Moduli space of paired punctures, cyclohedra and particle pairs on a circle

A bstract In this paper, we study a new moduli space ℳ n + 1 c , which is obtained from ℳ 0 , 2 n + 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 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.