Skyrme fields, multi-instantons and the -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 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 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 -Toda equation.