Local Nahm transform and singularity formation of ASD connections

TL;DR

This paper introduces a local version of the ADHM construction for ASD instantons on bounded domains in Euclidean 4-space and analyzes how these instantons develop singularities.

Contribution

It develops a local analogue of the ADHM construction for ASD instantons and describes their degeneration into singularities.

Findings

Constructs a local ADHM-like framework for ASD instantons.

Describes the degeneration process of instantons into curvature singularities.

Provides a mathematical characterization of singularity formation in ASD connections.

Abstract

This paper develops a local analogue of the ADHM construction, which characterises ASD instantons defined over smooth bounded domains inside Euclidean $\mathbb{R}^4$ diffeomorphic to the 4-ball, in terms of infinite dimensional Hilbert spaces and bounded Hermitian linear operators satisfying an analogue of the ADHM equation. Morever, we describe the degeneration of this construction when a family of instantons develops a curvature singularity at the origin.