Gravitational Instantons from Gauge Theory

TL;DR

This paper demonstrates that noncommutative U(1) instantons correspond to ALE gravitational instantons via the Seiberg-Witten map, linking NC gauge theories to hyper-Kähler geometries and gravitational instantons.

Contribution

It establishes a direct correspondence between NC U(1) instantons and ALE gravitational instantons through the exact Seiberg-Witten map, revealing a novel geometric interpretation.

Findings

NC U(1) instantons map to ALE gravitational instantons

NC gauge theory of U(1) instantons is equivalent to hyper-Kähler geometries

ALE gravitational instantons can arise from NC photon condensates

Abstract

A gauge theory can be formulated on a noncommutative (NC) spacetime. This NC gauge theory has an equivalent dual description through the so-called Seiberg-Witten (SW) map in terms of an ordinary gauge theory on a commutative spacetime. We show that all NC U(1) instantons of Nekrasov-Schwarz type are mapped to ALE gravitational instantons by the exact SW map and that the NC gauge theory of U(1) instantons is equivalent to the theory of hyper-Kaehler geometries. It implies the remarkable consequence that ALE gravitational instantons can emerge from local condensates of purely NC photons.