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

TL;DR

This paper demonstrates that a deformed Lorentzian type IIB matrix model, simulated with complex Langevin methods up to size 128, exhibits emergent (3+1)-dimensional expanding spacetime.

Contribution

It introduces a deformation to the Lorentzian type IIB matrix model and shows that this leads to the emergence of a smooth, real, expanding (3+1)-dimensional spacetime in simulations.

Findings

Emergence of (3+1)-dimensional expanding spacetime in the deformed model.

Successful simulation of the model up to matrix size 128.

Avoidance of the singular drift problem via model deformation.

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.