Moduli Space of Paired Punctures, Cyclohedra and Particle Pairs on a Circle
Zhenjie Li Chi Zhang

a,b]Zhenjie Li
a,b], Chi Zhang
[a]CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
[b]School of Physical Sciences, University of Chinese Academy of Sciences, No.19A Yuquan Road, Beijing 100049, China
lizhenjie@itp.ac.cn
zhangchi@itp.ac.cn

In this paper, we study a new moduli space ℳ_n+1^c, which is obtained from ℳ_0,2n+2 by identifying pairs of punctures. We find that this space is tiled by 2^n-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 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.

	Statistical Modeling of Atmospheric Mean Temperature in sub-Sahel West Africa
    [
    December 30, 2023
=============================================================================



§ 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 <cit.> or CHY formalism <cit.>. 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

    ∑_j≠ ik_i· k_j/z_i-z_j = 0        for i∈{1,…,n}.
   
This map itself has been studied in detail in a new framework <cit.> and recast as a pushforward from differential forms on the moduli space ℳ_0,n, which are called as worldsheet forms, to differential forms on the kinematic space of n massless particles, which are called as scattering forms. The combinatorial and geometrical aspects of moduli space and kinematic space become hence crucial in this context. Such idea has been further developed by considering extended logarithmic differential forms as in <cit.> and introducing subspace in kinematic space as in <cit.>.

In another closely-related approach to amplitudes—intersection theory <cit.>, the combinatorial and geometrical properties of moduli space also play an important role. In this formalism, scattering amplitudes and related physical qualities are understood as intersection numbers of various twisted cycles and(or) cocycles which characterize the boundary information of the moduli space concerned. In the case of ℳ_0,n, a basis of twisted cocycles turn out to be the Parke-Taylor forms which are precisely the most important worldsheet forms considered in <cit.>.  The combinatorial structure of some physical qualities, for example, the planar bi-adjoint ϕ^3 amplitudes in <cit.> and the inverse of KLT matrix m_α^'(α|β) in <cit.>, are governed by the boundary structure of ℳ_0,n(ℝ), which turn out to be the so-called assocaihedra <cit.>.   


In this article, we will use these two tools—positive geometry <cit.> and intersection theory—to study another interesting moduli space, which is obtained from ℳ_0,2n+2 by imposing identifications for pairs of particles, such as z_i=-z_i+n+1, equipped with the corresponding Koba-Nielsen factor. This is a n-dimensional space and denoted as ℳ_n+1^c in the following, and the corresponding Koba-Nielsen factor can be inherited from the usual Koba-Nielsen factor for 2n+2 particles by imposing the above identifications. In particular, the real moduli space ℳ_n+1^c(ℝ) is tiled by n-dimensional cyclohedra W_n. The main result of this article is to recover this structure in kinematic space via intersection theory and positive geometry. To this end, we first introduce the ingredients in section <ref>, i.e., Parke-Taylor forms and scattering equations, and study their properties further. To do that, We will give the precise definition of ℳ_n+1^c(ℝ) and ℳ_n+1^c+(ℝ) and show:  (i) ℳ_n+1^c(ℝ) is tiled by 2^n-1n! ℳ_n+1^c+(ℝ) which can be identified with a cyclohedron W_n, (ii) the corresponding Parke-Taylor form is the canonical form on ℳ_n+1^c+(ℝ),
and (iii) the scattering equations as the stationary points of a particle system have 2^n-1n! solutions. 

With this knowledge of moduli space ℳ_n+1^c, we move to the kinematic space. The way through intersection theory is quite straightforward, the intersection number of two twisted cocycles α and β tell us the result and the structure of m_α^'(α|β), and shown in section <ref>. While in section <ref>, we will first construct a cyclohedron in the kinematic space, then rewriting the scattering equations as a map from the moduli cyclohedron to the kinematic cyclohedron, as done in <cit.>. As a result of this pushforward, the amplitudes m(α|β) can be extracted from the scattering form by pullbacking to the corresponding subspace. It becomes obvious that m(α|β) is given by m_α^'(α|β) in the limit of α^'→ 0.

Last but not the least, we will consider the natural integrals on such moduli space that are analogs of the so-called Z-integrals for open-string case <cit.>.  Interestingly, the Koba-Nielsen factor for integrals of open strings and one closed-string insertion is a special limit of our case. However, this limit needs to be taken carefully since some integrals don't behave well under this limit. Here we just give a brief survey on its α^'-expansion and number theoretical properties before taking this limit.

	

§ MODULI SPACE OF PAIRS OF PUNCTURES ON THE RIEMANN SPHERE
 

In this section, we consider the moduli space of distinct pairs of punctures (z,z') on the Riemann sphere ℂℙ^1, 
where two punctures in each pair are related by z'=-z. 

It's clear that an arbitrary SL(2,) transformation on the Riemann sphere 
will not keep two punctures z and z' in a pair still satisfying the relation that z'=-z. One can check that the surviving transformation is

    z↦λ z   and   
        z↦1/z,

where λ is a non-zero complex number. Therefore, the remaining SL(2,) 
effectively is the group (-{0}) ⋊_2 whose elements are transformations

    (λ,a):z↦λ z^a,

where λ∈ (-{0}) and a=± 1.[ 
It's the simi-product of (-{0}) and _2 since the multiplication rule is (λ_1,a_1)· (λ_2,a_2)=(λ_1λ_2^a_1,a_1a_2).]

Now, suppose there are n+1 distinct such pairs of two punctures
{(z_i,z'_i) :  0≤ i≤ n} on ℂℙ^1. 
Our complex moduli space is the quotient of this configurations space, modulo the remaining redundancy (-{0}) ⋊_2. 
Let's denote it by ℳ^c_n+1(ℂ). 
For future convenience, it's useful to enlarge the
index set to _2n+2={0,…,n,n+1,…,2n+1} and
identify z'_i with z_i+n+1 or z_,
where =i+n+1∈_2n+2.



 §.§ Real Moduli Space M(n+1) and Cyclohedron W(n)
 

In the context of string theory, the moduli space ℳ_0,n() of
n distinct punctures on the Riemann sphere ℂℙ^1 and its real part ℳ_0,n() of n distinct punctures 
on the circle (i.e. the boundary of the disk) ℝℙ^1=S^1
are used to calculate the tree-level amplitude of closed strings and open strings respectively. 

Similarly, we can consider the real part of our complex moduli space ℳ_n+1^c(ℂ)
by setting all these pairs of punctures on the unit circle of . 

We denote this new real moduli space by ℳ^c_n+1(ℝ).
As in the complex case, the SL(2,ℝ) redundancy now
reduces to U(1)⋊_2, which is generated by

    z↦exp(θ)z
          and  
        z↦ 1/z=z,

where θ∈ and |z|=1. Since all punctures now live on the unit circle, it's very convenient to use their arguments 
θ_i to coordinate them, i.e., z_i=exp(θ_i). 
Besides, it's also convenient to introduce another set of useful real variables

    x_i=tan(θ_i/2)= 1-z_i/1+z_i.

In fact, it's the natrual coordinates of another possible great circle ℝ∪{∞} on the Riemann sphere where pairs of punctures can live.


A lot of works on ℳ_0,n() has revealed some 
hidden structures of this moduli space. For example, 
its Fulton-MacPherson compactification can be tiled by associahedra <cit.>.
In the rest of this subsection, we will show some similar structures on ℳ^c_n+1(ℝ).

Firstly, we fix the U(1)⋊_2 redundancy
by setting z_0=-z_0̃=1 and limiting z_1 on the upper semi-circle S^1_+. 
Since either z or -z is on the upper semi-circle for any pairs of punctures (z,-z)≠ (1,-1) on the unit circle,
one can use the one living on the S^1_+ to represent this pair.
Therefore, after the gauge fixing, 
ℳ^c_n+1(ℝ) is described by

    ℳ^c_n+1(ℝ)={(z_1,± z_2,…, ± z_n)
        ∈ (S_+^1)^n}-Δ,

where 

    Δ={(z_1,± z_2,…, ± z_n)∈ (S_+^1)^n : ∃  i,j  s.t.  z_i^2=z_j^2  or 
    z_i^2=1}.


One should exclude Δ because that points in this subset represent some pairs of punctures on the unit circle collide.

Since there are two choices z_i and -z_i for all 2≤ i≤ n, 
ℳ_n+1^c(ℝ) is a 2^n-1 copies of the following space

    {(p_1,p_2,…, p_n)∈ (S_+^1)^n}-Δ
        ={(1,p^2_1,p^2_2,…, p^2_n)∈ (S^1)^n+1}-Δ',

where 

    Δ'={(1,q_1,q_2,…, q_n)∈ (S^1)^n+1 : ∃  i,j  s.t. q_i=q_j or q_i=1},

This is a known moduli space F(S^1,n+1)/S^1 which has been well investigated in <cit.>. Therefore,
our real moduli space ℳ_n+1^c(ℝ) is 
2^n-1F(S^1,n+1)/S^1.


The real Fulton-MacPherson compactification of
F(S^1,n+1)/S^1, denoted by 𝒵^n+1, can be tiled by n!
cyclohedra <cit.>, so the same compactification of our moduli space
can be tiled by 2^n-1n! cyclohedra.
Suppose 𝗉𝗋𝗈𝗃:𝒵^n+1→F(S^1,n+1)/S^1 
is the compactification map, every cyclohedron is given by

    W_n(σ)=
        𝗉𝗋𝗈𝗃^-1(
        {0<φ_σ(1)<φ_σ(2)<⋯<φ_σ(n)<2π}),

where φ_i is the central angle of p^2_i and σ∈ S_n is a permutation of indices set {1,…,n}. In our case, every cyclohedron is given by

    W_n(σ,{𝗌_i})   =
    𝗉𝗋𝗈𝗃^-1(
    {0<θ^𝗌_σ(1)_σ(1)<θ^𝗌_σ(2)_σ(2)<⋯<θ^𝗌_σ(n)_σ(n)<π}), 
       =:𝗉𝗋𝗈𝗃^-1(ℳ_n+1^c+(σ,{𝗌_i}))

where 𝗌_i=± 1 and 

    θ^+_i=θ_i,  θ^-_i=θ_i-π.


From this construction, we can conclude that facets of the cyclohedron ℳ_n+1^c+(σ,{𝗌_i})
(before compactification) lay on hyperplanes defined by equations
𝗌_σ(i) z_σ(i)-𝗌_σ(i+1) z_σ(i+1)=0.
In terms of {θ_i}, these hyperplane equations are

    0≤θ_i < 2π  and  θ_i-θ_j=
            0,          if 𝗌_i-𝗌_j=0   2,
    ±π,    if 𝗌_i-𝗌_j=± 1.


For example, hyperplanes for n=2 are shown in the following diagram:


    [scale=2]
        [fill=green!20] (0,0) – (0,2) – (1,2) – (1,0)
        – (0,0);
        [thick,->] (0,0) – (2.5,0) node[right] θ_1;
        [thick,->] (0,0) – (0,2.5) node[above] θ_2;
        [thick] (0,0) – (2,2);
        [thick] (0,0) node[below] 0
        – (1,0) node[below=2pt] π
        – (2,0) node[below] 2π
        – (2,2)
        – (0,2) node[left] 2π
        – (0,1) node[left] π
        – (0,0);
        [thick] (1,0) – (2,1) – (0,1)
        – (1,2) – (1,0);
    


with the identification (θ_1,θ_2)∼ (2π-θ_1,2π-θ_2) due to the _2 redundancy.

As before, we can fix this redundancy by setting θ_1 < π,
so there are 4=2^1× 2! chambers (green ones in the above
diagram) corresponding to the cyclohedron before compactification.
After compactification, each chamber becomes a hexagon, the 2-dimensional cyclohedron.
For example, the triangle 0< θ_1 <θ_2<π actually becomes


    [scale=2]
        [thick] (0,0.2) – (0,1.8) – (0.2,2) –
        (1.8,2) – (2,1.8) – (0.2,0) – (0,0.2);
        (1,1) node[below=10pt,right=10pt] θ_1^+=θ_2^+;
        (1,2) node[above] θ_2^-=0;
        (0,1) node[left] θ_1^+=0;
        [thick,red] (0,1.8) – (0.2,2);
        [thick,red] (1.8,2) – (2,1.8);
        [thick,red] (0.2,0) – (0,0.2);
        [thick,red,->] (-0.3,2.1) node[left,black] θ_1^+=θ_2^-=0
        .. controls (-0.2,2.2) and (-0.1,2.1) .. (0.05,1.95);
        [thick,red,->] (-0.3,-0.1) node[left,black] θ_1^+=θ_2^+=0
        .. controls (-0.2,-0.2) and (-0.1,-0.1) .. (0.05,0.05);
        [thick,red,->] (2.3,2.1) node[right,black] θ_1^-=θ_2^-=0
        .. controls (2.1,2.2) and (2.05,2.1) .. (1.95,1.95);
    


where three red facets are the result of compactification.
Notice that, every vertex should be blowed-up in the compactification such that no two cyclohedra intersect at a point. 

For general n, we can construct W_n(σ,{𝗌_i}) from the corresponding ℳ_n+1^c+(σ,{𝗌_i}) and the compactification of the whole space ℳ^c_n+1(ℝ) by the following steps:

    
  * For every chamber before compactification, 
        label (ij) on 
        each codimension-one face (facet) H defined by z_i=z_j. 
    
  * For the face H with higher codimension k defined 
        by z_i_1=⋯=z_i_k of these chambers, 
        truncate it and label 
        (i_1i_2⋯ i_k) on the new created facet. 
        Repeat this operation from low-dimensional faces to high-dimensional faces. 
    
  * Glue the faces with the same label in different cyclohedron,
        they will be the same face after compactification.


Here we give a direct combinatorial definition of cyclohedron. 
For any polytope P, there's a natural partial order of faces on it: for 
face a and b,

    a≤ b   if and only if   a⊂ b,

then P defined a partially ordered set (or poset for short) (P,≤). 
Conversely, we can use a poset to define a polytope whose associated
poset of faces is isomorphic to the given poset. 




[Cyclohedron]
Let Cyc(n) be the poset of all centrally symmetric dissections 
of a convex (2n+2)-gon using non-intersecting diagonals, 
ordered such that b ≤  b' if  b is obtained from b' by 
adding (non-intersecting) diagonals. The n-dimensional cyclohedron W_n is 
convex polygon whose face poset is isomorphic to Cyc(n). 


For example, W_2 is a hexagon.


    #1#2
        
            [draw,minimum size=0.6cm,regular polygon,regular polygon sides=6] (a) ;
            (a.corner #1) – (a.corner #2);
        
    
    #1#2#3#4
        
            [draw,minimum size=0.6cm,regular polygon,regular polygon sides=6] (a) ;
            (a.corner #1) – (a.corner #2);
            (a.corner #3) – (a.corner #4);
        
    
    #1#2#3#4
        
            [draw,minimum size=0.6cm,regular polygon,regular polygon sides=6] (a) ;
            (a.corner #1) – (a.corner #2);
            (a.corner #3) – (a.corner #4);
            (a.corner #1) – (a.corner #3);
        
    
    [baseline=([yshift=-.5ex]current bounding box.center)]
        [draw,thick,minimum size=3cm,regular polygon,regular polygon sides=6] (a) ;
        (p1) at (60:2cm);
        (p2) at (60*2:2cm);
        (p3) at (60*3:2cm);
        (p4) at (60*4:2cm);
        (p5) at (60*5:2cm);
        (p6) at (60*6:2cm);
        (p1) node 1346;
        (p2) node 3164;
        (p3) node 3562;
        (p4) node 5326;
        (p5) node 5124;
        (p6) node 1542;
        ((p1)!0.5!(p2)) node 1346;
        ((p2)!0.5!(p3)) node 36;
        ((p3)!0.5!(p4)) node 3562;
        ((p4)!0.5!(p5)) node 52;
        ((p5)!0.5!(p6)) node 5124;
        ((p6)!0.5!(p1)) node 14;
        [draw,minimum size=0.6cm,regular polygon,regular polygon sides=6] (a) ;
    



The definitive property of cyclohedron is its geometric factorization <cit.> that every facet of a n-dimensional cyclohedron W_n is the product of a (n-i)-dimensional cyclohedron W_n-i and a (i-1)-dimensional associahedron A_i-1, i.e.

    H≅ W_n-i× A_i-1,

where H is a facet of W_n and i is a positive integer.
In the following sections, we will see it again and again in different forms.



 §.§ Parke-Taylor Forms as Canonical Forms of Moduli Space Cyclohedra


As shown in the last subsection, the interior of a cyclohedron W_n(σ,{𝗌_i}) is a 
simplex whose facets lay on the hyperplanes defined by equations 

    𝗌_σ(i) z_σ(i)- 𝗌_σ(i+1) z_σ(i+1)=0.

Therefore, we can write down the Parke-Taylor form
as the canonical form <cit.> of this simplex that

    𝖯𝖳_n(
        σ(1)^𝗌_σ(1),…,
        σ(n)^𝗌_σ(n)):= 2z_0  (𝗌_1z_1)∧⋯∧ (𝗌_nz_n)/(z_0- 𝗌_σ(1) z_σ(1))(𝗌_σ(1) z_σ(1)- 𝗌_σ(2) z_σ(2))⋯(𝗌_σ(n) z_σ(n)+z_0) ,

The appearance of z_0 on the numerator makes 𝖯𝖳_n invariant under








(λ,a):z↦λ z^a for
all (λ,a)∈ (-{0})⋊_2.
As revealed in <cit.>, Parke-Taylor forms can be rewritten as

    𝖯𝖳_n   (σ(1)^𝗌_σ(1),…,
    σ(n)^𝗌_σ(n))
       =(-1)^nsgn(σ) log( z_0-𝗌_σ(1) z_σ(1)/𝗌_σ(1) z_σ(1)- 𝗌_σ(2) z_σ(2))∧⋯∧log( 𝗌_σ(n-1) z_σ(n-1)- 𝗌_σ(n) z_σ(n)/𝗌_σ(n) z_σ(n)+z_0),

so they are log forms.







What's more, Parke-Taylor forms 
still work well after compactification (locally it's a series of blowing-ups), so
they are also the canonical forms of cyclohedra
[
More precisely, it is the pull-back of Parke-Taylor forms according to the compactification map that are the canonical forms of cyclohedra.
]
. Since for every facet
of a cyclohedron, it can be factorized into a cyclohedron and an associahedron, the residue of 
the canonical form of cyclohedron at the facet should also be
factorized into the corresponding canonical forms.

To see it, consider the facet of 
W_n(id,{+}) whose label is 
(i⋯ j). It corresponds to the blowing-up
of the pinch of the coordinates {z_i,…,z_j} of the moduli space.
Let z_k=z_i+ty_k for i+1≤ k≤ j, where [y_i+1,…,y_j] 
are projective coordinates of ℙ^j-i-1, then 
the residue of 𝖯𝖳_n(1^+2^+⋯ n^+) at t=0 
can be written as

    Res_t=0𝖯𝖳_n(1^+2^+⋯ n^+)   =Res_t=0[(-1)^nlog( z_0- z_1/ z_1-  z_2)∧⋯∧log(  z_n-1- z_n/ z_n+z_0)]
       =(±)  𝖯𝖳_n-(j-i)(1^+2^+⋯ i^+(j+1)^+⋯ n^+)∧Ω,

here 𝖯𝖳_n-(j-i) is the canonical form of a cyclohedron, while

    Ω=log( 0- y_i+1/y_i+1- y_i+2)∧⋯∧log( y_j-2-y_j-1/y_j-1-y_j)

is the canonical form of an associahedron up to a sign since it can be rewritten as

    Ω=(±)  y_i+1∧⋯∧ y_j-1/(0- y_i+1)(y_i+1- y_i+2)⋯ (y_j-1-1).

in the piece of ℙ^j-i-1 where y_j=1.

Apparently, there are 2^nn! different Parke-Taylor forms 
corresponding to different cyclohedra. Like the case of associahedra, there are some linear relations between these Parke-Taylor forms. A simple relation is

    𝖯𝖳_n(
        σ(1)^𝗌_σ(1),…,
        σ(n)^𝗌_σ(n))
        =(-1)^n𝖯𝖳_n(
        σ(n)^-𝗌_σ(n),…,
        σ(1)^-𝗌_σ(1)),

some not so trivial relations are

    𝖯𝖳_n   (1^+2^𝗌_2… n^𝗌_n)
            +𝖯𝖳_n(2^𝗌_21^+… n^𝗌_n)
            +⋯ +
            𝖯𝖳_n(2^𝗌_2… n^𝗌_n1^+)+
            
       𝖯𝖳_n(1^-2^𝗌_2… n^𝗌_n)
            +𝖯𝖳_n(2^𝗌_21^-… n^𝗌_n)
            +⋯ +
            𝖯𝖳_n(2^𝗌_2… n^𝗌_n1^-)=0

and its permutations. The second kind of relations are analogues of 
Kleiss-Kuijf relations in the case of associahedron. We conjecture that
the rank of these Parke-Taylor forms is (2n-1)!! because of these relations.

To end this subsection, we express Parke-Taylor forms
in terms of {x_i} and {θ_i}. 
They live on the real line, so it's more convenient
for some purpose or other. In terms of {x_i}, 
Parke-Taylor forms are written as

    𝖯𝖳(
        σ(1)^𝗌_σ(1),…,
        σ(n)^𝗌_σ(n))=
        (x_0+x_0^-1)  (𝗌_1x^𝗌_1_1)∧⋯∧ (𝗌_nx^𝗌_n_n)/
        (x_0-𝗌_σ(1)x_σ(1)^𝗌_σ(1))⋯ 
        (𝗌_σ(n)x_σ(n)^𝗌_σ(n)+x_0^-1)
        ,

where s_i=± 1.











Then it's direct to carry out Parke-Taylor forms
in terms of {θ_i^𝗌_i} by using 
𝗌_ix_i^𝗌_i=tan(θ^𝗌_i/2),

    𝖯𝖳(
        σ(1)^𝗌_σ(1),…,
        σ(n)^𝗌_σ(n))=
        sin(θ^𝗌_0,n+1/2)  θ_1∧⋯∧θ_n/sin(θ^𝗌_0,σ(1)/2)
        ⋯sin(θ^𝗌_σ(n-1),σ(n)/2)sin(θ^𝗌_σ(n),n+1/2),

where θ^𝗌_i,j:=θ_i^𝗌_i-θ_j^𝗌_j.



 §.§ Koba-Nielsen Factor and Scattering Equations: Particle Pairs on a Circle


In order to connect to the kinematic space, we should introduce 
`Mandelstam variables' s_ij for any pair of punctures with the usual properties, i.e. s_ii=0, s_ij=s_ji 
and the conservation of momentum

    ∑_j=0^2n+1s_ij=0,

but with additional identifications s_=s_ij since z_i-z_j=0 and z_-z_=0 are the same hyperplane in the moduli space. It's natural to write down a factor

    ℐ=∏_0≤ i<j≤ 2n+1(z_i-z_j)^α' s_ij

as the analogue of the so-called Koba-Nielsen factor in string theory. However, with the new identification s_ij=s_ of Mandelstam variables, this factor ℐ can be rewritten to drop this redundancy. For future use, it's convenient to rewrite it as

    ℐ=∏_0≤ i<j≤ nsin(θ_i-θ_j/2)^2α' s_ijcos(θ_i-θ_j/2)^2α' s_i

in terms of {θ_i}, or,

    ℐ   =  ∏_0≤ i<j≤ n((x_i-x_j)^2/(1+x_i^2)(1+x_j^2))^s_ij((1+x_ix_j)^2/(1+x_i^2)(1+x_j^2))^s_i

in terms of {x_i}.

As shown in <cit.>, the number of stationary points of the Koba-Nielsen factor
equals to the number of associahedra in the moduli space ℳ_0,n(ℝ), which is a highly non-trivial result because of the Poincaré dual. Therefore, 
it's believed that the number of stationary points of our new Koba-Nielsen
factor eq.(<ref>)
should equal the number of cyclohedra 2^n-1n!. In the rest of this
subsection, we will prove that it's indeed the number of stationary points.

The stationary points of Koba-Nielsen factor is determined by 
the equation log(ℐ)=0,
then it gives a set of equations in z-variables, which are 
called scattering equations,

    E_i:=∂/∂ z_ilog(ℐ)
    =∑_0≤ j≤ 2n+1
     j≠ is_ij/z_i-z_j=0 ,

for i=1, …, n. Or in terms of θ-variables, 
from eq.(<ref>), the scattering equations are

    ∑_0≤ j≤ n
     j≠ i(s_ij(θ_ij/2)θ_i-s_itan(θ_ij/2))=0 ,

where θ_ij=θ_i-θ_j.











This number 2^n-1n! can be recursively verified by taking soft limits <cit.>. However, here we use another strategy proposed by Cachazo, Mizera and Zhang <cit.>: each solution of scattering equation can be regarded as one stationary configuration of some particular particle system on some region of kinematic space 𝒦_+.  


In our case, the region of kinematic space 𝒦_+ is defined by all s_ij and s_i with j≠ i are positive, this particle system is a 2-dimensional system and consists of n+1 pairs of particles connected by n+1 light rigid rods of unit length (see figure <ref>), where the centers of all rods are in the same position and each pair of particles connected by a rod have the same charge. If we denote the i^th pair of particles as i and  and the charge product of particles i and j as s_ij, then the potential of this system is simply

    V(θ)= -2∑_0≤ i<j≤ n( s_ijlog|sin(θ_ij/2) |+ s_ilog|cos(θ_ij/2) |)
     .

Here we take no account of the all contribution from particle pairs connected by the rigid rods, and the factor 2 arise from the fact of s_ij=s_ and s_i=s_j. Since all particles mutually repel on the region 𝒦_+, it is obvious that the solutions of scattering equation eq.(<ref>) are stationary points of the potential eq.(<ref>) on this region, and all these solutions in terms of θ-variables hence are real.

Next, we take a simple counting of stationary configurations of this system. A naive counting gives 2^n+1(n+1)! since these rods have two distinct endpoints and different orderings of these rods correspond different stationary configurations. However the symmetry U(1)⋊_2 mentioned above will reduce this number to (2^nn!)/2. (One rod can be taken as the reference point, and only one-half of all these orderings are indeed different since this is a central symmetric system.)

	

§ INTERSECTION NUMBERS
 

In this section, we consider the new geometric structures of 
the moduli space ℳ_n+1^c() after introducing the 
Koba-Nielsen factor. 

Since Koba-Nielsen factor 
eq.(<ref>) or eq.(<ref>) is not a well-defined 
single-valued function on ℳ_n+1^c() 
whose branch points are the hyperplains defined by z_i-z_j=0 for
all 0≤ i<j≤ 2n+1, one should assign a branch of Koba-Nielsen factor on each chamber
for the sake of certainty. Therefore, 
we can define twisted chambers
Δ⊗ℐ_Δ as our basic objects, where 
Δ is a chamber and ℐ_Δ is a given branch of ℐ on Δ. The same consideration still works for 
the compactification of ℳ_n+1^c(). 

Generally, one can consider a twisted m-simplex
σ⊗ℐ_σ,
where σ is a topological m-simplex in ℳ_n+1^c(),
then there is a natural inherited boundary map ∂_ℐ for
any twisted simplex. 
For example, suppose σ=⟨ 01⋯ m⟩ is a topological m-simplex, where ⟨ 01⋯ m⟩ is the standard notation <cit.>, the traditional boundary map is 

    ∂^m σ=∑_k=0^m (-1)^k⟨ 01⋯k̂⋯ m⟩,

and the boundary map ∂^m_ℐ for
any twisted simplex σ⊗ℐ_σ is

    ∂_ℐ^m (σ⊗ℐ_σ)=∑_k=0^m (-1)^k⟨ 01⋯k̂⋯ m⟩⊗ I_σ|_⟨ 01⋯k̂⋯ m⟩.

It's not hard to check that 
∂^m-1_ℐ∂^m_ℐ=0, so there's a 
natural homology group defined by 

    H_m(ℳ_n+1^c(),∂_ℐ)
    	=∂^m-1_ℐ/im∂^m_ℐ.


Kita and Yoshida <cit.> investigated these objects for a general space and
a general factor. They developed a homology theory according to these
twisted simplexes and defined a new intersection number of two twisted 
cycles. Loosely speaking, the intersection number is a pairing

    H_n(ℳ_n+1^c(),∂_ℐ)×
    	H_n(ℳ_n+1^c(),∂_ℐ)
    	→ℂ.

Sabastian <cit.> showed that 
the intersection numbers of associahedra (with chosen
branches of Koba-Nielsen factor) on the moduli space form
the Kawai-Lewellen-Tye (KLT) kernel <cit.> which relates
the tree-level open string amplitude and closed string amplitude. What's 
more, its α'→ 0 limit gives the amplitude of 
bi-adjoint φ^3-theory  <cit.> which relates to the 
Cachazo-He-Yuan (CHY) formula. In this section, 
we consider intersection numbers of cyclohedra. 

However, we will not go into the details of the whole geometric theory. Instead of defining intersection number step by step <cit.>, we directly give a formula eq.(<ref>) of two cyclohedra with the chosen branches |ℐ|.

Suppose Δ(α) and Δ(β) are two 
cyclohedra, K is their shared face with
the lowest codimension k, and they are on the different side
of K. Suppose K = H_1 ∩⋯∩  H_k is the 
intersection of k facets, and the chosen branches of ℐ
on Δ(α) and Δ(β) is 

    |ℐ| = f(z) ∏_0≤ i<j≤ n|z_i-z_j|^2α' s_ij|z_i-z_|^2α' s_i,

where the factor f(z) will not bother us since it never vanishes on ℳ_n+1^c(). 
Then the intersection number <cit.> of Δ(α) and Δ(β) is defined by 

    ⟨Δ(α),Δ(β) ⟩ :=
    	(1/2)^k∏_i=1^k1/sin(α'π s_H_i)∑_faces H of K1/d_H

where s_H_i=2s_i_1,…,i_k=2∑_i_1≤ p<q≤ i_ks_pq for H_i whose 
label is (i_1⋯ i_k) and 

    d_⋂_i H_i=∏_i (
    		exp(2πα' s_H_i)-1)   and   d_Δ(α)=1.





For example, 

    Δ(1^+)   =
    	[scale=1,
    		baseline=([yshift=-.5ex]current bounding box.center)]
    	[thick] (0,0) – (2,0);
    	fill=black, inner sep=1pt,
    	[red,thick] (0,0) circle (1pt)
    	node[left,black] (01);
    	[red,thick] (2,0) circle (1pt)
    	node[right,black] (01̃);
    
    Δ(1^+2^+)   =
    	[scale=1,
    		baseline=([yshift=-.5ex]current bounding box.center)]
    		[thick] (0,0.2) – (0,1.8) – (0.2,2) –
    		(1.8,2) – (2,1.8) – (0.2,0) – (0,0.2);
    		 (1,1) node[below=8pt,right=8pt] (12);
    		 (1,2) node[above] (02̃);
    		 (0,1) node[left] (01);
    		[thick,red] (0,1.8) – (0.2,2);
    		[thick,red] (1.8,2) – (2,1.8);
    		[thick,red] (0.2,0) – (0,0.2);
    		[thick,red,->] (-0.3,2.1) node[left,black] (012̃)
    		.. controls (-0.2,2.2) and (-0.1,2.1) .. (0.05,1.95);
    		[thick,red,->] (-0.3,-0.1) node[left,black] (012)
    		.. controls (-0.2,-0.2) and (-0.1,-0.1) .. (0.05,0.05);
    		[thick,red,->] (2.3,2.1) node[right,black] (0̃12)
    		.. controls (2.1,2.2) and (2.05,2.1) .. (1.95,1.95);
    	  Δ(1^+2^-)=
    	[scale=1,
    		baseline=([yshift=-.5ex]current bounding box.center)]
    		[thick] (0,0.2) – (0,1.8) – (0.2,2) –
    		(1.8,2) – (2,1.8) – (0.2,0) – (0,0.2);
    		 (1,1) node[below=8pt,right=8pt] (12̃);
    		 (1,2) node[above] (02);
    		 (0,1) node[left] (01);
    		[thick,red] (0,1.8) – (0.2,2);
    		[thick,red] (1.8,2) – (2,1.8);
    		[thick,red] (0.2,0) – (0,0.2);
    		[thick,red,->] (-0.3,2.1) node[left,black] (012)
    		.. controls (-0.2,2.2) and (-0.1,2.1) .. (0.05,1.95);
    		[thick,red,->] (-0.3,-0.1) node[left,black] (012̃)
    		.. controls (-0.2,-0.2) and (-0.1,-0.1) .. (0.05,0.05);
    		[thick,red,->] (2.3,2.1) node[right,black] (01̃2)
    		.. controls (2.1,2.2) and (2.05,2.1) .. (1.95,1.95);

and the self intersection number of Δ(1^+) is

    ⟨Δ(1^+),Δ(1^+)⟩   =
    	1+1/exp(2πα' 2s_01)-1+
    	1/exp(2πα' 2s_01̃)-1
       =1/2(1/tan(α'π 2s_01)
    	+1/tan(α'π 2s_01̃)).

For Δ(1^+2^+) and Δ(1^+2^-), they intersect at 
the face labeled by 2s_01, so

    ⟨Δ(1^+2^+),Δ(1^+2^-)⟩   =1/21/sin(α'π 2s_01)(
    	1+1/exp(2πα' 2s_012)-1+
    1/exp(2πα' 2s_012̃)-1)
       =1/(2)^21/sin(α'π 2s_01)(
    1/tan(α'π 2s_012)
    	+1/tan(α'π 2s_012̃)).


Finally, define the `KLT inverse matrix' by 

    m_α'(α|β)=(2)^n⟨Δ(α),Δ(β)⟩,

where n=Δ(α)=Δ(β), and define

    m(α|β)=lim_α'→ 0(2α'π)^n
    	m_α'(α|β) .

For example,









    m(1^+2^+,1^+2^-)
    =1/s_01(
    	1/s_012+1/s_012̃).

In the next section, we will use CHY formula to calculate m(α|β), which gives another representation.

At the end of this section, we briefly comment on the dual of H_n(ℳ_n+1^c(),∂_ℐ), which is the cohomology group on ℳ_n+1^c() defined by a flat connection

    ∇= + ω_ℐ∧,

with the property ∇^2=0, where 

    ω_ℐ=logℐ=
    	∑_i=1^n E_i  z_i=
    	∑_i=1^n∑_0≤ j≤ 2n+1
     j≠ is_ij/z_i-z_j z_i

is a well-defined single-valued form on ℳ_n+1^c(). 
Let's denote these cohomology groups by 
H^∙(ℳ_n+1^c(),ω_ℐ).
As shown in <cit.>, H^n(ℳ_n+1^c(),ω_ℐ) is generated
by log forms

    log( z_i_1± z_j_1/z_k_1± z_l_1)∧⋯∧log( z_i_n± z_j_n/z_k_n± z_l_n),

where 0≤ i_r,j_r,k_r,l_r≤ n for 
all 1≤ r≤ n, which should be invariant under z↦ 1/z.
Parke-Taylor forms eq.(<ref>) are in these forms.
Unlike the case of ℳ_0,n(ℝ), the independent 
Parke-Taylor forms can no longer generate the whole space
H^n(ℳ_n+1^c(),ω_ℐ). One has to consider more forms, 
for example,

    log(z_1/z_0)∧log(z_2/z_0)
    	= z_1∧ z_2/z_1z_2.

It is invariant under the transformation z↦ 1/z,
and then belongs to H^2(ℳ_3^c(),ω_ℐ). However, it is not the canonical form of a cyclohedron anymore.
    

§ KINEMATIC CYCLOHEDRA AND CHY FORMULA
 
In <cit.>, Arkani-Hamed, Bai, He and Yan have constructed associahedra in kinematic space. They found a set of equations and inequalities
of X-variables

    X_ij:=s_i,i+1,…,j-1=∑_i≤ k<l≤ j-1s_kl,

where the interior of an associahedron is given by {X_ij>0} and every facet represented by X_ij is laying on the hyperplane defined by X_ij=0. They also found that the scattering equations can be interpreted as a map between worldsheet associahedra and kinematic associahedra.
Here, we generalize these stories to cyclohedra. 



 §.§ Kinematic Cyclohedra


Suppose P_n is a (2n+2)-gon, for every diagonal with vertices i and j, we put a variable X_ij on it, 
where labels are living in _2n+2. Thanks to momentum conservation and our identification of 
s_ij and s_, X_ij=X_ji=X_.
Usually, it's more convenient to use X_
=X_ij to represent a pair of centrally symmetric
diagonals (i,j) and (,), where we 
think the longest diagonals (i,)=(,i) as `degenerate' pairs. Therefore, there are n(n+1)
independent X-variables. 
The inverse map from X-variables to
s-variables is given by the following 
identity

    -s_ij=X_ij+X_i+1,j+1-X_i+1,j-X_i,j+1.

For future use, we generalize the above identity to

    -∑_i≤ a<j<k≤ b<ls_ab=X_ik+X_jl-X_jk-X_il.


In order to construct a n-dimensional object from n^2+n
independent variables, we need n^2 constraints, which can be chosen like the old story <cit.>: let

    c_ij=-s_ij=X_i,j+X_i+1,j+1-X_i,j+1-X_i+1,j

be positive constants for 0≤ i<n and i+1<j≤ 2n, 
where we still require that c_i and c_j are the same, so the rank of these constraints is
n^2.
It's very convenient to take into account equations of c_ij
for n< i≤ 2n by identifying c_ 
and c_ij. All in all, we require that s_ij=-c_ij<0 for 

    0≤ i<j-1<j≤ 2n,    i≠ n   and   
        j≠ñ.

Our polytope 𝒲_n is given by inequalities

    X_ij≥ 0   for all 0≤ i<j≤ 2n+1 ,

and setting eq.(<ref>) to be positive constants. Such construction can also obtained from C_n cluster algebra by using the method proposed in <cit.>.
We will show that it is a cyclohedron after a short example.

For n=2, independent variables are
{
        X_02,X_00̃,X_01̃,X_11̃,
        X_12̃, X_22̃},
and constraints are

    X_02+X_01̃-X_00̃   =c_02,
    
        X_00̃+X_11̃-2X_01̃   =c_00̃,
    
        X_01̃+X_12̃-X_11̃   =c_01̃,
    
        X_11̃+X_22̃-2X_12̃   =c_11̃,

where the factor 2 arise form identification and the equation for c_10̃ is the same as the equation for c_01̃. 
Thus, 𝒲_2 is the polytope defined by inequalities

    X_02≥ 0,   X_00̃≥ 0,
           X_01̃≥ 0,   X_11̃≥ 0,
           X_12̃≥ 0,    X_22̃≥ 0.

It's shown in figure <ref> by using X_00̃ and X_02 as the coordinates. Therefore, 𝒲_2 is 
a hexagon, that is a 2 dimensional cyclohedron. Now let's consider the 𝒲_n for general n. 




    𝒲_n is the n-dimensional cyclohedron.



In our construction, every facet of the polytope 𝒲_n is given by setting some
particular X_ij→ 0. However, it's impossible to set arbitrary two X-variables approaching to 0 together. Indeed, recall eq.(<ref>), for j≤ n and l≤ñ,

    X_jk+X_il=X_ik+X_jl-∑_i≤ a<j<k≤ b<lc_ab,

if X_ik, X_jl→ 0, the right-hand side of above equation will
be negative, but the left-hand side will never be negative, so there is a contradiction. 
It follows that we cannot reach a lower dimensional boundary of 𝒲_n by setting two intersecting diagonals to zero. 
Conversely, lower dimensional boundaries of 𝒲_n can only be reached by setting non-intersecting diagonals to zero. 
On the other hand,
the dissections of (2n+2)-gon by setting non-intersecting diagonals to zero are always centrally symmetric since X_ij=X_. Therefore, 𝒲_n is a n-dimensional cyclohedron according to Definition <ref>.


It's very natural to construct other cyclohedra 𝒲(σ,{𝗌_i}) in kinematic space. The data {𝗌_1,𝗌_2,…,𝗌_n} and permutation σ define an centrally symmetric ordering in S^2n+2, and vice versa. For example,
(2^+1^-3^+) ↔ (021̃30̃2̃ 13̃).
Therefore, we identify these two notations of centrally symmetric orderings. Now, we can defined X-variables in ordering α by setting

    X_α(i),α(j)=s_α(i),α(i+1),…,α(j-1).

Then 𝒲_n(α) in kinematic space is similarly given by setting 

    H_n(α)={c_α(i)α(j) =positive constant |  0≤ i<j-1<j≤ 2n, i≠ n  and j≠ñ}

and

    X_α(i),α(j)≥ 0   for all  i<j.





Cyclohedra are simple polytopes, i.e. every vertex of n-dimensional cyclohedron W_n is adjacent to n facets. It's direct from the 
Definition <ref> of cyclohedron. Since every centrally symmetric triangulation of a (2n+2)-gon, which
corresponds to a vertex of W_n, has n pairs of centrally symmetric diagonals, which correspond to the
adjacent facets of this vertex. 
Therefore the canonical form <cit.> of 
𝒲_n is 

    Ω(𝒲_n)=∑_vertex Zsign(Z)⋀_a=1^n
        log X_i^Z_a,j^Z_a,

where vertex Z corresponds to a centrally symmetric triangulation of this (2n+2)-gon, X_i^Z_a,j^Z_a is the X-variable which corresponds to the pair of diagonals (i^Z_a,j^Z_a) and (^Z_a,^Z_a) in this triangulation, and sign(Z) is evaluated on
the ordering of the facets in the wedge product
to ensure Ω(𝒲_n) is a projective form <cit.>, for example,

    Ω(𝒲_1)    =  X_00̃/X_00̃-  X_11̃/X_11̃ ,  
    Ω(𝒲_2)    =  X_02∧ X_00̃/X_02X_00̃ - X_02∧ X_22̃/X_02X_22̃ + X_12̃∧ X_22̃/X_12̃X_22̃
         -  X_12̃∧ X_11̃/X_12̃X_11̃+  X_01̃∧ X_11̃/X_01̃X_11̃ - X_01̃∧ X_00̃/X_01̃X_00̃ .


On the other aspect, every vertex of 𝒲_n also corresponds to the dual graph of each centrally symmetric triangulation of a (2n+2)-gon, see figure <ref>. 

If we further think dual graphs as planar Feynman diagrams with scalar
propagators, they share the same poles with the term corresponding to the vertex Z 
in eq.(<ref>)

    ⋀_a=1^n
        log X_i^Z_a,j^Z_a=⋀_a=1^n X_i^Z_a,j^Z_a/∏_a=1^n X_i^Z_a,j^Z_a.

For example, for the triangulation in the left-hand side of figure <ref>, poles of this planar Feynman diagram are s_12=s_1̃2̃
and s_012=s_0̃1̃2̃, and the corresponding scattering form is

    X_02∧ X_03/X_02X_03= s_01∧ s_012/s_01s_012.

Note that, 1/(s_01s_012) is not the amplitude determined by this Feynman diagram.
Besides, we can even read sign(Z) from this viewpoint. Indeed, if Z and Z' are full centrally symmetric triangulations related by a mutation of a pair of diagonals, then
sign(Z)=-sign(Z').

We can extend this viewpoint of dual graph to other orderings. For an ordering α, define
𝖼𝗌𝗉(α) to be the set of all dual graphs of full centrally symmetric triangulations in
the ordering α (see the right-hand side of figure.<ref> for an example). 
If a planar graph g belongs to 𝖼𝗌𝗉(α), we say that this graph is compatible with
α.
Therefore, 

    Ω(𝒲_n(α))=∑_g∈𝖼𝗌𝗉(α)sign(g)⋀_a=1^n
        log X_i^g_a,j^g_a.

Now for any graph g, we can pullback it to subspace H(α),
then <cit.>

    .(sign(g)⋀_a=1^n
         X_i^g_a,j^g_a)|_H(α)=
        ^n X(α)    if g∈𝖼𝗌𝗉(α),
    
            0   otherwise,

where we assume that sign(g) for a compatible graph in different
ordering are equal. One can check that ^n X(α) is independent of choice of g, so it's the volume form of 𝒲_n(α).

Now we define m(α|β) by the pullback of Ω(𝒲_n(α))
at the space H_n(β):

    Ω(𝒲_n(α))|_H_n(β) =: m(α|β)  ^n X(β),

or more concretely by eq.(<ref>),

    m(α|β)=∑_g∈𝖼𝗌𝗉(α)∩𝖼𝗌𝗉(β)∏_a=1^n1/X_i^g_a,j^g_a .

    
To this point, one can easily verify eq.(<ref>) by checking the combinatoric structure of both sides.











 §.§ Scattering Equations as a Map Between Cyclohedra

Up to now, we have constructed cyclohedra in 
moduli space and kinematic space. Here we claim
that the connection between these two cyclohedra
is given by the so-called scattering equation map <cit.> which is 
the solution of s_ij of the scattering equations eq.(<ref>).

Recall that, in our construction of kinematic cyclohedron, 

    s_01,s_12,…,s_n-1,n,s_0ñ,s_1ñ,…,s_nñ

are not constants. We can eliminate s_iñ by the conservation of momentum, 




then scattering equations eq.(<ref>) are linear equations
of {s_i,i+1 | 0≤ i≤ n-1}, whose solution is 




    s_k,k+1=∑_0≤ i≤ k<k+1≤ j≤ 2n
     j≠ i+1sin(θ_iñ/2) sin(θ_jñ/2) sin(θ_k,k+1/2)/sin(θ_ij/2) sin(θ_kñ/2) sin(θ_k+1,ñ/2)c_ij




for k=0,…,n-1. Eq.(<ref>) defines a map φ:z↦ X from moduli space to kinematic space. 
It's direct from eq.(<ref>) that X_k,k+2=s_k,k+1>0 when 
0<θ_1<⋯<θ_n<π. 
The other X_ij's can also be computed by the
definition of X-variables and one can check 
that all these X-variables are positive.
Therefore, this map further maps the worldsheet cyclohedra W_n(id,{+}) into the kinematic cyclohedra 𝒲_n.

What's more, the scattering equation map is a map between the interiors of these two cyclohedra. 
It's equivalent to say that φ(z) is in the boundary of 𝒲_n if and only if 
z is in the boundary of W_n(id,{+}). 
This is straightforward from the scattering equation map eq.(<ref>) or scattering equations eq.(<ref>). 
For example, suppose θ_p=θ_r+tx_p for r<p≤ s,
when t goes to 0, 

    s_k,k+1→∑_i=r^k-1x_k+1-x_k/x_k+1-x_ic_i,k+1+∑_i=r^k∑ _j=k+2^sx_k+1-x_k/x_j-x_ic_ij,

for r≤ k<s. Therefore,

    ∑_k=r^s-1s_k,k+1→∑_i=r^s∑_j=i+2^sc_ij,

and then

    X_r,s+1=s_r,…,s=∑_k=r^s-1s_k,k+1-∑_i=r^s∑_j=i+2^sc_ij→ 0.

Conversely, one can use eq.(<ref>) to rewrite eq.(<ref>) in X and 
z-variables, then when X_ij goes to zero, the corresponding z_k-z_i goes to 
zero for i<k<j. Actually, it does not depend on the condition c_ij>0 at all. We conjecture that 
φ is a one-to-one map after imposing these conditions.













































The pushforward <cit.>
of scattering equation map 
connects the canonical forms of these two cyclohedra, i.e. 

    ∑_sol. z𝖯𝖳_n(1^+⋯ n^+)=Ω(𝒲_n(id))|_H_n(id)
        =m(id|id) ^n X(id),

where z are the solution of scattering equations. One can consider 
other ordering pairs α, β and the scattering equation map φ^α, then

    ∑_sol. z𝖯𝖳_n(β)=Ω(𝒲_n(α))|_H_n(β)
        =m(α|β) ^n X(β).

We can rewrite pushforward in delta function form: 

    m(α|β)   =∫𝖯𝖳_n(β)∏_a=1^n δ(X_α(i_a),α(j_a)-φ^α_a(z))
       =∫𝖯𝖳_n(β)𝖯𝖳_n(α)∏_a=1^n δ(E_a)
       =: m_CHY(α|β).

    

§ Z-INTEGRALS ON THE MODULI SPACE M(R)
 

It is the place to consider some basic prototypes of string-like integrals on such moduli space since the Parke-Taylor factors and Koba-Nielsen factors on this space have been defined.  
In analogy with the story in ℳ_0,n, it's natural to define the so-called `Z-integrals' <cit.> on ℳ_n+1^c(ℝ) as 

    Z_α(β)=∫_ℭ(α)𝖯𝖳(β) ∏_0≤ i<j≤ n(-(z_α(i)-z_α(j))^2/4z_α(i)z_α(j))^α^'s_ij((z_α(i)+z_α(j))^2/4z_α(i)z_α(j))^α^'s_i

where 𝖯𝖳(β) is the Parke-Taylor factor defined by eq.(<ref>) with respect to the ordering β, ℭ(α) denotes the integration ordering on a circle which is defined by θ_α(0)< θ_α(1)<⋯<θ_α(n)<θ_α(0̃) (recall that θ_i=(z_i)), and the form of Koba-Nielsen factor is directly derived form eq.(<ref>) by a change of variables and hence ensure its reality. Obviously, these integral satisfy KK-like relations

    Z_α(1^+2^𝗌_2… n^𝗌_n)
            +   Z_α(2^𝗌_21^+… n^𝗌_n)
            +⋯ +
            Z_α(2^𝗌_2… n^𝗌_n1^+)+
            
       Z_α(1^-2^𝗌_2… n^𝗌_n)
            +Z_α(2^𝗌_21^-… n^𝗌_n)
            +⋯ +
            Z_α(2^𝗌_2… n^𝗌_n1^-)=0
 In what follows, we will limit ourselves to the case of ℭ(012⋯ n) since all the other cases just differ by a relabelling of indices, and the subscript α on Z will also be omitted since the integration orderings have been fixed. Further more, we fix z_0 as 0 (then θ_0=0 and θ_0̃=π) to mod out the U(1) redundancy.

In this section, we will take α^'-expansion for two simple examples of integrals (<ref>) to give a taste of number theoretical properties. We will see, not only multiple zeta values, but also alternating Euler sums appear in the α^'-expansion. Before proceeding, it's worth mentioning an important limit of Koba-Nielson factor (<ref>)  

    ℐ^'   =  ∏_0≤ i<j≤ n((x_i-x_j)^2/(1+x_i^2)(1+x_j^2))^s_ij
           = ∏_0≤ i<j≤ n (x_i-x_j)^2α^'s_ij∏_i=0^n| x_i-|^2α^'s_i

which is obtained by taking all s_i with i≠ j  to be vanish, where s_i now is -∑_j=0^ns_i j. Note that, eq.(<ref>) is proportional to the Koba-Nielsen factor for n+1 open strings and one closed string <cit.>

    ℐ_n+1,1=| z-z̅|^α^'q_∥^2∏_i<j| x_i-x_j|^2α^'p_i· p_j∏_i| x_i-z|^2α^'p_i· q

under the gauge fixing z=. However, some Z-integrals we defined don't have a good behaviour under this limit, and a comprehensive treatment of such integrals lie somewhat out the main line of this paper, we leave it to the future work.

Let us first consider the simplest cases, n=1. In this case, there is actually only one kind of Parke-Taylor factor since 𝖯𝖳(1^+)=-𝖯𝖳(1^-), and the Z-integral is simply

    Z(1^+)   =∫_0^∞ x_1/-2x_1(x_1^2/1+x_1^2)^α^'s_01(1/1+x_1^2)^α^'s_01̃
       =-Γ(α^'s_01)Γ(α^'s_01̃)/4Γ(α^'s_01+α^'s_01̃) .

Here the x-parameterization is used. The result is the usual Beta function, hence the α^'-expansion can be trivially obtained. The reader can check this result is divergent under the limit s_01̃→ 0.


The first nontrivial case is n=2. In this case, the result of integral can no longer be expressed as the special functions we are familiar with, like the usual hypergeometric function appearing in string tree amplitudes. Thus, we turn to seek its α^'-expansion.  As pointed out in the paper <cit.>, the obstacle to this expansion is singularities in Mandelstam variables pole. Putting it manifestly, the leading order of the integral (<ref>), which can be obtained as a CHY formula

    Z_α(β)∼ (α^')^-nm(α|β)+⋯ ,
     
can not appear in a Taylor expansion in α^'. An approach to this problem is pole subtractions introduced in <cit.>. The basic idea of this method is to subtract the integrand by several terms which can be integrated out easier and contain the pole contributions. There are four Z-integrals for n=2 which are related by 

    Z(1^+2^+)=-Z(2^+1^+)-Z(1^-2^+)-Z(2^+1^-) ,
   
and the pole subtractions of individual terms on R.H.S. are easier than Z(1^+2^+), which can be seen from leading contributions of these integrals. Further more, Z(1^-2^+) and Z(2^+1^-) can be obtained from Z(2^+1^+) through shifting particle index by -1 and +1, respectively. Thus, the remaining task is to attack the integral Z(2^+1^+).

We omit the detail of pole subtractions since this process is technical and tedious. The α^'-expansion of remainder term after pole subtractions can be obtained by using the Maple program HyperInt <cit.> with an appropriate choice of the integration ordering. The result is simply

    -Z(2^+1^+)   =1/4 α^' 2(1/ s_12 s_012+1/s_12s_0̃12) 
       +ζ_2/4(4-4 s_0̃2/s_0̃12-s_0̃12+s_012/s_12-4 s_01/s_012) 
          + α^'ζ_3/2((s_0̃12+s_012)^2/2 s_12+4 s_01 (s_12+s_01)/s_012+4 s_0̃2 (s_12+s_0̃2)/s_0̃12
                -2 (2 s_12+s_0̃2+s_01+s_012̃)) +O(α^' 2)  .
 
The next and higher orders of α^' are too long to record here, thus we just report the numbers appearing in the α^'-expansion: 

    α^'2:    ζ_2^2 , 
       α^'3:    ζ_2ζ_3, ζ_5
       α^'4:    ζ_1,-3ζ_2,ζ_2^3,ζ_3^2

where the convention for MZVs is

    ζ_n_1,…,n_r= Li_| n_1|,…,| n_r|(n_1/| n_1|,… ,n_r/| n_r|)

with the usual Goncharov's polylogarithms Li_n_1,…,n_r(z_1,…,z_r) of weight n=n_1+⋯+ n_r <cit.>.

Here we briefly remark the α^'-expansion of Z(2^+1^+) on the following points:

    
  * We do this integral beginning with x-parameterization. Then the result is not as simple as in eq.(<ref>). Where the numerical coefficients involving the letters {±} appear. (`Letters' mean the possible numbers appearing in iterated integral which can be translated into the input of polylogarithms.)
    
  * However, a huge simplification of the result is realized by a series of changes of integration variables, {y_1=x_1^2,y_2=x_1x_2} and {t_i=y_i/(1+y_i)| i=1,2}.
    In terms of the variables {t_i}, the Koba-Nielsen factor can be written as 
    
    ℐ=t_1^α^'s_012̃(1-t_1)^α^'s_01̃2  t_2^2α^'s_02(1-t_2)^α^'s_02̃(t_1-2t_1t_2+t_2^2)^α^'s_22̃(t_2-t_1)^2α^'s_12 .

    In its present form, the linear reducibility of α^'-expansion of the Koba-Nielsen factor is manifest, then a similar analysis as in <cit.> will tell us that only the letters {-1,0,1} can appear, namely, the numbers appearing in the α^'-expansion can only be the usual multiple zeta values and alternating Euler sums, like log 2 and ζ_1,-3 and so on. Note that, it is the introduction of y-variables such that the linear reducibility better.
   
  * There is no contribution of order α^'-1 although the number log 2=ζ_-1 of weight 1 does exist. 
  

To obtain the α^'-expansion of such Z-integrals, it will be convenient to introduce simplicial coordinates by setting

    t_i=x_i/1+x_i    ∀ i∈{1,…,n} .

In terms of simplicial coordinates, Z(2^+1^+) is expressed as

    Z(2^+1^+) =-∫_0<t_1<t_2<1 t_1 t_2/2(1-t_1)(t_2-t_1)t_2 ℐ_2

where

    ℐ_2   =(t_1^2/1-2t_1+2t_1^2)^α^'s_01(t_2^2/1-2t_2+2t_2^2)^α^'s_02((t_2-t_1)^2/(1-2t_1+2t_1^2)(1-2t_2+2t_2^2))^α^'s_12
         ×((1-t_1)^2/1-2t_1+2t_1^2)^α^'s_01̃((1-t_2)^2/1-2t_2+2t_2^2)^α^'s_02̃

is the corresponding Koba-Nielsen factor in terms of simplicial coordinates. In its present form, Z(2^+1^+) can be further decomposed into the pole basis integral (cite 1304.7267),

    -2Z(2^+1^+) = ∫_Δ^2 t_1 t_2/t_2(t_2-t_1) ℐ_2+∫_Δ^2 t_1 t_2/(1-t_1)(t_2-t_1) ℐ_2 +∫_Δ^2 t_1 t_2/t_2(t_1-1) ℐ_2 ,

where the leading order of each term contains one pole at most. Note that, the second term will have the same form as the first term after change of variables t_1→ 1-t_2 and t_2→ 1-t_1, while the last term contains no pole at its leading order and hence can be expanded directly

    ∫_Δ^2 t_1 t_2/t_2(t_1-1) ℐ_2=
  
It is easy to see that 𝖯𝖳(2^+1^+) only has one singularity along the boundary x_1=x_2 of the entire integral domain. A natural way to remove this singularity is simply to subtract from ℐ_2 by a quality ℐ_2^' which is defined as

    ℐ_2^':=ℐ_2|_x_1→ x_2=(x_2^2/1+x_2^2)^α^'(s_01+s_02)((x_2-x_1)^2/(1+x_2^2)(1+x_2^2))^α^'s_12(1/1+x_2^2)^α^'(s_01̃+s_02̃) .

Apparently, ∫𝖯𝖳(2^+1^+)(ℐ_2-ℐ_2^') has no singularity along the boundary x_1=x_2 and can be expanded as a Taylor series in α^' , while ∫𝖯𝖳(2^+1^+)ℐ_2^' can be directly integrated out

    ∫_0<x_1<x_2<∞ x_1 x_2/2x_2(x_2-x_1) ℐ_2^' =
        2^-3 Γ(α^'s_012)Γ(α^'s_0̃12)/α^'s_12Γ(α^'(s_012+s_0̃12))



§ OUTLOOK


In this article, we study a new moduli space ℳ_n+1^c and the corresponding kinematic space via two mathematic tools, intersection theory and positive geometry. We have shown that ℳ_n+1^c(ℝ) is tiled by 2^n-1n! copies of the  cyclohedron W_n, and each of which corresponds to a stationary point of the system with n+1 pairs of particles on a circle. We construct the `inverse KLT matrix' for this case by considering intersection numbers of twisted cycles. At the same time, we construct another cyclohedron 𝒲_n in kinematic space
and prove the scattering equations provide a map from W_n to 𝒲_n. It seems that everything works well as in the old story ℳ_0,n. However, there are several problems we have not taken up yet, for instance, how to triangulate the kinematic cyclohedra themselves as in <cit.>, from which we expect to obtain a recursion relation for m(α|β). 

Meanwhile, there are two aspects are obviously different from the case for ℳ_0,n. One is the mismatching between the number of independent Parke-Taylor factors, which is conjectured to be (2n-1)!!, and the number of solutions of scattering equations, which is proved to be (2n)!!/2. The reason for this mismatching, as pointed out at the end of section <ref>, 
is that there are log forms beyond Parke-Taylor forms in the whole twisted cohomology group. One consequence is the lack of BCJ-like relations as in <cit.> which first appear in gauge theory amplitudes and is crucial for the color-kinematics duality <cit.>. Therefore, an important and interesting question is how to find other log forms to complete the basis of the whole twisted cohomology group.

Another direction is obviously the Z-integrals we defined in section <ref>, which suppose to be a special case of a very rich class of `integrals over cluster associahedron' <cit.>, which will be explored (together with many other topics) further. We just showed the α^'-expansion for two simple examples, however, new numerical coefficients comparing with the Z-integrals over ℳ_0,n have started to appear. So, two natural questions are which kind of numbers can appear in the α^'-expansion of such integrals, do these α^'-expansions have the similar structure as in <cit.>? Meanwhile, it is also interesting to find the α^'-expansion without integrating as in <cit.>.

Apart from all questions mentioned above, another interesting question is the connection with the scattering of open strings and one closed string. Although the corresponding Koba-Nielsen factor emerges through a specific limit of Mandelstam variables, other ingredients of such string integrals are still missing in our treatment. 
What's more, it's important to find the connection between the moduli spaces of such string integrals, i.e. one puncture in the interior and n+1 punctures at the boundary of a disk, and our case. The connection between these two moduli spaces may be a clue of all missing parts.



Apart from all questions mentioned above, what we concern most is whether the underlying field or string theory behind this picture exist. One suspicion is the string theories based on ℝℙ^2, which is obtained from ℂℙ^1 by identifying z with -z̅^-1. It would be fascinating to find any connection between this story and the known physical theories.





§ ACKNOWLEDGEMENT
The original idea for this work came from the study of positive geometries, canonical forms and integrals related to cluster associahedra, by Nima Arkani-Hamed, Song He and Hugh Thomas, to whom we are grateful for suggesting the project, sharing ideas and many valuable inputs. We thank Gongwang Yan for collaborations in an early stage of the project. We also thank Giulio Salvatori for inspiring discussions, and Erik Panzer for his kind instruction on the nice program HyperInt.

	
	
	
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