Double self-dual and anti-self-dual structure of N=1 supergravity

A unified definition of the dual operation acting on the elements of osp(1|4) is introduced in N=1 supergravity. Under this definition the self-dual and anti-self-dual operations are identified with the chiral projections. Then these operations are extended to forms on the spacetime manifold. Using these operations the osp(1|4)-valued connection one form and its curvature are decomposed into the self-dual and anti-self-dual parts with respect to both the internal and the spacetime indices. As a result four basic invariants which differ from each other in their dualities are obtained. The linear combinations of them constitute all of the possible invariants quadratic in curvature and then can be used to construct various theories of N=1 supergravity of Yang-Mills type.