Notes on Noncommutative Instantons

TL;DR

This paper analyzes the ADHM construction of noncommutative U(N) instantons, highlighting potential issues with the completeness condition, and provides explicit new solutions ensuring regularity and integer topological charge.

Contribution

It identifies conditions where the ADHM construction fails and introduces explicit regular instanton solutions on noncommutative spaces, ensuring correct topological charge.

Findings

Completeness condition can be invalidated in certain ADHM constructions.

Existing solutions may violate the completeness condition and be incorrect.

Explicit regular U(N) instanton solutions are constructed on noncommutative spaces.

Abstract

We study in detail the ADHM construction of U(N) instantons on noncommutative Euclidean space-time R_{NC}^4 and noncommutative space R_{NC}^2 x R^2. We point out that the completeness condition in the ADHM construction could be invalidated in certain circumstances. When this happens, regular instanton configuration may not exist even if the ADHM constraints are satisfied. Some of the existing solutions in the literature indeed violate the completeness condition and hence are not correct. We present alternative solutions for these cases. In particular, we show for the first time how to construct explicitly regular U(N) instanton solutions on R_{NC}^4 and on R_{NC}^2 x R^2. We also give a simple general argument based on the Corrigan's identity that the topological charge of noncommutative regular instantons is always an integer.