Noncommutative Gauge Theory on Fuzzy Sphere from Matrix Model

TL;DR

The paper builds noncommutative gauge theories on the fuzzy sphere from three-dimensional matrix models by expanding around a fuzzy-sphere classical background with a Chern-Simons term and, in one model, a Majorana mass. It establishes mappings from matrices to functions via noncommutative spherical harmonics, yielding $U(1)$ and $U(n)$ theories with a CS-induced gauge mass and two key large-$N$ limits: a commutative sphere and a decompactified NC plane with the Moyal product. Dirac operator structure and one-loop stability analyses show that the fuzzy-sphere background can be perturbatively stable for certain regimes, though SUSY properties differ between the two models (one preserves full SUSY in the commuting sector but not on the fuzzy sphere, the other yields a 1/2-BPS fuzzy sphere with enhanced SUSY). A second SUSY-preserving matrix model with modified transformations illustrates a robust ${\cal N}=1$/${\cal N}=2$ NC gauge theory on the fuzzy sphere, with the fuzzy background acting as a BPS state and suppressing perturbative corrections. The work highlights star- and Berezin-product mappings as a path to extending NC gauge theories to more general curved backgrounds.

Abstract

We derive a noncommutative U(1) and U(n) gauge theory on the fuzzy sphere from a three dimensional matrix model by expanding the model around a classical solution of the fuzzy sphere. Chern-Simons term is added in the matrix model to make the fuzzy sphere as a classical solution of the model. Majorana mass term is also added to make it supersymmetric. We consider two large $N$ limits, one corresponding to a gauge theory on a commutative sphere and the other to that on a noncommutative plane. We also investigate stability of the fuzzy sphere by calculating one-loop effective action around classical solutions. In the final part of this paper, we consider another matrix model which gives a supersymmetric gauge theory on the fuzzy sphere. In this matrix model, only Chern-Simons term is added and supersymmetry transformation is modified.