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arxiv: 2604.19836 · v1 · submitted 2026-04-21 · ✦ hep-th · gr-qc· hep-lat

Recognition: unknown

The emergence of (3+1)-dimensional expanding spacetime from complex Langevin simulations of the Lorentzian type IIB matrix model with deformations

Konstantinos N. Anagnostopoulos , Takehiro Azuma , Mitsuaki Hirasawa , Jun Nishimura , Stratos Papadoudis , Asato Tsuchiya
Authors on Pith no claims yet

Pith reviewed 2026-05-10 02:36 UTC · model grok-4.3

classification ✦ hep-th gr-qchep-lat
keywords Lorentzian type IIB matrix modelcomplex Langevin methodemergent spacetimesuperstring theorynumerical simulationsdimensional reductionexpanding universe
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0 comments X

The pith

Simulations of the deformed Lorentzian type IIB matrix model produce a phase with emergent (3+1)-dimensional expanding spacetime that is smooth and real.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The Lorentzian type IIB matrix model is studied numerically as a nonperturbative formulation of superstring theory, with the eigenvalue distributions of its bosonic matrices expected to generate spacetime dynamically in the large-N limit. Complex Langevin simulations are performed up to matrix size 128 after introducing a supersymmetry-inspired deformation to avoid the singular drift problem caused by the Pfaffian. The deformed model is found to enter a phase in which (3+1)-dimensional expanding spacetime appears, with both the spatial and temporal directions remaining smooth and real. A sympathetic reader would care because this supplies a concrete dynamical mechanism by which the model could select the observed dimensionality and signature from higher-dimensional matrix degrees of freedom.

Core claim

The authors perform numerical simulations of the deformed Lorentzian type IIB matrix model using the complex Langevin method up to matrix size N=128. They find that the deformed model exhibits a phase in which (3+1)-dimensional expanding spacetime emerges with both space and time being smooth and real.

What carries the argument

The eigenvalue distribution of the N by N bosonic matrices A_mu, which encodes the emergent spacetime geometry in the large-N limit and is evolved under the complex Langevin dynamics with the supersymmetry-inspired deformation.

If this is right

  • The model's large-N dynamics can select three spatial dimensions plus time without external tuning.
  • The expansion arises internally from the matrix evolution and therefore carries built-in cosmological behavior.
  • The smoothness of the emergent geometry implies that classical spacetime can appear without disruptive quantum fluctuations at the simulated scales.
  • The reality of both space and time supports emergence of a Lorentzian metric directly from the model.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the deformation can be removed in future work while preserving the phase, the result would apply more directly to the original undeformed Lorentzian model.
  • Larger-N runs could reveal whether the expansion rate or dimensionality changes with system size, offering a testable signature.
  • The approach might be combined with other matrix-model techniques to extract additional observables such as correlation functions of the emergent geometry.

Load-bearing premise

The deformation introduced to avoid the singular drift problem does not distort the physical content of the original Lorentzian model in a way that artificially produces the observed (3+1)-dimensional phase.

What would settle it

A simulation of the undeformed model at comparable or larger N that shows no (3+1)-dimensional expanding phase, or that shows the phase only for deformation parameters that significantly alter the original action, would indicate the emergence is an artifact of the deformation.

Figures

Figures reproduced from arXiv: 2604.19836 by Asato Tsuchiya, Jun Nishimura, Konstantinos N. Anagnostopoulos, Mitsuaki Hirasawa, Stratos Papadoudis, Takehiro Azuma.

Figure 1
Figure 1. Figure 1: The expectation values of 1 N Tr(A0) 2 (Left) and 1 N Tr(Ai) 2 (Right) are plotted in the complex plane for the Euclidean (E) and Lorentzian (L) models. The relative complex phase for these quantities are − 3π 4 and π 4 , respectively, which agree with Eq. (2.11) [70, 71]. Setting S˜ γ = −iSγ so that it appears in the partition function as e −S˜γ , we obtain S˜ γ = N 2 γ {e − i 4 π(2+3u)Tr(A˜ 0) 2 + e i 4 … view at source ↗
Figure 2
Figure 2. Figure 2: The eigenvalues of the matrix M obtained for N = 16, γ = 6, ˜d = 6, ξ = 18 and mf = 2, where the parameters ˜d and ξ are introduced in Eq.(3.12). in (2.5). Since the drift term includes the following contribution10 ∂ ∂(Ai)ba {− log PfM} = − 1 2 Tr  ∂M ∂(Ai)ba M−1  , (3.10) near-zero eigenvalues of M can lead to the singular drift problem. Here we avoid this problem by adding a fermionic mass term [9] Smf… view at source ↗
Figure 3
Figure 3. Figure 3: (Left) The expectation values ⟨αa⟩ of the eigenvalues of the temporal matrix A0 are plotted in the complex plane for the bosonic model with N = 96, n = 6 and γ = 4. The solid line represents the behavior (2.19) expected for the original Lorentzian model (γ = 0). (Right) The real parts of ⟨λi(t)⟩, the expectation values of the eigenvalues of Tij (t), are plotted against time for the same model. as the simul… view at source ↗
Figure 4
Figure 4. Figure 4: (Left) ⟨Xi(t)⟩ defined by Eq. (4.1) is plotted against time for the bosonic model with N = 96, n = 6 and γ = 4 before the Lorentz transformations. Different colors correspond to different indices i. (Right) ⟨Xi(t)⟩ is plotted against time for the same model after the Lorentz transformations. -6 -4 -2 0 2 4 6 -2.5-2.0-1.5-1.0-0.50.0 0.5 1.0 1.5 2.0 2.5 Im 〈α〉 Re 〈α〉 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 0.1… view at source ↗
Figure 5
Figure 5. Figure 5: The same as Figure [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: (Left) The expectation values ⟨αa⟩ of the eigenvalues of the temporal matrix A0 are plotted in the complex plane for the bosonic model with N = 128, n = 12, γ = 4, ˜d = 3 and ξ = 10. The solid line represents the behavior (2.19) expected for the original Lorentzian model (γ = 0). (Right) The real parts of ⟨λi(t)⟩, the expectation values of the eigenvalues of Tij (t), are plotted against time for the same m… view at source ↗
Figure 7
Figure 7. Figure 7: (Left) The expectation values ⟨αa⟩ of the eigenvalues of the temporal matrix A0 are plotted in the complex plane for the bosonic model (Top), the model with fermionic contributions at mf = 10 (Middle) and at mf = 6 (Bottom), with N = 128, n = 6, γ = 4, ˜d = 5 and ξ = 12. The solid line represents the behavior (2.19) expected for the original Lorentzian model (γ = 0). (Right) The real parts of ⟨λi(t)⟩, the … view at source ↗
Figure 8
Figure 8. Figure 8: (Left) The phase of space θs(t) is plotted for the bosonic model (Top) and for the model with fermionic contributions at mf = 10 (Middle) and at mf = 6 (Bottom), with N = 128, n = 6, γ = 4, ˜d = 5 and ξ = 12. The horizontal line shows the value π 8 ∼ 0.39 for the original Lorentzian model with γ = 0, which is equivalent to the Euclidean model. (Right) The real parts of ⟨qp(t)⟩, the expectation values of th… view at source ↗
Figure 9
Figure 9. Figure 9: The quantity |Aab|, defined by Eq. (2.16), is plotted for the bosonic model (Top￾Left) and for the model with fermionic contributions at mf = 10 (Top-Right) and at mf = 6 (Bottom), with N = 128, n = 6, γ = 4, ˜d = 5 and ξ = 12. exhibit rapid growth. This result provides an evidence for a phase in which an expanding (3+1)-dimensional spacetime emerges at late times. As we increase mf adiabatically from this… view at source ↗
read the original abstract

The Lorentzian type IIB matrix model is a promising candidate for a nonperturbative formulation of superstring theory. In this model, the eigenvalue distribution of the $N\times N$ bosonic matrices $A_\mu$ $(\mu = 0 , \ldots , 9)$ represents an emergent spacetime, which is determined by the dynamics of the model in the large-$N$ limit. Here we perform numerical simulations of the model overcoming the sign problem by the complex Langevin method with the matrix size $N$ up to $128$. In order to avoid the singular drift problem due to the Pfaffian, which appears after integrating out the fermionic matrices, we deform the model in a manner inspired by the supersymmetric deformation, which is used to define the ``polarized type IIB matrix model'' in the Euclidean case. We find that the deformed model exhibits a phase in which (3+1)-dimensional expanding spacetime emerges with both space and time being smooth and real.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript reports complex Langevin simulations of the Lorentzian type IIB matrix model with a deformation introduced to avoid the singular drift problem from the Pfaffian. With matrix sizes up to N=128, the authors claim to identify a phase in which (3+1)-dimensional expanding spacetime emerges, characterized by smooth and real eigenvalue distributions for both spatial and temporal directions.

Significance. If the deformation can be shown to become irrelevant and the numerical evidence is robust, the result would supply non-perturbative support for emergent realistic spacetime in the Lorentzian IIB matrix model, strengthening its status as a candidate for a non-perturbative formulation of superstring theory. The approach of combining complex Langevin dynamics with a supersymmetry-inspired deformation addresses a longstanding technical obstacle in the field.

major comments (2)
  1. [deformation and results sections] The central claim that the observed (3+1)D expanding phase belongs to the original Lorentzian model (rather than being induced by the regulator) requires an explicit scaling study in the deformation parameter. No such limit is reported, leaving open the possibility that the deformation biases the reality and smoothness of the eigenvalues or the effective dimensionality.
  2. [numerical results] The extraction of dimensionality from the eigenvalue distributions of the bosonic matrices A_mu, together with convergence diagnostics and statistical error bars for N up to 128, is not described. Without these, the reliability of the reported phase cannot be assessed.
minor comments (1)
  1. [model definition] The precise definition of the deformation term and its relation to the Euclidean polarized model should be given explicitly (e.g., as an additional term in the action) to allow direct comparison.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below.

read point-by-point responses

  1. Referee: [deformation and results sections] The central claim that the observed (3+1)D expanding phase belongs to the original Lorentzian model (rather than being induced by the regulator) requires an explicit scaling study in the deformation parameter. No such limit is reported, leaving open the possibility that the deformation biases the reality and smoothness of the eigenvalues or the effective dimensionality.

    Authors: We agree that an explicit scaling study with respect to the deformation parameter is essential to demonstrate that the emergent (3+1)D phase is not an artifact of the regulator. Our simulations employ a small fixed deformation parameter, chosen on the basis of supersymmetry-inspired regularization to suppress the singular drift while preserving the large-N dynamics of the Lorentzian model. In the revised manuscript we will add results for at least one smaller deformation value, together with a discussion of the observed stability of the phase, to indicate the approach to the undeformed limit. revision: partial

  2. Referee: [numerical results] The extraction of dimensionality from the eigenvalue distributions of the bosonic matrices A_mu, together with convergence diagnostics and statistical error bars for N up to 128, is not described. Without these, the reliability of the reported phase cannot be assessed.

    Authors: We will revise the manuscript to include a dedicated subsection that details the extraction of effective dimensionality from the eigenvalue distributions of the A_μ matrices, the precise criteria used to identify the three extended spatial directions and the compact time direction, the convergence diagnostics applied to the complex Langevin trajectories, and the statistical error estimation (via jackknife or bootstrap resampling) for the N=128 data. revision: yes

Circularity Check

0 steps flagged

No circularity: results are direct numerical observations from Monte Carlo sampling

full rationale

The paper reports an observed phase in complex Langevin simulations of a deformed Lorentzian type IIB matrix model, where (3+1)-dimensional expanding spacetime emerges for N up to 128. This is a direct output of the dynamics under the chosen regularization, not an analytic derivation that reduces by construction to fitted inputs, self-definitions, or self-citations. The deformation is explicitly introduced as a technical fix for the Pfaffian-induced singular drift (inspired by Euclidean polarized models), and the central claim is the numerical finding itself rather than a prediction forced by parameter fitting or renaming. No load-bearing uniqueness theorems or ansatze from prior self-work are invoked to derive the spacetime emergence; the result stands as an independent simulation outcome. This matches the low circularity expected for non-analytic Monte Carlo studies.

Axiom & Free-Parameter Ledger

1 free parameters · 0 axioms · 0 invented entities

The central claim rests on the assumption that the complex Langevin algorithm with the chosen deformation correctly samples the target distribution and that the eigenvalue distribution of the bosonic matrices can be interpreted as emergent spacetime geometry.

free parameters (1)
  • deformation parameter
    A tunable deformation strength is introduced to suppress singular drift from the Pfaffian; its specific value is chosen to enable stable simulations.

pith-pipeline@v0.9.0 · 5514 in / 1181 out tokens · 34102 ms · 2026-05-10T02:36:23.257765+00:00 · methodology

discussion (0)

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Forward citations

Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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  2. Path integral for the closed superstring and the matrix model

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  3. Quantum spacetime and quantum fluctuations in the IKKT model at weak coupling

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  4. Impact of supersymmetry on the dynamical emergence of the spacetime in the type IIB matrix model with the Lorentz symmetry "gauge fixed"

    hep-lat 2026-04 unverdicted novelty 4.0

    Numerical investigation of supersymmetry's role in spacetime emergence within the Lorentz gauge-fixed type IIB matrix model using Complex Langevin dynamics.

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