Scattering Amplitudes from Intersection Theory

TL;DR

The paper develops a new localization formula for intersection numbers of twisted cocycles associated to hyperplane arrangements, derived via Picard–Lefschetz theory and applicable even at singular configurations. In the special case where the arrangement yields the moduli space M_{0,n}, the framework recovers CHY-type scattering amplitudes, with Parke–Taylor cocycles forming a natural basis and the scattering equations encoding the twist ω. The results connect intersection theory to a spectrum of amplitudes (bi-adjoint scalars, Yang–Mills, gravity) and provide a rigorous foundation for KLT-type relations in the CHY context, while suggesting paths to higher-loop and non-generic kinematic generalizations. Overall, the work establishes a mathematically precise bridge between twisted cohomology, localization, and modern amplitude techniques.

Abstract

We use Picard-Lefschetz theory to prove a new formula for intersection numbers of twisted cocycles associated to a given arrangement of hyperplanes. In a special case when this arrangement produces the moduli space of punctured Riemann spheres, intersection numbers become tree-level scattering amplitudes of quantum field theories in the Cachazo-He-Yuan formulation.