Noncommutative instantons: a new approach

TL;DR

This paper presents a new, more rigorous and transparent approach to studying instantons on noncommutative four-dimensional Euclidean space, simplifying the derivation of ADHM equations.

Contribution

It introduces a novel method for analyzing noncommutative instantons that simplifies existing techniques and provides clearer derivations of key equations.

Findings

Simplified derivation of ADHM equations for noncommutative instantons

A more rigorous and transparent framework for noncommutative instanton analysis

Establishment of a connection between algebraic structures and geometric instanton properties

Abstract

We discuss instantons on noncommutative four-dimensional Euclidean space. In commutative case one can consider instantons directly on Euclidean space, then we should restrict ourselves to the gauge fields that are gauge equivalent to the trivial field at infinity. However, technically it is more convenient to work on four-dimensional sphere. We will show that the situation in noncommutative case is quite similar. One can analyze instantons taking as a starting point the algebra of smooth functions vanishing at infinity, but it is convenient to add a unit element to this algebra (this corresponds to a transition to a sphere at the level of topology). Our approach is more rigorous than previous considerations ; it seems that it is also simpler and more transparent. In particular, we obtain the ADHM equations in a very simple way.