Instantons with continuous conformal symmetries: Hyperbolic and singular monopoles and more, oh my!

TL;DR

This paper develops a linear constraint approach to classify and construct a wide variety of instantons with continuous conformal symmetries, revealing new solutions and connections to monopoles and gauge-theoretic objects.

Contribution

It introduces a linear constraint framework that simplifies the study of symmetric instantons, enabling the discovery of new solutions and a classification scheme.

Findings

Found a large class of novel instantons with continuous conformal symmetries.

Proved the uniqueness of the basic instanton under certain symmetries.

Established connections between instantons, hyperbolic monopoles, and Higgs bundles.

Abstract

Throughout this paper, we comprehensively study instantons with every kind of continuous conformal symmetry. Examples of these objects are hard to come by due to non-linear constraints. However, by applying previous work on moduli spaces, we introduce a linear constraint, whose solution greatly simplifies these non-linear constraints. This simplification not only allows us to easily find a plethora of novel instantons with various continuous conformal symmetries and higher rank structure groups, it also provides a framework for classifying such symmetric objects. We also prove that the basic instanton is essentially the only instanton with two particular kinds of conformal symmetry. Additionally, we discuss the connections between instantons with continuous symmetries and other gauge-theoretic objects: hyperbolic and singular monopoles as well as hyperbolic analogues to Higgs bundles and Nahm data.