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The type IIB matrix model is a promising candidate for a nonperturbativeformulation of superstring theory. As such, it is expected to explain theorigin of space--time and matter at the same time. This has been partiallydemonstrated by the previous Monte Carlo studies on the Lorentzian version ofthe model, which suggested the emergence of (3+1)-dimensional expandingspace--time. Here we investigate the same model by solving numerically theclassical equation of motion, which is expected to be valid at late times sincethe action becomes large due to the expansion of space. Many solutions areobtained by the gradient descent method starting from random matrixconfigurations, assuming a quasi-direct-product structure for the(3+1)-dimensions and the extra 6 dimensions. We find that these solutionsgenerally admit the emergence of expanding space--time and a block-diagonalstructure in the extra dimensions, the latter being important for the emergenceof intersecting D-branes. For solutions corresponding to D-branes withappropriate dimensionality, the Dirac operator is shown to acquire a zero modein the limit of infinite matrix size.

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