- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Letters

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

Monte Carlo Studies of the Emergent Spacetime in the Polarized Type IIB Matrix Model

Chien-Yu Chou*, Jun Nishimura†, and Cheng-Tsung Wang‡

  • *Contact author: ccy@post.kek.jp
  • †Contact author: jnishi@post.kek.jp
  • ‡Contact author: ctwang@post.kek.jp
PDF

Phys. Rev. Lett. 135, 221601 – Published 24 November, 2025

DOI: https://doi.org/10.1103/y1rm-n85b

Abstract

The IKKT model (or the type IIB matrix model) has been investigated as a promising nonperturbative formulation of superstring theory. One of the recent developments concerning this model is the discovery of the dual supergravity solution corresponding to the model obtained after supersymmetry-preserving mass deformation, which is dubbed the polarized IKKT model. Here we perform Monte Carlo simulations of this model in the case of matrix size N=2 for a wide range of the deformation parameter Ω. While we reproduce precisely the known result for the partition function obtained by the localization method developed for supersymmetric theories, we also calculate the observables, which were not accessible by the previous work, in order to probe the spacetime structure emergent from the dominant matrix configurations. In particular, we find that the saddle point corresponding to the original IKKT model is smoothly connected to the saddle represented by the fuzzy sphere dominant at large Ω, whereas the dominant configurations become diverging commuting matrices at small Ω.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (53)

  1. N. Ishibashi, H. Kawai, Y. Kitazawa, and A. Tsuchiya, A large N reduced model as superstring, Nucl. Phys. B498, 467 (1997).
  2. H. Aoki, S. Iso, H. Kawai, Y. Kitazawa, A. Tsuchiya, and T. Tada, IIB matrix model, Prog. Theor. Phys. Suppl. 134, 47 (1999).
  3. T. Eguchi and H. Kawai, Reduction of dynamical degrees of freedom in the large N gauge theory, Phys. Rev. Lett. 48, 1063 (1982).
  4. G. Parisi, A simple expression for planar field theories, Phys. Lett. 112B, 463 (1982).
  5. D. J. Gross and Y. Kitazawa, A quenched momentum prescription for large N theories, Nucl. Phys. B206, 440 (1982).
  6. A. Gonzalez-Arroyo and M. Okawa, The twisted Eguchi-Kawai model: A reduced model for large N lattice gauge theory, Phys. Rev. D 27, 2397 (1983).
  7. E. Witten, Bound states of strings and p-branes, Nucl. Phys. B460, 335 (1996).
  8. M. B. Green and M. Gutperle, Effects of D instantons, Nucl. Phys. B498, 195 (1997).
  9. H. Aoki, S. Iso, H. Kawai, Y. Kitazawa, and T. Tada, Space-time structures from IIB matrix model, Prog. Theor. Phys. 99, 713 (1998).
  10. T. Hotta, J. Nishimura, and A. Tsuchiya, Dynamical aspects of large N reduced models, Nucl. Phys. B545, 543 (1999).
  11. S.-W. Kim, J. Nishimura, and A. Tsuchiya, Expanding (3+1)-dimensional universe from a Lorentzian matrix model for superstring theory in (9+1)-dimensions, Phys. Rev. Lett. 108, 011601 (2012).
  12. S.-W. Kim, J. Nishimura, and A. Tsuchiya, Expanding universe as a classical solution in the Lorentzian matrix model for nonperturbative superstring theory, Phys. Rev. D 86, 027901 (2012).
  13. S.-W. Kim, J. Nishimura, and A. Tsuchiya, Late time behaviors of the expanding universe in the IIB matrix model, J. High Energy Phys. 10 (2012) 147.
  14. H. C. Steinacker, Cosmological space-times with resolved Big Bang in Yang-Mills matrix models, J. High Energy Phys. 02 (2018) 033.
  15. H. C. Steinacker, Quantized open FRW cosmology from Yang–Mills matrix models, Phys. Lett. B 782, 176 (2018).
  16. 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.
  17. K. Hatakeyama, A. Matsumoto, J. Nishimura, A. Tsuchiya, and A. Yosprakob, The emergence of expanding space–time and intersecting D-branes from classical solutions in the Lorentzian type IIB matrix model, Prog. Theor. Exp. Phys. 2020, 043B10 (2020).
  18. H. C. Steinacker, Gravity as a quantum effect on quantum space-time, Phys. Lett. B 827, 136946 (2022).
  19. S. Brahma, R. Brandenberger, and S. Laliberte, BFSS matrix model cosmology: Progress and challenges, arXiv:2210.07288.
  20. F. R. Klinkhamer, Emergent gravity from the IIB matrix model and cancellation of a cosmological constant, Classical Quantum Gravity 40, 124001 (2023).
  21. M. Hirasawa, K. N. Anagnostopoulos, T. Azuma, K. Hatakeyama, J. Nishimura, S. Papadoudis, and A. Tsuchiya, The effects of SUSY on the emergent spacetime in the Lorentzian type IIB matrix model, Proc. Sci. CORFU2023 (2024) 257 [arXiv:2407.03491].
  22. J. M. Maldacena, The large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2, 231 (1998).
  23. O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri, and Y. Oz, Large N field theories, string theory and gravity, Phys. Rep. 323, 183 (2000).
  24. T. Banks, W. Fischler, S. H. Shenker, and L. Susskind, M theory as a matrix model: A conjecture, Phys. Rev. D 55, 5112 (1997).
  25. G. W. Gibbons, M. B. Green, and M. J. Perry, Instantons and seven-branes in type IIB superstring theory, Phys. Lett. B 370, 37 (1996).
  26. H. Ooguri and K. Skenderis, On the field theory limit of D instantons, J. High Energy Phys. 11 (1998) 013.
  27. F. Ciceri and H. Samtleben, Holography for the IKKT matrix model, Phys. Rev. Lett. 135, 061601 (2025).
  28. S. A. Hartnoll and J. Liu, The polarised IKKT matrix model, J. High Energy Phys. 03 (2025) 060.
  29. S. Komatsu, A. Martina, J. Penedones, A. Vuignier, and X. Zhao, Einstein gravity from a matrix integral—Part I, arXiv:2410.18173.
  30. G. Bonelli, Matrix strings in pp wave backgrounds from deformed super Yang-Mills theory, J. High Energy Phys. 08 (2002) 022.
  31. H. Lin, O. Lunin, and J. M. Maldacena, Bubbling AdS space and 1/2 BPS geometries, J. High Energy Phys. 10 (2004) 025.
  32. H. Lin and J. M. Maldacena, Fivebranes from gauge theory, Phys. Rev. D 74, 084014 (2006).
  33. D. E. Berenstein, J. M. Maldacena, and H. S. Nastase, Strings in flat space and pp waves from N=4 Super Yang Mills, J. High Energy Phys. 04 (2002) 013.
  34. T. Azuma, S. Bal, K. Nagao, and J. Nishimura, Nonperturbative studies of fuzzy spheres in a matrix model with the Chern-Simons term, J. High Energy Phys. 05 (2004) 005.
  35. T. Azuma, K. Nagao, and J. Nishimura, Perturbative dynamics of fuzzy spheres at large N, J. High Energy Phys. 06 (2005) 081.
  36. R. C. Myers, Dielectric branes, J. High Energy Phys. 12 (1999) 022.
  37. T. J. Hollowood and S. P. Kumar, World sheet instantons via the Myers effect and N=1* quiver superpotentials, J. High Energy Phys. 10 (2002) 077.
  38. S. Komatsu, A. Martina, J. Penedones, A. Vuignier, and X. Zhao, Einstein gravity from a matrix integral—Part II, arXiv:2411.18678.
  39. V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313, 71 (2012).
  40. Y. Asano, G. Ishiki, T. Okada, and S. Shimasaki, Emergent bubbling geometries in the plane wave matrix model, J. High Energy Phys. 05 (2014) 075.
  41. G. W. Moore, N. Nekrasov, and S. Shatashvili, D particle bound states and generalized instantons, Commun. Math. Phys. 209, 77 (2000).
  42. W. Krauth, H. Nicolai, and M. Staudacher, Monte Carlo approach to M theory, Phys. Lett. B 431, 31 (1998).
  43. P. Austing and J. F. Wheater, Convergent Yang-Mills matrix theories, J. High Energy Phys. 04 (2001) 019.
  44. V. A. Kazakov, I. K. Kostov, and N. A. Nekrasov, D particles, matrix integrals and KP hierarchy, Nucl. Phys. B557, 413 (1999).
  45. S. A. Hartnoll and J. Liu, Statistical physics of the polarised IKKT matrix model, SciPost Phys. 19, 099 (2025).
  46. A. Kumar, A. Joseph, and P. Kumar, Complex Langevin study of spontaneous symmetry breaking in IKKT matrix model, Proc. Sci. LATTICE2022 (2023) 213 [arXiv:2209.10494].
  47. A. Kumar, A. Joseph, and P. Kumar, Investigating spontaneous SO(10) symmetry breaking in type IIB matrix model, Springer Proc. Phys. 304, 1201 (2024).
  48. T.-L. Chau, C.-Y. Chou, J. Nishimura, and C.-T. Wang (to be published).
  49. J. Nishimura, T. Okubo, and F. Sugino, Systematic study of the SO(10) symmetry breaking vacua in the matrix model for type IIB superstrings, J. High Energy Phys. 10 (2011) 135.
  50. K. N. Anagnostopoulos, T. Azuma, Y. Ito, J. Nishimura, T. Okubo, and S. K. Papadoudis, Complex Langevin analysis of the spontaneous breaking of 10D rotational symmetry in the Euclidean IKKT matrix model, J. High Energy Phys. 06 (2020) 069.
  51. K. N. Anagnostopoulos, T. Azuma, K. Hatakeyama, M. Hirasawa, Y. Ito, J. Nishimura, S. K. Papadoudis, and A. Tsuchiya, Progress in the numerical studies of the type IIB matrix model, Eur. Phys. J. Special Topics 232, 3681 (2023).
  52. 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).
  53. C.-Y. Chou, J. Nishimura, and A. Tripathi, Inequivalence between the Euclidean and Lorentzian versions of the type IIB matrix model from Lefschetz thimble calculations, Phys. Rev. Lett. 134, 211601 (2025).

Outline

Information

Phys. Rev. Lett. 135, 221601– Published 24 November, 2025
  • Received 1 August 2025
  • Revised 29 September 2025
  • Accepted 3 November 2025
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