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 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 360×360 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. arXiv:1906.05979v1 [hep-th]

Tropical Grassmannians, cluster algebras and scattering amplitudes

“…We describe here the tropical formulation of the Grassmannian spaces and how to select the positive region. We will see that this coincides with the criteria recently used in [1,2] to determine the generalised φ 3 amplitudes for Gr (3,6) and Gr (3,7).…”

“…9 Conclusions and outlook to Gr (4,8) In this paper we have utilised cluster algebra technology to construct tree-level biadjoint amplitudes on Gr(3, n) for n = 6, 7, 8. These amplitudes arise from scattering equations on the corresponding Grassmannians [1,2] and the relevance of cluster algebras for these amplitudes arises from the interpretation of these amplitudes as volumes of certain geometric objects. In the cases we studied in this paper these objects are polyhedra in (k − 1)(n − k − 1) − 1 dimensions, where k = 3.…”

“…Recently a very interesting connection between scattering amplitudes and tropical geometry has been uncovered [1,2]. The connection outlined so far is for tree-level biadjoint φ 3 amplitudes, which can be related to the series of tropical Grassmannians Gr(2, n) and a generalisation to higher Grassmannians Gr(k, n).…”

“…The positive region of the tropical computation should then equal (3.8) for the canonical ordering α = β = I.Let us consider explicit examples of the tropical Grassmannian[14]. The simplest case is Gr(2,4), defined by the single Plücker relation, p 12 p 34 − p 13 p 24 + p 14 p 23 = 0 . (3.10) In this case the tropical hypersurface conditions have three solutions (modulo lineality), given by the three possibilities in (3.5) with {i, j, k, l} = {1, 2, 3, 4}.…”

Notes on polytopes, amplitudes and boundary configurations for Grassmannian string integrals

“…At leading order, the Grassmannian string integral is computed by the canonical function of a class of polytopes, which are closely related to the positive tropical Grassmannian [13], as shown in [14][15][16][17]). It has also been shown that the canonical function can also be obtained using a CHY formula which are higher-k generalization of the bi-adjoint φ 3 amplitudes (see also [18][19][20]).…”