Twistor sigma models for quaternionic geometry and graviton scattering

Abstract

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