Nonperturbative studies of fuzzy spheres in a matrix model with the Chern-Simons term

Fuzzy spheres appear as classical solutions in a matrix model obtained via dimensional reduction of 3-dimensional Yang-Mills theory with the Chern-Simons term. Well-defined perturbative expansion around these solutions can be formulated even for finite matrix size, and in the case of k coincident fuzzy spheres it gives rise to a regularized U(k) gauge theory on a noncommutative geometry. Here we study the matrix model nonperturbatively by Monte Carlo simulation. The system undergoes a first order phase transition as we change the coefficient (alpha) of the Chern-Simons term. In the small alpha phase, the large-N properties of the system are qualitatively the same as in the pure Yang-Mills model (alpha=0), whereas in the large alpha phase a single fuzzy sphere emerges dynamically. Various 'multi fuzzy spheres' are observed as meta-stable states, and we argue in particular that the k coincident fuzzy spheres cannot be realized as the true vacuum in this model even in the large-N limit. We also perform one-loop calculations of various observables for arbitrary k including k=1. Comparison with our Monte Carlo data suggests that higher order corrections are suppressed in the large-N limit.