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We reformulate the twistor construction for hyper- and quaternion-K\"ahlermanifolds, introducing new sigma models that compute scalar potentials for thegeometry. These sigma models have the twistor space of the quaternionicmanifold as their target and encode finite non-linear perturbations of the flatstructures. In the hyperk\"ahler case our twistor sigma models compute bothPlebanski fundamental forms (including the K\"ahler potential), while in thequaternion-K\"ahler setting the twistor sigma model computes the K\"ahlerpotential for the hyperk\"ahler structure on non-projective twistor space. Infour-dimensions, one of the models provides the generating functional oftree-level MHV graviton scattering amplitudes; perturbations of thehyperk\"ahler structure corresponding to positive helicity gravitons. The sigmamodel's perturbation theory gives rise to a sum of tree diagrams observedpreviously in the literature, and their summation via a matrix tree theoremgives a first-principles derivation of Hodges' formula for MHV gravitonamplitudes directly from general relativity. We generalise the twistor sigmamodel to higher-degree (defined in the first case with a cosmologicalconstant), giving a new generating principle for the full tree-level gravitonS-matrix.

Twistor sigma models for quaternionic geometry and graviton scattering