Spatially periodic instantons: Nahm transform and moduli

TL;DR

This paper proves a bijective Nahm transform between spatially periodic instantons and singular monopoles on dual tori, revealing deep geometric correspondences and establishing existence results.

Contribution

It rigorously establishes the bijection and links the Nahm transform to Fourier-Mukai transforms, advancing understanding of instanton-monopole dualities.

Findings

The Nahm transform is a bijection between instantons and monopoles.

The Nahm transform corresponds to a Fourier-Mukai transform via Kobayashi-Hitchin.

Existence and non-existence results for certain configurations are proved.

Abstract

This paper establishes that the Nahm transform sending spatially periodic instantons (instantons on the product of the real line and a three-torus) to singular monopoles on the dual three-torus is indeed a bijection as suggested by the heuristic. In the process, we show how the Nahm transform intertwines to a Fourier-Mukai transform via Kobayashi-Hitchin correspondences. We also prove existence and non-existence results.