Planar kinematic invariants, matroid subdivisions and generalized Feynman diagrams

Abstract

In recent work of Cachazo, Guevara, Mizera and the author, a generalization of the biadjoint scattering amplitude $m^{(k)}({\mathbb{I}}_n,{\mathbb{I}}_n)$ was introduced as an integral over the moduli space of $n$ points in ${\mathbb{CP}}^{k-1}$, with value a sum of certain rational functions on the kinematic space ${\mathcal{K}}_{k,n}$. It was shown there for $m^{(3)}({\mathbb{I}}_6,{\mathbb{I}}_6)$ and later by Cachazo and Rojas that collections of poles appearing in $m^{(3)}({\mathbb{I}}_7,{\mathbb{I}}_7)$ are compatible exactly when they are dual to collections of rays which generate the maximal faces of a polyhedral complex known as the (nonnegative) tropical Grassmannian. In this note, we derive a remarkable planar basis for the space of generalized kinematic invariants which coincides in the case $k=2$ with usual standard planar multi-particle basis for the kinematic space. We implement in Mathematica the action on formal linear combinations of planar matroid subdivisions of a boundary operator which, together with the planar basis, determines compatibility for any given poles appearing in the expansion of $m^{(k)}({\mathbb{I}}_n,{\mathbb{I}}_n)$, by computing a certain combinatorial non-crossing condition on the second hypersimplicial faces ${\Delta }_{2,n-(k-2)}$ of ${\Delta }_{k,n}$. The algorithms are implemented in an accompanying Mathematica notebook and are evaluated on existing tables of rays, in the form of tropical Plucker vectors, to tabulate the finest planar subdivisions of ${\Delta }_{3,8},{\Delta }_{3,9}$ and ${\Delta }_{4,8}$, or equivalently the set of maximal cones for the corresponding nonnegative tropical Grassmannians.

arXiv comment26 pages, 6 figures; v2: attached Mathematica notebook

References in corpus (6)

• Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet
• Weak Separation and Plabic Graphs
• Tropical Grassmannians, cluster algebras and scattering amplitudes
• Singular Solutions in Soft Limits
• A Soft Theorem for the Tropical Grassmannian
• The adjoint braid arrangement as a combinatorial Lie algebra via the Steinmann relations

Cited by in corpus (9)

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• Minimal Kinematics: An All $k$ and $n$ Peek into ${\mathrm{Trop}}^{+}\mathrm{G}(k,n)$
• Notes on Worldsheet-Like Variables for Cluster Configuration Spaces
• Color-Dressed Generalized Biadjoint Scalar Amplitudes: Local Planarity
• Weighted blade arrangements and the positive tropical Grassmannian
• Planarity in Generalized Scattering Amplitudes: PK Polytope, Generalized Root Systems and Worldsheet Associahedra
• Species-theoretic foundations of perturbative quantum field theory