Perturbative dynamics of fuzzy spheres at large N

TL;DR

This paper investigates the perturbative expansion around fuzzy-sphere solutions in matrix models, revealing conditions under which one-loop calculations suffice and demonstrating the finite convergence radius of the expansion.

Contribution

It clarifies the role of one-loop dominance in fuzzy-sphere matrix models and introduces a method to compute observables to all orders via a shifted expansion point.

Findings

Effective action saturation at one-loop in large-N limit.

One-loop dominance applies to effective action but not all observables.

Perturbative expansion has a finite radius of convergence.

Abstract

We clarify some peculiar aspects of the perturbative expansion around a classical fuzzy-sphere solution in matrix models with a cubic term. While the effective action in the large-N limit is saturated at the one-loop level, we find that the ``one-loop dominance'' does not hold for generic observables due to one-particle reducible diagrams. However, we may exploit the one-loop dominance for the effective action and obtain various observables to all orders from one-loop calculation by simply shifting the center of expansion to the ``quantum solution'', which extremizes the effective action. We confirm the validity of this method by comparison with the direct two-loop calculation and with Monte Carlo results in the 3d Yang-Mills-Chern-Simons matrix model. From the all order result we find that the perturbative expansion has a finite radius of convergence.