Instantons, Monopoles and Toric HyperKaehler Manifolds

TL;DR

This paper explicitly constructs the toric hyperKaehler metric on the moduli space of SU(n) calorons with arbitrary holonomy, connecting monopole phases and Nahm data, and confirms the Nahm construction's isometric property.

Contribution

It provides an explicit metric construction for SU(n) calorons with arbitrary holonomy, confirming the Lee-Yi conjecture and analyzing the role of monopole phases.

Findings

Explicit toric hyperKaehler metric derived for caloron moduli space

Confirmation of the Nahm construction's isometric property

Discussion of massless monopoles effects

Abstract

In this paper, the metric on the moduli space of the k=1 SU(n) periodic instanton -or caloron- with arbitrary gauge holonomy at spatial infinity is explicitly constructed. The metric is toric hyperKaehler and of the form conjectured by Lee and Yi. The torus coordinates describe the residual U(1)^{n-1} gauge invariance and the temporal position of the caloron and can also be viewed as the phases of n monopoles that constitute the caloron. The (1,1,..,1) monopole is obtained as a limit of the caloron. The calculation is performed on the space of Nahm data, which is justified by proving the isometric property of the Nahm construction for the cases considered. An alternative construction using the hyperKaehler quotient is also presented. The effect of massless monopoles is briefly discussed.