  The IKKT matrix model has been conjectured to provide a promising
nonperturbative formulation of superstring theory. In this model, spacetime
emerges dynamically from the microscopic matrix degrees of freedom in the
large-N limit, and Monte Carlo simulations of the Lorentzian version provide
evidence of an emergent (3+1)-dimensional expanding space-time. In this talk,
we discuss the Euclidean version of the IKKT matrix model and provide evidence
of dynamical compactification of the extra dimensions via the spontaneous
symmetry breaking (SSB) of the 10D rotational symmetry. We perform numerical
simulations of a system with a severe complex action problem by using the
complex Langevin method (CLM). The CLM suffers from the singular-drift problem
and we deform the model in order to avoid it. We study the SSB pattern as we
vary the deformation parameter and we conclude that the original model has an
SO(3) symmetric vacuum, in agreement with previous calculations using the
Gaussian expansion method (GEM). We employ the GEM to the deformed model and we
obtain results consistent with the ones obtained by CLM.
