The Chirotropical Grassmannian

TL;DR

This paper explores the structure of the chirotropical Grassmannian and Dressian, proving their equality for certain parameters, developing algorithms for their computation, and analyzing their geometric properties and limitations.

Contribution

It introduces the concept of chirotropical Grassmannian and Dressian, proves their equality for specific cases, and provides algorithms to compute and analyze these structures.

Findings

Proved $ ext{Trop}^ ext{chi} ext{G}(3,n) = ext{Dr}^ ext{chi}(3,n)$ for n=6,7,8.

Developed algorithms to compute chirotropical Grassmannians from rays.

Showed the failure of equality for (k,n) = (4,8).

Abstract

Recent developments in particle physics have revealed deep connections between scattering amplitudes and tropical geometry. From the heart of this relationship emerged the chirotropical Grassmannian $\text{Trop}^\chi \text{G}(k,n)$ and the chirotropical Dressian $\text{Dr}^\chi(k,n)$, polyhedral fans built from uniform realizable chirotopes that encode the combinatorial structure of Generalized Feynman Diagrams. We prove that $\text{Trop}^\chi \text{G}(3,n) = \text{Dr}^\chi(3,n)$ for $n = 6,7,8$, and develop algorithms to compute these objects from their rays modulo lineality. Using these algorithms, we compute all chirotropical Grassmannians $\text{Trop}^\chi \text{G}(3,n)$ for $n = 6,7,8$ across all isomorphism classes of chirotopes. We prove that each chirotopal configuration space $X^\chi(3,6)$ is diffeomorphic to a polytope and propose an associated canonical logarithmic differential form. Finally, we show that the equality between chirotropical Grassmannian and Dressian fails for $(k,n) = (4,8)$.