ADHM Construction of Noncommutative Instantons

TL;DR

This paper explores the ADHM construction of noncommutative U(N) instantons, establishing a correspondence with moduli spaces and deriving the construction via Nahm transformation, advancing understanding of instanton solutions in noncommutative geometry.

Contribution

It proves the one-to-one correspondence between noncommutative instanton moduli spaces and ADHM data, including the U(1) case, and derives the ADHM construction from Nahm transformation.

Findings

Established correspondence between moduli spaces and ADHM data.

Derived ADHM construction from Nahm transformation.

Clarified instanton number origin for U(1).

Abstract

We discuss the Atiyah-Drinfeld-Hitchin-Manin (ADHM) construction of U(N) instantons in noncommutative (NC) space and prove the one-to-one correspondence between moduli spaces of the noncommutative instantons and the ADHM data, together with an origin of the instanton number for U(1). We also give a derivation of the ADHM construction from the viewpoint of the Nahm transformation of instantons on four-tori. This article is a composite version of [23] and [24].