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 (α) of the Chern-Simons term. In the small α phase, the large N properties of the system are qualitatively the same as in the pure Yang-Mills model (α = 0), whereas in the large α 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.

It has been known that the dynamics of k coincident D-branes in string theory is described effectively by U(k) Yang-Mills theory at low energy. While these configurations appear as classical solutions in matrix models, it was not clear whether it is possible to realize the k = 1 case as the true vacuum. The massive Yang-Mills-Chern-Simons matrix model has classical solutions corresponding to all the representations of the SU(2) algebra, and provides an opportunity to address the above issue on a firm ground. We investigate the phase structure of the model, and find in particular that there exists a parameter region where O(N ) copies of the spin-1/2 representation appear as the true vacuum, thus realizing a nontrivial gauge group dynamically. Such configurations are analogous to the ones that are interpreted in the BMN matrix model as coinciding transverse 5-branes in M-theory.

We study a matrix model with a cubic term, which incorporates both the fuzzy S 2 × S 2 and the fuzzy S 2 as classical solutions. Both of the solutions decay into the vacuum of the pure Yang-Mills model (even in the large-N limit) when the coefficient of the cubic term is smaller than a critical value, but the large-N behavior of the critical point is different for the two solutions. The results above the critical point are nicely reproduced by the all order calculations in perturbation theory. By comparing the free energy, we find that the true vacuum is given either by the fuzzy S 2 or by the "pure Yang-Mills vacuum" depending on the coupling constant. In Monte Carlo simulation we do observe a decay of the fuzzy S 2 × S 2 into the fuzzy S 2 at moderate N , but the decay probability seems to be suppressed at large N . The above results, together with our previous results for the fuzzy CP 2 , reveal certain universality in the large-N dynamics of four-dimensional fuzzy manifolds realized in a matrix model with a cubic term.

Fuzzy CP 2 ", which is a four-dimensional fuzzy manifold analogous to the fuzzy 2-sphere (S 2 ), appears as a classical solution in the dimensionally reduced 8d Yang-Mills model with a cubic term involving the structure constant of the SU(3) Lie algebra. Although the fuzzy S 2 , which is also a classical solution of the same model, has actually smaller free energy than the fuzzy CP 2 , Monte Carlo simulation shows that the fuzzy CP 2 is stable even nonperturbatively due to the suppression of tunneling effects at large N as far as the coefficient of the cubic term (α) is sufficiently large. As α is decreased, both the fuzzy CP 2 and the fuzzy S 2 collapse to a solid ball and the system is essentially described by the pure Yang-Mills model (α = 0). The corresponding transitions are of first order. The gauge group generated dynamically above the critical point turns out to be of rank one for both CP 2 and S 2 cases. Above the critical point, we also perform perturbative calculations for various quantities to all orders, taking advantage of the one-loop saturation of the effective action in the large-N limit. By extrapolating our Monte Carlo results to N = ∞, we find excellent agreement with the all order results.