CP^n Model on Fuzzy Sphere

TL;DR

This paper constructs and analyzes the CP^n model on a fuzzy sphere, demonstrating that BPS solitons saturate the Bogomolny bound, with moduli space dimensions matching classical cases and zero modes linked to topological charge.

Contribution

It introduces the CP^n model on fuzzy spheres, constructs BPS solutions, and establishes the relation between zero modes and topological charge, extending classical results to noncommutative geometry.

Findings

BPS solitons saturate the Bogomolny bound on fuzzy sphere.

Moduli space dimension matches classical sphere and plane cases.

Number of Dirac zero modes equals the topological charge.

Abstract

We construct the CP^n model on fuzzy sphere. The Bogomolny bound is saturated by (anti-)self-dual solitons and the general solutions of BPS equation are constructed. The dimension of moduli space describing the BPS solution on fuzzy sphere is exactly the same as that of the commutative sphere or the (noncommutative) plane. We show that in the soliton backgrounds, the number of zero modes of Dirac operator on fuzzy sphere, Atiyah-Singer index, is exactly given by the topological charge of the background solitons.