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

TL;DR

This paper introduces a new Grassmannian integral formula for tree-level scattering amplitudes in N=8 supergravity, revealing hidden symmetries and connecting to Hodges' and KLT formulas.

Contribution

It presents a novel Grassmannian integral representation for all tree-level amplitudes in N=8 supergravity, making R-symmetry and permutation invariance manifest.

Findings

The new formula reproduces known amplitudes for up to 8 particles.

It establishes the equivalence between Hodges' MHV formula and the KLT relations.

Provides explicit proof of orthogonality in the Veronese embedding for small particle numbers.

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.