Comments on the U(2) Noncommutative Instanton

TL;DR

This paper analyzes the 't Hooft ansatz for noncommutative U(2) instantons, revealing that certain extensions lead to non-self-dual solutions, while proposing a method to obtain self-dual solutions with non-Hermitian gauge fields.

Contribution

It demonstrates that extending the 't Hooft ansatz in noncommutative space results in non-self-dual configurations and proposes a new approach to achieve self-dual solutions with specific topological charge.

Findings

Extension of the ansatz yields non self-dual solutions.

A new method can produce selfdual solutions with topological charge Q=1.

Selfdual solutions involve non-Hermitian gauge fields.

Abstract

We discuss the 't Hoof ansatz for instanton solutions in noncommutative U(2) Yang-Mills theory. We show that the extension of the ansatz leading to singular solutions in the commutative case, yields to non self-dual (or self-antidual) configurations in noncommutative space-time. A proposal leading to selfdual solutions with Q=1 topological charge (the equivalent of the regular BPST ansatz) can be engineered, but in that case the gauge field and the curvature are not Hermitian (although the resulting Lagrangian is real).