Notes on biadjoint amplitudes, 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 CP2. 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 ×360 biadjoint amplitudes obtained by using the facets of 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.