A $U(3)$ Gauge Theory on Fuzzy Extra Dimensions

TL;DR

This paper investigates a $U(3)$ gauge theory on fuzzy spheres, deriving low-energy effective models that resemble abelian Higgs theories with vortex solutions, and explores generalizations to higher $U(n)$ groups.

Contribution

It provides a detailed equivariant parametrization of gauge fields on fuzzy spheres and derives the resulting low-energy abelian Higgs-like models with vortex solutions.

Findings

Low-energy effective action is abelian Higgs type with $U(1) imes U(1)$ symmetry.

Vortex solutions are found and analyzed on ${\mathbb R}^2$.

Framework is extended to $U(n)$ gauge theories.

Abstract

In this article, we explore the low energy structure of a $U(3)$ gauge theory over spaces with fuzzy sphere(s) as extra dimensions. In particular, we determine the equivariant parametrization of the gauge fields, which transform either invariantly or as vectors under the combined action of $SU(2)$ rotations of the fuzzy spheres and those $U(3)$ gauge transformations generated by $SU(2) \subset U(3)$ carrying the spin $1$ irreducible representation of $SU(2)$. The cases of a single fuzzy sphere $S_F^2$ and a particular direct sum of concentric fuzzy spheres, $S_F^{2 \, Int}$, covering the monopole bundle sectors with windings $\pm 1$ are treated in full and the low energy degrees of freedom for the gauge fields are obtained. Employing the parametrizations of the fields in the former case, we determine a low energy action by tracing over the fuzzy sphere and show that the emerging model is abelian Higgs type with $U(1) \times U(1)$ gauge symmetry and possess vortex solutions on ${\mathbb R}^2$, which we discuss in some detail. Generalization of our formulation to the equivariant parametrization of gauge fields in $U(n)$ theories is also briefly addressed.