$G_2$-monopoles with singularities (examples)

TL;DR

This paper explores examples of $G_2$-monopoles with various singularities on noncompact and compact manifolds, providing existence results and insights into their singularity types for gauge theory applications.

Contribution

It presents explicit examples of non-Abelian $G_2$-monopoles with different singularities and establishes an existence result for Abelian monopoles with Dirac type singularities on compact manifolds.

Findings

Examples of non-Abelian monopoles with Dirac singularities

Examples of monopoles with non-Dirac singularities

Existence of Abelian monopoles with Dirac singularities on compact manifolds

Abstract

$G_2$-monopoles are solutions to gauge theoretical equations on $G_2$-manifolds. If the $G_2$-manifolds under consideration are compact, then any irreducible $G_2$-monopole must have singularities. It is then important to understand which kind of singularities $G_2$-monopoles can have. We give examples (in the noncompact case) of non-Abelian monopoles with Dirac type singularities, and examples of monopoles whose singularities are not of that type. We also give an existence result for Abelian monopoles with Dirac type singularities on compact manifolds. This should be one of the building blocks in a gluing construction aimed at constructing non-Abelian ones.