Minimal Kinematics: An All and Peek into Trop+G( , )

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2021
• Main Authors: Cachazo, Freddy, Early, Nick
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2021
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/211345
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Minimal Kinematics: An All and Peek into Trop+G( , ). Freddy Cachazo and Nick Early. SIGMA 17 (2021), 078, 22 pages

In this note, we present a formula for the Cachazo-Early-Guevara-Mizera (CEGM) generalized biadjoint amplitudes for all and n on what we call the minimal kinematics. We prove that on the minimal kinematics, the scattering equations on the configuration space of points on ℂℙᵏ⁻¹ have a unique solution, and that this solution is in the image of a Veronese embedding. The minimal kinematics is a generalization of the one recently introduced by Early for = 2 and uses a choice of cyclic ordering. We conjecture an explicit formula for ⁽ᵏ⁾ₙ( , ), which we have checked analytically through = 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 C⁽¹’²’ ˙˙˙’ ᵏ⁻²⁾ₙ. Generalized biadjoint amplitudes can also be evaluated using the positive tropical Grassmannian Tr+G( , ) 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 Tr+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 = 2 and uses the planar basis recently introduced by Early for all . Based on numerical computations through = 8 for all k, we conjecture that on the planar-basis kinematics ⁽ᵏ⁾ₙ( , ) evaluates to the multidimensional Catalan numbers, suggesting the possibility of novel combinatorial interpretations. For = 2, these are the standard Catalan numbers.