Moduli Space of Paired Punctures, Cyclohedra and Particle Pairs on a Circle

Abstract

In this paper, we study a new moduli space ${\mathcal{M}}_{n+1}^{\mathrm{c}}$, which is obtained from ${\mathcal{M}}_{0,2n+2}$ by identifying pairs of punctures. We find that this space is tiled by $2^{n-1}n!$ cyclohedra, and construct the canonical form for each chamber. We also find the corresponding Koba-Nielsen factor can be viewed as the potential of the system of $n+1$ pairs of particles on a circle, which is similar to the original case of ${\mathcal{M}}_{0,n}$ where the system is $n-3$ particles on a line. We investigate the intersection numbers of chambers equipped with Koba-Nielsen factors. Then we construct cyclohedra in kinematic space and show that the scattering equations serve as a map between the interior of worldsheet cyclohedron and kinematic cyclohedron. Finally, we briefly discuss string-like integrals over such moduli space.

23 pages, 7 figures

References in corpus (3)

• Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals
• Combinatorics and Topology of Kawai-Lewellen-Tye Relations
• Non-abelian $Z$-theory: Berends-Giele recursion for the ${\alpha }^{\prime }$-expansion of disk integrals

Cited by in corpus (2)

• Kinematic Jacobi Identity is a Residue Theorem: Geometry of Color-Kinematics Duality for Gauge and Gravity Amplitudes
• Stringy canonical forms and binary geometries from associahedra, cyclohedra and generalized permutohedra