Minimal Kinematics: An All $k$ and $n$ Peek into ${\mathrm{Trop}}^{+}\mathrm{G}(k,n)$

Abstract

In this note we present a formula for the Cachazo-Early-Guevara-Mizera (CEGM) generalized biadjoint amplitudes for all $k$ and $n$ on what we call the minimal kinematics. We prove that on the minimal kinematics, the scattering equations on the configuration space of $n$ points on ${\mathbb{CP}}^{k-1}$ has a unique solution, and that this solution is in the image of a Veronese embedding. The minimal kinematics is an all $k$ generalization of the one recently introduced by Early for $k=2$ and uses a choice of cyclic ordering. We conjecture an explicit formula for $m_n^{(k)}(\mathbb{I},\mathbb{I})$ which we have checked analytically through $n=10$ for all $k$. The answer is a simple rational function which has only simple poles; the poles have the combinatorial structure of the circulant graph ${\mathrm{C}}_n^{(1,2,\dots ,k-2)}$. Generalized biadjoint amplitudes can also be evaluated using the positive tropical Grassmannian ${\mathrm{Tr}}^{+}\mathrm{G}(k,n)$ in terms of generalized planar Feynman diagrams. We find perfect agreement between both definitions for all cases where the latter is known in the literature. In particular, this gives the first strong consistency check on the $90608$ planar arrays for ${\mathrm{Tr}}^{+}\mathrm{G}(4,8)$ recently computed by Cachazo, Guevara, Umbert and Zhang. We also introduce another class of special kinematics called planar-basis kinematics which generalizes the one introduced by Cachazo, He and Yuan for $k=2$ and uses the planar basis recently introduced by Early for all $k$. Based on numerical computations through $n=8$ for all $k$, we conjecture that on the planar-basis kinematics $m_n^{(k)}(\mathbb{I},\mathbb{I})$ evaluates to the multidimensional Catalan numbers, suggesting the possibility of novel combinatorial interpretations. For $k=2$ these are the standard Catalan numbers.

References in corpus (6)

• Tropical fans, scattering equations and amplitudes
• Singular Solutions in Soft Limits
• Schubert calculus and representations of general linear group
• The positive tropical Grassmannian, the hypersimplex, and the m=2 amplituhedron
• Planar kinematic invariants, matroid subdivisions and generalized Feynman diagrams
• Planar Matrices and Arrays of Feynman Diagrams

Cited by in corpus (5)

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• Notes on Worldsheet-Like Variables for Cluster Configuration Spaces
• Weighted blade arrangements and the positive tropical Grassmannian
• Planarity in Generalized Scattering Amplitudes: PK Polytope, Generalized Root Systems and Worldsheet Associahedra