Noncommutative Gauge Theory on Fuzzy Four-Sphere and Matrix Model

TL;DR

This paper develops a noncommutative gauge theory on a fuzzy four-sphere using a matrix model with a Chern-Simons term, revealing higher spin fields and a flat space limit with Heisenberg noncommutativity.

Contribution

It introduces a matrix model framework for gauge theory on fuzzy four-sphere with extra degrees of freedom and higher spin fields, extending previous noncommutative geometries.

Findings

Constructed a matrix model with a fifth-rank Chern-Simons term.

Expanded matrices around the fuzzy four-sphere as a classical solution.

Derived a noncommutative gauge theory on a four-dimensional plane with Heisenberg-type noncommutativity.

Abstract

We study a noncommutative gauge theory on a fuzzy four-sphere. The idea is to use a matrix model with a fifth-rank Chern-Simons term and to expand matrices around the fuzzy four-sphere which corresponds to a classical solution of this model. We need extra degrees of freedom since algebra of coordinates does not close on the fuzzy four-sphere. In such a construction, a fuzzy two sphere is added at each point on the fuzzy four-sphere as extra degrees of freedom. It is interesting that fields on the fuzzy four-sphere have higher spins due to the extra degrees of freedom. We also consider a theory around the north pole and take a flat space limit. A noncommutative gauge theory on four-dimensional plane, which has Heisenberg type noncommutativity, is considered.