JHEP Journal of High Energy Physics 1029-8479 JHEP05(2019)029 10.1007/JHEP05(2019)029 Moduli space of paired punctures, cyclohedra and particle pairs on a circle Zhenjie Li lizhenjie@itp.ac.cn Chi Zhang zhangchi@itp.ac.cn 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 06052019 2019 05 029 22022019 26042019 OPEN ACCESS, © The Authors 2019 This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. 1812.10727

In this paper, we study a new moduli space , which is obtained from by identifying pairs of punctures. We find that this space is tiled by 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 pairs of particles on a circle, which is similar to the original case of where the system is 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

Article funded by SCOAP3

Introduction

Moduli space of pairs of punctures on the Riemann sphere

Real moduli space \texorpdfstring{$\mathcal{M}^{\mathrm{c}}_{n+1}(\mathbb{R})$}{M(n+1)} and cyclohedron \texorpdfstring{$W_{n}$}{W(n)}

] (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. ]]>

Parke-Taylor forms as canonical forms of moduli space cyclohedra

Koba-Nielsen factor and scattering equations: particle pairs on a circle

Intersection numbers

] (-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. ]]>

Kinematic cyclohedra and CHY formula

0\}$and the facet represented by$X_{ij}$lays on the hyperplane defined by$X_{ij}=0$. They also found that the scattering equations can be interpreted as a map between worldsheet associahedra and kinematic associahedra. Here, we generalize these stories to cyclohedra. ]]> Kinematic cyclohedra ] (0,0) -- (5,0) node[right] {$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}$}; \end{tikzpicture} ]]> Scattering equations as a map between cyclohedra 0$ 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 $r0$ 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} ]]>

\texorpdfstring{\boldmath $Z$}{Z}-integrals on the moduli space \texorpdfstring{\boldmath $\mathcal{M}^{\mathrm{c}}_{n+1}(\mathbb{R})$}{M(R)}

Outlook

Acknowledgments

https://doi.org/10.1007/s00220-004-1187-3 https://doi.org/10.1103/PhysRevD.70.026009 https://doi.org/10.1103/PhysRevD.90.065001 https://doi.org/10.1103/PhysRevLett.113.171601 https://doi.org/10.1007/JHEP07%282014%29033 https://doi.org/10.1007/JHEP05%282018%29096 https://doi.org/10.1007/JHEP08%282018%29040 https://arxiv.org/abs/1806.01842 https://doi.org/10.1007/JHEP08%282017%29097 https://doi.org/10.1103/PhysRevLett.120.141602 https://doi.org/10.1007/JHEP06%282017%29084 https://doi.org/10.2307/1993608 https://doi.org/10.2307/1993609 https://doi.org/10.1007/JHEP11%282017%29039 https://doi.org/10.1007/JHEP01%282017%29031 https://arxiv.org/abs/9807010 https://doi.org/10.1007/s00454-002-2810-8 K. Aomoto, M. Kita, T. Kohno and K. Iohara, Theory of hypergeometric functions, Springer (2011). https://doi.org/10.1007/JHEP03%282017%29151 M. Nakahara, Geometry, topology and physics, CRC Press (2003). https://doi.org/10.1002/mana.19941660122 https://doi.org/10.1002/mana.19941680111 https://doi.org/10.1016/0550-3213%2886%2990362-7 https://doi.org/10.1007/JHEP06%282018%29153 https://doi.org/10.1007/BF01232040 https://arxiv.org/abs/1808.09986 https://arxiv.org/abs/0907.2211 https://doi.org/10.1002/prop.201300019 https://doi.org/10.1088/1751-8121/aaea14 https://doi.org/10.1016/j.cpc.2014.10.019 https://doi.org/10.4310/MRL.1998.v5.n4.a7 https://arxiv.org/abs/0606419 https://arxiv.org/abs/0910.0114 https://arxiv.org/abs/1506.07243 https://arxiv.org/abs/1810.08508 https://arxiv.org/abs/1206.5970 https://doi.org/10.1103/PhysRevD.78.085011 N. Arkani-Hamed, S. He and H. Thomas, to appear. https://doi.org/10.1088/1751-8113/46/47/475401 https://doi.org/10.1016/j.nuclphysb.2014.02.005 https://arxiv.org/abs/1810.07682 https://doi.org/10.1103/PhysRevD.89.066014