Planar Kinematics: Cyclic Fixed Points, Mirror Superpotential, k-Dimensional Catalan Numbers, and Root Polytopes

TL;DR

This paper explores the geometric and combinatorial structure of solutions to generalized scattering equations in planar kinematics, revealing connections to Catalan numbers, root polytopes, and mirror superpotentials, with implications for biadjoint amplitudes.

Contribution

It establishes a link between cyclic fixed points in configuration spaces and solutions to generalized scattering equations, introduces new polytopal neighborhoods, and verifies conjectures relating amplitudes to Catalan numbers.

Findings

Solutions inject into aperiodic k-element subsets, bounded by Lyndon words.

Generalized biadjoint amplitudes evaluate to k-dimensional Catalan numbers on PK.

Volumes of root polytopes relate to multi-dimensional Catalan numbers.

Abstract

In this paper we prove that points in the space $X(k,n)$ of configurations of $n$ points in $\mathbb{CP}^{k-1}$ which are fixed under a certain cyclic action are the solutions to the generalized scattering equations on planar kinematics (PK). In the first part, we give a constructive upper bound: we show that these solutions inject into certain aperiodic k-element subsets of $\{1,\ldots, n\}$, and consequently that their number is bounded above by the number of Lyndon words with k one's and n-k zeros. The proof uses a somewhat surprising connection between the superpotential of the mirror of $G(n-k,n)$ and the generalized CHY potential on $X(k,n)$. We also check the recent conjecture that generalized biadjoint amplitudes evaluate to $k$-dimensional Catalan numbers on PK for several examples including $k=3$ and $n\leq 40$ and $(k,n)=(6,13)$. We then reformulate the CEGM generalized biadjoint scalar amplitude directly as a Laplace transform-type integral over ${\rm Trop}^+ G(k,n)$ and we use it to evaluate the amplitude on PK with the purpose of exhibiting how GFD's glue together. We initiate the study of two minimal lattice polytopal neighborhoods of the planar kinematics point. One of these, the rank-graded root polytope $\mathcal{R}_{k,n}$, in the case $k=2$, is a projection of the standard type A root polytope. The other, denoted $\Pi_{k,n}$, in the case $k=2$, is a degeneration of the associahedron. We check up to and including $\mathcal{R}_{3,9}$ and $\mathcal{R}_{4,9}$ that the relative volume of $\mathcal{R}_{k,n}$ is the multi-dimensional Catalan number $C^{(k)}_{n-k}$, hinting towards the possibility of deeper geometric and combinatorial interpretations of $m^{(k)}(\mathbb{I}_n,\mathbb{I}_n)$ near the PK point.