Instantons on Gravitons

TL;DR

This paper develops a new framework for constructing and analyzing Yang-Mills instantons on ALF gravitational instantons using matrix-valued functions organized into a bow, extending previous work on ALE spaces and exploring the moduli space geometry.

Contribution

It introduces the concept of a bow and its representation, providing a new method to construct instantons on ALF spaces and studying their moduli spaces.

Findings

The Nahm transform maps bow solutions to instantons on ALF spaces.

The map preserves complex structures, suggesting it is an isometry.

Analysis of the moduli space asymptotics for instantons on ALF spaces.

Abstract

Yang-Mills instantons on ALE gravitational instantons were constructed by Kronheimer and Nakajima in terms of matrices satisfying algebraic equations. These were conveniently organized into a quiver. We construct generic Yang-Mills instantons on ALF gravitational instantons. Our data are formulated in terms of matrix-valued functions of a single variable, that are in turn organized into a bow. We introduce the general notion of a bow, its representation, its associated data and moduli space of solutions. For a judiciously chosen bow the Nahm transform maps any bow solution to an instanton on an ALF space. We demonstrate that this map respects all complex structures on the moduli spaces, so it is likely to be an isometry, and use this fact to study the asymptotics of the moduli spaces of instantons on ALF spaces.