The search for a theory of the S-Matrix over the past five decades has revealed
surprising geometric structures underlying scattering amplitudes ranging from the
string worldsheet to the amplituhedron, but these are all geometries in auxiliary
spaces as opposed to the kinematical space where amplitudes actually live. Motivated
by recent advances providing a reformulation of the amplituhedron and planar

Article funded by SCOAP3

}0$} \end{overpic} ]]>

0$ is a constant.]]>

0$ is a positive constant. Note that the intersection of this line with the positive region $s,t>0$ is a line segment with two boundaries at $s=0$ and $t=0$, which is a one-dimensional positive geometry (See figure~\ref{fig:4pt}). Furthermore, quite beautifully, pulling back our scattering one-form to this one-dimensional subspace accomplishes two things: (1) this pulled-back form is also the canonical form of the positive geometry of the interval; (2) given that $-u = s+t = c$, we have $ds + dt = 0$ on the line, and so the pullback of the form can be written as e.g.\ $ds/s - dt/t = ds (1/s + 1/t)$, whereby factoring out the top form $ds$ on the line segment leaves us with the amplitude! \begin{figure} \centering \qquad \begin{overpic}[width=2.5cm] {CT2.pdf} \put(47,102){$1$} \put(86,86){$2$} \put(101,32){$3$} \put(84,4){$4$} \put(5,7){$5$} \put(-10,45){$6$} \put(165,45){\large $\xrightarrow[\text{as a diffeomorphism}]{\text{ scattering equations}}$} \end{overpic} \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \begin{overpic}[width=4cm] {6ptreg.pdf} \end{overpic} ]]>

0$ are positive constants. These equalities pick out an $(n-3)$-dimensional hyperplane in kinematic space whose intersection with the positive region is the associahedron polytope. A picture of $n{=}5,6$ associahedra can be seen in figure~\ref{fig:assoc_intro}. As we saw for four points, when the scattering form is pulled back to this subspace, it is revealed to be the canonical form with logarithmic singularities on all the boundaries of this associahedron! The computation of scattering amplitudes then reduces to triangulating the associahedron. Quite nicely one natural choice of triangulation directly reproduces the Feynman diagram expansion, but other triangulations are of course also possible. As a concrete example, for $n{=}5$ the Feynman diagrams express the amplitude as the sum over 5 cyclically rotated terms: \begin{equation} \frac{1}{s_{12} s_{123}} +\frac{1}{s_{23} s_{234}} +\frac{1}{s_{34} s_{345}} +\frac{1}{s_{45} s_{451}} +\frac{1}{s_{51} s_{512}} \end{equation} But there is another triangulation of the $n{=}5$ associahedron that yields a surprising 3-term expression: \begin{equation} \frac{s_{12}+ s_{234}} {s_{12} s_{34} s_{234}} +\frac{s_{12}+s_{234}} {s_{12} s_{234} s_{23}} +\frac{s_{12}-s_{123}+s_{23}} {s_{12} s_{23} s_{123}} \end{equation} which can not be obtained by any recombination of the Feynman diagram terms. Indeed, we will see that the form enjoys a symmetry that is destroyed by individual terms in the Feynman diagram triangulation and restored only in the full sum. In contrast, this new representation comes from a simple triangulation that keeps this symmetry manifest, much as ``BCFW triangulations'' of the amplituhedron~\cite{Hodges-ml-2009hk,ArkaniHamed-ml-2012nw} make manifest the dual conformal/Yangian symmetries of planar ${\cal N}=4$ SYM that are not seen in the usual Feynman diagram expansion. Beyond these parallels to the story of the amplituhedron, the picture of scattering forms on kinematic space appears to have a fundamental role to play in the physics of scattering amplitudes in more general settings. For instance, string theorists have long known of an important associahedron, associated with the open string worldsheet; this raises a natural question: is there a natural diffeomorphism from the (old) worldsheet associahedron to the (new) kinematic space associahedron? The answer is yes, and the map is precisely provided by the scattering equations! This correspondence gives a one-line conceptual proof of the CHY formulas for bi-adjoint amplitudes~\cite{Cachazo-ml-2013iea} as a ``pushforward'' from the worldsheet ``Parke-Taylor form'' to the kinematic space scattering form. The scattering forms also give a strikingly simple and direct connection between kinematics and color! This is seen at two levels. First, we can define very general scattering forms as a sum over all possible cubic graphs $g$ in a ``big kinematic space'', with each graph given by the wedge of the $d\log$ of all its propagator factors weighted with ``kinematic coefficients'' $N(g)$. The first important observation is that the projectivity of the form on this big kinematic space forces the kinematic coefficients $N(g)$ to satisfy the same Jacobi relations as color factors; in other words, projectivity of the scattering form provides a deep geometric origin for and interpretation of the BCJ \emph{color-kinematics duality}~\cite{Bern-ml-2008qj,Bern-ml-2010ue} But there is a second, more startling connection to color made apparent by the scattering forms---``Color is Kinematics''. More precisely, as a simple consequence of momentum conservation and on-shell conditions, the wedge product of the $d$(propagator) factors associated with any cubic graph satisfies exactly the same algebraic identities as the color factors associated with the same graph, as indicated in figure~\ref{fig:5ptdual} for a $n=5$ example. \begin{figure} \centering \begin{overpic}[width=5.4cm] {CA5ptL.pdf} \put(-2.5,20){$1$} \put(21,42){$2$} \put(47,42){$3$} \put(73,42){$4$} \put(98,20){$5$} \put(24,25){$f^{a_1a_2b}$} \put(51,25){$f^{b a_3 c}$} \put(77,25){$f^{c a_4 a_5}$} \put(40,5){$f^{a_1 a_2 b}f^{b a_3 c}f^{c a_4 a_5}$} \put(113,5){$\leftrightarrow$} \end{overpic} \qquad\qquad \begin{overpic}[width=5.4cm] {CA5ptR.pdf} \put(-2.5,20){$1$} \put(21,42){$2$} \put(47,42){$3$} \put(73,42){$4$} \put(98,20){$5$} \put(32,25){$s_{12}$} \put(58,25){$s_{45}$} \put(20,5){$\mathrm{d}s_{12}\wedge\mathrm{d}s_{45}$} \end{overpic} ]]>

3$ is lower than the kinematic space dimension. We resolve this by restricting to a $(n{-}3)$-subspace $H_n\subset \K_n$ defined by a set of constants:
\ba\label{eq:cXXXX}
\text{Let $c_{ij}:=X_{i,j}+X_{i{+}1,j{+}1}-X_{i,j{+}1}-X_{i{+}1,j}$ be a \emph{positive constant }}\nonumber\\
\text{for every pair of \emph{non-adjacent} indices $1\leq i

0$ to be a positive constant. \item The mutual triangulation (assuming $d$ diagonals) subdivides the $n$-gon into $(d{+}1)$ sub-polygons, and we impose the non-adjacent constant conditions eq.~\eqref{eq:cXXXX} to each sub-polygon. \end{enumerate} For the last step, it is necessary to omit an edge from each sub-polygon when imposing the non-adjacent constants. By convention, we omit edges corresponding to the diagonals of the mutual triangulation as well as edge $n$ of the $n$-gon so that no two sub-polygons omit the same element. A moment's thought reveals that there is only one way to do this. Finally, we define the kinematic polytope $\A[\alpha|\beta]:=H[\alpha|\beta]\cap\Delta[\alpha]$. In particular, for the standard ordering $\alpha=\beta=(1,\ldots, n)$, we recover $(\Delta[\alpha],H[\alpha|\beta],\A[\alpha|\beta])=(\Delta_n,H_n,\A_n)$. Let us get some intuition for the shape of the kinematic polytope. Clearly $\A[\alpha|\alpha]$ is just the associahedron with boundaries relabeled by $\alpha$. For general $\alpha,\beta$, we can think of the mutual partial triangulation (with $d$ diagonals) as a partial triangulation corresponding to some codimension $d$ boundary of the associahedron $\A[\alpha|\alpha]$. Now imagine ``zooming in'' on the boundary by pushing all non-adjacent boundaries to infinity. The non-adjacent boundaries precisely correspond to partial triangulations of the $\alpha$-ordered $n$-gon that cross at least one diagonal of the mutual partial triangulation. This provides the correct intuition for the ``shape'' of the kinematic polytope $\A[\alpha|\beta]$. Said in another way, the polytope $\A[\alpha|\beta]$ is again an associahedron but with incompatible boundaries pushed to infinity. \begin{figure} \centering \begin{subfigure}{0.33\linewidth} \centering \includegraphics[width=3cm]{4pt_line_u.jpg} ]]>

0$ const. \\ $d\log t$.]]>

0$ const. \\ $d\log s$.]]>

0\\ b_{3,5}&:=&X_{3,5}>0\\ c_{14}&:=&X_{1,4}+X_{2,5}-X_{2,4}>0 \ea and the inequalities are given by \ba X_{2,4}&\geq& 0\\ X_{2,5}&\geq& 0\\ X_{1,4}&\geq& 0 \ea Finally we plot this region in the basis $(X_{2,4},X_{2,5})$ as shown in figure~\ref{fig:5pt_13245} where the first two inequalities simply give the positive quadrant while the last inequality gives the diagonal boundary $X_{1,4}=c_{14}-X_{2,5}+X_{2,4}\geq 0$. \begin{figure} \centering \begin{overpic}[width=4.5cm]{pentagon_13245.jpg} \put(16,0){$1$} \put(-5,55){$2$} \put(45,100){$3$} \put(100,55){$4$} \put(80,0){$5$} \put(45,53){$X_{2,4}$} \end{overpic} \qquad\qquad \begin{overpic}[width=4.5cm]{5pt_13245_faded.jpg} \put(-15,20){$X_{2,4}$} \put(15,80){$X_{1,4}$} \put(40,-10){$X_{2,5}$} \end{overpic} ]]>

3$ is of lower than top rank. This scattering form is fully determined by its association with a positive geometry living in the kinematic space defined in the following way. First, there is a top-dimensional ``positive region'' in the kinematic space given by $X_{i,j}\geq 0$ whose boundaries are associated with all the poles of the planar graphs. Next, there is a family of $(n{-}3)$-dimensional linear subspaces defined by $X_{i,j} + X_{i+1, j+1} - X_{i ,j+1} - X_{i+1, j} = c_{ij}$. With appropriate positivity constraints on the constants $c_{ij}>0$, this subspace intersects the ``positive region'' in a positive geometry --- the kinematic associahedron $\A_n$. Furthermore, the scattering form $\Omega^{(n-3)}_n$ on the full kinematic space is fully determined by the property of pulling back to the canonical form of the associahedron on this family of subspaces. Hence, the physics of on-shell tree-level bi-adjoint $\phi^3$ amplitudes are completely determined by the positive geometry not in any auxiliary space but directly in kinematic space. Furthermore, there is a striking similarity between this description of bi-adjoint $\phi^3$ scattering amplitudes and the description of planar $\mathcal{N}=4$ super Yang-Mills (SYM) with the amplituhedron as the positive geometry~\cite{Arkani-Hamed-ml-2017vfh}. Indeed the general structure is identical. There is once again a kinematic space, which for planar ${\cal N} = 4$ SYM is given by the momentum-twistor variables $Z_i\in \mathbb{P}^3(\mathbb{R})$ for $i=1,\ldots, n$, and a differential form $\Omega_n^{(4 k)}$ of rank $4\times k$ (for N$^k$MHV) on kinematic space that is fully determined by its association with a positive geometry. We again begin with a ``positive region'' in the kinematic space which enforces positivity of all the poles of planar graphs via $\langle Z_i Z_{i+1} Z_j Z_{j+1} \rangle \geq 0$; however, also required is a set of topological ``winding number'' conditions enforced by a particular ``binary code'' of sign-flip patterns for the momentum-twistor data. This is a top-dimensional subspace of the full kinematic space. There is also a canonical $4 \times k$ dimensional subspace of the kinematic space, corresponding to an affine translation of a given set of external data ${\bf }Z_*$ in the direction of a fixed $k$-plane $\Delta$ in $n$ dimensions; this subspace is thus specified by a $(4+k) \times n$ matrix ${\cal Z} := (Z_*, \Delta)^T$. Provided the condition that all ordered $(4{+}k)\times(4{+}k)$ minors of ${\cal Z}$ are positive, this subspace intersects the ``positive region'' in a positive geometry --- the (tree) amplituhedron. The form $\Omega_n^{(4 k)}$ on the full space is fully determined by the property of pulling back to the canonical form of the amplituhedron found on this family of subspaces. Once again this connection between scattering forms and positive geometry is seen directly in ordinary momentum-twistor space, without any reference to the auxiliary Grassmannian spaces where amplituhedra were originally defined to live. The nature of the relationship between ``kinematic space'', ``positive region'', ``positive family of subspaces'' and ``scattering form'' is literally identical in the two stories. We say therefore that ``the associahedron is the amplituhedron for bi-adjoint $\phi^3$ theory''. Of course there are some clear differences as well. Most notably, the scattering form $\Omega_n^{(4k)}$ is directly the super-amplitude with the differentials $dZ^I_i$ interpreted as Grassmann variables $\eta_i^I$, whereas for the bi-adjoint $\phi^3$ theory we have forms on the space of Mandelstam variables with no supersymmetric interpretation. While the planar ${\cal N}=4$ scattering forms are unifying different helicities into a single natural object, what are the forms in Mandelstam space doing? As we have already seen in the bi-adjoint example, and with more to come in later sections, these forms are instead \emph{geometrizing color factors}, as established in section~\ref{sec:color}. ]]>

}{(Y\cdot W_*)\prod_{a=1}^{n{-}3}(Y\cdot W_{i_a,j_a})}
\ea
where $\sign(Z)$ is the orientation of the adjacent vertices $W_{i_1,j_1},\ldots, W_{i_{n{-}3},j_{n{-}3}}$ (in that order) relative to the inherited orientation. Note that the antisymmetry of $\sign(Z)$ is compensated by the antisymmetry of the determinant $\left<\cdots\right>$ in the numerator, and the sum is independent of the choice of reference point $W_*$. Furthermore, the $\sign(Z)$ here is equivalent to the $\sign(Z)$ appearing in eq.~\eqref{eq:canon_assoc} where $Z$ denotes the corresponding \emph{vertex} of $\A_n$. In fact, we now argue that for an appropriate choice of reference point $W_*$, the Feynman diagram expansion eq.~\eqref{eq:amp_form} is term-by-term equivalent to the expression eq.~\eqref{eq:dual_vol}, where each $Z$ is associated with its corresponding planar cubic graph $g$.
With the benefit of hindsight, we set the reference point to $W_*=(1,0\ldots,0)$, which is particularly convenient because the numerators in eq.~\eqref{eq:dual_vol} are now equivalent for all $Z$. Indeed, since $X_{i_a,j_a}=Y\cdot W_{i_a,j_a}$, we have
\be
\left

\rightarrow
\left(\frac{X_{i,j}^0}{X_{i,j}}\right)d^{n{-}3}X
\ee
where $X$ denotes the vector $Y$ with the initial component chopped off, and the angle brackets denote the determinant $\left

0}(2,n)$ modded out by the torus action $\mathbb{R}_{>0}^n$. More precisely, we consider the set of all $2\times n$ matrices $(C_1,\ldots, C_n)$ with positive Pl\"{u}cker coordinates $(ab):=\det(C_a,C_b)>0$ for $1\leq a

3$ is higher than the dimension $n(n{-}3)/2$ of the small kinematic space $\K_n$. Nonetheless, the latter can be recovered by imposing a \emph{7-term identity} which we now~describe. \begin{figure} \centerline{ \begin{overpic}[width=.97\linewidth]{ggst_gr7.pdf}\label{ggst7} \put(12,26){$I_1$}\put(26,26){$I_2$}\put(26,12){$I_3$}\put(12,12){$I_4$} \put(41,26){$I_1$}\put(55,26){$I_4$}\put(55,12){$I_2$}\put(41,12){$I_3$} \put(70,26){$I_1$}\put(84,26){$I_3$}\put(84,12){$I_4$}\put(70,12){$I_2$} \put(22,18){$S_{I_1I_2}$}\put(51,18){$S_{I_2I_3}$}\put(80,18){$S_{I_1I_3}$} \put(18,4){$g_s$}\put(46,4){$g_t$}\put(76,4){$g_u$} \end{overpic}} ]]>

0}(2,n)$ (the twistor-string worldsheet) to the ``amplituhedron'' in momentum space; its canonical form, or the pullback of $\Omega_n^{(2n{-}4)}$ to the subspace where it lives, is then given by the pushforward of the cyclic form of $G_{>0}(2,n)$~\cite{KITP, strings}. We leave the study of these exciting questions for future investigations. ]]>

0\right\}
\ee
Note of course that the coefficients generically form a redundant representation of the interior. Furthermore, the polytope can be cut out by linear equations of the form $Y\cdot W_j\geq 0$ for some collection of dual vectors $W_j$. The facets of the polytope are therefore given by $Y\cdot W_j=0$.
Furthermore, we construct the \emph{dual polytope} $\A^*$ as the convex polytope in the \emph{dual projective space} whose vertices are given by the dual vectors $W_j$. It follows that the interior of $\A^*$ is the set of all positive linear combinations of the dual vectors:
\be
\mathcal{A}^*=\left\{\sum_j C_jW_j \;\;|\;\; C_j>0\right\}
\ee
It can be shown that $\A^*$ is precisely the set of all points $W$ cut out by the inequalities $W\cdot Z_i\geq 0$, implying that the facets of the dual polytope are given by $W\cdot Z_i=0$. This leads us to an important fact about the duality of polytopes:
\be
\text{The facets of $\A$ are dual to the vertices of $\A^*$, and vice versa.}
\ee
More generally, we have:
\begin{enumerate}
\item
\text{The codim-$d$ boundaries of $\A$ correspond to the $(d{-}1)$-boundaries of the dual $\A^*$.}
\item
\text{Any two boundaries of $\A$ differing by one dimension are adjacent precisely if}\\
\text{their duals are adjacent.}
\end{enumerate}
It follows that the dual of every simple polytope is simplicial, and vice versa. Recall that a polytope of dimension $m$ is called \emph{simple} if every vertex is adjacent to exactly $m$ facets (or equivalently $m$ edges); and a polytope is called \emph{simplicial} if every facet is a simplex. We leave the derivation as an exercise for the reader.
Having established the dual polytope $\A^*$, we find a direct connection to the canonical form of the original polytope $\A$ --- the canonical form is determined by the volume of the dual. For any $Y$ on the interior of $\A$, we define a $Y$-dependent measure on the dual space:
\be\label{eq:dVol}
d\text{Vol} := \frac{\left

0$, and our claim has been proven for all dimensions less than $m$. It suffices to argue that~\eqref{eq:canon_simple} has the correct first order poles and residues, since the canonical form is uniquely defined by such properties. Clearly, it has poles on the facets of the polytope, as required. Furthermore, for any facet $F$ given by $Y\cdot W=0$, the residue of eq.~\eqref{eq:canon_simple} along $Y\cdot W=0$ is \be \sum_{Z'} \sign(Z')\bigwedge_{a=1}^{m{-}1}d\log(Y\cdot W_a) \ee where we sum over all vertices $Z'$ adjacent to the facet $F$. But by the induction hypothesis this is the required canonical form $\Omega(F)$, thus completing the derivation. ]]>

$ from $\hat{\Omega}(F)$ and replace it by a different form:
\be\label{eq:num_replace}
\left<*Yd^{m{-}1}Y\right>\rightarrow \frac{(Z_*\cdot W)}{(Y\cdot W)}\left