Spinor-Helicity Varieties
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.
We provide the code for computing the ideal \(I_{k,n,0}\) of the Spinor-Helicity variety for a given \(k, n,\) and \(r=0\). The first part of the code is a function that computes the ideal \(I_{k,n,0}\) for a given \(n\) and \(k\). The second part showcases some properties (e.g., multidegree) for the example \(n=7\) and \(k=3\). The reader may change the values of \(k, n,\) and the commands to explore other cases and properties.
We report the code for Proposition 6.9. The result of the provided Macaulay code is a Julia script. It can then be copied and pasted in Julia using the HomotopyContinuation.jl package to compute the number of irreducible components of the Scattering Correspondence. The first part of the code is a routine that outputs Julia code. In the second part of the code, we showcase this computation for \(n=7, k=3\) and \(r=1\). The reader may change the values of \(k, n,\) and \(r\) to explore other cases.
Project page created: 24/06/2024
Project contributors: Yassine El Maazouz, Anaëlle Pfister and Bernd Sturmfels
Corresponding author of this page: Anaëlle Pfister, anaelle.pfister@mis.mpg.de
Software used: Julia (Version 1.5.2), Macaulay2 (v1.20)
License for code of this project page: MIT License (https://spdx.org/licenses/MIT.html)
License for all other content of this project page (text, images, …): CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/)
Last updated 27/06/2025.