The Chirotropical Grassmannian
Dario Antolini, Nick Early
TL;DR
This work defines and studies the chirotropical Grassmannian $\text{Trop}^\chi \text{G}(k,n)$ and the chirotropical Dressian $\text{Dr}^\chi(k,n)$, linking tropical geometry with generalized biadjoint amplitudes. The authors prove $\text{Trop}^\chi \text{G}(3,n) = \text{Dr}^\chi(3,n)$ for $n=6,7,8$, and develop ray-based algorithms to compute these spaces from their rays modulo lineality, enabling complete computations of all $\text{Trop}^\chi \text{G}(3,n)$ for $n=6,7,8$ across all isomorphism classes of chirotopes. They also show each chirotopal configuration space $X^\chi(3,6)$ is diffeomorphic to a polytope and provide canonical logarithmic differential forms, supporting a positive-geometric interpretation. The paper further demonstrates that the equality fails for $(k,n)=(4,8)$, and discusses realizability questions, explicit parameterizations for rank-3 cases, and future directions toward Puiseux real tropicalizations and higher-rank chirotopes.
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 $\text{Trop}^χ\text{G}(k,n)$ and the chirotropical Dressian $\text{Dr}^χ(k,n)$, polyhedral fans built from uniform realizable chirotopes that encode the combinatorial structure of Generalized Feynman Diagrams. We prove that $\text{Trop}^χ\text{G}(3,n) = \text{Dr}^χ(3,n)$ 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 $\text{Trop}^χ\text{G}(3,n)$ 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 $(k,n) = (4,8)$.