In this paper, we study a new moduli space

Article funded by SCOAP3

] (0,0) -- (2.5,0) node[right] {$\theta_1$}; \draw[thick,->] (0,0) -- (0,2.5) node[above] {$\theta_2$}; \draw[thick] (0,0) -- (2,2); \draw[thick] (0,0) node[below] {$0$} -- (1,0) node[below=2pt] {$\pi$} -- (2,0) node[below] {$2\pi$} -- (2,2) -- (0,2) node[left] {$2\pi$} -- (0,1) node[left] {$\pi$} -- (0,0); \draw[thick] (1,0) -- (2,1) -- (0,1) -- (1,2) -- (1,0); \end{tikzpicture} \end{equation*} with the identification $(\theta_1,\theta_2)\sim (2\pi-\theta_1,2\pi-\theta_2)$ due to the $\zz_2$ redundancy. As before, we can fix this redundancy by setting $\theta_1 < \pi$, so there are $4=2^1\times 2!$ chambers (green ones in the above diagram) corresponding to the cyclohedra before compactification. After compactification, each chamber becomes a hexagon, the 2-dimensional cyclohedron. For example, the triangle $0< \theta_1 <\theta_2<\pi$ actually becomes \begin{center} \begin{tikzpicture}[scale=2] \draw[thick] (0,0.2) -- (0,1.8) -- (0.2,2) -- (1.8,2) -- (2,1.8) -- (0.2,0) -- (0,0.2); \draw (1,1) node[below=10pt,right=10pt] {$\theta_1^+=\theta_2^+$}; \draw (1,2) node[above] {$\theta_2^-=0$}; \draw (0,1) node[left] {$\theta_1^+=0$}; \draw[thick,red] (0,1.8) -- (0.2,2); \draw[thick,red] (1.8,2) -- (2,1.8); \draw[thick,red] (0.2,0) -- (0,0.2); \draw[thick,red,->] (-0.3,2.1) node[left,black] {$\theta_1^+=\theta_2^-=0$} .. controls (-0.2,2.2) and (-0.1,2.1) .. (0.05,1.95); \draw[thick,red,->] (-0.3,-0.1) node[left,black] {$\theta_1^+=\theta_2^+=0$} .. controls (-0.2,-0.2) and (-0.1,-0.1) .. (0.05,0.05); \draw[thick,red,->] (2.3,2.1) node[right,black] {$\theta_1^-=\theta_2^-=0$} .. controls (2.1,2.2) and (2.05,2.1) .. (1.95,1.95); \end{tikzpicture} \end{center} where three red facets are the result of compactification. Note that, every vertex should be blowed-up in the compactification so that no two cyclohedra intersect at a point. For a general $n$, we can construct $W_{n}(\sigma,\{\mathsf{s}_{i}\})$ from the corresponding $\mathcal{M}_{n+1}^{\mathrm{c}+}(\sigma,\{\mathsf{s}_{i}\})$ and the compactification of the whole space $\mathcal{M}^{\mathrm{c}}_{n+1}(\mathbb R)$ by the following steps: \begin{enumerate}[(1)] \item For every chamber before compactification, label $(ij)$ on each codimension-one face (facet) $H$ defined by $z_{i}=z_j$. \item For the face $H$ with higher codimension $k$ defined by $z_{i_1}=\cdots=z_{i_k}$ of these chambers, truncate it and label $(i_1i_2\cdots i_k)$ on the new created facet. Repeat this operation from low-dimensional faces to high-dimensional faces. \item Glue the faces with the same label in different cyclohedra, they will be the same face after compactification. \end{enumerate} Here we give a direct combinatorial definition of the cyclohedron. For any polytope $P$, there's a natural partial order of faces on it: for face $a$ and $b$, \[ a\leq b \quad \text{if and only if}\quad a\subset b, \] then $P$ defines a partially ordered set (or poset for short) $(P,\leq)$. Conversely, we can use a poset to define a polytope whose associated poset of faces is isomorphic to the given poset. \begin{defi}[Cyclohedron]\label{def-1} Let $\operatorname{Cyc}(n)$ be the poset of all centrally symmetric dissections of a convex $(2n+2)$-gon using non-intersecting diagonals, ordered such that $b \leq b'$ 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 $\operatorname{Cyc}(n)$. \end{defi} For example, $W_2$ is a hexagon. \begin{center} \def\fp#1#2{ \begin{tikzpicture} \node[draw,minimum size=0.6cm,regular polygon,regular polygon sides=6] (a) {}; \draw (a.corner #1) -- (a.corner #2); \end{tikzpicture} } \def\fpo#1#2#3#4{ \begin{tikzpicture} \node[draw,minimum size=0.6cm,regular polygon,regular polygon sides=6] (a) {}; \draw (a.corner #1) -- (a.corner #2); \draw (a.corner #3) -- (a.corner #4); \end{tikzpicture} } \def\fpoo#1#2#3#4{ \begin{tikzpicture} \node[draw,minimum size=0.6cm,regular polygon,regular polygon sides=6] (a) {}; \draw (a.corner #1) -- (a.corner #2); \draw (a.corner #3) -- (a.corner #4); \draw (a.corner #1) -- (a.corner #3); \end{tikzpicture} } \begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)}] \node[draw,thick,minimum size=3cm,regular polygon,regular polygon sides=6] (a) {}; \coordinate (p1) at (60:2cm); \coordinate (p2) at (60*2:2cm); \coordinate (p3) at (60*3:2cm); \coordinate (p4) at (60*4:2cm); \coordinate (p5) at (60*5:2cm); \coordinate (p6) at (60*6:2cm); \draw (p1) node {\fpoo{1}{3}{4}{6}}; \draw (p2) node {\fpoo{3}{1}{6}{4}}; \draw (p3) node {\fpoo{3}{5}{6}{2}}; \draw (p4) node {\fpoo{5}{3}{2}{6}}; \draw (p5) node {\fpoo{5}{1}{2}{4}}; \draw (p6) node {\fpoo{1}{5}{4}{2}}; \draw ($(p1)!0.5!(p2)$) node {\fpo{1}{3}{4}{6}}; \draw ($(p2)!0.5!(p3)$) node {\fp{3}{6}}; \draw ($(p3)!0.5!(p4)$) node {\fpo{3}{5}{6}{2}}; \draw ($(p4)!0.5!(p5)$) node {\fp{5}{2}}; \draw ($(p5)!0.5!(p6)$) node {\fpo{5}{1}{2}{4}}; \draw ($(p6)!0.5!(p1)$) node {\fp{1}{4}}; \node[draw,minimum size=0.6cm,regular polygon,regular polygon sides=6] (a) {}; \end{tikzpicture} \end{center} The definitive property of a cyclohedron is its geometric factorization~\cite{devadoss2002space} 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. \[ H\cong W_{n-i}\times A_{i-1}, \] 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. ]]>

] (-0.3,2.1) node[left,black] {$(01\tilde 2)$} .. controls (-0.2,2.2) and (-0.1,2.1) .. (0.05,1.95); \draw[thick,red,->] (-0.3,-0.1) node[left,black] {$(012)$} .. controls (-0.2,-0.2) and (-0.1,-0.1) .. (0.05,0.05); \draw[thick,red,->] (2.3,2.1) node[right,black] {$(\tilde 012)$} .. controls (2.1,2.2) and (2.05,2.1) .. (1.95,1.95); \end{tikzpicture}\quad \Delta(1^+2^-)= \begin{tikzpicture}[scale=1, baseline={([yshift=-.5ex]current bounding box.center)}] \draw[thick] (0,0.2) -- (0,1.8) -- (0.2,2) -- (1.8,2) -- (2,1.8) -- (0.2,0) -- (0,0.2); \draw (1,1) node[below=8pt,right=8pt] {$(1\tilde 2)$}; \draw (1,2) node[above] {$(02)$}; \draw (0,1) node[left] {$(01)$}; \draw[thick,red] (0,1.8) -- (0.2,2); \draw[thick,red] (1.8,2) -- (2,1.8); \draw[thick,red] (0.2,0) -- (0,0.2); \draw[thick,red,->] (-0.3,2.1) node[left,black] {$(012)$} .. controls (-0.2,2.2) and (-0.1,2.1) .. (0.05,1.95); \draw[thick,red,->] (-0.3,-0.1) node[left,black] {$(01\tilde 2)$} .. controls (-0.2,-0.2) and (-0.1,-0.1) .. (0.05,0.05); \draw[thick,red,->] (2.3,2.1) node[right,black] {$(0\tilde 12)$} .. controls (2.1,2.2) and (2.05,2.1) .. (1.95,1.95); \end{tikzpicture} \end{aligned} \] and the self intersection number of $\Delta(1^+)$ is \[ \begin{aligned} \langle \Delta(1^+),\Delta(1^+)\rangle &= 1+\frac{1}{\exp(2\pi \ii \alpha' 2s_{01})-1}+ \frac{1}{\exp(2\pi \ii \alpha' 2s_{0\tilde 1})-1}\\ &=\frac{1}{2\ii}\left(\frac{1}{\tan(\alpha'\pi 2s_{01})} +\frac{1}{\tan(\alpha'\pi 2s_{0\tilde 1})}\right). \end{aligned} \] For $\Delta(1^+2^+)$ and $\Delta(1^+2^-)$, they intersect at the face labeled by $(01)$, so \[ \begin{aligned} \langle \Delta(1^+2^+),\Delta(1^+2^-)\rangle &=\frac{1}{2\ii}\frac{1}{\sin(\alpha'\pi 2s_{01})} \left( 1+\frac{1}{\exp(2\pi \ii \alpha' 2s_{012})-1}+ \frac{1}{\exp(2\pi \ii \alpha' 2s_{01\tilde 2})-1} \right)\\ &=\frac{1}{(2\ii)^2}\frac{1}{\sin(\alpha'\pi 2s_{01})} \left( \frac{1}{\tan(\alpha'\pi 2s_{012})} +\frac{1}{\tan(\alpha'\pi 2s_{01\tilde 2})} \right). \end{aligned} \] Finally, define the `KLT inverse matrix' by \[ m_{\alpha'}(\alpha\vert \beta)=(2\ii)^{n} \langle \Delta(\alpha),\Delta(\beta)\rangle, \] where $n=\dim \Delta(\alpha)=\dim \Delta(\beta)$, and define \begin{equation} m(\alpha\vert \beta)=\lim_{\alpha'\to 0}(2\alpha'\pi)^n m_{\alpha'}(\alpha\vert\beta)\:. \label{eqvof2m} \end{equation} For example, \[ m(1^+2^+,1^+2^-) =\frac{1}{s_{01}}\left( \frac{1}{s_{012}}+\frac{1}{s_{01\tilde 2}} \right). \] In the next section, we will use CHY formula to calculate $m(\alpha\vert\beta)$, which gives another representation. At the end of this section, we briefly comment on the dual of $H_n(\mathcal M_{n+1}^{c}(\rr),\partial_{\mathcal I})$, which is the cohomology group on $\mathcal M_{n+1}^{c}(\rr)$ defined by a flat connection \[ \nabla=\dd + \omega_{\mathcal I}\wedge, \] with the property $\nabla^2=0$, where \[ \omega_{\mathcal I}=\dd \log\mathcal{I}= \sum_{i=1}^n E_i \dd z_i= \sum_{i=1}^n\sum_{\substack{0\leq j\leq 2n+1\\ j\neq i}} \frac{s_{ij}}{z_i-z_{j}}\dd z_i \] is a well-defined single-valued form on $\mathcal M_{n+1}^{\mathrm{c}}(\rr)$. Let's denote these cohomology groups by $H^{\bullet}(\mathcal M_{n+1}^{\mathrm{c}}(\rr),\omega_{\mathcal I})$. As shown in~\cite{esnault1992cohomology}, $H^{n}(\mathcal M_{n+1}^{\mathrm{c}}(\rr),\omega_{\mathcal I})$ is generated by $\dd \log$ forms \[ \dd \log \left( \frac{z_{i_1}\pm z_{j_1}}{z_{k_1}\pm z_{l_1}} \right)\wedge \cdots \wedge \dd \log \left( \frac{z_{i_n}\pm z_{j_n}}{z_{k_n}\pm z_{l_n}} \right), \] where $0\leq i_r,j_r,k_r,l_r\leq n$ for all $1\leq r\leq n$, which should be invariant under \mbox{$z\mapsto 1/z$}. Parke-Taylor forms eq.~\eqref{dlogPT} belong to this class. Unlike the case of $\mathcal M_{0,n}(\mathbb R)$, the independent Parke-Taylor forms can \emph{no longer} generate the whole space $H^n(\mathcal M_{n+1}^{\mathrm{c}}(\rr),\omega_\mathcal I)$. One has to consider more forms, for example, \[ \dd \log \left(\frac{z_1}{z_0}\right)\wedge \dd \log \left(\frac{z_2}{z_0}\right) =\frac{\dd z_1\wedge \dd z_2}{z_1z_2}. \] It is invariant under the transformation $z\mapsto 1/z$, and then belongs to $H^2(\mathcal M_{3}^{c}(\rr),\omega_{\mathcal I})$. However, it is not the canonical form of a cyclohedron anymore. ]]>

0\}$ and the facet represented by $X_{ij}$ lays 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. ]]>

] (0,0) -- (5,0) node[right] {$X_{0\tilde 0}$}; \draw[thick,->] (0,0) -- (0,3.5) node[above] {$X_{02}$}; \draw (2,0) node[below] {$X_{02}$}; \draw (2,2.6) node[above] {$X_{1\tilde 1}$}; \draw (3.5,3) node[above] {$X_{1\tilde 2}$}; \draw (4,1.5) node[right] {$X_{2\tilde 2}$}; \draw (0.4,1.5) node[above] {$X_{0\tilde 1}$}; \draw (0,0.5) node[left] {$X_{0\tilde 0}$}; \end{tikzpicture} ]]>

0$ when $0<\theta_1<\cdots<\theta_n<\pi$. 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 $W_{n}(\operatorname{id},\{+\})$ into the kinematic cyclohedra $\mathcal{W}_{n}$. What's more, the scattering equation map is \emph{a map between the interiors of these two cyclohedra}. It's equivalent to say that $\varphi(z)$ is in the boundary of $\mathcal{W}_n$ if and only if $z$ is in the boundary of $W_n(\operatorname{id},\{+\})$. This is straightforward from the scattering equation map eq.~\eqref{sm} or scattering equations eq.~\eqref{scatteringequation}. For example, suppose $\theta_p=\theta_r+tx_p$ for $r

0$ at all. We conjecture that $\varphi$ is a one-to-one map after imposing these conditions. The pushforward~\cite{Arkani-Hamed-ml-2017mur} of scattering equation map connects the canonical forms of these two cyclohedra, i.e. \[ \sum_{\text{sol. }z}\mathsf{PT}_n(1^+\cdots n^+)=\Omega(\mathcal{W}_n(\operatorname{id}))|_{H_n(\operatorname{id})} =m(\operatorname{id}\vert\operatorname{id})\,\dd^n X(\operatorname{id}), \] where $z$ are solutions of scattering equations. One can consider other ordering pairs $\alpha$, $\beta$ and the scattering equation map $\varphi^\alpha$, then \[ \sum_{\text{sol. }z}\mathsf{PT}_n(\beta)=\Omega(\mathcal{W}_n(\operatorname{\alpha}))|_{H_n(\beta)} =m(\alpha\vert \beta)\,\dd^n X(\beta). \] We can rewrite this pushforward in terms of delta function: \[ \begin{aligned} m(\alpha\vert\beta)&=\int \mathsf{PT}_n(\beta)\prod_{a=1}^n \delta(X_{\alpha(i_a),\alpha(j_a)}-\varphi^\alpha_a(z))\\ &=\int \mathsf{PT}_n(\beta)\mathsf{PT}_n(\alpha)\prod_{a=1}^n \delta(E_a)\\ &=: m_{\text{CHY}}(\alpha\vert \beta). \end{aligned} \] ]]>