Dynamical aspects of the fuzzy CP$^{2}$ in the large $N$ reduced model with a cubic term

TL;DR

This paper investigates the stability and phase transitions of fuzzy CP^2 and S^2 manifolds in a reduced 8d Yang-Mills model with a cubic term, combining Monte Carlo simulations and analytical calculations.

Contribution

It provides a nonperturbative analysis of fuzzy CP^2 stability, phase transitions, and gauge group dynamics in a large-N reduced model with a cubic term.

Findings

Fuzzy CP^2 remains stable at large N due to tunneling suppression.

Both fuzzy CP^2 and S^2 undergo first-order collapse transitions as the cubic term coefficient decreases.

Analytical all-order calculations agree well with Monte Carlo extrapolations at large N.

Abstract

``Fuzzy CP^2'', which is a four-dimensional fuzzy manifold extension of the well-known fuzzy 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 ($\alpha$) is sufficiently large. As \alpha 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 (\alpha = 0). The corresponding transitions are of first order and the critical points can be understood analytically. 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=\infty, we find excellent agreement with the all order results.