Higher Order Terms in the Improved Mean Field Approximation for the IIB Matrix Model and the Emergence of Four-Dimensional Space-Time

The spontaneous breakdown of SO(10) symmetry in the IIB matrix model has been studied using the improved mean field approximation (IMFA). In this report, the eighth-order contribution to the improved perturbative series is obtained, which involves the evaluation of 20410 planar two-particle irreducible vacuum diagrams. We consider SO(d)-preserving configurations as an ansatz (d = 4,7). The development of a plateau, representing the condition of self-consistency, is seen in both ansatz. A large ratio of the space-time extent of the d-dimensional part to that of the remaining (10 − d)-dimensional part is obtained for the SO(4) ansatz evaluated at representative points of the plateau. This is interpreted as the emergence of four-dimensional space-time in the IIB matrix model.