Twistor sigma models for quaternionic geometry and graviton scattering

We reformulate the twistor construction for hyper- and quaternion-K\"ahler manifolds, introducing new sigma models that compute scalar potentials for the geometry. These sigma models have the twistor space of the quaternionic manifold as their target and encode finite non-linear perturbations of the flat structures. In the hyperk\"ahler case our twistor sigma models compute both Plebanski fundamental forms (including the K\"ahler potential), while in the quaternion-K\"ahler setting the twistor sigma model computes the K\"ahler potential for the hyperk\"ahler structure on non-projective twistor space. In four-dimensions, one of the models provides the generating functional of tree-level MHV graviton scattering amplitudes; perturbations of the hyperk\"ahler structure corresponding to positive helicity gravitons. The sigma model's perturbation theory gives rise to a sum of tree diagrams observed previously in the literature, and their summation via a matrix tree theorem gives a first-principles derivation of Hodges' formula for MHV graviton amplitudes directly from general relativity. We generalise the twistor sigma model to higher-degree (defined in the first case with a cosmological constant), giving a new generating principle for the full tree-level graviton S-matrix.