Combinatorics

Physics

Scattering amplitude

Grassmannian

Pure mathematics

Mathematics

Mathematical physics

Gravitational singularity

Adjacency list

Mathematical analysis

Polytope

Artificial intelligence

Cluster algebra

Scattering

Class (philosophy)

Computer science

Ising model

Quantum mechanics

Journal of High Energy Physics

A bstract 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 ϕ 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 $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 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.

Tropical fans, scattering equations and amplitudes