Moduli Space of SU(2) Singular Monopole

TL;DR

This paper analyzes the moduli space of $SU(2)$ singular monopoles, constructing its metric via zero modes, and demonstrates it is isometric to the Taub-NUT space, confirming the Nahm data correspondence.

Contribution

It explicitly constructs the monopole moduli space metric using zero modes and confirms its isometry with the Nahm data moduli space for singular monopoles.

Findings

The moduli space is the Taub-NUT space.

Zero modes are constructed using the Nahm transform.

The isometry between monopole and Nahm data moduli spaces is proven.

Abstract

The $SU(2)$ monopole with a Dirac singularity was constructed in Durcan (2007) and Cherkis and Durcan (2007). We study its moduli space by identifying the tangent direction to the moduli space. The tangent vectors to the moduli space are composed of the monopole's phase and translational zero modes. We construct the phase and translational zero modes using the Nahm transform. These zero modes are then used to construct the metric components $g_{00}$ and $g_{0q}$ of the $SU(2)$ singular monopole moduli space, where $0$ denotes the gauge coordinate and q denotes the translational coordinates. We find the moduli space to be the Taub-NUT space. It has been shown by Nakajima (1990) and Maciocia (1991) that there is an isomorphism between the monopole moduli space of regular monopoles and the moduli space of the Nahm data used to construct them. We compute the moduli space of the Nahm data of a singular monopole using the hyperk\"ahler quotient construction as described in Gibbons and Rychenkova (1997). We find it to be isometric to the moduli space of the singular monopole. Finally we make an explicit connection between the moduli space of the Nahm data and the monopole moduli space using Corrigan's inner product formula (Osborn 1981) independently proving this isomorphism.