The Scattering Variety

TL;DR

This work treats the scattering equations for massless particle kinematics as algebraic varieties arising from irreducible Möbius-algebra representations. It demonstrates that the resulting affine threefolds are affine Calabi–Yau spaces, with projective counterparts forming singular K3 and Fano surfaces, and reveals regular Hilbert-series patterns linked to Mahonian numbers and cyclotomic polynomials. Through Betti-number resolutions, the authors show a Pascal-triangle structure, establishing arithmetically Cohen–Macaulay modules and systematic dimensional shifts between the varieties ${\cal V}_\varphi$ and ${\cal V}^*_\varphi$. The geometry concretizes into familiar spaces such as the conifold, a K3 surface, and del Pezzo-type Fano bases, while the singular loci suggest deep connections to the Möbius representation constraints and potential physical interpretations in scattering theory.

Abstract

The so-called Scattering Equations which govern the kinematics of the scattering of massless particles in arbitrary dimensions have recently been cast into a system of homogeneous polynomials. We study these as affine and projective geometries which we call Scattering Varieties by analyzing such properties as Hilbert series, Euler characteristic and singularities. Interestingly, we find structures such as affine Calabi-Yau threefolds as well as singular K3 and Fano varieties.