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

TL;DR

This paper investigates the nonperturbative behavior of fuzzy spheres in a matrix model with a Chern-Simons term using Monte Carlo simulations, revealing phase transitions and the emergence of fuzzy spheres as stable configurations.

Contribution

It provides the first nonperturbative analysis of fuzzy spheres in this matrix model, showing phase transitions and the dynamics of multiple fuzzy spheres.

Findings

First order phase transition as Chern-Simons coefficient varies

Emergence of a single fuzzy sphere in the large alpha phase

Multiple fuzzy spheres observed as metastable states

Abstract

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.