Self-duality in superspace

The absence of local divergences in the vacuum functional of supergravity at all loop orders in a self-dual gravitational or supergravitational background is discussed. It is also shown that the first variational derivative of the generating functional of supergravity has no gauge independent divergences in a self-dual background, in particular that the gauge independent part of the energy momentum tensor is finite in a self-dual background. Simple expressions defining the corresponding half-flat superspace are presented.