Twistor sigma models for quaternionic geometry and graviton scattering

Abstract

We reformulate the twistor construction for hyper-and quaternion-Kähler 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 hyperkähler case our twistor sigma models compute both Plebanski fundamental forms (including the Kähler potential), while in the quaternion-Kähler setting the twistor sigma model computes the Kähler potential for the hyperkähler 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 hyperkähler 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.

Mapping SYK to the Sky

“…The conformal primary wavefunctions with positive helicity are exact (complex) self-dual spacetimes of this type [42]. The authors of [20] explore this w 1+∞ symmetry in the context of twistor sigma models sigma models [43]. We will return to the connection between these area preserving diffeomorphisms and the celestial dictionary in the context of our 2D SYK model in section 4.…”

“…4 The above discussion closely follows the non-linear graviton description of classical self-dual geometries in 4D gravity [41]. To make this relationship explicit we need to lift the story to 4D spacetime via the twistor correspondence [43]. We define twistor variables (λ α , ζ α) via…”

“…As shown in [43], one can write down a sigma model which produces this relation between Z α and h as its equation of motion…”

“…Upon minimizing the action with respect to the dynamical field Z α(x, z), and after integrating over CP 1 , the only remaining dependence is on the spacetime point x. The tetrad of the corresponding self-dual geometry is then obtained via [43] e a α = (dx α,…”

“…The conformal primary wavefunctions with positive helicity are examples of self-dual Ricci-flat spacetimes, pertinent to the non-linear graviton construction [42,43]. Let us use this to make the connection between the area preserving diffeomorphisms of our model and the celestial w 1+∞ symmetry explicit.…”