Planar kinematics: cyclic fixed points, mirror superpotential, -dimensional Catalan numbers, and root polytopes

  • Freddy Cachazo

    Perimeter Institute for Theoretical Physics, Waterloo, Canada
  • Nick Early

    Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany
Planar kinematics: cyclic fixed points, mirror superpotential, $k$-dimensional Catalan numbers, and root polytopes cover

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Abstract

In this paper, we prove that points in the space of configurations of points in 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 -element subsets of , and consequently that their number is bounded above by the number of Lyndon words with ones and zeros. The proof uses a somewhat surprising connection between the superpotential of the mirror of and the generalized CHY potential on . We also check the recent conjecture that generalized biadjoint amplitudes evaluate to -dimensional Catalan numbers on PK for several examples including and and . We then reformulate the CEGM generalized biadjoint scalar amplitude directly as a Laplace transform-type integral over , and we use it to evaluate the amplitude on PK with the purpose of exhibiting how generalized Feynman diagrams 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 , in the case , is a projection of the standard type A root polytope. The other, denoted by , in the case , is a degeneration of the associahedron. We check up to and including and that the relative volume of is the multi-dimensional Catalan number , hinting towards the possibility of deeper geometric and combinatorial interpretations of near the PK point.

Cite this article

Freddy Cachazo, Nick Early, Planar kinematics: cyclic fixed points, mirror superpotential, -dimensional Catalan numbers, and root polytopes. Ann. Inst. Henri Poincaré Comb. Phys. Interact. (2024), published online first

DOI 10.4171/AIHPD/185