The Lorentzian version of the type IIB (IKKT) matrix model — a conjectured nonperturbative definition of superstring theory in (9+1) dimensions — was shown by Monte Carlo simulation to generate time and space dynamically, with three of nine spatial directions beginning to expand after a critical time. The open problem left by that program is whether the full quantum ensemble of the model is dominated, at late times, by a classical solution representing smooth commutative (3+1)-dimensional expanding spacetime, and whether the conventional universe — commutative spacetime, its local effective field theory and particle content, and the physics of the rapid expansion (inflation, dark energy) — can be derived from the matrix dynamics rather than only suggested by it. Later work sharpened the problem substantially: the phase-quenched approximation (replacing the complex integration weight by its modulus) used in the original simulations was shown to distort the emergent space, the naive large-N limit of the original model was found to yield Euclidean rather than Lorentzian signature, and properly regularized or Lorentz-gauge-fixed definitions with the sign problem treated now show a smooth (3+1)-dimensional expanding phase at finite N — but the identification of the dominant saddle, the quantitative commutative late-time limit, the derivation of the Standard Model, and a first-principles account of inflation and dark energy all remain open, as stated explicitly in the most recent (2026) literature.

Background

The type IIB matrix model (also called the IKKT model, after Ishibashi, Kawai, Kitazawa and Tsuchiya, 1996) is a conjectured nonperturbative definition of type IIB superstring theory: a zero-dimensional supersymmetric integral over ten N×NN\times N traceless Hermitian matrices AμA_\mu (μ=0,…,9\mu=0,\ldots,9) and sixteen fermionic matrices, with action given by the dimensional reduction of ten-dimensional super-Yang-Mills theory. The eigenvalues of the bosonic matrices are interpreted as the coordinates of an emergent spacetime; because the matrices generally do not commute, the emergent space is generically noncommutative (coordinates fail to commute as operators), and ordinary 'commutative' space corresponds to configurations in which the relevant matrices can be simultaneously diagonalized. Nishimura and collaborators studied a Lorentzian version of this model, in which the action is real-time rather than Wick-rotated; since the Lorentzian action is unbounded, the model must be regularized, and the original work introduced infrared cutoffs on the quadratic moments of the matrices (the partition function is otherwise divergent, partly because the Lorentz symmetry group is noncompact). Monte Carlo simulations of this regularized model (Kim, Nishimura and Tsuchiya, Phys. Rev. Lett. 108 (2012) 011601) showed that time as well as space emerges dynamically, and that after a 'critical time' the SO(9)SO(9) rotational symmetry of the nine spatial directions is spontaneously broken (a symmetry of the action broken by the typical configurations in the ensemble) down to SO(3)SO(3): exactly three spatial directions expand rapidly, suggesting a dynamical origin for the observed (3+1)-dimensional spacetime and possibly for inflation. Integrating out the fermions produces a Pfaffian (the antisymmetric-matrix analogue of a determinant) in the partition function; in the Lorentzian model the Pfaffian is real but can flip sign, while the bosonic factor eiSbe^{iS_b} carries a genuinely complex phase: together these constitute the sign problem, and the original simulations replaced the complex weight by its modulus, an approximation called phase quenching. On the classical side, the equations of motion are expected to become reliable at late times because the action grows as space expands; infinitely many solutions representing commutative (3+1)-dimensional spacetime are known to exist, one simple expanding solution was found to address the cosmological constant problem (the extreme smallness of the vacuum energy) at the classical level (Kim, Nishimura and Tsuchiya, JHEP 2012), and a procedure to extract the local effective field theory of the massless modes around such solutions yields a grand unified theory with gauge group SU(k)SU(k) and adjoint matter (Nishimura and Tsuchiya, PTEP 2013). What is not known is whether any such solution — or any smooth (3+1)-dimensional expanding configuration — actually dominates the partition function of the full quantum model at late times, whether matrix noncommutativity provably dies out so that a continuum local field theory applies, and how realistic particle physics, inflation, and dark energy follow from the matrix dynamics. Subsequent developments sharpened the problem: phase quenching was shown to artifactually singularize the expanding space; the conventional large-N limit of the original model was found to produce Euclidean rather than Lorentzian signature, motivating Lorentz-invariant mass-term regularizations and a nonperturbative 'gauge fixing' of the Lorentz symmetry; and complex Langevin simulations (a stochastic method that complexifies the fields so that complex weights can be sampled) of deformed versions of the model recently found a phase with smooth, real (3+1)-dimensional expanding spacetime at matrix sizes up to N=128N=128.

Problem Statement

The problem is to derive the late-time universe from the Lorentzian type IIB matrix model. Its common scientific objective is to determine the dominant late-time phase of the model — in the formulation of the 2013 source article, whether a unique classical solution representing smooth, commutative, expanding (3+1)-dimensional spacetime dominates the partition function at late times — and to establish, from the matrix dynamics, that this phase is our universe. Concretely, the central open question is the selection problem: identify which phase or classical solution of the properly defined model dominates the quantum ensemble at late times, with the Lorentz-symmetry divergence regularized in a controlled way and the phase of the complex weight (sign problem) treated without phase quenching — as the 2026 literature restates it, whether the most dominant saddle point (a stationary point of the action that dominates the path integral in a semiclassical evaluation) is the (3+1)-dimensional expanding spacetime. Directly tied to it is the commutative limit: show quantitatively that noncommutativity of the three expanding spatial directions becomes negligible at late times, so that a continuum local field theory applies — whether the usual commutative spacetime appears at later times being, in the source's words, 'an important issue'. Beyond these, the source lists as interesting open directions along the same line the derivation of the local effective field theory around the dominant solution and its particle content — whether the Standard Model, or a unique low-energy theory containing it, emerges from perturbations around the dominant solution — and a first-principles microscopic account of the observed rapid expansion (its possible identification with inflation) and of the origin and dynamics of dark energy, from the matrix dynamics rather than by qualitative analogy; these related directions deepen a resolution but are not, in the source, additional jointly mandatory conditions for answering the dominance question. The large-NN and regulator-removal limits must be under control for the claimed results, since the original finite-NN, phase-quenched results are known to be method-sensitive. Numerical (Monte Carlo, complex Langevin, Lefschetz-thimble — a method that decomposes the complex integral into contributions associated with saddle points), analytical (classical solutions, semiclassical saddle comparison, effective-field-theory derivation), or hybrid approaches are all admissible.

References

  1. Kim, Sang-Woo; Nishimura, Jun; Tsuchiya, Asato: 'Expanding (3+1)-dimensional universe from a Lorentzian matrix model for superstring theory in (9+1)-dimensions', Phys. Rev. Lett. 108 (2012) 011601, DOI: 10.1103/PhysRevLett.108.011601, arXiv:1108.1540, https://arxiv.org/abs/1108.1540
  2. Nishimura, Jun: '(3+1)-dimensional expanding universe from a Lorentzian matrix model for superstring theory in (9+1)-dimensions', Int. J. Mod. Phys. A 28 (2013) 1340002, DOI: 10.1142/S0217751X13400022, arXiv:1209.6386, https://arxiv.org/abs/1209.6386
  3. Kim, Sang-Woo; Nishimura, Jun; Tsuchiya, Asato: 'Late time behaviors of the expanding universe in the IIB matrix model', JHEP 2012 (10) (2012) 147, DOI: 10.1007/JHEP10(2012)147, arXiv:1208.0711, https://arxiv.org/abs/1208.0711
  4. Nishimura, Jun; Tsuchiya, Asato: 'Local field theory from the expanding universe at late times in the IIB matrix model', Prog. Theor. Exp. Phys. 2013 (2013) 043B03, DOI: 10.1093/ptep/ptt015, arXiv:1208.4910, https://arxiv.org/abs/1208.4910
  5. Ito, Yuta; Kim, Sang-Woo; Nishimura, Jun; Tsuchiya, Asato: 'Monte Carlo studies on the expanding behavior of the early universe in the Lorentzian type IIB matrix model', PoS LATTICE2013 (2014) 341, DOI: 10.22323/1.187.0341, arXiv:1311.5579, https://arxiv.org/abs/1311.5579
  6. Nishimura, Jun; Tsuchiya, Asato: 'Realizing chiral fermions in the type IIB matrix model at finite N', JHEP 2013 (12) (2013) 002, DOI: 10.1007/JHEP12(2013)002, arXiv:1305.5547, https://arxiv.org/abs/1305.5547
  7. Aoki, Hajime; Nishimura, Jun; Tsuchiya, Asato: 'Realizing three generations of the Standard Model fermions in the type IIB matrix model', JHEP 2014 (5) (2014) 131, DOI: 10.1007/JHEP05(2014)131, arXiv:1401.7848, https://arxiv.org/abs/1401.7848
  8. Ito, Yuta; Nishimura, Jun; Tsuchiya, Asato: 'Universality and the dynamical space-time dimensionality in the Lorentzian type IIB matrix model', JHEP 2017 (3) (2017) 143, DOI: 10.1007/JHEP03(2017)143, arXiv:1701.07783, https://arxiv.org/abs/1701.07783
  9. Aoki, Toshihiro; Hirasawa, Mitsuaki; Ito, Yuta; Nishimura, Jun; Tsuchiya, Asato: 'On the structure of the emergent 3d expanding space in the Lorentzian type IIB matrix model', Prog. Theor. Exp. Phys. 2019 (2019) 093B03, DOI: 10.1093/ptep/ptz092, arXiv:1904.05914, https://arxiv.org/abs/1904.05914
  10. Nishimura, Jun; Tsuchiya, Asato: 'Complex Langevin analysis of the space-time structure in the Lorentzian type IIB matrix model', JHEP 2019 (6) (2019) 077, DOI: 10.1007/JHEP06(2019)077, arXiv:1904.05919, https://arxiv.org/abs/1904.05919
  11. Hatakeyama, Kohta; Matsumoto, Akira; Nishimura, Jun; Tsuchiya, Asato; Yosprakob, Atis: '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 (2020) 043B10, DOI: 10.1093/ptep/ptaa042, arXiv:1911.08132, https://arxiv.org/abs/1911.08132
  12. Hirasawa, Mitsuaki; Anagnostopoulos, Konstantinos N.; Azuma, Takehiro; Hatakeyama, Kohta; Nishimura, Jun; Papadoudis, Stratos K.; Tsuchiya, Asato: 'The emergence of expanding space-time in a novel large-N limit of the Lorentzian type IIB matrix model', PoS LATTICE2022 (2023) 371, DOI: 10.22323/1.430.0371, arXiv:2212.10127, https://arxiv.org/abs/2212.10127
  13. Hirasawa, Mitsuaki; Anagnostopoulos, Konstantinos N.; Azuma, Takehiro; Hatakeyama, Kohta; Nishimura, Jun; Papadoudis, Stratos K.; Tsuchiya, Asato: 'The emergence of expanding space-time in the Lorentzian type IIB matrix model with a novel regularization', PoS CORFU2022 (2023) 309, DOI: 10.22323/1.436.0309, arXiv:2307.01681, https://arxiv.org/abs/2307.01681
  14. Asano, Yuhma; Nishimura, Jun; Piensuk, Worapat; Yamamori, Naoyuki: 'Defining the Type IIB Matrix Model without Breaking Lorentz Symmetry', Phys. Rev. Lett. 134 (2025) 041603, DOI: 10.1103/PhysRevLett.134.041603, arXiv:2404.14045, https://arxiv.org/abs/2404.14045
  15. Hirasawa, Mitsuaki; Anagnostopoulos, Konstantinos N.; Azuma, Takehiro; Hatakeyama, Kohta; Nishimura, Jun; Papadoudis, Stratos; Tsuchiya, Asato: 'The effects of SUSY on the emergent spacetime in the Lorentzian type IIB matrix model', PoS CORFU2023 (2024) 257, DOI: 10.22323/1.463.0257, arXiv:2407.03491, https://arxiv.org/abs/2407.03491
  16. Chou, Chien-Yu; Nishimura, Jun; Tripathi, Ashutosh: 'Inequivalence between the Euclidean and Lorentzian Versions of the Type IIB Matrix Model from Lefschetz Thimble Calculations', Phys. Rev. Lett. 134 (2025) 211601, DOI: 10.1103/PhysRevLett.134.211601, arXiv:2501.17798, https://arxiv.org/abs/2501.17798
  17. Anagnostopoulos, Konstantinos N.; Azuma, Takehiro; Hirasawa, Mitsuaki; Nishimura, Jun; Papadoudis, Stratos; Tsuchiya, Asato: 'The emergence of (3+1)-dimensional expanding spacetime from complex Langevin simulations of the Lorentzian type IIB matrix model with deformations', arXiv:2604.19836 (2026), https://arxiv.org/abs/2604.19836
  18. Anagnostopoulos, Konstantinos N.; Azuma, Takehiro; Hirasawa, Mitsuaki; Nishimura, Jun; Tsuchiya, Asato; Yamamori, Naoyuki: 'Impact of supersymmetry on the dynamical emergence of the spacetime in the type IIB matrix model with the Lorentz symmetry "gauge fixed"', arXiv:2604.25564 (2026), https://arxiv.org/abs/2604.25564