Metric in the moduli of SU(2) monopoles from spectral curves and Gauss-Manin connection in disguise

TL;DR

This paper demonstrates how to recover the metric of the moduli space of SU(2) 2-monopoles from spectral curves using the Gauss-Manin connection, advancing the inverse problem in monopole theory.

Contribution

It introduces a method to derive the metric of monopole moduli spaces from spectral curves via the Gauss-Manin connection, addressing a longstanding open problem.

Findings

Recovered the metric of $M^0_2$ from spectral curves.

Established a method for the inverse problem in monopole moduli spaces.

Provided insights towards generalizing to $M^0_k$.

Abstract

We show here that from the metric of the manifold $M^0_2$ , i.e., the reduced moduli of SU(2) 2-monopoles in Yang-Mills-Higgs theory, one can recover the respective moduli of spectral curves using the method Gauss-Manin connection in disguise. This work is a step towards creating a inverse process of finding the metric of any $M^0_k$ , from spectral curves. This is a thirty years old problem that we hope to shed some light in it.