The Nahm transform for calorons

TL;DR

This paper completes the proof of an equivalence between calorons and solutions to Nahm's equations, using twistors and complex geometry to encode the data, advancing the mathematical understanding of these gauge theory solutions.

Contribution

It finalizes the proof of the equivalence between calorons and Nahm's equations, incorporating complex geometric methods and twistors for a comprehensive encoding.

Findings

Completed the proof of the caloron-Nahm equivalence

Introduced a twistor-based encoding method

Enhanced understanding of boundary conditions for solutions

Abstract

In this paper, we complete the proof of an equivalence given by Nye and Singer of the equivalence between calorons (instantons on $S^1\times R^3$) and solutions to Nahm's equations over the circle, both satisfying appropriate boundary conditions. Many of the key ingredients are provided by a third way of encoding the same data which involves twistors and complex geometry.