Double complex (anti-)self-dual variables and a related double action in four-dimensional gravity

In this paper double dual transformations and double complex (anti-)self-dual variables in four-dimensional gravity are given by using the double complex function method. A double action with double complex anti-self-dual variables is proposed, hence the double complex (anti-) self-dual gravitational theory is establish. Finally, the analytic continuation from the double action to the Euclidean action is discussed, and some given results are here extended into a double form.