Gravitational Instantons from Gauge Theory

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 asymptotically locally Euclidean (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-Kähler geometries. It implies the remarkable consequence that ALE gravitational instantons can emerge from local condensates of purely NC photons.