Notes on Biadjoint Amplitudes, $\mathrm{Trop}G(3,7)$ and $X(3,7)$ Scattering Equations

Abstract

In these notes we use the recently found relation between facets of tropical Grassmannians and generalizations of Feynman diagrams to compute all "biadjoint amplitudes" for $n=7$ and $k=3$. We also study scattering equations on $X(3,7)$, the configuration space of seven points on ${\mathbb{CP}}^2$. We prove that the number of solutions is $1272$ in a two-step process. In the first step we obtain $1162$ explicit solutions to high precision using near-soft kinematics. In the second step we compute the matrix of $360\times 360$ biadjoint amplitudes obtained by using the facets of $\mathrm{Trop}G(3,7)$, subtract the result from using the $1162$ solutions and compute the rank of the resulting matrix. The rank turns out to be $110$, which proves that the number of solutions in addition to the $1162$ explicit ones is exactly $110$.

arXiv comment13 pages, 1 figure and 6 ancillary files; minor revision

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• Minimal Kinematics: An All $k$ and $n$ Peek into ${\mathrm{Trop}}^{+}\mathrm{G}(k,n)$
• One-loop integrand from generalised scattering equations
• Notes on Worldsheet-Like Variables for Cluster Configuration Spaces
• On kinetic theories with color and spin from amplitudes