Nonabelian gauge field and dual description of fuzzy sphere

TL;DR

This paper explores the duality between fuzzy spheres as noncommutative geometries in matrix models and their description via higher-dimensional D-branes with nonabelian gauge fields, revealing their equivalence in certain limits.

Contribution

It demonstrates that fuzzy $2k$-spheres and spherical D$2k$-branes with nonabelian gauge fields are equivalent objects in the large $n$ limit, linking noncommutative geometry to gauge theory.

Findings

Fuzzy spheres can be described by nonabelian gauge fields on D-branes.

The equivalence holds when the matrix size $n$ is large.

A connection to quantum Hall systems is discussed.

Abstract

In matrix models, higher dimensional D-branes are obtained by imposing a noncommutative relation to coordinates of lower dimensional D-branes. On the other hand, a dual description of this noncommutative space is provided by higher dimensional D-branes with gauge fields. Fuzzy spheres can appear as a configuration of lower dimensional D-branes in a constant R-R field strength background. In this paper, we consider a dual description of higher dimensional fuzzy spheres by introducing nonabelian gauge fields on higher dimensional spherical D-branes. By using the Born-Infeld action, we show that a fuzzy $2k$-sphere and spherical D$2k$-branes with a nonabelian gauge field whose Chern character is nontrivial are the same objects when $n$ is large. We discuss a relationship between the noncommutative geometry and nonabelian gauge fields. Nonabelian gauge fields are represented by noncommutative matrices including the coordinate dependence. A similarity to the quantum Hall system is also studied.