Skyrme fields, multi-instantons and the $SU(\infty)$-Toda equation

TL;DR

This paper constructs Skyrme fields from multi-Taub-NUT instantons' spin connections, analyzes their topological degrees, and links them to solutions of the $SU( olinebreak} ext{(} olinebreak ext{infinity} olinebreak ext{)}$-Toda equation, revealing gauge-dependent degrees and geometric structures.

Contribution

It introduces a method to construct and analyze Skyrme fields from multi-instantons, including degree calculations and connections to the $SU( olinebreak ext{(} olinebreak ext{infinity} olinebreak ext{)}$-Toda equation, with gauge considerations.

Findings

Different gauges yield Skyrme fields with varying topological degrees.

Provided an analytical method to compute degrees of Taub-NUT and Atiyah-Hitchin Skyrme fields.

Derived the induced Einstein-Weyl metric and its relation to the $SU( olinebreak ext{(} olinebreak ext{infinity} olinebreak ext{)}$-Toda equation.

Abstract

We construct Skyrme fields from holonomy of the spin connection of multi-Taub-NUT instantons with the centres positioned along a line in $\mathbb{R}^3.$ Our family of Skyrme fields includes the Taub-NUT Skyrme field previously constructed by Dunajski. However, we demonstrate that different gauges of the spin connection can result in Skyrme fields with different topological degrees. As a by-product, we present a method to compute the degrees of the Taub-NUT and Atiyah-Hitchin Skyrme fields analytically; these degrees are well defined as a preferred gauge is fixed by the $SU(2)$ symmetry of the two metrics. Regardless of the gauge, the domain of our Skyrme fields is the space of orbits of the axial symmetry of the multi-Taub-NUT instantons. We obtain an expression for the induced Einstein-Weyl metric on the space and its associated solution to the $SU(\infty)$-Toda equation.