Noncommutative deformation of the Ward metric

TL;DR

This paper investigates the moduli-space metric of a noncommutative CP^1 sigma model, providing explicit formulas and asymptotics, and explores different soliton regimes using perturbation and zeta-function methods.

Contribution

It offers the first detailed analysis of the noncommutative deformation of the Ward metric, including explicit expressions and asymptotic behaviors in various soliton regions.

Findings

Explicit expressions for the noncommutative Kähler potential are derived.

Asymptotic behaviors of the metric are analyzed in different moduli space regions.

The strong noncommutativity limit is well-understood, but the commutative limit remains challenging.

Abstract

The moduli-space metric in the static non-Abelian charge-two sector of the Moyal-deformed CP^1 sigma model in 1+2 dimensions is analyzed. After recalling the commutative results of Ward and Ruback and the zeta-regularized construction of the noncommutative K"ahler potential due to the second author, explicit expressions and asymptotics for it are presented and discussed in different regions of the moduli space. Along two curves in the moduli space the potential can be calculated analytically. In the region of solitons known as "ring-like", perturbation theory is used. In the region of "lump-like" solitons, both perturbation theory and the zeta-function approach are employed. While the strong noncommutativity limit is smooth and under control, the commutative limit in the two-lump region remains a semiclassical challenge.