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.