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Origin of the SO(9)→SO(3)×SO(6) symmetry breaking in the type IIB matrix model

Robert Brandenberger* and Julia Pasiecznik†

  • *Contact author: rhb@physics.mcgill.ca
  • †Contact author: Julia.Pasiecznik@mail.mcgill.ca
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Phys. Rev. D 112, 026006 – Published 2 July, 2025

DOI: https://doi.org/10.1103/bqzp-n28g

Abstract

We suggest a dynamical mechanism which explains why, in the supersymmetric Ishibashi-Kawai-Kitazawa-Tsuchiya matrix model, the SO(9) symmetry of the Lagrangian is spontaneously broken to SO(3)×SO(6), allowing only three large classical spatial dimensions to emerge. The argument relies on the identification of D-strings as stable excitations of the matrices about the cosmological background and the absence of forces between parallel D-string excitations. A spatial dimension can only become large if the D-strings winding in that direction can annihilate. In the absence of forces, this requires the string-world sheets to intersect, a process which is highly suppressed if more than three dimensions of space are large. Implications for a recently proposed matrix cosmology scenario are discussed.

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References (48)

  1. R. H. Brandenberger and C. Vafa, Superstrings in the early Universe, Nucl. Phys. B316, 391 (1989).
  2. M. Majumdar and A. Christine-Davis, Cosmological creation of D-branes and anti-D-branes, J. High Energy Phys. 03 (2002) 056.
  3. R. Durrer, M. Kunz, and M. Sakellariadou, Why do we live in 3+1 dimensions? Phys. Lett. B 614, 125 (2005).
  4. A. Karch and L. Randall, Relaxing to three dimensions, Phys. Rev. Lett. 95, 161601 (2005).
  5. R. H. Brandenberger, String Gas Cosmology, String Cosmology, edited by J. Erdmenger (Wiley, New York; Williams and Wilkins Baltimore, 2009), pp. 193–230; T. Battefeld and S. Watson, String gas cosmology, Rev. Mod. Phys. 78, 435 (2006).
  6. J. Kripfganz and H. Perlt, Cosmological impact of winding strings, Classical Quantum Gravity 5, 453 (1988).
  7. R. Hagedorn, Statistical thermodynamics of strong interactions at high-energies, Nuovo Cimento Suppl. 3, 147 (1965).
  8. J. Polchinski, String Theory. Vol. 1: An Introduction to the Bosonic String (Cambridge University Press, Cambridge, England, 2007); J. Polchinski, String Theory. Vol. 2: Superstring Theory and Beyond (Cambridge University Press, Cambridge, England, 2007).
  9. R. H. Brandenberger, S. Kanno, J. Soda, D. A. Easson, J. Khoury, P. Martineau, A. Nayeri, and S. P. Patil, More on the spectrum of perturbations in string gas cosmology, J. Cosmol. Astropart. Phys. 11 (2006) 009.
  10. G. F. R. Ellis and R. Maartens, The emergent Universe: Inflationary cosmology with no singularity, Classical Quantum Gravity 21, 223 (2004); G. F. R. Ellis, J. Murugan, and C. G. Tsagas, The emergent universe: An explicit construction, 21, 233 (2004).
  11. R. Easther, B. R. Greene, M. G. Jackson, and D. N. Kabat, String windings in the early Universe, J. Cosmol. Astropart. Phys. 02 (2005) 009; B. Greene, D. Kabat, and S. Marnerides, Dynamical decompactification and three large dimensions, Phys. Rev. D 82, 043528 (2010); On three dimensions as the preferred dimensionality of space via the Brandenberger-Vafa mechanism, 88, 043527 (2013).
  12. M. Sakellariadou, Numerical experiments on string cosmology, Nucl. Phys. B468, 319 (1996).
  13. S. Watson and R. H. Brandenberger, Isotropization in brane gas cosmology, Phys. Rev. D 67, 043510 (2003).
  14. S. Watson and R. Brandenberger, Stabilization of extra dimensions at tree level, J. Cosmol. Astropart. Phys. 11 (2003) 008; S. Watson, Moduli stabilization with the string Higgs effect, Phys. Rev. D 70, 066005 (2004); S. P. Patil and R. Brandenberger, Radion stabilization by stringy effects in general relativity, 71, 103522 (2005); S. P. Patil and R. H. Brandenberger, The cosmology of massless string modes, J. Cosmol. Astropart. Phys. 01 (2006) 005; R. Brandenberger, Y. K. E. Cheung, and S. Watson, Moduli stabilization with string gases and fluxes, J. High Energy Phys. 05 (2006) 025.
  15. S. Ferrara, L. Girardello, and H. P. Nilles, Breakdown of local supersymmetry through gauge fermion condensates, Phys. Lett. B 125, 457 (1983); I. Affleck, M. Dine, and N. Seiberg, Supersymmetry breaking by instantons, Phys. Rev. Lett. 51, 1026 (1983); M. Dine, R. Rohm, N. Seiberg, and E. Witten, Gluino condensation in superstring models, Phys. Lett. B 156, 55 (1985).
  16. R. J. Danos, A. R. Frey, and R. H. Brandenberger, Stabilizing moduli with thermal matter and nonperturbative effects, Phys. Rev. D 77, 126009 (2008).
  17. S. Mishra, W. Xue, R. Brandenberger, and U. Yajnik, Supersymmetry breaking and dilaton stabilization in string gas cosmology, J. Cosmol. Astropart. Phys. 09 (2012) 015.
  18. A. Nayeri, R. H. Brandenberger, and C. Vafa, Producing a scale-invariant spectrum of perturbations in a Hagedorn phase of string cosmology, Phys. Rev. Lett. 97, 021302 (2006).
  19. R. H. Brandenberger, A. Nayeri, S. P. Patil, and C. Vafa, Tensor modes from a primordial Hagedorn phase of string cosmology, Phys. Rev. Lett. 98, 231302 (2007); Closed string thermodynamics and a blue tensor spectrum, Phys. Rev. D 90, 067301 (2014).
  20. R. H. Brandenberger, Cosmology of the very early Universe, AIP Conf. Proc. 1268, 3 (2010); Alternatives to the inflationary paradigm of structure formation, Int. J. Mod. Phys. Conf. Ser. 01, 67 (2011).
  21. A. Bedroya and C. Vafa, Trans-Planckian censorship and the swampland, J. High Energy Phys. 09 (2020) 123; A. Bedroya, R. Brandenberger, M. Loverde, and C. Vafa, Trans-Planckian censorship and inflationary cosmology, Phys. Rev. D 101, 103502 (2020).
  22. M. Gasperini and G. Veneziano, Pre—big bang in string cosmology, Astropart. Phys. 1, 317 (1993).
  23. W. Siegel, Superspace duality in low-energy superstrings, Phys. Rev. D 48, 2826 (1993); C. Hull and B. Zwiebach, Double field theory, J. High Energy Phys. 09 (2009) 099; O. Hohm, C. Hull, and B. Zwiebach, Generalized metric formulation of double field theory, 08 (2010) 008; O. Hohm, W. Siegel, and B. Zwiebach, Doubled α′-geometry, 02 (2014) 065; O. Hohm and B. Zwiebach, Double field theory at order α′, 11 (2014) 075.
  24. R. Brandenberger, R. Costa, G. Franzmann, and A. Weltman, T-dual cosmological solutions in double field theory, Phys. Rev. D 99, 023531 (2019); H. Bernardo, R. Brandenberger, and G. Franzmann, T-dual cosmological solutions in double field theory. II., 99, 063521 (2019); O(d,d) covariant string cosmology to all orders in α′, J. High Energy Phys. 02 (2020) 178; String cosmology backgrounds from classical string geometry, Phys. Rev. D 103, 043540 (2021).
  25. T. Banks, W. Fischler, S. H. Shenker, and L. Susskind, M theory as a matrix model: A conjecture, Phys. Rev. D 55, 5112 (1997).
  26. N. Ishibashi, H. Kawai, Y. Kitazawa, and A. Tsuchiya, A large N reduced model as superstring, Nucl. Phys. B498, 467 (1997).
  27. W. Taylor, M(atrix) theory: Matrix quantum mechanics as a fundamental theory, Rev. Mod. Phys. 73, 419 (2001).
  28. B. Ydri, Review of M(atrix)-theory, type IIB Matrix model and Matrix string theory, arXiv:1708.00734.
  29. S. R. Das and A. Jevicki, String field theory and physical interpretation of D=1 strings, Mod. Phys. Lett. A 05, 1639 (1990).
  30. H. C. Steinacker, Quantum Geometry, Matrix Theory, and Gravity (Cambridge University Press, Cambridge, England, 2024).
  31. S. Brahma, R. Brandenberger, and S. Laliberte, Emergent metric spacetime from matrix theory, J. High Energy Phys. 09 (2022) 031.
  32. S. Brahma, R. Brandenberger, and S. Laliberte, Emergent cosmology from matrix theory, J. High Energy Phys. 03 (2022) 067.
  33. J. Nishimura, New perspectives on the emergence of (3+1)D expanding space-time in the Lorentzian type IIB matrix model, Proc. Sci. CORFU2019 (2020) 178 [arXiv:2006.00768]; M. Hirasawa, K. Anagnostopoulos, T. Azuma, K. Hatakeyama, Y. Ito, J. Nishimura, S. Papadoudis, and A. Tsuchiya, A new phase in the Lorentzian type IIB matrix model and the emergence of continuous space-time, LATTICE2021 (2022) 428 [arXiv:2112.15390]; K. Hatakeyama, K. Anagnostopoulos, T. Azuma, M. Hirasawa, Y. Ito, J. Nishimura, S. Papadoudis, and A. Tsuchiya, Relationship between the Euclidean and Lorentzian versions of the type IIB matrix model, LATTICE2021 (2022) 341 [arXiv:2112.15368]; Complex Langevin studies of the emergent space-time in the type IIB matrix model, arXiv:2201.13200; Y. Asano, J. Nishimura, W. Piensuk, and N. Yamamori, Defining the type IIB matrix model without breaking Lorentz symmetry, arXiv:2404.14045; 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].
  34. T. Hotta, J. Nishimura, and A. Tsuchiya, Dynamical aspects of large N reduced models, Nucl. Phys. B545, 543 (1999); J. Nishimura and G. Vernizzi, Spontaneous breakdown of Lorentz invariance in IIB matrix model, J. High Energy Phys. 04 (2000) 015; J. Ambjorn, K. N. Anagnostopoulos, W. Bietenholz, T. Hotta, and J. Nishimura, Monte Carlo studies of the IIB matrix model at large N, 07 (2000) 011; J. Nishimura and F. Sugino, Dynamical generation of four-dimensional space-time in the IIB matrix model, 05 (2002) 001; T. Aoyama, J. Nishimura, and T. Okubo, Spontaneous breaking of the rotational symmetry in dimensionally reduced super Yang-Mills models, Prog. Theor. Phys. 125, 537 (2011); 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); Y. Ito, J. Nishimura, and A. Tsuchiya, Power-law expansion of the Universe from the bosonic Lorentzian type IIB matrix model, J. High Energy Phys. 11 (2015) 070; T. Aoki, M. Hirasawa, Y. Ito, J. Nishimura, and A. Tsuchiya, On the structure of the emergent 3d expanding space in the Lorentzian type IIB matrix model, Prog. Theor. Exp. Phys. 2019, 093B03 (2019).
  35. M. Li, Strings from IIB matrices, Nucl. Phys. B499, 149 (1997).
  36. I. Chepelev, Y. Makeenko, and K. Zarembo, Properties of D-branes in matrix model of IIB superstring, Phys. Lett. B 400, 43 (1997); A. Fayyazuddin, Y. Makeenko, P. Olesen, D. J. Smith, and K. Zarembo, Towards a nonperturbative formulation of IIB superstrings by matrix models, Nucl. Phys. B499, 159 (1997); A. Fayyazuddin and D. J. Smith, P-brane solutions in IKKT IIB matrix theory, Mod. Phys. Lett. A 12, 1447 (1997).
  37. S. Brahma, R. Brandenberger, and S. Laliberte, BFSS matrix model cosmology: Progress and challenges, arXiv:2210.07288.
  38. J. Frohlich, G. M. Graf, D. Hasler, J. Hoppe, and S. T. Yau, Asymptotic form of zero energy wave functions in supersymmetric matrix models, Nucl. Phys. B567, 231 (2000).
  39. H. Aoki, S. Iso, H. Kawai, Y. Kitazawa, and T. Tada, Space-time structures from IIB matrix model, Prog. Theor. Phys. 99, 713 (1998).
  40. 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).
  41. M. Hirasawa, K. Anagnostopoulos, T. Azuma, K. Hatakeyama, Y. Ito, J. Nishimura, S. Papadoudis, and A. Tsuchiya, A new phase in the Lorentzian type IIB matrix model and the emergence of continuous space-time, Proc. Sci. LATTICE2021 (2022) 428 [arXiv:2112.15390]; K. Hatakeyama, K. Anagnostopoulos, T. Azuma, M. Hirasawa, Y. Ito, J. Nishimura, S. Papadoudis, and A. Tsuchiya, Relationship between the Euclidean and Lorentzian versions of the type IIB matrix model, LATTICE2021 (2022) 341 [arXiv:2112.15368]; Complex Langevin studies of the emergent space-time in the type IIB matrix model, arXiv:2201.13200.
  42. H. Kawai, S. Kawamoto, T. Kuroki, T. Matsuo, and S. Shinohara, Mean field approximation of IIB matrix model and emergence of four-dimensional space-time, Nucl. Phys. B647, 153 (2002); J. Nishimura, T. Okubo, and F. Sugino, Gaussian expansion analysis of a matrix model with the spontaneous breakdown of rotational symmetry, Prog. Theor. Phys. 114, 487 (2005); Convergent Gaussian expansion method: Demonstration in reduced Yang-Mills integrals, J. High Energy Phys. 10 (2002) 043; T. Aoyama, J. Nishimura, and T. Okubo, Spontaneous breaking of the rotational symmetry in dimensionally reduced super Yang-Mills models, Prog. Theor. Phys. 125, 537 (2011); 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.
  43. S. Brahma, R. Brandenberger, and S. Laliberte, Spontaneous symmetry breaking in the BFSS model: Analytical results using the Gaussian expansion method, Eur. Phys. J. C 83, 904 (2023).
  44. A. A. Tseytlin, Interactions between branes and matrix theories, Nucl. Phys. B, Proc. Suppl. 68, 99 (1998).
  45. D. P. Sorokin, Superbranes and superembeddings, Phys. Rep. 329, 1 (2000).
  46. S. Alexander, R. H. Brandenberger, and D. A. Easson, Brane gases in the early Universe, Phys. Rev. D 62, 103509 (2000).
  47. P. Agrawal, S. Gukov, G. Obied, and C. Vafa, Topological gravity as the early phase of our Universe, arXiv:2009.10077.
  48. H. Steinacker, Emergent gravity from noncommutative gauge theory, J. High Energy Phys. 12 (2007) 049; Emergent geometry and gravity from Matrix models: An introduction, Classical Quantum Gravity 27, 133001 (2010).

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Phys. Rev. D 112, 026006– Published 2 July, 2025
  • Received 21 December 2024
  • Accepted 10 June 2025
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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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