Cautionary remarks on the moduli space metric for multi-dyon simulations

TL;DR

This paper investigates the validity of a proposed moduli space metric for dyon models, finding it only positive definite at low densities and small numbers of dyons, highlighting limitations for unrestricted integrations.

Contribution

It provides a detailed numerical analysis showing the conditions under which the moduli space metric remains valid, emphasizing the importance of correlations in dyon models.

Findings

Metric is positive definite only at low dyon densities

Unrestricted integration over collective coordinates is limited

Correlations between dyons are essential for accurate modeling

Abstract

We perform a detailed numerical investigation of the approximate moduli space metric proposed by Diakonov and Petrov [arXiv:0704.3181] for a confining model of dyons. Our findings strongly indicate that only for a small number of dyons at sufficiently low density this metric is positive definite - and, therefore, a valid moduli space metric - throughout a considerable part of configuration space. This poses strong limitations on results obtained by an unrestricted integration over collective coordinates in this model. It also indicates that strong correlations between collective coordinates will be essential for the physical content of a dyon model, which could be exhibited by a suitable simulation algorithm.