Tropical fans, scattering equations and amplitudes

Abstract

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 $\mathrm{Gr}(k,n)$ there is a notion of a generalised ${\varphi }^3$ amplitude and an associated set of scattering equations which further generalise the $\mathrm{Gr}(k,n)$ 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 $\mathrm{Gr}(4,n)$ the tropical fans we describe are related to the singularities (or symbol letters) of loop amplitudes in planar $\mathcal{N}=4$ 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.

arXiv comment31 pages

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• Feynman Polytopes and the Tropical Geometry of UV and IR Divergences
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• Algebraic branch points at all loop orders from positive kinematics and wall crossing
• Bootstrapping elliptic Feynman integrals using Schubert analysis
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• Planar Matrices and Arrays of Feynman Diagrams: Poles for Higher $k$
• One-loop integrand from generalised scattering equations
• Notes on Worldsheet-Like Variables for Cluster Configuration Spaces
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• A nice two-loop next-to-next-to-MHV amplitude in $\mathcal{N}=4$ super-Yang-Mills
• Weighted blade arrangements and the positive tropical Grassmannian