Spinor-Helicity Varieties

TL;DR

This paper explores the geometric and algebraic structures arising from the spinor-helicity formalism in particle physics, focusing on varieties related to Grassmannians, flag varieties, and their tropical and positive geometries.

Contribution

It introduces a detailed study of spinor-helicity varieties, including their combinatorial, algebraic, tropical, and positive geometric aspects, connecting them to scattering amplitudes.

Findings

Identification of subvarieties in Grassmannian products

Analysis of Hadamard products leading to Mandelstam varieties

Connections between these varieties and scattering amplitudes

Abstract

The spinor-helicity formalism in particle physics gives rise to natural subvarieties in the product of two Grassmannians. These include two-step flag varieties for subspaces of complementary dimension. Taking Hadamard products leads to Mandelstam varieties. We study these varieties through the lens of combinatorics and commutative algebra, and we explore their tropicalization, positive geometry, and scattering correspondence.