Moduli space of paired punctures, cyclohedra and particle pairs on a circle
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Springer
Received: February 22, 2019 Accepted: April 26, 2019 Published: May 6, 2019
Zhenjie Li and Chi Zhang CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China School of Physical Sciences, University of Chinese Academy of Sciences, No. 19A Yuquan Road, Beijing 100049, China
E-mail: [email protected], [email protected] Abstract: In this paper, we study a new moduli space Mcn+1 , 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 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. Keywords: Scattering Amplitudes, Differential and Algebraic Geometry, Bosonic Strings ArXiv ePrint: 1812.10727
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP05(2019)029
JHEP05(2019)029
Moduli space of paired punctures, cyclohedra and particle pairs on a circle
Contents 1 Introduction
1
2 Moduli space of pairs of punctures on the Riemann sphere 2.1 Real moduli space Mcn+1 (R) and cyclohedron Wn 2.2 Parke-Taylor forms as canonical forms of moduli space cyclohedra 2.3 Koba-Nielsen factor and scattering equations: particle pairs on a circle
2 3 6 8 10
4 Kinematic cyclohedra and CHY formula 4.1 Kinematic cyclohedra 4.2 Scattering equations as a map between cyclohedra
13 13 16
5 Z-integrals on the moduli space Mcn+1 (R)
18
6 Outlook
20
1
Introduction
The scattering processes are closely linked to the Riemann surfaces with punctures since the birth of string theory. In particular, the scattering of n massless particles can be described as a localized integral over some moduli spaces in the context of Witten-RSV formalism [1, 2] or CHY formalism [3–5]. In other words, the tree-level S-matrix of massless particles can be computed based on a map from some moduli space to kinematic space, which in general dimension is provided by the so-called scattering equations X ki · kj =0 zi − z j
for i ∈ {1, . . . , n}.
j6=i
This map itself has been studied in detail in a new framework [6] and recast as a pushforward from differential forms on the moduli space M0,n , which are called worldsheet forms, to differential forms on the kinematic space of n massless particles, which are called scattering forms. The combinatorial and geometrical aspects of moduli space and kinematic space become hence crucial in this context. Such ideas have been further developed by considering extended logarithmic differen
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