A "Twistor String" Inspired Formula For Tree-Level Scattering Amplitudes in N=8 SUGRA

TL;DR

The paper presents a twistor-string inspired, Grassmannian-based formula for the complete tree-level S-matrix of N=8 supergravity, built as a GL(2)-invariant integral over G(2,n) with a Veronese embedding into G(k,n). Central to the construction are a Hodges-type determinant numerator H_n and a KLT-like denominator J_n, whose interplay yields manifest SU(8) R-symmetry and a polynomial, rather than factorial, growth in the number of terms. The authors establish a Hodges-KLT generalized MHV equivalence under an off-shell deformation and formulate an Orthogonality Conjecture for RSVW residues, supported by numerical evidence up to eight particles. By inserting RSVW into KLT and exploiting orthogonality, they derive a compact gravity formula and outline a path toward a GL(k)-invariant G(k,n) Grassmannian formulation, with future directions toward correlation-function interpretations, permutation-invariant forms, and twistor-space pictures.

Abstract

We propose a new formulation of the complete tree-level S-matrix of N = 8 supergravity. The new formula for n particles in the k R-charge sector is an integral over the Grassmannian G(2,n) and uses the Veronese map into G(k,n). The image of a point in G(2,n) is required to be in the "complement" of a 2|8-plane thus making the SU(8) R-symmetry manifest. The integrand is the ratio of two determinants. The numerator is an analog of Hodges' recent determinant formula for MHV amplitudes. The denominator is a 2(n+k-2) x 2(n+k-2) minor of a 2(n+k) x 2(n+k) matrix of rank 2(n+k-2). Just as Hodges' formula does for MHV amplitudes, our integrand makes the complete invariance under Sn manifest for all sectors. The validity of the new formula follows from two surprising facts. One is the equivalence of Hodges' MHV formula and the Kawai-Lewellen-Tye (KLT) formula when kinematic invariants are allowed to be off-shell in a novel way. We give a proof of this for any number of particles. The second fact is an orthogonality property of the solutions to the polynomial equations defining the Veronese embedding. Explicit proof of the orthogonality is given for all amplitudes in all R-charge sectors with eight or less particles thus providing non-trivial evidence for our proposal.