Singular Hermitian-Einstein monopoles on the product of a circle and a Riemann surface

TL;DR

This paper establishes a correspondence between the moduli space of singular Hermitian-Einstein monopoles on a circle times a Riemann surface and a moduli space of stable pairs on the surface, analyzing their complex geometry.

Contribution

It introduces a new correspondence linking monopoles with stable pairs on Riemann surfaces, and explores their complex geometric properties.

Findings

Moduli space of monopoles corresponds to stable pairs on the surface.

Computed dimensions of the moduli space from geometric and gauge perspectives.

Analyzed the complex geometry and singularity structure of the moduli space.

Abstract

In this paper, the moduli space of singular unitary Hermitian--Einstein monopoles on the product of a circle and a Riemann surface is shown to correspond to a moduli space of stable pairs on the Riemann surface. These pairs consist of a holomorphic vector bundle on the surface and a meromorphic automorphism of the bundle. The singularities of this automorphism correspond to the singularities of the singular monopole. We then consider the complex geometry of the moduli space; in particular, we compute dimensions, both from the complex geometric and the gauge theoretic point of view.