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 Trop^χG(k,n) and the chirotropical Dressian Dr^χ(k,n), polyhedral fans built from uniform realizable chirotopes that encode the combinatorial structure of Generalized Feynman Diagrams. We prove that Trop^χG(3,n) = Dr^χ(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 Trop^χG(3,n) for n = 6,7,8 across all isomorphism classes of chirotopes. We prove that each chirotopal configuration space X^χ(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).