O(3) Sigma model with Hopf term on Fuzzy Sphere

TL;DR

This paper formulates the O(3) sigma model with a Hopf term on a fuzzy sphere, classifies topological sectors, constructs BPS solitons, and demonstrates the Hopf term's value relates to the topological charge squared.

Contribution

It introduces a novel formulation of the O(3) sigma model with Hopf term on fuzzy spheres, including topological classification and soliton construction.

Findings

Topological charge Q ranges from -2j to 2j.

Hopf term value equals Q^2, matching the commutative case.

Field expansion uses Holstein-Primakoff SU(2) basis.

Abstract

We formulate the $O(3) \s-$ model on fuzzy sphere and construct the Hopf term. We show that the field can be expanded in terms of the ladder operators of Holstein-Primakoff realisation of SU(2) algebra and the corresponding basis set can be classified into different topological sectors by the magnetic quantum numbers. We obtain topological charge $Q$ and show that $-2j\le Q \le2j$. We also construct BPS solitons. Using the covariantly conserved current, we construct the Hopf term and show that its value is $Q^2$ as in the commutative case. We also point out the interesting relation of physical space to deformed SU(2) algebra.