Noncommutative self-dual gravity

Starting from a self-dual formulation of gravity, we obtain a noncommutative theory of pure Einstein theory in four dimensions. In order to do that, we use the Seiberg-Witten map. A procedure is outlined that allows one to find the solution of the noncommutative torsion constraint through the vanishing of the commutative one. Finally, the noncommutative corrections to the action are computed up to second order.