Gravity in Twistor Space and its Grassmannian Formulation

TL;DR

The paper proves that the complete tree-level S-matrix for $\mathcal{N}=8$ supergravity can be expressed as a contour integral over rational maps to twistor space, and that this representation satisfies the BCFW recursion with correct seeds and factorization properties.A central achievement is demonstrating the required $1/z^2$ falloff under large BCFW shifts, which clarifies gravity's unconventional high-energy behaviour within the Grassmannian framework.The authors further reformulate the construction as a Grassmannian integral over $G(k,n)$, introducing link variables and Veronese constraints to reveal a simple, parity-conjugate structure and a product form reminiscent of MHV/anti-MHV components.Together, these results provide a geometric, manifestly structured description of gravity amplitudes, offering both conceptual insight and practical computational frameworks for tree-level amplitudes.

Abstract

We prove the formula for the complete tree-level $S$-matrix of $\mathcal{N}=8$ supergravity recently conjectured by two of the authors. The proof proceeds by showing that the new formula satisfies the same BCFW recursion relations that physical amplitudes are known to satisfy, with the same initial conditions. As part of the proof, the behavior of the new formula under large BCFW deformations is studied. An unexpected bonus of the analysis is a very straightforward proof of the enigmatic $1/z^2$ behavior of gravity. In addition, we provide a description of gravity amplitudes as a multidimensional contour integral over a Grassmannian. The Grassmannian formulation has a very simple structure; in the N$^{k-2}$MHV sector the integrand is essentially the product of that of an MHV and an $\overline{\rm MHV}$ amplitude, with $k+1$ and $n-k-1$ particles respectively.

Figures (1)

• Figure 1: On twistor space, the residue of a factorization channel looks like a nodal curve with the location $\mathcal{Z}$ of the node integrated over.