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 viadimensional reduction of 3-dimensional Yang-Mills theory with the Chern-Simonsterm. Well-defined perturbative expansion around these solutions can beformulated even for finite matrix size, and in the case of $k$ coincident fuzzyspheres it gives rise to a regularized U($k$) gauge theory on a noncommutativegeometry. Here we study the matrix model nonperturbatively by Monte Carlosimulation. The system undergoes a first order phase transition as we changethe coefficient ($\alpha$) of the Chern-Simons term. In the small $\alpha$phase, the large $N$ properties of the system are qualitatively the same as inthe pure Yang-Mills model ($\alpha =0$), whereas in the large $\alpha$ phase asingle fuzzy sphere emerges dynamically. Various `multi fuzzy spheres' areobserved as meta-stable states, and we argue in particular that the $k$coincident fuzzy spheres cannot be realized as the true vacuum in this modeleven in the large $N$ limit. We also perform one-loop calculations of variousobservables for arbitrary $k$ including $k=1$. Comparison with our Monte Carlodata suggests that higher order corrections are suppressed in the large $N$limit.Comment: Latex 37 pages, 13 figures, discussion on instabilities refined, references added, typo corrected, the final version to appear in JHE