Topological Charge of U(1) Instantons

TL;DR

This paper demonstrates the existence of non-singular U(1) instantons on noncommutative R^4, linking their topological charge to projection operators and winding numbers, highlighting the role of space noncommutativity.

Contribution

It reveals that noncommutative geometry allows for non-singular U(1) instantons and clarifies their topological charge via projection operators and winding numbers.

Findings

Non-singular U(1) instantons exist on noncommutative R^4.

The instanton number corresponds to the winding number and projection dimension.

Noncommutativity enables instanton solutions absent in commutative space.

Abstract

Non-singular instantons are shown to exist on noncommutative R^4 even with a U(1) gauge group. Their existence is primarily due to the noncommutativity of the space. The relation between U(1) instantons on noncommutative R^4 and the projection operators acting on the representation space of the noncommutative coordinates is reviewed. The integer number of instantons on the noncommutative R^4 can be understood as the winding number of the U(1) gauge field as well as the dimension of the projection on the representation space.