Cluster Algebras in Kinematic Space of Scattering Amplitudes

TL;DR

This paper explores the natural cluster algebra structure within the kinematic space of scattering amplitudes, revealing its preservation in specific geometric configurations and its relation to cluster sub-algebras.

Contribution

It demonstrates the existence and preservation of a type A cluster algebra in the kinematic space and its substructures, connecting to scattering forms and cluster polylogarithms.

Findings

Identification of a residual type A cluster algebra in kinematic space

Preservation of cluster algebra in a hypercube necklace within the associahedron

Connection between cluster sub-algebras and scattering amplitude data

Abstract

We clarify the natural cluster algebra of type A that exists in a residual and tropical form in the kinematical space as suggested in 1711.09102 by the use of triangulations, mutations and associahedron on the definition of scattering forms. We also show that this residual cluster algebra is preserved in a hypercube (diamond) necklace inside the associahedron where cluster sub-algebras $(A_1)^n$ exist. This result goes in line with results with cluster poligarithms in 1401.6446 written in terms of $A_2$ and $A_3$ functions only and other works showing the primacy of $A$ cluster sub-algebras as data input for scattering amplitudes.