Scattering of instantons, monopoles and vortices in higher dimensions

TL;DR

This paper investigates the time evolution of instantons, monopoles, and vortices in higher-dimensional Yang-Mills theories, demonstrating that in the adiabatic limit their dynamics follow geodesic motion in their respective moduli spaces.

Contribution

It extends the understanding of soliton dynamics in higher dimensions by showing that their slow evolution approximates geodesic motion in moduli spaces, including instantons, monopoles, and vortices.

Findings

Instanton dynamics reduce to geodesic motion in moduli space in the adiabatic limit.

Similar geodesic motion results are briefly discussed for monopoles and vortices.

The study provides a unified framework for understanding soliton evolution in higher-dimensional gauge theories.

Abstract

We consider Yang-Mills theory on manifolds ${\mathbb R}\times X$ with a $d$-dimensional Riemannian manifold $X$ of special holonomy admitting gauge instanton equations. Instantons are considered as particle-like solutions in $d+1$ dimensions whose static configurations are concentrated on $X$. We study how they evolve in time when considered as solutions of the Yang-Millsequations on ${\mathbb R}\times X$ with moduli depending on time $t\in{\mathbb R}$. It is shown that in the adiabatic limit, when the metric in the $X$ direction is scaled down, the classical dynamics of slowly moving instantons corresponds to a geodesic motion in the moduli space $\cal M$ of gauge instantons on $X$. Similar results about geodesic motion in the moduli space of monopoles and vortices in higher dimensions are briefly discussed.