Tropical fans, scattering equations and amplitudes

We describe a family of tropical fans related to Grassmannian cluster algebras. These fans are related to the kinematic space of massless scattering processes in a number of ways. For each fan associated to the Grassmannian Gr(k, n) there is a notion of a generalised phi(3) amplitude and an associated set of scattering equations which further generalise the Gr(k, n) scattering equations that have been recently introduced. Here we focus mostly on the cases related to finite Grassmannian cluster algebras and we explain how face variables for the cluster polytopes are simply related to the scattering equations. For the Grassmannians Gr(4, n) the tropical fans we describe are related to the singularities (or symbol letters) of loop amplitudes in planar N = 4 super Yang-Mills theory. We show how each choice of tropical fan leads to a natural class of polylogarithms, generalising the notion of cluster adjacency and we describe how the currently known loop data fit into this classification.

Automated summary

This paper describes a family of tropical fans related to Grassmannian cluster algebras, which are connected to the kinematic space of massless scattering processes in various ways. The focus is on finite Grassmannian cluster algebras, explaining how face variables for the cluster polytopes are related to the scattering equations. For Grassmannians Gr(4, n), the described tropical fans are related to the singularities of loop amplitudes in planar N = 4 super Yang-Mills theory. Each choice of tropical fan leads to a natural class of polylogarithms, and the currently known loop data fit into this classification.