Resultants and Gravity Amplitudes

TL;DR

Cachazo investigates two tree-level formulations of the ${ m N}=8$ supergravity S-matrix expressed through holomorphic maps, showing that casting both into a parity-invariant bi-degree framework $(d, ilde d)$ drastically simplifies their structure. He proves that the CS formulation’s product of resultants $R(oldsymbolλ)R( ilde{oldsymbolλ})$ naturally appears, while the CG formulation’s integrand reduces to a ratio ${ m det}{oldsymbol H}/M$ with a map-dependent Jacobian $M$. Central to the work is the demonstration that ${ m det}{oldsymbol H}$ is divisible by both $R(oldsymbolλ)$ and $R( ilde{oldsymbolλ})$, enabling an exact equivalence between the two formulations after factoring. The results illuminate a deeper geometric role for the resultants and pave the way for further checks (e.g., BCFW) and potential worldsheet interpretations, offering a compact, parity-symmetric description of tree-level gravity amplitudes.

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.