A Twistor Action for N = 8 Self-dual Supergravity

We give holomorphic Chern-Simons-like action functionals on supertwistor space for self-dual supergravities in four dimensions, with and without gauged R-symmetry or cosmological constant. The gauge group is formally the group of holomorphic contact transformations of supertwistor space. In the case of maximal supersymmetry, we give a formulation in terms of a finite deformation of an integrable ∂̄-operator on a supertwistor space, i.e., on regions in CP3|8. For N = 0, we also give a formulation that does not require the choice of a background. The underlying geometry is related to a special six-dimensional complex contact geometry due to Cap and Eastwood, although our formulation is slightly different, being based on a complex differential one-form up to scale rather than a contact distribution. In this form, the geometry extends to dimension 4n + 2 for all n.