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

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.