Atiyah-Manton Construction of Skyrmions in Eight Dimensions

TL;DR

This paper extends the Atiyah-Manton construction to eight dimensions, demonstrating that eight-dimensional instantons can generate Skyrmions with high accuracy, suggesting a universal instanton/Skyrmion correspondence in 4k dimensions.

Contribution

It generalizes the Atiyah-Manton construction to eight dimensions, showing the instanton/Skyrmion correspondence holds beyond four dimensions.

Findings

High-accuracy agreement between instanton-derived and energy-extremized Skyrmions.

The construction indicates a universal property in 4k-dimensional spaces.

Provides a higher-dimensional analogue of the four-dimensional Skyrmion construction.

Abstract

We show that the eight-dimensional instanton solution, which satisfies the self-duality equation $F \wedge F = *_8 F \wedge F$, realizes the static Skyrmion configuration in eight dimensions through the Atiyah-Manton construction. The relevant energy functional of the Skyrme field is obtained by the formalism developed by Sutcliffe. By comparing the Skyrmion olution associated with the extreme of the energy, with the Atiyah-Manton solution constructed by the instantons, we find that they agree with high accuracy. This is a higher-dimensional analogue of the Atiyah-Manton construction of Skyrmions in four dimensions. Our result indicates that the instanton/Skyrmion correspondence seems to be an universal property in $4k \ (k=1, 2, \ldots)$ dimensions.