Resultants and 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.

21 pages

References in corpus (12)

• On the Tree-Level S-Matrix of Yang-Mills Theory
• The S-Matrix in Twistor Space
• Unification of Residues and Grassmannian Dualities
• Parity Invariance For Strings In Twistor Space
• Gravity from Rational Curves
• Gravity in Twistor Space and its Grassmannian Formulation
• A simple formula for gravitational MHV amplitudes
• Twistor Strings for N=8 Supergravity
• From Twistor String Theory To Recursion Relations
• A "Twistor String" Inspired Formula For Tree-Level Scattering Amplitudes in N=8 SUGRA
• Fundamental BCJ Relation in N=4 SYM From The Connected Formulation
• New Formulae for Gravity Amplitudes: Parity Invariance and Soft Limits

Cited by in corpus (12)

• Extensions of Theories from Soft Limits
• Scattering in Three Dimensions from Rational Maps
• N-Point Tree-Level Scattering Amplitude in the New Berkovits' String
• Perturbative gauge theory at null infinity
• Notes on Conformal Soft Theorems and Recursion Relations in Gravity
• Notes on Scattering Amplitudes as Differential Forms
• Graviton scattering in self-dual radiative space-times
• The momentum amplituhedron of SYM and ABJM from twistor-string maps
• Supersymmetric celestial OPEs and soft algebras from the ambitwistor string worldsheet
• Minitwistors and 3d Yang-Mills-Higgs theory
• Chiral Splitting and $\mathcal{N}=4$ Einstein--Yang--Mills Tree Amplitudes in 4d
• Minimal Basis in Four Dimensions and Scalar Blocks