Gravity, Twistors and the MHV Formalism

TL;DR

The paper derives gravitational MHV amplitudes from the anti self-dual (ASD) sector using the Plebanski chiral action, then recasts the construction in twistor space to obtain a generating function that reproduces BGK gravity amplitudes. It develops a twistorial action for MHV diagrams in gravity, showing how off-shell MHV vertices and propagators arise in a CSW-like gauge, and extends the framework to ${f N}=4$ and ${f N}=8$ supergravity. The work clarifies deep links between gravity’s chiral MHV structure, ASD integrability, and twistor theory, providing a coherent bridge between perturbative gravity and non-perturbative ASD geometry and pointing toward possible twistor-string realizations. Together, these results yield a constructive, geometrical path from ASD backgrounds to full gravitational MHV perturbation theory and its supersymmetric extensions.

Abstract

We give a self-contained derivation of the MHV amplitudes for gravity and use the associated twistor generating function to define a twistor action for the MHV diagram approach to gravity. Starting from a background field calculation on a spacetime with anti self-dual curvature, we obtain a simple spacetime formula for the scattering of a single, positive helicity linearized graviton into one of negative helicity. Re-expressing our integral in terms of twistor data allows us to consider a spacetime that is asymptotic to a superposition of plane waves. Expanding these out perturbatively yields the gravitational MHV amplitudes of Berends, Giele & Kuijf. We go on to take the twistor generating function off-shell at the perturbative level. Combining this with a twistor action for the anti self-dual background, we obtain a twistor action for the MHV diagram approach to perturbative gravity. We finish by extending these results to supergravity, in particular N=4 and N=8.

Figures (3)

• Figure 1: Reversing the momentum of one of the positive helicity particles leads to the interpretation of the MHV amplitude as measuring the helicity-flip of a single particle which traverses a region of ASD background curvature.
• Figure 2: Deformations of the complex structure induce deformations of the holomorphic curves. Identifying the four parameters $x$ on which $F^\alpha(x,\pi)$ depends with spacetime coordinates, the normal vector $\left(F^\alpha-{\rm i} x^{\alpha\dot\alpha}\pi_{\dot\alpha}\right)\partial/\partial\omega^\alpha$ on $L_x$ connects the original twistor line $\omega^\alpha={\rm i} x^{\alpha\dot\alpha}\pi_{\dot\alpha}$ to the deformed curve.
• Figure 3: Yang-Mills ( l) and gravitational ( r) MHV amplitudes are supported on holomorphic lines in twistor space. For gravity, the negative helicity gravitons arise from insertions of normal vector fields, giving a perturbative description of the deformation of the line.