A Link Representation for Gravity Amplitudes

TL;DR

The paper derives a link representation for all tree amplitudes in $\mathcal{N}=8$ supergravity by starting from the Cachazo–Skinner twistor-string–like formula. It reformulates the two determinant factors as products over trees via the Matrix-Tree theorem and introduces link variables $c_{I i}$ that connect external twistors and dual twistors, yielding momentum-space and contour forms with manifest parity and permutation symmetries. A GL$(k)$-invariant reformulation is presented, paving the way toward a Grassmannian contour integral and a potential description of leading singularities beyond tree level. The work provides a structurally rich framework that links gravity amplitudes to Grassmannian-like objects, with open questions about residues and full Grassmannian formulations. Overall, it offers a new, symmetry-rich decomposition of gravity amplitudes that may facilitate Grassmannian approaches to $\mathcal{N}=8$ supergravity.

Abstract

We derive a link representation for all tree amplitudes in N=8 supergravity, from a recent conjecture by Cachazo and Skinner. The new formula explicitly writes amplitudes as contour integrals over constrained link variables, with an integrand naturally expressed in terms of determinants, or equivalently tree diagrams. Important symmetries of the amplitude, such as supersymmetry, parity and (partial) permutation invariance, are kept manifest in the formulation. We also comment on rewriting the formula in a GL(k)-invariant manner, which may serve as a starting point for the generalization to possible Grassmannian contour integrals.