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

1. Problem Statement

The Lorentzian type IIB matrix model (IKKT model) is a leading candidate for a non-perturbative formulation of superstring theory, where spacetime, gauge fields, and matter are expected to emerge dynamically from $N \times N$ matrices in the large-$N$ limit. However, simulating the Lorentzian version has historically been hindered by two major obstacles:

1. The Sign Problem: The path integral contains a highly oscillatory phase factor $e^{iS}$, making standard Monte Carlo methods ineffective.
2. Equivalence to the Euclidean Model: Previous studies suggested that without modification, the Lorentzian model is mathematically equivalent to the Euclidean model via contour deformation. In the Euclidean model, the emergent spacetime is complex and lacks a real Lorentzian signature, failing to reproduce a realistic expanding universe.
3. Singular Drift in Complex Langevin: When applying the Complex Langevin Method (CLM) to overcome the sign problem, the integration of fermionic matrices yields a Pfaffian ($\text{Pf} M$). If eigenvalues of the matrix $M$ approach zero, the drift term in the Langevin equation becomes singular, causing the simulation to diverge or yield incorrect results.

2. Methodology

The authors employed the Complex Langevin Method (CLM) to simulate the model with matrix sizes up to $N=128$. To address the specific challenges of the Lorentzian model, they implemented several key technical strategies:

• Lorentz-Invariant Mass Term: To break the equivalence with the Euclidean model and allow for a real emergent spacetime, they added a Lorentz-invariant mass term to the action:
  $$S_\gamma = -\frac{N}{2}\gamma \text{Tr}(A_\mu A^\mu)$$
  With $\gamma > 0$, the equivalence to the Euclidean model is broken, and classical solutions suggest an expanding spacetime.

• Deformation to Avoid Singular Drift: To prevent the singular drift problem caused by near-zero eigenvalues of the fermion matrix $M$, they introduced two deformations:

  1. Fermionic Mass Term ($S_{mf}$): A mass term for fermions was added to shift eigenvalues of $M$ away from the origin.
  2. SUSY-Inspired Anisotropic Deformation: Inspired by the "polarized" type IIB matrix model, they modified the bosonic mass term to suppress fluctuations in specific spatial directions:
     $$S_\gamma = \frac{1}{2N\gamma} \left( \text{Tr}(A_0)^2 - \sum_{i=1}^{\tilde{d}} \text{Tr}(A_i)^2 - \xi \sum_{j=\tilde{d}+1}^{9} \text{Tr}(A_j)^2 \right)$$
     Here, $\xi \geq 1$ suppresses fluctuations in $(9-\tilde{d})$ directions, breaking $SO(9,1)$ to $SO(\tilde{d}, 1)$.

• Lorentz Boost Removal: Since the model retains Lorentz symmetry, simulation configurations undergo random Lorentz boosts, obscuring the physical time evolution. The authors developed a procedure to iteratively apply Lorentz transformations to minimize $\text{Tr}(A_0^\dagger A_0)$, effectively removing the "center-of-mass motion" artifact to extract the intrinsic spacetime structure.

• Observables: They analyzed:

  • Eigenvalues of the temporal matrix $A_0$ to determine the reality of time.
  • The "moment of inertia tensor" $T_{ij}(t)$ to detect spontaneous symmetry breaking (SSB) of spatial rotational symmetry.
  • The eigenvalues of $Q(t) = \sum (\bar{A}_i(t))^2$ to check for smoothness (absence of singular structures like Pauli matrices).

3. Key Contributions

• Overcoming the Sign Problem: Successfully applied CLM to the Lorentzian type IIB matrix model with fermions up to $N=128$, a scale previously unattainable due to the sign problem.
• Resolution of Singular Drift: Demonstrated that a specific deformation (fermionic mass + anisotropic bosonic mass) stabilizes the CLM simulation by avoiding the singular drift problem while preserving the physical mechanism for dimensional reduction.
• Artifact Removal: Established a robust method to remove Lorentz boost artifacts from CLM configurations, allowing for the correct interpretation of time evolution and spatial expansion.

4. Key Results

• Bosonic Model (No Fermions): Even with the mass term, the bosonic model exhibited expansion but no spontaneous symmetry breaking (SSB) of the $SO(9)$ rotational symmetry. All spatial directions expanded similarly.
• Fermionic Model with Deformation:
  • Emergence of Real Time: The eigenvalues of $A_0$ aligned with the real axis at late times, confirming the emergence of real time (unlike the complex time in the Euclidean case).
  • Spontaneous Symmetry Breaking: For specific parameters ($N=128, \gamma=4, \tilde{d}=5, \xi=12, m_f=6$), the system exhibited SSB from $SO(\tilde{d})$ to $SO(3)$. Three spatial directions began to expand significantly, while the other six remained small.
  • Smooth and Real Spacetime: The spatial matrices exhibited a band-diagonal structure, and the eigenvalues of the spatial extent operator were densely distributed. This indicates the emergence of a smooth, non-singular (3+1)-dimensional expanding spacetime, contrasting with previous findings of singular structures.
  • Phase Transition: The system showed a phase boundary; as the fermion mass $m_f$ was increased adiabatically, the simulation became unstable around $m_f=8$, suggesting a transition between the expanding phase and a non-expanding phase.

5. Significance

This work provides the first non-perturbative numerical evidence that the Lorentzian type IIB matrix model can dynamically generate a realistic (3+1)-dimensional expanding universe with smooth space and time.

• Cosmological Implications: It supports the scenario where our universe emerges naturally from string theory without requiring ad-hoc compactification parameters. The extra dimensions remain small due to the model's dynamics.
• Methodological Breakthrough: It validates the Complex Langevin Method as a viable tool for Lorentzian quantum field theories and matrix models, provided that singular drifts are managed via appropriate deformations.
• Future Directions: The authors note that while their results are strong, the Lorentz symmetry was not fixed non-perturbatively (using Faddeev-Popov) during the simulation. Future work aims to fix this symmetry and explore the Lefschetz thimble method to ensure the sampled configurations correspond to the dominant saddle points.

In summary, the paper demonstrates that with specific deformations to stabilize the simulation, the Lorentzian type IIB matrix model successfully reproduces the dynamical emergence of a smooth, expanding (3+1)-dimensional universe, offering a promising pathway for understanding the origin of spacetime in string theory.