The Plancherel theorem for Fourier--Laplace--Nahm transform for connections on the projective line

TL;DR

This paper proves that the Fourier--Laplace--Nahm transform acts as a hyper-Kähler isometry on connections over the projective line, establishing a deep geometric symmetry.

Contribution

It demonstrates that the Fourier--Laplace--Nahm transform preserves hyper-Kähler structures for connections on the projective line, a novel geometric result.

Findings

The transform is a hyper-Kähler isometry.

Preservation of geometric structures under the transform.

New insights into the geometry of connections on the projective line.

Abstract

We prove that the Fourier--Laplace--Nahm transform for connections on the projective line is a hyper-K\"ahler isometry.