Singular $G$-monopoles on $S^1\times \Sigma$

TL;DR

This paper generalizes a correspondence between singular $G$-monopoles on $S^1\times \Sigma$ and stable meromorphic pairs on $\Sigma$ from unitary to arbitrary gauge groups, and discusses spectral decomposition.

Contribution

It extends the functorial correspondence to all compact, connected gauge groups, broadening the scope of previous results.

Findings

Generalization of the correspondence theorem to arbitrary gauge groups

Clear distinctions between unitary and non-unitary cases

Discussion of spectral decomposition for $G$-monopoles

Abstract

This article provides an account of the functorial correspondence between irreducible singular $G$-monopoles on $S^1\times \Sigma$ and $\vec{t}$-stable meromorphic pairs on $\Sigma$. The main theorem of [1] is thus generalized here from unitary to arbitrary compact, connected gauge groups. The required distinctions and similarities for unitary versus arbitrary gauge are clearly outlined and many parallels are drawn for easy transition. Once the correspondence theorem is complete, the spectral decomposition is addressed.