Lorentzian Type IIB Matrix Model Research Articles

The emergence of expanding space–time and intersecting D-branes from classical solutions in the Lorentzian type IIB matrix model

Abstract The type IIB matrix model is a promising candidate for a nonperturbative formulation of superstring theory. As such, it is expected to explain the origin of space–time and matter at the same time. This has been partially demonstrated by the previous Monte Carlo studies on the Lorentzian version of the model, which suggested the emergence of (3+1)-dimensional expanding space–time. Here we investigate the same model by solving numerically the classical equation of motion, which is expected to be valid at late times since the action becomes large due to the expansion of space. Many solutions are obtained by the gradient descent method starting from random matrix configurations, assuming a quasi-direct-product structure for the (3+1)-dimensions and the extra 6 dimensions. We find that these solutions generally admit the emergence of expanding space–time and a block-diagonal structure in the extra dimensions, the latter being important for the emergence of intersecting D-branes. For solutions corresponding to D-branes with appropriate dimensionality, the Dirac operator is shown to acquire a zero mode in the limit of infinite matrix size.

On the structure of the emergent 3D expanding space in the Lorentzian type IIB matrix model

Abstract The emergence of (3+1)D expanding space-time in the Lorentzian type IIB matrix model is an intriguing phenomenon that has been observed in Monte Carlo studies of this model. In particular, this may be taken as support for the conjecture that the model is a nonperturbative formulation of superstring theory in (9+1) dimensions. In this paper we investigate the space-time structure of the matrices generated by simulating this model and its simplified versions, and find that the expanding part of the space is described essentially by the Pauli matrices. We argue that this is due to an approximation used in the simulation to avoid the sign problem, which actually amounts to replacing ${e}^{iS_{\rm b}}$ by ${e}^{\beta S_{\rm b}}$ ($\beta>0$) in the partition function, where $S_{\rm b}$ is the bosonic part of the action. We also discuss the possibility of obtaining a regular space-time with the (3+1)D expanding behavior in the original model with the correct ${e}^{iS_{\rm b}}$ factor.

Complex Langevin analysis of the space-time structure in the Lorentzian type IIB matrix model

The Lorentzian type IIB matrix model has been studied as a promising candidate for a nonperturbative formulation of superstring theory. In particular, the emergence of (3+1)D expanding space-time was observed by Monte Carlo studies of this model. It has been found recently, however, that the matrix configurations generated by the simulation is singular in that the submatrices representing the expanding 3D space have only two large eigenvalues associated with the Pauli matrices. This problem has been attributed to the approximation used to avoid the sign problem in simulating the model. Here we investigate the model using the complex Langevin method to overcome the sign problem instead of using the approximation. Our results indicate a clear departure from the Pauli-matrix structure, while the (3+1)D expanding behavior is kept intact.

A new method for probing the late-time dynamics in the Lorentzian type IIB matrix model

The type IIB matrix model has been investigated as a possible nonperturbative formulation of superstring theory. In particular, it was found by Monte Carlo simulation of the Lorentzian version that the 9-dimensional rotational symmetry of the spatial matrices is broken spontaneously to the 3-dimensional one after some "critical time". In this paper we develop a new simulation method based on the effective theory for the submatrices corresponding to the late time. Using this method, one can obtain the results for $N\times N$ matrices by simulating matrices typically of the size $O(\sqrt{N})$. We confirm the validity of this method and demonstrate its usefulness in simplified models.

Universality and the dynamical space-time dimensionality in the Lorentzian type IIB matrix model

The type IIB matrix model is one of the most promising candidates for a nonperturbative formulation of superstring theory. In particular, its Lorentzian version was shown to exhibit an interesting real-time dynamics such as the spontaneous breaking of the 9-dimensional rotational symmetry to the 3-dimensional one. This result, however, was obtained after regularizing the original matrix integration by introducing “infrared” cutoffs on the quadratic moments of the Hermitian matrices. In this paper, we generalize the form of the cutoffs in such a way that it involves an arbitrary power (2p) of the matrices. By performing Monte Carlo simulation of a simplified model, we find that the results become independent of p and hence universal for p ≳ 1.3. For p as large as 2.0, however, we find that large-N scaling behaviors do not show up, and we cannot take a sensible large-N limit. Thus we find that there is a certain range of p in which a universal large-N limit can be taken. Within this range of p, the dynamical space-time dimensionality turns out to be (3 + 1), while for p = 2.0, where we cannot take a sensible large-N limit, we observe a (5+1)d structure.