Scalar Multi-Solitons on the Fuzzy Sphere

TL;DR

This paper investigates scalar multi-solitons on the fuzzy sphere, analyzing their moduli space, interactions, and limits, revealing smooth configurations and connections to noncommutative plane results.

Contribution

It constructs and analyzes multi-soliton configurations on the fuzzy sphere, showing the moduli space structure and its reduction in certain limits, and relates findings to noncommutative plane results.

Findings

Moduli space is the Grassmannian Gr(k,2j+1) and reduces to CP^k in the large j limit.

Solitons attract each other, but attraction vanishes as j becomes large.

The moduli space of multi-solitons is smooth with no singularities when solitons coalesce.

Abstract

We study solitons in scalar theories with polynomial interactions on the fuzzy sphere. Such solitons are described by projection operators of rank k, and hence the moduli space for the solitons is the Grassmannian Gr(k,2j+1). The gradient term of the action provides a non-trivial potential on Gr(k,2j+1), thus reducing the moduli space. We construct configurations corresponding to well-separated solitons, and show that although the solitons attract each other, the attraction vanishes in the limit of large j. In this limit, it is argued that the moduli space is CP^k. For the k-soliton bound state, the moduli space is simply CP^1, all other moduli being lifted. We find that the moduli space of multi-solitons is smooth and that there are no singularities as several solitons coalesce. When the fuzzy S^2 is flattened to a noncommutative plane, we find agreement with the known results, modulo some operator-ordering ambiguities. This suggests that the fuzzy sphere is a natural way to regulate the noncommutative plane both in the ultraviolet and infrared.