Noncommutative 't Hooft Instantons

TL;DR

This paper constructs noncommutative 't Hooft multi-instantons on R^4 using twistor methods, addressing self-duality issues with a Murray-von Neumann transformation, and provides explicit solutions in different gauges.

Contribution

It introduces a novel approach to noncommutative 't Hooft instantons via twistor methods and resolves self-duality problems with a specific transformation.

Findings

Explicit noncommutative 't Hooft multi-instanton solutions in singular and regular gauges.

Identification of self-duality failure in naive solutions and its correction.

Development of a twistor-based construction for noncommutative instantons.

Abstract

We employ the twistor approach to the construction of U(2) multi-instantons `a la 't Hooft on noncommutative R^4. The noncommutative deformation of the Corrigan-Fairlie-'t Hooft-Wilczek ansatz is derived. However, naively substituting into it the 't Hooft-type solution is unsatisfactory because the resulting gauge field fails to be self-dual on a finite-dimensional subspace of the Fock space. We repair this deficiency by a suitable Murray-von Neumann transformation after a specific projection of the gauge potential. The proper noncommutative 't Hooft multi-instanton field strength is given explicitly, in a singular as well as in a regular gauge.