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

  • Zhenjie Li
  • Chi ZhangEmail author
Open Access
Regular Article - Theoretical Physics


In this paper, we study a new moduli space \( {\mathrm{\mathcal{M}}}_{n+1}^{\mathrm{c}} \), which is obtained from \( {\mathrm{\mathcal{M}}}_{0,2n+2} \) by identifying pairs of punctures. We find that this space is tiled by 2n − 1n! 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 \( {\mathrm{\mathcal{M}}}_{0,n} \) 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.


Scattering Amplitudes Differential and Algebraic Geometry Bosonic Strings 


Open Access

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Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.CAS Key Laboratory of Theoretical Physics, Institute of Theoretical PhysicsChinese Academy of SciencesBeijingChina
  2. 2.School of Physical SciencesUniversity of Chinese Academy of SciencesBeijingChina

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