ALE spaces from noncommutative U(1) instantons via exact Seiberg-Witten map

TL;DR

This paper explores how noncommutative U(1) instantons relate to ALE spaces through the exact Seiberg-Witten map, revealing a connection to Eguchi-Hanson metrics and a non-quantized instanton number.

Contribution

It provides an explicit solution for U(1) instantons under the exact SW map and links these solutions to gravitational instantons like Eguchi-Hanson.

Findings

The effective metric from a single U(1) instanton is related to Eguchi-Hanson space.

The instanton number depends on an integration constant, not quantized.

The results confirm a non-perturbative breakdown of the SW map.

Abstract

The exact Seiberg-Witten (SW) map of a noncommutative (NC) gauge theory gives the commutative equivalent as an ordinary gauge theory coupled to a field dependent effective metric. We study instanton solutions of this commutative equivalent whose self-duality equation turns out to be the exact SW map of NC instantons. We derive general differential equations governing U(1) instantons and we explicitly get an exact solution corresponding to the single NC instanton. Remarkably the effective metric induced by the single U(1) instanton is related to the Eguchi-Hanson metric - the simplest gravitational instanton. Surprisingly the instanton number is not quantized but depends on an integration constant. Our result confirms the expected non-perturbative breakdown of the SW map. However, the breakdown of the map arises in a consistent way: The instanton number plays the role of a parameter giving rise to a one-parameter family of Eguchi-Hanson metrics.