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

In this paper, we study a new moduli space Mn+1c</mml:msubsup>, which is obtained from M0,2n+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 <mml:msub>M0,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.