gravity (Cambridge University Press, Cambridge, England, 2024).
  3. M. Sperling and H. C. Steinacker, Covariant cosmological quantum space-time, higher-spin and gravity in the IKKT matrix model, J. High Energy Phys. 07 (2019) 010.
  4. M. Sperling and H. C. Steinacker, The fuzzy 4-hyperboloid Hn4 and higher-spin in Yang–Mills matrix models, Nucl. Phys. B941, 680 (2019).
  5. H. C. Steinacker, Quantized open FRW cosmology from Yang–Mills matrix models, Phys. Lett. B 782, 176 (2018).
  6. A. Manta and H. C. Steinacker, Minimal covariant quantum space-time, J. Phys. A 58, 175204 (2025).
  7. A. Manta, H. C. Steinacker, and T. Tran, hs-extended gravity from the IKKT matrix model, J. High Energy Phys. 02 (2025) 031.
  8. Emmanuele Battista and Harold C. Steinacker, On the propagation across the big bounce in an open quantum FLRW cosmology, Eur. Phys. J. C 82, 909 (2022).
  9. C. Gaß and H. C. Steinacker, Spatially flat FLRW spacetimes with a Big Bang from matrix geometry, Nucl. Phys. B1018, 116988 (2025).
  10. N. Ishibashi, H. Kawai, Y. Kitazawa, and A. Tsuchiya, A Large N reduced model as superstring, Nucl. Phys. B498, 467 (1997).
  11. A. Manta and H. C. Steinacker, Dynamical covariant quantum spacetime with fuzzy extra dimensions in the IKKT model, arXiv:2509.24753.
  12. M. Buric, D. Latas, and L. Nenadovic, Fuzzy de Sitter space, Eur. Phys. J. C 78, 953 (2018).
  13. J. P. Gazeau, J. Mourad, and J. Queva, Fuzzy de Sitter space-times via coherent states quantization, in 26th International Colloquium on Group Theoretical Methods in Physics (2006).
  14. J. P. Gazeau and F. Toppan, A natural fuzzyness of de Sitter space-time, Classical Quantum Gravity 27, 025004 (2010).
  15. A. Chaney, Lei Lu, and A. Stern, Matrix model approach to cosmology, Phys. Rev. D 93, 064074 (2016).
  16. S. Brahma, R. Brandenberger, and S. Laliberte, Emergent early universe cosmology from BFSS matrix theory, Int. J. Mod. Phys. D 31, 2242004 (2022).
  17. S. Brahma, R. Brandenberger, and S. Laliberte, Emergent cosmology from matrix theory, J. High Energy Phys. 03 (2022) 067.
  18. Y. Asano, J. Nishimura, W. Piensuk, and N. Yamamori, Defining the Type IIB matrix model without breaking Lorentz symmetry, Phys. Rev. Lett. 134, 041603 (2025).
  19. J. Nishimura and A. Tsuchiya, Complex Langevin analysis of the space-time structure in the Lorentzian Type IIB matrix model, J. High Energy Phys. 06 (2019) 077.
  20. Chien-Yu Chou and Jun Nishimura, Monte Carlo studies of quantum cosmology by the generalized Lefschetz thimble method, J. High Energy Phys. 05 (2025) 142.
  21. S. Fernando and M. Gunaydin, Minimal unitary representation of SU(2,2) and its deformations as massless conformal fields and their supersymmetric extensions, J. Math. Phys. 51, 082301 (2010).
  22. S. A. Hartnoll and J. Liu, The polarised IKKT matrix model, J. High Energy Phys. 03 (2025) 060.
  23. S. Komatsu, Ad. Martina, J. Penedones, A. Vuignier, and X. Zhao, Einstein gravity from a matrix integral—Part II, arXiv:2411.18678.
  24. A. Martina, Massive deformations of supersymmetric Yang-Mills matrix models, arXiv:2507.17813.
  25. S. Komatsu, A. Martina, J. Penedones, A. Vuignier, and X. Zhao, Einstein gravity from a matrix integral—Part I, arXiv:2410.18173.
  26. C.-Y. Chou, J. Nishimura, and C.-T. Wang, Monte Carlo studies of the emergent spacetime in the polarized IKKT model, arXiv:2507.18472.
  27. H. C. Steinacker, On the quantum structure of space-time, gravity, and higher spin in matrix models, Classical Quantum Gravity 37, 113001 (2020).
  28. H. C. Steinacker, Quantum (Matrix) geometry and quasi-coherent states, J. Phys. A 54, 055401 (2021).
  29. H. C. Steinacker, One-loop effective action and emergent gravity on quantum spaces in the IKKT matrix model, J. High Energy Phys. 05 (2023) 129.
  30. E. Battista and H. C. Steinacker, One-loop effective action of the IKKT model for cosmological backgrounds, J. High Energy Phys. 01 (2024) 125.
  31. H. C. Steinacker, Higher-spin kinematics & no ghosts on quantum space-time in Yang–Mills matrix models, Adv. Theor. Math. Phys. 25, 1025 (2021).
  32. H. C. Steinacker, Modified gravity at large scales on quantum spacetime in the IKKT model, arXiv:2601.08031.
  33. P.-M. Ho, H. Kawai, and H. C. Steinacker, General relativity in IIB matrix model, arXiv:2509.06646.
  34. H. C. Steinacker and T. Tran, Quantum hs-Yang-Mills from the IKKT matrix model, Nucl. Phys. B1005, 116608 (2024).
  35. H. Aoki, N. Ishibashi, S. Iso, H. Kawai, Y. Kitazawa, and T. Tada, Noncommutative Yang-Mills in IIB matrix model, Nucl. Phys. B565, 176 (2000).
  36. E. Battista and H. C. Steinacker, Fermions on curved backgrounds of matrix models, Phys. Rev. D 107, 046021 (2023).
  37. Askold Perelomov, Generalized Coherent States and Their Applications, Theoretical and Mathematical Physics (Springer, New York, 1986).

Outline

Information

Phys. Rev. D 113, 024050– Published 23 January, 2026
  • Received 30 October 2025
  • Accepted 5 January 2026
Crossmark - Check for updates button
Creative Commons Logo

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.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation