Adiabatic dynamics of instantons on $S ^4 $

TL;DR

This paper computes the $L^2$ metric on the moduli space of circle-invariant 1-instantons on $S^4$, analyzes geodesic behavior, and explores connections to hyperbolic monopoles and Euclidean instantons.

Contribution

It defines and analyzes the $L^2$ metric on a specific moduli space of instantons, revealing geodesic incompleteness and linking to hyperbolic monopoles and Euclidean instantons.

Findings

The moduli space is four-dimensional with $SO(3) imes U(1)$ symmetry.

The metric is geodesically incomplete.

Connections to hyperbolic monopoles and Euclidean instantons are established.

Abstract

We define and compute the $L^2$ metric on the framed moduli space of circle invariant 1-instantons on the 4-sphere. This moduli space is four dimensional and our metric is $SO(3) \times U(1)$ symmetric. We study the behaviour of generic geodesics and show that the metric is geodesically incomplete. Circle-invariant instantons on the 4-sphere can also be viewed as hyperbolic monopoles, and we interpret our results from this viewpoint. We relate our results to work by Habermann on unframed instantons on the 4-sphere and, in the limit where the radius of the 4-sphere tends to infinity, to results on instantons on Euclidean 4-space.