Resultants and Gravity Amplitudes

TL;DR

This paper simplifies two known formulations of N=8 Einstein supergravity tree-level amplitudes by expressing them as integrals over holomorphic maps of bi-degree, revealing new polynomial and determinant structures.

Contribution

It demonstrates that both formulations can be written in a parity-invariant form using bi-degree maps, uncovering their algebraic structures involving resultants and determinants.

Findings

Both formulations are simplified to integrals over bi-degree (d, n-d-2) maps.

The integrand in one formulation is the product of resultants of the maps.

The other formulation's integrand includes a polynomial determinant containing resultants.

Abstract

Two very different formulations of the tree-level S-matrix of N=8 Einstein supergravity in terms of rational maps are known to exist. In both formulations, the computation of a scattering amplitude of n particles in the k R-charge sector involves an integral over the moduli space of certain holomorphic maps of degree d=k-1. In this paper we show that both formulations can be simplified when written in a manifestly parity invariant form as integrals over holomorphic maps of bi-degree (d,n-d-2). In one formulation the full integrand becomes directly the product of the resultants of each of the two maps defining the one of bi-degree (d,n-d-2). In the second formulation, a very different structure appears. The integrand contains the determinant of a (n-3)x(n-3) matrix and a 'Jacobian'. We prove that the determinant is a polynomial in the coefficients of the maps and contains the two resultants as factors.