Kinematic Jacobi Identity is a Residue Theorem: Geometry of Color-Kinematics Duality for Gauge and Gravity Amplitudes

TL;DR

This work casts tree-level gauge and gravity amplitudes as intersection numbers on the moduli space $\mathcal{M}_{0,n}$, where color and kinematic numerators arise as residues at boundaries corresponding to trivalent diagrams. The key insight is that the kinematic Jacobi identity emerges from a residue theorem, making color-kinematics duality manifest in this geometric framework. By constructing explicit twisted forms $\varphi_{\pm}$ for color, gauge, and scalar theories and computing their intersection numbers, the authors provide a constructive route to Jacobi-satisfying numerators and hence BCJ representations, including connections to KLT relations. The paper also discusses extensions to higher loops, the role of the parameter $\Lambda$, and open problems related to moduli-space geometry at higher genus.

Abstract

We give a geometric interpretation of color-kinematics duality between tree-level scattering amplitudes of gauge and gravity theories. Using their representation as intersection numbers we show how to obtain Bern-Carrasco-Johansson numerators in a constructive way as residues around boundaries of the moduli space. In this language the kinematic Jacobi identity between each triple of numerators is a residue theorem in disguise.