The Chirotropical Grassmannian

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 $(k,n) = (4,8)$.