A note on the (1, 1,..., 1) monopole metric

TL;DR

The paper studies the hyperkähler metric on the moduli space of maximal symmetry-breaking $(1,\dots,1)$ monopoles for $SU(n+1)$. It constructs the Nahm-data space ${\cal N}$ as a hyperkähler quotient, reduces to a finite-dimensional model ${\cal A}_0/{\cal G}_0$, and computes the metric on the centered moduli ${\cal N}_c$ using Hitchin-type techniques, obtaining the explicit form with $K^{-1}=P^{-1}+X^{-1}$. It then identifies the centered metric with the asymptotic metric proposed by Lee, Weinberg and Yi (via $\mu_{ij}=P^{-1}_{ij}$) and shows that ${\cal N}$ is isometric to the monopole moduli space up to a $\\mathbb{Z}$–quotient, thereby supporting the conjecture that the asymptotic form extends globally. The result provides a rigorous hyperkähler construction linking Nahm data to the full monopole moduli space metric for maximal symmetry breaking, validating the conjectured global exactness in this class. The work advances the understanding of monopole metrics beyond SU(2), leveraging hyperkähler quotients and explicit matrix relations to connect Nahm data with the physical monopole system.

Abstract

Recently K. Lee, E.J. Weinberg and P. Yi in CU-TP-739, hep-th/9602167, calculated the asymptotic metric on the moduli space of (1, 1, ..., 1) BPS monopoles and conjectured that it was globally exact. I lend support to this conjecture by showing that the metric on the corresponding space of Nahm data is the same as the metric they calculate.