The ADHM Construction of Instantons on Noncommutative Spaces

TL;DR

This paper extends the classical ADHM construction of instantons to noncommutative spaces using deformation techniques, providing a new framework for understanding instantons in noncommutative geometry.

Contribution

It introduces a deformation of the classical ADHM construction to noncommutative spaces, illustrating the method on Moyal-Groenewold and Connes-Landi planes.

Findings

Constructs families of instantons on noncommutative spaces

Deforms classical ADHM equations via cocycle twisting

Provides explicit examples on specific noncommutative planes

Abstract

We present an account of the ADHM construction of instantons on Euclidean space-time $\mathbb{R}^4$ from the point of view of noncommutative geometry. We recall the main ingredients of the classical construction in a coordinate algebra format, which we then deform using a cocycle twisting procedure to obtain a method for constructing families of instantons on noncommutative space-time, parameterised by solutions to an appropriate set of ADHM equations. We illustrate the noncommutative construction in two special cases: the Moyal-Groenewold plane $\mathbb{R}^4_\hbar$ and the Connes-Landi plane $\mathbb{R}^4_\theta$.