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

TL;DR

This paper introduces a new moduli space formed by pairing punctures on a sphere, revealing its tiling by cyclohedra, and explores its geometric and physical implications for particle systems on a circle.

Contribution

It constructs the moduli space $ ext{M}_{n+1}^{c}$, identifies its tiling by cyclohedra, and relates it to particle systems and scattering equations, extending the understanding of moduli spaces in string theory.

Findings

The moduli space is tiled by $2^{n-1}n!$ cyclohedra.

The Koba-Nielsen factor acts as a potential for particle pairs on a circle.

Scattering equations map the worldsheet cyclohedron to kinematic cyclohedron.

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.