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+\pdfoutput=1
+
+\RequirePackage{ifpdf}
+\documentclass[12pt,nohyper]{JHEP3}
+
+\usepackage{epsfig}
+\usepackage{float}
+\usepackage{amsmath}
+\usepackage{array}
+\usepackage{cite}
+\usepackage{arydshln}
+
+%\widowpenalty=500
+%\clubpenalty=1000
+\let\normalcolor\relax
+%\textheight=8in
+%\textwidth=5.95in
+
+\newcommand{\smallminus}{{\rm\rule[2.4pt]{6pt}{0.65pt}}}
+\newcommand{\smallplus}{\hspace{0.5pt}\text{{\small+}}\hspace{-0.5pt}}
+\newcommand{\mi}{\smallminus}
+\newcommand{\pl}{\smallplus}
+\newcommand{\la}{\langle}
+\newcommand{\ra}{\rangle}
+\newcommand{\figBox}[4]{\mbox{\hspace{#1 cm}\raisebox{#2 cm}{\includegraphics[scale=#3]{#4}}}}
+
+\title{\hspace{-0.0cm}{\LARGE Into the Amplituhedron}}
+\author{\vspace{-.5cm}Nima Arkani-Hamed$^{a}$ and Jaroslav Trnka$^{b}$\\
+{\footnotesize{\it $^{a}$ School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA}\\
+{\it $^{b}$ California Institute of Technology, Pasadena, CA 91125,
+USA}}\vspace{-.5cm}} \preprint{2013}
+
+\abstract{We initiate an exploration of  the physics and geometry of
+the amplituhedron, starting with  the simplest case of the integrand
+for four-particle scattering in planar ${\cal N} = 4$ SYM.  We show
+how the textbook structure of the unitarity double-cut  follows from
+the positive geometry. We also use the geometry to expose the
+behavior of the multicollinear limit,  providing a direct motivation
+for studying the logarithm of the amplitude. In addition to
+computing the two and three-loop integrands, we explore various
+lower-dimensional faces of the amplituhedron, thereby computing
+non-trivial cuts of the integrand to all loop orders. }
+
+\preprint{CALT-68-2873}
+
+
+\begin{document}
+
+\newpage
+\section{Geometry and Physics of the Amplituhedron}
+
+In \cite{P1}, we introduced a new geometric object--the
+Amplituhedron--underlying the physics of scattering amplitudes for
+${\cal N}=4$ SYM in the planar limit. At tree level, the
+amplituhedron is a natural generalization of
+``the inside of a convex polygon".  Loops arise by extending the
+geometry to incorporate the idea of ``hiding particles" in the only
+natural way possible.
+
+The amplituhedron ${\cal A}_{n,k,L}$ for $n$-particle N$^k$MHV
+amplitudes at $L$ loops, lives in $G(k,k+4;L)$, which is the space
+of $k$-planes $Y$ in $k+4$ dimensions, together with $L$ 2-planes
+${\cal L}_1, \cdots, {\cal L}_L$  in the $4$ dimensional complement
+of $Y$. The external data are given by a collection of $n$ $(k+4)$
+dimensional vectors $Z_a^I$. Here $a=1, \cdots n$, and $I = 1,
+\cdots, (k+4)$. This data is taken to be ``positive", in the sense
+that all the ordered $(k+4) \times (k+4)$ determinants $\langle
+Z_{a_1} \cdots Z_{a_{k+4}} \rangle > 0$ for $a_1 < \cdots <
+a_{k+4}$. The subspace of ${\cal A}_{n,k,L}$ of $G(k,k+4;L)$ is
+determined by a ``positive" linear combination of the (positive)
+external data.  The $k$-plane is $Y_{\alpha}^I$, and the 2-planes
+are ${\cal L}_{\gamma (i)}^I$, where $\gamma = 1,2$ and  $i = 1,
+\dots, L$ . The amplituhedron is the space of all $Y, {\cal
+L}_{(i)}$ of the form
+\begin{equation}
+Y_{\alpha}^I = C_{\alpha a} Z_a^I, \qquad {\cal L}_{\gamma (i)}^{I}
+= D_{\gamma a (i)} Z_a^I
+\end{equation}
+where the $C_{\alpha a}$ specifies a $k$-plane in $n$-dimensions,
+and the $D_{\gamma a (i)}$ are $L$ 2-planes living in the $(n-k)$
+dimensional complement of $C$, with the positivity property that for
+any $0 \leq l \leq L$, all the ordered maximal minors of the $(k + 2
+l) \times n$ matrix
+\begin{equation}
+\left(\begin{array}{ccc} & D_{(i_1)} & \\
+\hdashline & \vdots &  \\ \hdashline & D_{(i_l)} & \\ \hdashline & C
+\end{array} \right)
+\end{equation}
+are positive.
+
+There is a canonical rational form $\Omega_{n,k;L}$
+associated with ${\cal A}_{n,k;L}$, with the property of having
+logarithmic singularities on all  the lower-dimensional boundaries
+of ${\cal A}_{n,k;L}$. The loop integrand form for the
+super-amplitude is naturally extracted from $\Omega_{n,k;L}$
+\cite{P1}.
+
+The amplituhedron can be defined in a few lines, as we have just
+done. But the resulting geometry is incredibly rich and
+intricate--as it must be, to generate all the structure  found in
+planar ${\cal N} = 4$ SYM scattering amplitudes to all loop orders!
+For instance, the singularity structure of the amplitude is
+reflected in the geometry of the various boundaries of the
+amplituhedron; studying this geometry in some of the simplest cases
+allows us to see the emergence of locality and unitarity from
+positive geometry.
+
+Even just the tree amplituhedron generalizes the positive
+Grassmannian $G_+(k,n)$ \cite{alex}. A complete understanding of $G_+(k,n)$
+revealed many surprising connections to other structures, from the
+fundamentally combinatorial backbone of affine permutations, to
+cluster algebras, to the physical connection with on-shell processes
+\cite{alex, FG, positive}.  It is natural to expect the full amplituhedron
+${\cal A}_{n,k,L}$ to have a much richer structure. A complete
+understanding of the full geometry of the amplituhedron, at the same
+level as our understanding of the positive Grassmannian, will likely
+involve further physical and mathematical ideas. Our goal in this
+note is to begin laying the groundwork for this exploration, by
+looking at various simple aspects of amplituhedron geometry in the
+simplest non-trivial case of clear physical interest.
+
+While the tree amplituhedron generalizes the positive Grassmannian
+in a direct way, extending the notion of positivity to external
+data, the extension of positivity associated with ``hiding
+particles" which gives rise to loops is more novel and interesting.
+The very simplest case of four-particle scattering has $k=0, n=4$.
+Here, we don't have the additional structure of Grassmann components
+for the external data \cite{P1}, the external data are just the
+ordinary bosonic momentum-twistor\cite{A1} variables
+$Z^I_1,Z^I_2,Z^I_3,Z^I_4$, for $I=1, \cdots, 4$. Furthermore, the
+constraint of positivity for external data is trivial in this case;
+indeed using a $GL(4)$ transformation we can set the $4 \times 4$
+matrix $(Z_1, \cdots, Z_4)$ to identity.  The loop variables are
+just lines in momentum-twistor space (or better, two-planes in
+four-dimensions), which correspond to points in the (dual)
+space-time. Having set the $Z$ matrix to the identity, each $2
+\times 4$ matrix for the lines ${\cal L}^I_{\gamma (a)}$ is simply
+identified with the $D$ matrices $D_{(i)}$.
+
+The amplituhedron positivity
+constraints are that the all the ordered minors of each $D_{(i)}$
+matrix are positive
+\begin{equation}
+(12)_i, (13)_i, (14)_i, (23)_i, (24)_i,(34)_i > 0
+\end{equation}
+We also have mutual positivity, that the $4 \times 4$ determinant
+$\langle D_{(i)} D_{(j)} \rangle > 0$, which tells us that
+\begin{align}
+(12)_i (34)_j + (23)_i (14)_j + (34)_i
+(12)_j + (14)_i (23)_j - (13)_i (24)_j - (24)_i (13)_j > 0
+\end{align}
+We can also express these conditions in a convenient gauge, where
+\begin{equation}
+D_{(i)} = \left(\begin{array}{cccc} 1 & x_i & 0 & -w_i \\ 0 &
+y_i & 1 & z_i
+\end{array} \right)
+\end{equation}
+Then the positivity of each $D_{(i)}$ simply tells us that
+\begin{equation}
+x_i,y_i,z_i,w_i > 0
+\end{equation}
+while the mutual positivity conditions become
+\begin{equation}
+(x_i - x_j)(z_i - z_j) + (y_i - y_j)(w_i - w_j)<0
+\end{equation}
+
+In this note we study various aspects of  the geometry defined by these
+inequalities, as well as the corresponding canonical form $\Omega$,
+which directly gives us the loop integrand for four-particle scattering. Of course the
+four-particle amplitude has been an object of intensive study
+for many years \cite{Bern:2006ew, Bern:2005iz, Bern:2007ct,
+Bourjaily:2011hi, Eden:2012tu}, with loop integrand now available
+through seven loops. But our approach will be fundamentally
+different from previous works. We will not begin by drawing planar
+diagrams made out of ``boxes", we will make no mention of recursion
+relations, we will not make ansatze for the integrand which are
+checked against cuts, and we will make no mention of physical
+constraints from exponentiation of infrared divergences etc.
+Instead, we will discover all the known general properties of the
+loop integrand, and many other properties besides, directly by
+studying the positive geometry of the amplituhedron.
+
+We will start with a lightning review of the one-loop geometry,
+which is just that of $G_+(2,4)$, mostly to define some notation and
+nomenclature. We then do some warm-up exercises for associating
+canonical forms $\Omega$ with spaces specified by particularly
+simple inequalities, which will come in handy in later sections. The
+first non-trivial case with mutual positivity is obviously two
+loops, and we show how to triangulate the space and extract the
+loop integrand, matching the well-known result given as a
+sum of two double-boxes. Interestingly, while our triangulation of
+the two-loop amplituhedron is manifestly ``positive", the sum of
+double-boxes is not, with each term having singularities outside the amplituhedron
+that only cancel in the sum.
+
+We then make some general observation on the structure of certain
+cuts of the amplitude, which correspond to various boundaries of the
+amplituhedron. In particular, the textbook understanding of
+unitarity as following from the break-up of the loop integrand into two
+parts sewed together on the ``unitarity cut" follows in a beautiful way from
+positive geometry. These general
+results and some further explicit triangulations also allow us to
+determine the three-loop integrand. We move on to exploring another
+natural set of cuts that take the amplitude into the multi-collinear
+region. This exposes a fascinating property of cuts of the
+multi-loop integrand: the residues depend not only on
+the final cut geometry, but also on the path taken to reach that
+geometry. Studying the combinatorics of this path dependence naturally
+motivates looking at the logarithm
+of the amplitude, and explains why the log has such good IR
+behavior.
+
+From our new perspective, the determination of the integrand to all loop orders
+requires a complete understanding of the full
+amplituhedron geometry. We have not yet achieved this yet, but we
+believe that a systematic approach to this problem is
+possible. As a prelude,  we give a survey of
+some of the lower-dimensional ``faces" of amplituhedron. We can
+explicitly triangulate these faces and find their corresponding
+canonical forms, which give us cuts of the full integrand. This
+already gives us highly non-trivial all-loop order information about
+the integrand, in many cases not readily available from any other approach.
+
+
+\section{One Loop Geometry}
+
+At one loop we have a single line ${\cal L}_1 {\cal L}_2$, which we
+often also called ``$(AB)$".  The geometry is given by the positive
+Grassmannian $G_+(2,4)$. The external data form a polygon in
+$\mathbb{P}^3$ with vertices $Z_1$, $Z_2$, $Z_3$, $Z_4$ and edges
+$Z_1Z_2$, $Z_2Z_3$, $Z_3Z_4$, $Z_1Z_4$.
+$$
+\includegraphics[scale=.65]{pic00.pdf}
+$$
+The line $AB={\cal L}_1 {\cal L}_2$ is parametrized as
+\begin{equation}
+{\cal L}^I_\gamma = D_{\gamma a}Z_a^I
+\end{equation}
+where $\gamma=1,2$ and $a,I=1,\dots,4$. The matrix $D$ represents a
+cell of positive Grassmannian $G_+(2,4)$, in the generic case it is a
+top cell. In one particularly convenient gauge-fixing we can write
+\begin{equation}
+D = \left(
+      \begin{array}{cccc}
+        1 & x &0 & -w \\
+        0 & y & 1 & z \\
+      \end{array}
+    \right)\label{Cmatrix}
+\end{equation}
+where $x,y,z,w>0$. This gauge-fixing of the $D$ matrix covers all
+boundaries by sending variables $x,y,z,w$ to zero or infinity.
+
+The form with logarithmic singularities on the boundaries of the
+space is trivially
+\begin{equation}
+\Omega = \frac{dx}{x}\frac{dy}{y}\frac{dw}{w}\frac{dz}{z}
+\end{equation}
+The  boundaries occur when one of the variables approaches $0$ or
+$\infty$. We can easily translate this expression back to
+momentum twistor space by solving two linear equations:
+\begin{equation}
+Z_A = Z_1 + xZ_2 - w Z_4,\qquad Z_B = yZ_2 + Z_3 + zZ_4
+\end{equation}
+which gives
+\begin{equation}
+\Omega = \frac{\la AB\,d^2Z_A\ra\la AB\,d^2Z_B\ra\la1234\ra^2} {\la
+AB12\ra\la AB23\ra\la AB34\ra\la AB14\ra}
+\end{equation}
+
+We now describe the boundaries of this space in detail--these are
+nothing but all the cells of $G_+(2,4)$, which have also been
+described at length in e.g. \cite{positive}. We describe them in
+detail here since the same geometry will arise repeatedly in the context of cuts of
+the multiloop amplitudes. At the level of the form they correspond
+to logarithmic singularities. In giving co-ordinates for the boundaries,
+we will freely use different gauge-fixings as convenient for
+any given case, with all parameters positive. They will  always be
+trivially related to boundaries of (\ref{Cmatrix}).
+
+The first boundaries occur when line
+$AB$ intersects one of the lines $Z_1Z_2$, $Z_2Z_3$, $Z_3Z_4$ or
+$Z_1Z_4$. In the gauge-fixing (\ref{Cmatrix}) this sets one of the
+variables to $x,y,z,w$ to $0$. In particular,
+\begin{equation}
+\la AB12\ra =
+w,\quad \la AB23\ra = z,\quad \la AB34\ra = y,\quad \la AB14\ra = x
+\end{equation}
+where we suppressed $\la1234\ra$. For cutting $Z_1Z_2$, $\la AB12\ra
+= w=0$ we get
+$$
+\begin{minipage}[c]{0.23\textwidth}
+\includegraphics[scale=.65]{pic0.pdf}
+\end{minipage}
+\begin{minipage}[c]{0.2\textwidth}
+$$\left(
+      \begin{array}{cccc}
+        1 & x & 0 & 0 \\
+        0 & y & 1 & z \\
+      \end{array}
+    \right)$$
+\end{minipage}
+$$
+In all four cases the form is the dlog of remaining three variables;
+we will suppress writing it explicitly.
+
+The second boundaries occur when the line $AB$
+intersects two lines $Z_iZ_{i\pl1}$ and $Z_j Z_{j \pl 1}$. There are two
+distinct cases. If we cut two non-adjacent lines $Z_1Z_2$, $Z_3Z_4$
+or $Z_2Z_3$, $Z_1Z_4$ there is just one solution. For the first one
+$\la AB12\ra=\la AB34\ra=0$ we have
+$$
+\begin{minipage}[c]{0.23\textwidth}
+\includegraphics[scale=.65]{pic1.pdf}
+\end{minipage}
+\begin{minipage}[c]{0.2\textwidth}
+$$\left(
+      \begin{array}{cccc}
+        1 & x & 0 & 0 \\
+        0 & 0 & 1 & z \\
+      \end{array}
+    \right)\qquad
+%\Omega = \frac{dx}{x}\frac{dy}{y}
+$$
+\end{minipage}
+$$
+In the second case we intersect two adjacent lines. Let us cut
+$Z_1Z_2$, $Z_2Z_3$ (the other three cases are cyclically related), ie.
+$\la AB12\ra=\la AB23\ra=0$. There are two different solutions --
+either the line $AB$ passes through $Z_2$ or the line $AB$ lies in
+the plane $(Z_1Z_2Z_3)$.
+$$
+\begin{minipage}[c]{0.23\textwidth}
+\includegraphics[scale=.65]{pic2.pdf}
+\end{minipage}
+\begin{minipage}[c]{0.2\textwidth}
+$$\left(
+      \begin{array}{cccc}
+        1 & 0 & 0 & 0 \\
+        0 & y & 1 & z \\
+      \end{array}
+    \right)
+%\qquad \Omega = \frac{dx}{x}\frac{dy}{y}
+$$
+\end{minipage}\qquad
+\begin{minipage}[c]{0.23\textwidth}
+\includegraphics[scale=.65]{pic3.pdf}
+\end{minipage}
+\begin{minipage}[c]{0.2\textwidth}
+$$
+\left(
+      \begin{array}{cccc}
+        1 & x & 0 & 0 \\
+        0 & y & 1 & 0 \\
+      \end{array}
+    \right)
+%\qquad \Omega = \frac{dx}{x}\frac{dy}{y}
+$$
+\end{minipage}
+$$
+There are two different types of third boundaries. The first type is
+a triple cut -- the line $AB$ intersects three of four lines. One
+representative is $\la AB12\ra = \la AB23\ra= \la AB34\ra = 0$.
+There are two solutions to this problem. Either $AB$ passes through
+$Z_2$ and intersects the line $Z_3Z_4$ or $AB$ passes through $Z_3$
+and intersects the line $Z_1Z_2$.
+$$
+\begin{minipage}[c]{0.23\textwidth}
+\includegraphics[scale=.65]{pic4.pdf}
+\end{minipage}
+\begin{minipage}[c]{0.2\textwidth}
+$$\left(
+      \begin{array}{cccc}
+        0 & 1 & 0 & 0 \\
+        0 & 0 & 1 & \alpha \\
+      \end{array}
+    \right)
+%\qquad \Omega=\frac{dx}{x}
+$$
+\end{minipage}\qquad
+\begin{minipage}[c]{0.23\textwidth}
+\includegraphics[scale=.65]{pic4b.pdf}
+\end{minipage}
+\begin{minipage}[c]{0.2\textwidth}
+$$\left(
+      \begin{array}{cccc}
+        0 & 0 & 1 & 0 \\
+        -\alpha & -1 & 0 & 0 \\
+      \end{array}
+    \right)
+%\qquad \Omega=\frac{dx}{x}
+$$
+\end{minipage}
+$$
+There is also a ``composite" cut when we cut only two lines while
+imposing three constraints. We can pass $AB$ through $Z_2$  while lying in the plane
+plane $(Z_1Z_2Z_3)$.
+$$
+\begin{minipage}[c]{0.23\textwidth}
+\includegraphics[scale=.65]{pic5.pdf}
+\end{minipage}
+\begin{minipage}[c]{0.2\textwidth}
+$$\left(
+      \begin{array}{cccc}
+        0 & 1 & 0 & 0 \\
+        -\alpha & 0 & 1 & 0 \\
+      \end{array}
+    \right)
+%\qquad \Omega= \frac{dx}{x}
+$$
+\end{minipage}
+$$
+Finally, for the quadruple cuts we can either cut all four lines
+which localizes $AB$ to $AB=Z_1Z_3$ or $AB=Z_2Z_4$, or we can
+consider the "composite'' cut $AB=Z_1Z_2$ (and cyclically related)
+which cuts only three lines (not $Z_3Z_4$) while still imposing four
+constraints.
+$$
+\begin{minipage}[c]{0.23\textwidth}
+\includegraphics[scale=.65]{pic6.pdf}
+\end{minipage}
+\begin{minipage}[c]{0.2\textwidth}
+$$\left(
+      \begin{array}{cccc}
+        1 & 0 & 0 & 0 \\
+        0 & 0 & 1 & 0 \\
+      \end{array}
+    \right)
+%\qquad \Omega=1
+$$
+\end{minipage}\qquad
+\begin{minipage}[c]{0.23\textwidth}
+\includegraphics[scale=.65]{pic7.pdf}
+\end{minipage}
+\begin{minipage}[c]{0.2\textwidth}
+$$\left(
+      \begin{array}{cccc}
+        1 & 0 & 0 & 0 \\
+        0 & 1 & 0 & 0 \\
+      \end{array}
+    \right)
+%\qquad \Omega=1
+$$
+\end{minipage}
+$$
+
+\section{Warmup Exercises}
+
+The amplituhedron is defined by various positivity conditions. We
+will shortly be ``triangulating" the spaces defined by these inequalities and finding the
+canonical form $\Omega$ associated with them. But it will be helpful
+to practice on some simpler cases, which will also later be useful to
+determining amplitudes and cuts of amplitudes.
+
+Let us start with a trivial example; suppose we have
+\begin{equation}
+a < x < b
+\end{equation}
+It's obvious that the form is $\frac{1}{x - a} - \frac{1}{x - b}$,
+but lets reproduce this in a heavy-handed way, from the viewpoint
+using ``positive co-ordinates". In this case, we can write
+\begin{equation}
+x = a + (b - a)  \frac{\alpha}{1 + \alpha}
+\end{equation}
+Note that for $\infty > \alpha > 0$, we cover the entire range of $a < x < b$. The canonical form is just
+$\frac{d \alpha}{\alpha}$, which can be re-written in the original co-ordinates as
+\begin{equation}
+\frac{d \alpha}{\alpha} = \frac{dx}{x -a} - \frac{dx}{x - b} =
+\frac{(a - b) dx}{(x - a)(x - b)}
+\end{equation}
+
+Next, consider $0 < x_1 < x_2$. Once again, we can use positive
+variables
+\begin{equation}
+x_1 = \alpha_1, \qquad x_2 = \alpha_1 +
+\alpha_2
+\end{equation}
+and the form is quite trivially
+\begin{equation}
+\frac{d \alpha_1}{\alpha_1} \frac{d \alpha_2}{\alpha_2} = \frac{dx_1
+dx_2}{x_1 (x_2 - x_1)}
+\end{equation}
+We will henceforth skip the step of parametrization with
+positive variables, and also omit the measure factor in
+presenting results.
+
+Next consider $0 < x_1 < x_2 < a$, the form
+is
+\begin{equation}
+\left(\frac{1}{x_1} - \frac{1}{x_1 - a}\right)\left(\frac{1}{x_2 -
+x_1} - \frac{1}{x_2 - a}\right) = \frac{a}{x_1 (x_2 - x_1) (a -
+x_1)}
+\end{equation}
+This extends trivially to e.g. $0 < x_1 < x_2 < a < x_3 < x_4 < b$,
+for which the form is
+\begin{equation}
+\frac{a b}{x_1 (x_2 - x_1) (a - x_2) (x_3 - a) (x_4 - x_3)(b - x_4)}
+\end{equation}
+We will find it convenient to use a notation to represent these
+forms. Consider a chain of inequalities of the form $0 < X_1
+< X_2 \cdots < X_N$. Some of the $X$'s are the variables our form
+depends on, and some are constants like $a,b$ in our previous
+examples. We will represent the constants by underlining the
+corresponding $X$'s. In this notation, the form accompanying our two
+examples above are denoted as $[x_1, x_2, \underline{a}]$ and
+$[x_1,x_2,\underline{a},x_3,x_4,\underline{b}]$. As yet another
+example,
+
+\begin{align} [x_1, \underline{a},
+\underline{b},x_2,x_3,\underline{c},x_4] = \left(\frac{1}{x_1} -
+\frac{1}{x_1 - a}\right)\left(\frac{1}{x_2 - b} - \frac{1}{x_2 -
+c}\right)\left(\frac{1}{x_3 - x_2} - \frac{1}{x_3 - c}\right)\left(
+\frac{1}{x_4 - c}\right)
+\end{align}
+
+Next, suppose we have $x_i, y$ with $y > x_i$
+for all $i$. Then, if the $x's$ are ordered so that $x_1 < \cdots, <
+x_n$, we have $y> x_n$, and the form is
+\begin{equation}
+[x_1, \cdots, x_n, y] = \frac{1}{x_1} \frac{1}{x_2 - x_1} \cdots
+\frac{1}{x_n - x_{n-1}} \frac{1}{y - x_n}
+\end{equation}
+Then we simply sum over all the permutations
+\begin{equation}
+\sum_\sigma [x_{\sigma_1}, \cdots, x_{\sigma_n}, y]
+\end{equation}
+Note that individual terms in this sum have spurious poles $(x_i -
+x_j)$, which cancel in the sum. Indeed, in this simple case, it is
+trivial to do the sum explicitly, and find
+\begin{equation}
+\frac{y^{n-1}}{(y - x_1)(y-x_2) \cdots (y - x_n) x_1 \cdots x_n}
+\end{equation}
+Extremely naively, we may have expected the product in the
+denominator, but why is there is a numerator factor? The reason is
+that otherwise, the form would not have only logarithmic
+singularities! For instance, the residues on $x_1, \cdots, x_n \to
+0$ would give $1/y^n$; it is the numerator that makes this $1/y$.
+We can extend this to $y_I > x_i$ for a collection of $m$ $y$'s. This
+means that the smallest $y$ is larger than the largest $x$. Thus the
+form is
+\begin{equation}
+\sum_{\sigma, p}  [x_{\sigma_1}, \cdots, x_{\sigma_n},y_{p_1}
+\cdots, y_{p_m}]
+\end{equation}
+Again the spurious poles cancel in the sum, but the forms are more
+interesting. In the simplest new case where $n=3, m=2$ the form is
+\begin{equation}
+\frac{x_1 x_2 x_3 y_1 + x_1 x_2 x_3 y_2 - x_1 x_2 y_1 y_2 - x_1 x_3
+y_1 y_2 - x_2 x_3 y_1 y_2 + y_1^2 y_2^2}{x_1 x_2 x_3 (y_1 - x_1)(y_1
+- x_2)(y_1 - x_3)(y_2 - x_1)(y_2 - x_2)(y_3 - x_3)}
+\end{equation}
+
+Let us now consider the inequality $x, y > 0$ and
+also $x+ y < 1$, or $x + y > 1$. The first case is just the inside
+of a triangle, while the second case is a quadrilateral:
+$$
+\includegraphics[scale=.75]{pic8.pdf}
+$$
+Obviously the form in the first case $x + y < 1$ is
+\begin{equation}
+\frac{-1}{x y (x + y - 1)}
+\end{equation}
+For the second case, the region can be broken into two pieces in
+obvious ways. For instance, if $x > a$, there is no further
+restriction on $y$, while if $x<1$, we must have $y > 1 - x$
+$$
+\includegraphics[scale=.75]{pic9.pdf}
+$$
+The form is then
+\begin{equation}
+\frac{1}{x - 1} \frac{1}{y} + \frac{1}{x (1 - x)} \frac{1}{y + x
+- 1} = \frac{x + y}{x y (x + y - 1)}
+\end{equation}
+
+This form could have also been derived without any triangulation.
+The denominator reflects all the inequalities as it should. However,
+with a random numerator, we would have non-vanishing residue at the
+origin $x=y=0$, which is clearly not in the space. The numerator
+kills that residue, and the resulting form has logarithmic
+singularities on the boundary of our space. We could
+have also arrived at this form in another way. We know the form for $x +
+y < 1$. Since the form with no restriction (other than positivity)
+on $x,y$ is just $1/(xy)$, we conclude that the form for $x + y > 1$
+is
+\begin{equation}
+\frac{1}{x y } - \frac{-a}{x y (x + y - 1)} = \frac{x + y}{x y (x +
+y - 1)}
+\end{equation}
+
+As a final example, let us consider $x,y,a_1,b_1,a_2,b_2 > 0$, together with the two constraints
+\begin{equation}
+\frac{x}{a_1} + \frac{y}{b_1} > 1, \quad \frac{x}{a_2} +
+\frac{y}{b_2}
+> 1
+\end{equation}
+We will find the form by triangulating the space in two different
+ways. In the first triangulation, begin by ordering $a_1 < a_2$ without
+loss of generality; the final form will be obtained by symmetrizing
+$1 \leftrightarrow 2$. The shape of the allowed region $x,y$ space
+depends on whether $b_1 < b_2$ or $b_1 > b_2$:
+$$
+\includegraphics[scale=.83]{pic9a.pdf}
+$$
+If $b_1 < b_2$, then the space is essentially the same as the
+quadrilateral we just studied. The associated form obtained by
+breaking it up into the two regions $x> a_2$, and $0<x<a_2$, is
+given by
+\begin{equation}
+[a_1,a_2][b_1,b_2] \left([\underline{a_2},x] \frac{1}{y} + [x, \underline{a_2}] \frac{1}{y + \frac{b_2 x}{a_2} -b_2} \right)
+\end{equation}
+
+If $b_1 > b_2$, we have a pentagonal shape. We can break this up into three regions, where $x>a_2$, $a_2 > x > a_{12}$ and $a_{12}> x > 0$. Here
+$a_{12} = \frac{a_1 a_2 (b_1 - b_2)}{a_2 b_1 - a_1 b_2}$. The associated form is
+\begin{equation}
+[a_1,a_2] [b_2,b_1] \left([\underline{a_2},x] \frac{1}{y} + [\underline{a_{12}},x,\underline{a_2}] \frac{1}{y+ \frac{b_2 x}{a_2} - b_2} +
+[x, \underline{a_{12}}] \frac{1}{y + \frac{b_1 x}{a_1} - b_1} \right)
+\end{equation}
+
+Summing these forms and symmetrizing in $1 \leftrightarrow 2$, all
+the spurious poles cancel and we find for the final form
+\begin{equation}
+\frac{(\frac{x}{a_1} + \frac{y}{b_1})(\frac{x}{a_2} +
+\frac{y}{b_2})}{x y a_1 b_1 a_2 b_2 (\frac{x}{a_1} + \frac{y}{b_1} -
+1)(\frac{x}{y_2} + \frac{y}{b_2} - 1)}
+\end{equation}
+
+Note that we could also have arrived at this result in another
+simpler way, by thinking of the constraints in $(a_1,b_1)$ and
+$(a_2,b_2)$ spaces separately. For fixed $x,y$, if we redefine $A_i
+= x/a_i$ and $B_i = y/b_i$, we just have $A_1 + B_1 > 1, A_2 + B_2 >
+1$. We then get for the form
+\begin{equation}
+\frac{1}{x y} \times \frac{A_1 + B_1}{A_1 B_1 (A_1 + B_1 - 1)}
+\times\frac{(A_2 + B_2)}{A_2 B_2 (A_2 + B_2 - 1)}
+\end{equation}
+which, including the trivial Jacobian factors from the change of
+variables, reduces immediately to our above result obtained using
+triangulation.
+
+\section{Two Loops}
+
+We now move on to studying the inequalities defining the
+amplituhedron for four-particle scattering, starting at  two-loops,
+where we just have a single mutual positivity condition to deal
+with, simply
+\begin{equation}
+(x_1 - x_2)(z_1 - z_2) + (y_1 - y_2)(w_1 - w_2) < 0\label{twoloop}
+\end{equation}
+Without loss of generality we can take $x_1 < x_2$. Then we have
+\begin{equation}
+z_1 - z_2 > \frac{(y_1 - y_2)(w_1 - w_2)}{x_2 - x_1}
+\end{equation}
+If either $y_1> y_2, w_1 > w_2$ or$y_1 < y_2, w_1 < w_2$, we have
+$(y_1 - y_2)(w_1 - w_2) > 0$; the form is then
+\begin{equation}
+[x_1, x_2] \frac{1}{z_2} \frac{1}{z_1 - z_2 - \frac{(y_1 - y_2)(w_1
+- w_2)}{x_2 - x_1}} \left([y_1,y_2][w_1,w_2] + [y_2,y_1][w_2,w_1]
+\right)
+\end{equation}
+But if $y_1 < y_2, w_1 > w_2$ or $y_1> y_2, w_1 < w_2$, we have
+\begin{equation}
+z_2 - z_1 < - \frac{(y_1 - y_2)(w_1 - w_2)}{x_2 - x_1}
+\end{equation}
+Then the form is
+\begin{align}
+\frac{1}{x_1} \frac{1}{x_2 - x_1} \frac{1}{z_1} \left(\frac{1}{z_2} - \frac{1}{z_2 - z_1 + \frac{(y_1 - y_2)(w_1 - w_2)}{x_2
+- x_1}}\right) \left([y_1,y_2] [w_2,w_1] + [y_2,y_1][w_1,w_2]
+\right)
+\end{align}
+Finally, we just have to swap $1 \leftrightarrow 2$. The sum of
+these terms is then
+\begin{equation}
+\frac{x_1 z_2 + x_2 z_1 + y_1 w_2 + y_2 w_1}{x_1 x_2 y_1 y_2 z_1 z_2
+w_1 w_2 [(x_1 - x_2)(z_1 - z_2) + (y_1 - y_2)(w_1 -
+w_2)]}\label{twoloopform}
+\end{equation}
+
+We can expand it as a sum of four terms by canceling terms in
+numerator and denominator,
+\begin{align}
+\left(\frac{1}{x_2 y_1 y_2 z_1 w_1 w_2 [(x_1 - x_2)(z_1 - z_2) +
+(y_1 - y_2)(w_1 - w_2)]} + 1 \leftrightarrow 2 \right) \nonumber\\ +
+\left(\frac{1}{x_1 x_2 y_2 z_1 z_2 w_1 [(x_1 - x_2)(z_1 - z_2) +
+(y_1- y_2)(w_1 - w_2)]}  + 1 \leftrightarrow 2
+\right)\label{doublebox}
+\end{align}
+We can solve for all variables in terms of momentum twistors, finding
+\begin{align}
+\left[\frac{\la1234\ra^3}{\begin{array}{c}\la AB12\ra\la AB23\ra\la
+AB34\ra\la ABCD\ra\\ \la CD34\ra\la CD14\ra\la CD12\ra\end{array}}+
+\frac{\la1234\ra^3}{\begin{array}{c}\la AB23\ra\la AB34\ra\la
+AB14\ra\la ABCD\ra\\ \la CD14\ra\la CD12\ra\la
+CD23\ra\end{array}}\right] + {\rm symmetrization} \label{doublebox2}
+\end{align}
+where by symmetrization we mean adding another two terms where we swap
+$(AB)\leftrightarrow(CD)$. The expression
+(\ref{doublebox2}) is the integrand for two double boxes
+$$
+\includegraphics[scale=.98]{pic10a.pdf}
+$$
+which is the standard representation of the two-loop amplitude. Note
+that our approach gives the fully symmetrized (in
+$(AB)\leftrightarrow(CD)$) integrand, so we get four terms
+instead of two.
+
+It is natural to ask whether some other triangulation of the space
+may have directly given us this local expansion, but it is easy to
+see that this is impossible: each double box individually has a cut which
+is not allowed by the positivity conditions and is therefore
+``outside" the amplituhedron. The cut is a simple one: suppose
+we double cut one loop variable so that $D_{(1)}$ passes through the
+point 1, while $D_{(2)}$ passes through the point $3$. The $D$
+matrices on this cut have the form
+\begin{equation}
+D_{(1)} = \left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & y & 1 & z
+\end{array} \right),\quad D_{(2)} = \left(\begin{array}{cccc} 1 &
+x & 0 & -w \\ 0 & 0 & 1 & 0 \end{array}\right)
+\end{equation}
+But note that the mutual positivity condition between $D_{(1)}$ and
+$D_{(3)}$ is automatically satisfied,
+\begin{equation}
+\langle D_{(1)} D_{(2)} \rangle = x z + y w > 0
+\end{equation}
+Because of this, we conclude that taking a further residue where
+$(AB)_1 (AB)_2$ is cut must vanish, since there is no way to set
+this to zero without further setting one of $x,z$, and $y,w$, to
+zero.
+
+This is a very simple and striking prediction of positivity, which
+is true at any loop order: if we single out any two loops $(AB)_1,
+(AB)_2$, and consider double cutting each so one line passes through
+$1$ and the other through $3$, then the residue cutting $(AB)_1
+(AB)_2$ vanishes. The vanishing of this cut is not manifest from
+the local expansion. Even at two loops, each double box individually
+obviously has support on this 5-cut, but the residue cancels in the
+sum
+$$
+\includegraphics[scale=.98]{pic10.pdf}
+$$
+By contrast, obviously each term in our triangulation is compatible
+with all positivity conditions. This mirrors familiar features of
+the BCFW expansion for tree amplitudes: they correspond to
+triangulations that are ``inside" the amplituhedron and manifestly
+consistent with positivity properties (and therefore also the
+symmetries of the theory), at the expense of manifest locality.
+
+\section{Generalities on Cuts}
+
+Before starting our more detailed exploration of multiloop
+amplitudes, let us make some general observations about  cuts of the
+integrand.
+
+\subsection*{Reconstruction from Single Cuts}
+
+We are familiar with reconstructing the integrand from BCFW shifts
+of the external data \cite{BCFWloop}.  For instance, if we shift $Z_1 \to \hat{Z}_1 =
+Z_1 + \alpha Z_4$, the integrand at $\alpha = 0$ is (the negative
+of) the residues of the single cuts where $\langle (AB)_i\,\hat{1} 2
+\rangle \to 0$. This is trivially reflected from positivity.  We can
+divide the $w_i$ space into the pieces where $w_1$ is smallest,
+$w_2$ is smallest and so on. Suppose $w_1$ is smallest; then we can
+set $w_i = w_1 + \hat{w}_i$. The remaining positivity conditions are
+then exactly the same as the same as the computation of the single
+cut where $w_1 \to 0$. And we have to sum over the single cuts for
+setting each of the $w_i \to 0$.
+
+Obviously, we can extend this to all the variables $(x,y,z,w)$. We
+can always take sum $x_{i_x}, y_{i_y}, z_{i_z},w_{i_w}$ to be
+smallest, and sum over all possible $i_x,i_y,i_z,i_w$. Then, we can
+compute the integrand directly by summing over all these 4-cuts.
+This naturally corresponds to using a residue theorem using an
+extended BCFW deformation under which $Z_1 \to \hat{Z}_1 = Z_1 + \alpha
+Z_4 + \beta Z_2, \, Z_3 \to \hat{Z}_3 = Z_3 + \gamma Z_2 + \rho
+Z_4$.
+
+\subsection*{Emergent Planarity and Leading Singularities}
+Let us now consider the opposite extreme, and look at the
+zero-dimensional faces of the amplituhedron. Here, each $D_{(i)}$ is
+taken to be one of the zero dimensional cells of $G(2,4)$, where the
+columns $i,j$ can be set to the identity and the remaining entries
+are zero. From the mutual positivity of equation (\ref{twoloop}), we
+learn something very simple right away: the configuration will
+satisfy positivity in all cases except one: we can't have the $(13)$
+cells and $(24)$ cells at the same time. It is trivial to see that
+this fact extends to all $n$ MHV amplitudes at all loop
+orders. If all the $AB_i = (ab)_i$ are drawn as chords on a disk,
+then a configuration with lines that don't cross is allowed, but
+a configuration with lines that cross violates positivity and must
+have vanishing residue. Examples of allowed and non-allowed
+configurations are shown below:
+$$
+\includegraphics[scale=.63]{pic11.pdf}
+$$
+
+
+The fact that our form can ultimately be expressed as a sum over
+planar local integrands is not obviously built into the
+geometry, but of course does emerge from it. We see this planarity
+very vividly in the above structure of leading
+singularities--clearly planar diagrams can only give us leading
+singularities of the allowed type, while all other objects can give
+us the illegal ``crossing" configurations. Indeed there are many
+meaningful local integrands, compatible with the cyclic structure on
+external data, which can nonetheless not be considered as ``planar".
+A simple example is the square of one-loop amplitude, whose
+integrand can be written in momentum twistor space as
+\begin{equation}
+\frac{\la1234\ra^4}{\la AB12\ra\la AB23\ra\la AB34\ra\la AB14\ra\la
+CD12\ra\la CD23\ra\la CD34\ra\la CD14\ra}
+\end{equation}
+This integrand has an obvious leading singularity where e.g. $AB =
+13$ and $CD = 24$, which cross in index space. This is a ``not
+allowed" leading singularity that is incompatible with positivity.
+Thus, we see that the planar structure of the integrand is not a
+trivial consequence of cyclically ordered external data, but
+actually emerges from the positive geometry of the amplituhedron. Note that
+``planarity"  is not an obvious invariant
+property of the full integrand, but is only a natural statement
+about a particular expansion of the integrand in terms of (local)
+Feynman diagrams. It is thus perhaps not surprising that planarity
+should be one of many derived properties of the integrand from the
+amplituhedron point of view.
+
+\section{Unitarity from Positivity}
+
+In much of the recent work on scattering amplitudes in planar ${\cal
+N}=4$ SYM, the unitarity of loop amplitudes has been directly
+associated with correctly matching the single cut of the loop
+integrand \cite{BCFWloop}, determined by the the forward limit \cite{Simon} of the lower-loop
+amplitude. However there is an even simpler manifestation of
+unitarity, familiar from the textbooks, in the double-cut (or
+``unitarity cut") of the integrand, which is given by sewing
+together two lower loop integrand.
+$$
+\includegraphics[scale=.75]{pic12.pdf}
+$$
+This is easy to translate to momentum-twistor language. Starting
+with the $L$-loop integrand,  we take  one loop variable, $(AB)_L$,
+to cut $(12)$ and $(34)$. The corresponding $D$ matrix is then of the form
+\begin{equation}
+D_L = \left(
+        \begin{array}{cccc}
+          1 & x & 0 & 0 \\
+          0 & 0 & y & 1 \\
+        \end{array}
+      \right)
+\end{equation}
+If we compute the residue of the integrand on this configuration,
+unitarity tells us that the result must be
+\begin{equation}
+\frac{dx}{x} \frac{dy}{y} \times \hspace{-0.2cm}\sum_{L_1+L_2=L-1}
+M_4^{L_1}(Z_1-xZ_2,Z_2,Z_3,Z_4-yZ_3)\,\,
+M_4^{L_2}\left(Z_1,Z_2-\frac{1}{x}Z_1,Z_3-\frac{1}{y}Z_4,Z_4\right)
+\end{equation}
+We will now see that this result follows in a simple and beautiful
+way from the positive geometry of the amplituhedron.
+
+On the unitarity cut, the positivity conditions are the usual ones
+for the $(L-1)$ loop variables. For $D_L$ we just have that $x,y >
+0$. The mutual positivity between $D_L$ and the remaining $D_i$ just
+tells us that
+\begin{equation}
+(23)_i + xy(14)_i - x (13)_i - y(24)_i > 0\label{eq1}
+\end{equation}
+This condition also tells us that
+\begin{align}
+&[(13) - y(14)][(24) - x(14)] =(13)(24) - x(13)(14) - y(14)(24) + x y (14)^2 \\
+&\hspace{3cm}= (12)(34) + (14)[(23) + x y (14) - x(13) - y(24)]
+> (12)(34) > 0\nonumber
+\end{align}
+where in the second line we used $(13)(24) = (12)(34) + (23)(14)$.
+Now, obviously we can divide the space of each $D_i$ into ones where
+$(13)_i - (14)_i> 0$, and $(13)_i - y (14)_i< 0$, similarly $(24)_i
+- x(14)_i > 0$ or $(24)_i - x (14)_i <0$. However, if the product of
+these two factors is negative it is impossible to satisfy equation
+(\ref{eq1}). Thus, for each $i$, we have {\it either} that
+\begin{equation}
+(13) - y (14) > 0 \quad {\rm and} \quad (24) - x (14) > 0
+\end{equation}
+or
+\begin{equation}
+(13) - y (14) > 0, \quad {\rm and} \quad (24) - x (14) > 0
+\end{equation}
+
+Let us say that $L_1$ of the lines $D_{a}$ satisfy the first
+inequality and the remaining $L_2 = L - 1 - L_1$ lines $D_{A}$
+satisfy the second inequality.  Explicitly, in the first case the
+region is represented by positivity conditions
+$$
+(12)_a>0,(13)_a-y(14)_a>0,(14)_a>0,(23)_a>0,(24)_a-x(14)_a>0,(34)_a>0
+$$
+\begin{equation}
+(23)_a + xy(14)_a - x (13)_a - y(24)_a> 0
+\end{equation}
+Let us define shifted columns
+\begin{equation}
+(\hat{3})_a = (3)_a- y(4)_a,\qquad (\hat{2})_a = (2)_a-x(1)_a
+\end{equation}
+Thus the set of positivity conditions become
+\begin{equation}
+(1\hat{2})_a>0,(1\hat{3})_a>0,(14)_a>0,(\hat{2}\hat{3})_a>0,(\hat{2}4)_a>0,(\hat{3}4)_a>0
+\end{equation}
+In the second region we have
+$$
+(12)_A>0,y(14)_A-(13)_A>0,(14)_A>0,(23)_A>0,x(14)_A-(24)_A>0,(34)_A>0
+$$
+\begin{equation}
+(23)_A + xy(14)_A - x (13)_A - y(24)_A > 0
+\end{equation}
+Let us define shifts
+\begin{equation}
+(\hat{1})_A = (1)_A - \frac{1}{x}(2)_A,\qquad (\hat{4})_A =
+(4)_A-\frac{1}{y}(3)_A
+\end{equation}
+Then the set of positivity conditions become
+\begin{equation}
+(\hat{1}2)_A>0,(\hat{1}3)_A>0,(\hat{1}\hat{4})_A>0,(23)_A>0,(2\hat{4})_A>0,(3\hat{4})_A>0
+\end{equation}
+
+Now, we come to the positivity conditions internal to the $D_{a}$'s, internal to the $D_{A}$'s, and also the ones between $D_{a}$ and $D_{A}$'s. Actually quite strikingly, the $D_a$'s and $D_A$'s are
+automatically mutually positive! We look at
+\begin{equation}
+(12)_a(34)_A+(23)_a(14)_A+(34)_a(12)_A+(14)_a(23)_A-(13)_a(24)_A-(24)_a(13)_A \label{pos4}
+\end{equation}
+Rewriting this in terms of the natural shifted variables
+\begin{equation}
+(2)_a =(\hat{2})_a + x(1)_a,\quad (3)_a = (\hat{3})_a+y(4)_a \quad
+(1)_A = (\hat{1})_A + \frac{1}{x}(2)_A,\quad (4)_A=
+(\hat{4})_A+\frac{1}{y}(3)_A
+\end{equation}
+and plugging  into (\ref{pos4}) we find
+\begin{align}
+&(1\hat{2})_a(3\hat{4})_A+[(\hat{2}\hat{3})_a+xy(14)_a+y(\hat{2}4)_a+x(1\hat{3})_a]
+\left[\frac{1}{xy}(23)_A+(\hat{1}\hat{4})_A+\frac{1}{x}(2\hat{4})_A+\frac{1}{y}(\hat{1}3)_A\right]\nonumber\\
+&\hspace{-0.7cm}+(\hat{3}4)_a(\hat{1}2)_A+(14)_a(23)_A -
+[(1\hat{3})_a+y(14)_a]\left[(2\hat{4})_A+\frac{1}{y}(23)_A\right]-[(\hat{2}4)_a+x(14)_a]\left[(\hat{1}3)_A+\frac{1}{x}(23)_A\right]\nonumber\\
+&=(1\hat{2})_a(3\hat{4})_A+(\hat{3}4)_a(\hat{1}2)_A+\frac{1}{y}[(\hat{2}\hat{3})_a+x(1\hat{3})_a](\hat{1}3)_A
++x[(1\hat{3})_a+y(14)_a](\hat{1}\hat{4})_A\nonumber\\
+&+[(\hat{2}\hat{3})_a+y(\hat{2}4)_i](14)_A+\frac{1}{x}[(\hat{2}\hat{3})_a+y(\hat{2}4)_i](2\hat{4})_A
++\frac{1}{xy}(\hat{2}\hat{3})_a(23)_A>0
+\end{align}
+
+The positivity here is quite non-trivial; the expression many terms
+with plus and minus signs that cancel each other, leaving only
+pluses.
+
+The mutual positivity internally for the $D_a$'s (or the $D_A$'s) are
+exactly the same for the shifted and unshifted columns, since the
+$(4 \times 4)$ determinants are unchanged in shifting a column by a
+multiple of another. These can be easily translated in shifts of
+external twistors. Under ${\cal A}_\gamma = D_{\gamma a} \cdot Z_a$,
+we have for the first shift
+\begin{align}
+A = D\cdot Z &= (1)Z_1+(\hat{2})Z_2+(\hat{3})Z_3
++(4)Z_4\nonumber\\&=
+(1)Z_1+(2)Z_2-x(1)Z_2+(3)Z_3-y(4)Z_3+(4)Z_4\nonumber\\ &=
+(1)\hat{Z_1}+(2)Z_2+(3)Z_3+(4)\hat{Z_4}
+\end{align}
+Thus, the form for the $L_1$ lines is
+\begin{equation}
+M_4^{L_1}(Z_1-xZ_2,Z_2,Z_3,Z_4-yZ_3)
+\end{equation}
+and analogously the form for the $L_2$ lines is
+\begin{equation}
+M_4^{L_2}\left(Z_1,Z_2-\frac{1}{x}Z_1,Z_3-\frac{1}{y}Z_4,Z_4\right)
+\end{equation}
+
+Thus, we conclude that the unitarity cut is
+\begin{equation}
+\frac{dx}{x} \frac{dy}{y} \times\hspace{-0.3cm} \sum_{L_1+L_2=L-1}
+M_4^{L_1}(Z_1-xZ_2,Z_2,Z_3,Z_4-yZ_3)\,\,
+M_4^{L_2}\left(Z_1,Z_2-\frac{1}{x}Z_1,Z_3-\frac{1}{y}Z_4,Z_4\right)
+\end{equation}
+precisely as needed to enforce unitarity.
+
+\section{Three Loops}
+Having established these general results, let us turn to the three-loop
+amplitude. Recall from our general discussion  that it
+suffices to look at various cuts of the amplitude, coming from
+taking $x_{\sigma_1}, y_{\sigma_2}, z_{\sigma_3}, w_{\sigma_4}$ to
+be smallest. For the case of three loops, at least one pair of
+$\sigma_1, \sigma_2, \sigma_3, \sigma_4$ will correspond to the same
+loop, thus, to compute the full three-loop integrand, it suffices to
+compute the cut of the integrand where one loop is double-cut. We
+have already verified that the unitarity double-cut is correctly
+reproduced at any loop order. It thus suffices to compute the
+remaining double cuts, which we call the ``corner cuts": where the
+line passes through one of the points $Z_i$, or its parity
+conjugate, where the line lies in the plane $(Z_{i-1} Z_i Z_{i+1})$.
+Since these are parity conjugate, it is enough to compute one of
+them, which we take to be the cut where the line corresponding to
+the third loop passes through point 4. It will be convenient to use
+a different gauge-fixing for the third loop
+\begin{equation}
+D_{(3)} = \left(\begin{array}{cccc} a & 1 & b & 0 \\ 0 & 0 & 0 & 1
+\end{array} \right)
+\end{equation}
+If we further rescale the variables for the remaining loop variables
+as $w_i \to w_i/b, y_i \to b y_i; z_i \to z_i/a, x_i \to a x_i$, the
+remaining positivity conditions become
+\begin{equation}
+x_i + y_i > 1, \quad (x_1 - x_2)(z_1 - z_2) + (y_1 - y_2)(w_1 - w_2)
+< 0
+\end{equation}
+We can assume that $x_1 < x_2$, so that just as for two-loops we
+have then sum at the end over $1 \leftrightarrow 2$. Let us also
+define
+\begin{equation}
+Z_+ = \frac{1}{z_2} \frac{1}{z_1 - z_2 - \frac{(y_1 - y_2)(w_1 -
+w_2)}{x_2 - x_1}}, \quad Z_- = \frac{1}{z_1} \left(\frac{1}{z_2} -
+\frac{1}{z_2 - z_1 - \frac{(y_2 - y_1)(w_1 - w_2)}{x_2 - x_1}}
+\right)
+\end{equation}
+Then, by dividing the space into pieces much as we did at 2-loops,
+we find that the form is
+\begin{align}
+&[x_1, x_2, \underline{1}] ( [\underline{1 - x_1}, y_1, y_2]
+([w_1,w_2] Z_+ + [w_2,w_1]
+Z_-)\\
+&+([\underline{1 - x_2}, y_2, \underline{1 - x_1}, y_1] + [
+\underline{1 - x_1},y_2, y_1]) ([w_2,w_1] Z_+ + [w_1,w_2]
+Z_-))\nonumber\\
+&+ [x_1, \underline{1}, x_2]( [\underline{1 - x_1}, y_1, y_2] ([w_1,
+w_2] Z_+ + [w_2,w_1] Z_-)\nonumber\\
+&+ ([y_2, \underline{1 - x_1}, y_1] + [\underline{1 - x_1}, y_2,
+y_1])([w_2,w_1] Z_+ + [w_1,w_2] Z_-))\nonumber\\
+&+ [\underline{1}, x_1, x_2] ([y_1, y_2]([w_1,w_2] Z_+ + [w_2,w_1]
+Z_-) + [y_2,y_1]([w_2,w_1] Z_+ + [w_1,w_2] Z_-)\nonumber
+\end{align}
+Adding $1 \leftrightarrow 2$, all spurious poles cancel and we
+obtain
+\begin{align} \frac{\Bigg\{
+\begin{array}{c}
+w_2 x_1 x_2 y_1 + w_2 x_2 y_1^2 + w_1 x_1 x_2 y_2 - w_1 y_1 y_2 -
+w_2 y_1 y_2 + w_2 x_1 y_1 y_2 + w_1 x_2 y_1 y_2 \\
++ w_2 y_1^2 y_2 +w_1 x_1 y_2^2 + w_1 y_1 y_2^2 - x_1 x_2 z_1 + x_1
+x_2^2 z_1 + x_2^2 y_1 z_1 + x_1 x_2 y_2 z_1 \\
++ x_2 y_1 y_2 z_1 - x_1 x_2 z_2 + x_1^2 x_2 z_2+ x_1 x_2 y_1 z_2 +
+x_1^2 y_2 z_2 + x_1 y_1 y_2 z_2
+\end{array}\Bigg\}}{abx_1 x_2 y_1 y_2 z_1 z_2 w_1 w_2 (x_1 + y_1 - 1) (x_2 + y_2 - 1)
+((x_2 - x_1)(z_1 - z_2) + (y_2 - y_1)(w_1 - w_2))}
+\end{align}
+This matches what we get from the familiar local expansion, as a sum
+over ladders and ``tennis court" diagrams:
+$$
+\includegraphics[scale=.9]{pic18.pdf}
+$$
+
+\section{Multi-Collinear Region}
+
+We have seen that a particular double cut of a single loop--the unitarity cut-- is
+simply expressed in terms of (shifted) lower-loop objects. It is thus
+natural to look at the other two kinds of double cuts. Let us
+consider the cut where the $L^{th}$ line passes through 2. It will
+be convenient to use a different gauge-fixing for this last line
+\begin{equation}
+D_{(L)} = \left( \begin{array}{cccc}0 & 1 & 0 & 0 \\ -\alpha & 0 & 1
+& \gamma \end{array} \right)
+\end{equation}
+Then the mutual positivity conditions between $D_{(L)}$ and the
+other lines is simply
+\begin{equation}
+\alpha w_i + z_i > \gamma
+\end{equation}
+It is amusing that from the point of view of the
+lower-loop problem, we are simply putting a simple additional
+restriction on the allowed region for $w_i,z_i$.
+
+Despite the apparent simplicity of this deformation of the $L-1$
+loop problem, unlike the unitarity cut, this double cut can't be
+determined in terms of shifts of lower-loop problems in a
+straightforward way. However, there is a further, triple cut, which
+does have a very simple interpretation. Consider the limit where
+$\beta \to 0$. This is the collinear region, where the line passes
+through the point 2 while lying in the plane (123) \cite{localintegrand}. Note that the
+positivity condition is now automatically satisfied, and so the cut
+is trivial:
+\begin{equation}
+A_L^{{\rm coll.}} = \frac{d \alpha}{\alpha} \times A_{L-1}
+\end{equation}
+
+In this discussion we assumed that all the lines but one are
+generic. We now investigate what happens when $l$ lines are sent
+into the collinear region. The most general way this can happen is
+to start with $L_{12}$ lines cutting $(12)$ and $L_{23}$ lines
+cutting $(23)$. Let us gauge-fix in a convenient way, and write for
+the two sets of lines
+\begin{equation}
+D_{(i)} = \left(\begin{array}{cccc} \beta_i & 1 & 0 & 0 \\ -\alpha_i
+& 0 & 1 & \gamma_i \end{array} \right) \quad D_{(I)} =
+\left(\begin{array}{cccc} 0 & 1 & \rho_I & 0
+\\ \alpha_I & 0 & 1 & \gamma_I \end{array} \right)
+\end{equation}
+
+In order to reach the collinear limit, we must send $\beta_i,
+\gamma_i \to 0$, and $\rho_I, \delta_I \to 0$. We can send these to
+zero in different ways, but let us focus on one for definiteness,
+the other cases can be treated similarly. For the lines cutting
+$(12)$, we first take them to pass through $2$, and then move them
+into the collinear region where they lie in $(123)$; in other words,
+we first send $\beta_i \to 0$, and then $\gamma_i \to 0$. Similarly
+for the lines intersecting $(23)$, we first send them to pass
+through $2$, then to lie in $(123)$, so that we put the $\rho_I \to
+0$, then send $\gamma_I \to 0$. Now, the positivity conditions
+between these lines are just
+\begin{align}
+(\beta_i -\beta_j)(\gamma_i - \gamma_j) > 0, \quad (\rho_I -
+\rho_J)\left(\frac{\gamma_I}{\alpha_I} -
+\frac{\gamma_J}{\alpha_J}\right) > 0, \quad (\beta_i - \alpha_I
+\rho_I) (\gamma_i - \gamma_I) > 0
+\end{align}
+
+Collectively, these tell us something simple. Suppose we take the
+lines to pass through $2$ in some particular order, say by first
+taking $\beta_ 1 \to 0$, then $\beta_2 \to 0$, then $\rho_1 \to 0$,
+then $\rho_2 \to 0$, then $\beta_3 \to 0$ etc. Then, the cut
+vanishes unless the lines are taken into the collinear limit in
+exactly the same order! In this case, the cut is just
+\begin{equation}
+\prod_{a=1}^{l} \frac{d \alpha_a}{\alpha_a} \times M^{L-l}
+\end{equation}
+
+\section{Log of the Amplitude}
+
+Scattering amplitudes have well-known double-logarithmic infrared
+divergences, arising precisely from loop integration in the
+collinear region. At $L$ loops, we have a log$^{2L}$ divergence,
+which exponentiates in a well-known way; the logarithm of the
+amplitude only has a log$^2$ divergence. This is a
+motivation for looking at the log of the amplitude from a
+physical point of view. But as we have just seen, the loop integrand
+form also has an extremely simple behavior in the multicollinear
+limit. We will now see that this behavior, together with some very
+simple combinatorics, already motivates looking at the logarithm of
+the amplitude directly at the level of the integrand. While the
+amplitude itself has a non-vanishing residue when one loop momentum
+is brought into the collinear region, we will see that the log of
+the amplitude vanishes in the multicollinear region, unless {\it
+all} $L$ loop momenta are taken into the collinear region. The
+residue depends in a non-trivial way on the specific path taken into
+the collinear region. Furthermore, we will see that the log of the
+amplitude naturally leads us to consider {\it all} the natural
+``positive regions" we can think of related to amplituhedron
+geometry.
+
+Let us start by introducing a generating function combining together
+the amplitude at all loop order, otherwise known as the amplitude
+itself:
+\begin{equation}
+M = 1 + g M_1 + g^2 M_2 + \cdots
+\end{equation}
+Now, consider for any function $f$, the expansion for $f(M)$.
+Suppose that
+\begin{equation}
+f(1 + x) = x + a_2 x^2 + a_3 x^3 + \cdots
+\end{equation}
+then
+\begin{eqnarray}
+f(M) &=& (g M_1 + g^2 M_2 + \cdots) + a_2(g^2 A_1^2 + 2 g^3 M_1 M_2
++ \cdots) + a_3 g^3 M_3 + \cdots \nonumber \\ &=& g M_1 + g^2(M_2 +
+a_2 M_1^2) + g^3(M_3 + 2 a_2 M_1 M_2 + a_3M_1^3) + \cdots
+\end{eqnarray}
+
+We'd now like to extract the permutation-invariant integrand from
+this expression at $L$ loops. For instance,
+\begin{eqnarray}
+M_3 &=& \int d^4 x_1 d^4 x_2 d^4 x_3\,\, M_3(x_1,x_2,x_3) \nonumber \\
+M_1 M_2 &=& \int d^4 x_1 d^4 x_2 d^4 x_3\,\, \left[(M_1(x_1) M_2(x_2,x_3) + M_1(x_2) M_2(x_1,x_3) + M_1(x_3) M_2(x_1,x_2)  \right] \nonumber \\
+M_1^3 &=& \int d^4 x_1 d^4 x_2  d^4 x_3\,\, M_1(x_1)M_1(x_2)M_1(x_3)
+\end{eqnarray}
+Actually, for the combinatorics, the ``$\int d^4 x$" are irrelevant.
+Instead, we define a generating function
+\begin{equation}
+M = 1 + \sum_i x_i (i) + \sum_{i<j} x_i x_j (ij) + \sum_{i<j<k} x_i
+x_j x_k (ijk) + \cdots
+\end{equation}
+Here $``(1)"$ stands for $M_1(x_1)$, $``(134)"$ stands for
+$M_3(x_1,x_3,x_4)$ and so on. Then, the integrand for $f(M)$ at $L$
+loops is just the coefficient of $(x_1 \cdots x_L)$ in the expansion
+of $f(A)$, or put another way
+\begin{equation}
+f(M)^{L-loop} = \partial_{x_1 \cdots x_L} f(M)|_{x_1 = \cdots =
+x_L = 0}
+\end{equation}
+
+Obviously, if we are only interested in $L$ loops, we can truncate
+the $x_i$ to just $x_1, \cdots, x_L$ if we like. Thus, explicitly,
+for $L=3$, we have
+\begin{align}
+M = 1 + x_1 (1) + x_2 (2) + x_3(3) + x_1 x_2 (12) + x_1 x_3 (13) +
+x_2 x_3 (23) + x_1 x_2 x_3 (123)
+\end{align}
+and foreseeing our future interest, for $f(M) =$ log$(M)$, we have
+for the 3-loop log of the amplitude
+\begin{equation}
+({\rm log} M)^{3-loop} = (123)+ 2 (1) (2) (3)-[(1) (23) + (2)(13) +
+(3)(12)] \label{log3loop}
+\end{equation}
+%\begin{eqnarray}
+%({\rm log} M)^{3-loop} &=& M_3(x_1,x_2,x_3)+ 2 M_1(x_1) M_1(x_2)
+%M_1(x_3)   \nonumber \\ &-& [M_1(x_1) M_2(x_2,x_3) + M_1(x_2)
+%M_2(x_1,x_3) - M_1(x_3) M_2(x_1,x_2)]\label{log3loop}
+%\end{eqnarray}
+
+We would now like to compute the cut of $f(M)$ in the
+multi-collinear limit. Suppose $L$ lines are
+sent to pass through $2$ in some order $(1,\cdots, L)$. We already
+know that the cut of the amplitude in the multi-collinear limit
+depends on the order in which the lines are then sent to the
+collinear region--indeed for the amplitude the cut vanishes unless
+the lines are sent to the collinear limit in the same order
+$(1,\cdots, L)$. This will not in general be true for $f(M)$.
+Suppose that a set of $l$ lines are moved into the collinear limit
+in some order $\sigma = \{\sigma_1, \cdots, \sigma_l\}$. For
+instance for $L=3$, we could take $\sigma = \{2\}$ or $\sigma=
+\{13\}$ or $\sigma = \{231\}$. Then, it is easy to see that the
+multi-collinear cut of $f(M)$ can be computed as follows.
+
+We first
+play the following game, to produce a new generating function
+$M^{\sigma}$: (I) if an ordered subset of $\sigma$ occurs out of
+order in the brackets of $M$, we drop that term. (II) We then delete
+all the labels in $\sigma$. Let's illustrate this for $L=3$ with
+$\sigma = \{31\}$. The terms $x_1 x_3 (13)$ and $x_1 x_2 x_3 (123)$
+in $M$ are dropped, and the rest are kept. Then, we drop the
+indices $3,1$, and are left with
+\begin{equation}
+M^{\{31\}} = 1 + x_1 + x_2 (2) + x_3 + x_1 x_2 (2) + x_2 x_3 (2)
+\end{equation}
+
+The multi-collinear cut of $f(M)$ is easily seen to be just
+\begin{equation}
+\prod \frac{d \alpha_{\sigma_i}}{\alpha_{\sigma_i}} \times
+\partial_{x_1, \cdots x_L} f(M^\sigma)|_{x_1 = \cdots = x_L = 0}
+\end{equation}
+
+We are now ready to see why the logarithm of the amplitude is so
+natural from a purely combinatorial point of view. Let us return to
+looking at $M^{\{ 31 \}}$. Observe that while $M$ is itself and
+irreducible polynomial, $M^{\{31\}}$ factorizes as
+\begin{equation}
+M^{\{31\}} = [1+ x_1 + x_3][1 + x_2 (2)]
+\end{equation}
+Note that the second factor is just the $M$ polynomial made of the
+undeleted variables, while the first factor is the generating
+function made of the variables with only terms in correct order
+kept.
+
+This is a general statement. For any $\sigma$, let
+$\bar{\sigma}$ be the complementary set. Then
+\begin{equation}
+M^{\sigma} = \left(1 + \sum_i x_{\sigma_i} + \sum_{i<j; \sigma_i <
+\sigma_j} x_{\sigma_i} x_{\sigma_j} + \cdots\right) \times
+M^{\bar{\sigma}}
+\end{equation}
+To illustrate with a more non-trivial example, say for $L=5$ and
+$\sigma = \{415\}$, we have
+\begin{align}
+M^{\{415\}} = (1 + x_1 + x_4 + x_5 + x_1 x_5 + x_4 x_5) \times (1 +
+x_2(2) + x_3(3) + x_2 x_3 (23))
+\end{align}
+
+Note that if $\sigma$ is anything other than the empty set, the
+factorization in non-trivial. Because of this factorization, it is
+natural to consider the log of the object. Then, we see that for any
+string $\sigma$ of length $0 < l < L$,
+\begin{equation}
+\partial_{x_1\cdots x_L} {\rm log} M^{\sigma} = 0
+\end{equation}
+
+We thus learned something remarkable: if we take the log of the
+amplitude, then the cut taking any number $l < L$ of the loop
+variables into the collinear region vanishes! Only if all $L$ lines
+are taken into the collinear region together, can we get something
+non-zero. This explains why the log of the amplitude only has a
+log$^2$ divergence. We get a log$^2$ divergence from each loop
+momentum brought into this collinear region. For the amplitude
+itself, we can bring all $L$ lines into the collinear region one at
+a time, and thus we get the log$^{2L}$ IR divergence. However for
+the log of the amplitude, since all $L$ lines must be brought in the
+collinear region together we just get a single overall log$^2$
+divergence. (Note that had even this limit given us no residue, the
+log would have have been completely IR divergence free!).
+
+Very interestingly, the logarithm of the amplitude doesn't have the
+property, familiar for the amplitude itself, of having ``unit
+leading singularities". If all $L$ lines are taken into the
+collinear region in an order $\sigma = (\sigma_1, \cdots,
+\sigma_L)$, then the residue is
+\begin{equation}
+\partial_{x_1 \cdots x_L} {\rm log}\left(1 + \sum_i x_i + \hspace{-0.2cm}\sum_{i<j; \sigma_i < \sigma_j} x_i x_j + \cdots\right)|_{x_i=0}
+\end{equation}
+
+Note that unlike the amplitude itself, which is only non-vanishing
+for $\sigma = (1,2,\cdots, n)$, the log of the amplitude vanishes in
+this case, since
+\begin{equation}
+\left(1 + \sum_i x_i + \hspace{-0.2cm}\sum_{i<j; \sigma_i <
+\sigma_j} x_i x_j + \cdots\right) = (1 +x_1) \cdots (1+x_n)
+\end{equation}
+maximally factorizes! In the other extreme, if $\sigma = (n,
+\cdots, 1)$ is oppositely ordered to $(1, \cdots, n)$, then we have
+\begin{equation}
+\partial_{x_1 \cdots x_L} (1 + x_1 + \cdots x_L)|_{x_i = 0}  = (L-1)!
+\end{equation}
+In general, we can find residues ranging from $1$ to $(L-1)!$. For instance, at 4 loops,
+we have the non-vanishing residues 1,2,3, 4 and 6, coming from the following paths:
+\begin{align}
+&1: (2,3,4,1)(2,4,1,3)(3,1,4,2)(4,1,2,3)\quad
+2:(2,4,3,1)(3,2,4,1)(4,1,3,2)(4,2,1,3) \nonumber\\
+&3:(3,4,1,2) \quad4: (3,4,2,1)(4,2,3,1)(4,3,1,2) \quad 6:(4,3,2,1)
+\quad 0:\mbox{other}
+\end{align}
+
+The log of the amplitude has another fascinating feature, which we
+can see already starting at 2-loops, where
+\begin{equation}
+{\rm log} M_2 = (12) - (1)(2)
+\end{equation}
+
+Note that the 2-loop amplitude puts the positivity restriction
+$\langle D_{(1)} D_{(2)} \rangle > 0$ on the lines, but
+``$1-$loop$^2$" part does not put any positivity restrictions on
+them. Indeed, we can think of this as the sum over two regions, with
+$\langle D_{(1)} D_{(2)} \rangle > 0$ and $\langle D_{(1)} D_{(2)}
+\rangle < 0$. Thus, the sum that gives the log is the form
+associated with the region where $\langle D_{(1)} D_{(2)} \rangle <
+0$ ! The pattern continues at all higher loops. At 3-loops we have
+three positivity conditions involving
+\begin{equation}
+\{\la D_{(1)} D_{(2)}\ra,\la D_{(1)} D_{(3)}\ra,\la
+D_{(2)}D_{(3)}\ra\}
+\end{equation}
+For the amplitude they are all positive, $M_3 =  \{+++\}$ while for
+the log of the amplitude (\ref{log3loop}) we get a sum of terms
+\begin{equation}
+({\rm log}\,M)^{3-loop} = \{+--\} \oplus \{-+-\} \oplus \{--+\}
+\oplus 2\{---\}
+\end{equation}
+
+At 4 loops we have 6 positivity conditions,
+\begin{equation}
+\{\la D_{(1)}D_{(2)}\ra,\la D_{(1)} D_{(3)}\ra,\la D_{(1)}D_{(4)}
+\ra, \la D_{(2)}D_{(3)}\ra, \la D_{(2)}D_{(4)}\ra, \la
+D_{(3)}D_{(4)}\ra\}
+\end{equation}
+For the amplitude we have $M_4=\{++++++\}$. The log is
+\begin{align}
+&({\rm log}\,M)^{4-loop}=
+(1234) -[(12)(34) + (13)(24) + (14)(23)] \nonumber\\
+&\hspace{1cm}+[(12)(3)(4) + (13)(2)(4) + (14)(2)(3) + (23)(1)(4) +
+(24)(1)(3)+(34)(1)(2)] \nonumber\\
+&\hspace{1cm}+2\,[(123)(4) + (124)(3) + (134)(2) + (234)(1)] -
+6\,(1)(2)(3)(4) \label{log4}
+\end{align}
+%\begin{align}
+%&({\rm log}\,M)^{4-loop}=
+%M_4(x_1,x_2,x_3,x_4)-6M_1(x_1)M_1(x_2)M_1(x_3)M_1(x_4)\label{log4}\\
+%&+2[M_3(x_1,x_2,x_3)M_1(x_4)+M_3(x_1,x_2,x_4)M_1(x_3)+M_3(x_1,x_3,x_4)M_1(x_2)+M_3(x_2,x_3,x_4)M_1(x_1)]\nonumber\\
+%&-[M_2(x_1,x_2)M_2(x_3,x_4)+M_2(x_1,x_3)M_2(x_2,x_4)+M_2(x_1,x_4)M_2(x_2,x_3)]\nonumber\\
+%&+M_2(x_1,x_2)M_1(x_3)M_1(x_4)+M_2(x_1,x_3)M_1(x_2)M_1(x_4)+M_2(x_1,x_4)M_1(x_2)M_1(x_3)\nonumber\\
+%&+M_2(x_2,x_3)M_1(x_1)M_1(x_4)+M_2(x_2,x_4)M_1(x_1)M_1(x_3)+M_2(x_3,x_4)M_1(x_1)M_1(x_2)\nonumber
+%\end{align}
+and can be decomposed into a sum of regions as
+\begin{equation}
+({\rm log}\,M)^{4-loop} = R_1 \oplus 2R_2 \oplus 3R_3 \oplus
+4R_4\oplus 6R_6
+\end{equation}
+where
+\begin{align*}
+R_1 =&
+\{---+++\}\oplus\{--++-+\}\oplus\{--+++-\}\oplus\{-+--++\}\\&\oplus\{-+-++-\}\oplus\{-++--+\}\oplus\{-++-+-\}\oplus\{-+++--\}\\
+&\oplus\{+---++\}\oplus\{+--+-+\}\oplus\{+-+--+\}\oplus\{+-+-+-\}\\&\oplus\{+-++--\}\oplus\{++---+\}\oplus\{++--+-\}\oplus\{++-+--\}\\
+R_2=&\{----++\}\oplus\{---+-+\}\oplus\{---++-\}\oplus\{--+--+\}\\&\oplus\{--+-+-\}\oplus\{-+---+\}\oplus\{-+-+--\}\oplus\{-++---\}\\
+&\oplus\{+---+-\}\oplus\{+--+--\}\oplus\{+-+---\}\oplus\{++----\}\\
+R_3=&\{--++--\}\oplus\{-+--+-\}\oplus\{+----+\}\\
+R_4=&\{-----+\}\oplus\{----+-\}\oplus\{---+--\}\oplus\{--+---\}\\
+&\oplus\{-+----\}\oplus\{+-----\}\\
+R_6= &\{------\}
+\end{align*}
+While the expansion of the logarithm itself includes terms with both plus and minus signs,
+remarkably, in all cases we get a sum over regions, with all
+positive integer coefficients, reflecting the allowed leading
+singularities for different orderings of approaching the collinear
+region.
+
+\newpage
+
+\section{Some Faces of the Amplituhedron}
+
+In this section, we study a few classes of lower-dimensional faces
+of the amplituhedron, that are particularly easy to triangulate. The
+canonical form associated with these faces computes corresponding
+cuts of the full integrand.
+
+\subsection*{Ladders and Next-to-Ladders}
+Already in \cite{P1}, we discussed a set of faces that are
+extremely easy to understand. Let's take all $L$ loops to cut
+the line $(12)$, by sending all the $w_i \to 0$. The positivity
+conditions just become $(x_i - x_j)(z_i - z_j) < 0$. In whatever
+configuration of $x$'s we have, they are ordered in some way, say
+$x_1 < \dots < x_L$, and this condition tells us that the $z$'s are
+oppositely ordered $z_1 > \dots
+> z_L$. The $y_i$ just have to be positive. The associated form is
+then trivially
+\begin{equation}
+\frac{1}{y_1} \dots \frac{1}{y_L} \frac{1}{x_1} \frac{1}{x_2 - x_1}
+\dots \frac{1}{x_L - x_{L-1}} \frac{1}{z_L} \frac{1}{z_{L-1} - z_L}
+\dots \frac{1}{z_1 - z_2}
+\end{equation}
+which corresponds to the unique ``ladder" local diagrams that can
+contribute to this cut; to see this propagator structure explicitly,
+we simply regroup the terms in the product as $1/(y_1 \cdots y_L)$
+multiplying
+\begin{equation}
+\frac{1}{x_1} \times \frac{1}{(x_2 - x_1)(z_1 - z_2)} \times
+\cdots \times \frac{1}{(x_L - x_{L-1})(z_{L-1} -z_L)} \times
+\frac{1}{z_L}
+\end{equation}
+
+We can move on to consider ``next-to-ladder" cuts. Suppose for
+instance that $(L-1)$ of the loop variables cutting $(12)$, while
+the $L$'th loop cuts $(34)$ so that $y_L \to 0$. The positivity for
+the $(L-1)$ lines is simply $x_1<x_2< \cdots < x_{L-1}$ and $z_1 >
+z_2 > \cdots> z_{L-1}$ as above. The mutual positivity conditions
+are just that
+\begin{equation}
+w_L y_i > (x_i - x_L)(z_i - z_L)
+\end{equation}
+
+The canonical form is very easy to determine. We simply consider all
+the for $L$ orderings of the $x$'s for which $x_1 < \cdots, <
+x_{L-1}$, i.e. the orderings $[x_1, \cdots, x_{L-1}, x_L]$, $[x_1,
+\cdots, x_L, x_{L-1}]$, $\cdots$, $[x_L, x_1, \cdots, x_{L-1}]$;
+similarly, we consider all the analogous orderings of the $z$'s:
+$[z_L, z_{L-1},\cdots, z_1]$, $[z_{L-1},z_L,\cdots,z_1]$, $\cdots$,
+$[z_{L-1}, \cdots, z_1, z_L]$, Then if in the ordering, either both
+$x_k > x_L$ , $z_k > z_L$ or $x_k < x_L, z_k < z_L$, we have $y_k >
+(x_k - x_L)(z_k - z_L)/w_L$, otherwise we just have $y_k > 0$. The
+corresponding form is
+
+\begin{align}
+&\sum_{\sigma_1 < \cdots < \sigma_{L-1}, \rho_1
+> \cdots > \rho_{L-1}} [x_{\sigma^{-1}_1}, \cdots,
+x_{\sigma^{-1}_L}] [z_{\rho^{-1}_1}, \cdots, z_{\rho^{-1}_L}]\\
+&\hspace{0.5cm}\times \prod_{k = 1}^{L-1} \left\{
+\begin{array}{c} [y_k - \frac{1}{w_L}(x_k - x_L)(z_k - z_L)]^{-1} \,
+\sigma_k > \sigma_L,
+\rho_k > \rho_L \quad {\rm or}\quad \sigma_k < \sigma_L, \rho_k < \rho_L\\
+y_k^{-1} \,\,\, {\rm otherwise} \end{array} \right\}\nonumber
+\end{align}
+
+This expression sums the cuts for local diagrams of the form
+$$
+\includegraphics[scale=.85]{pic19.pdf}
+$$
+
+\subsection*{Corner Cuts}
+We can systematically approach the faces of the amplituhedron where
+every line is one of the double-cut configurations. We already know
+what happens with the unitarity double-cut on general grounds. So we
+are left with the ``corner cuts", where any line either passes
+through $Z_i$, or lies in the plane $(Z_{i-1} Z_i Z_{i+1})$. We use
+different convenient gauge fixings: for the case of lines passing
+through $1$, and lines in the plane $(412)$, we use
+\begin{equation}
+D_{{\rm through \, 1}} = \left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\
+0 & y & 1 & z
+\end{array} \right), \quad D_{{\rm in \, (412)}} =
+\left(\begin{array}{cccc} u & 1 & 0 & 0 \\ - v & 0 & 0 & 1
+\end{array} \right)
+\end{equation}
+
+Note that
+\begin{equation}
+\langle D_{{\rm through\, 1}} D_{{\rm in \,(412)}} \rangle = - 1
+\end{equation}
+is negative, and so we immediately learn that it is impossible to
+have lines of both types in one corner! We can either have a
+collection of lines passing through 1, {\it or} a collection of
+lines lying in the plane $(412)$. Suppose we approach the
+configuration where all the lines path through $1$, by starting with
+all the lines intersecting $(41)$, and sending the lines into the
+corner in some order, first $w_1 \to 0, \cdots,$ then $w_L > 0$.
+This orders $w_1 < \cdots < w_L$ and so $y_1 > \cdots > y_L$, thus
+the form on this final corner cut is just $[y_L, \cdots, y_1]$. Note
+that we see again something we have observed already a number of
+times: the form on the cut depends not just on the geometry of the
+ultimate configuration of lines, but also on the path taken to that
+configuration.
+
+We can easily determine completely general
+corner cuts where all the lines are of one type or the other. For
+instance, suppose we start with $L_1$ lines cutting $(14)$, and
+$L_2$ lines cutting $(12)$, and that we send these lines to pass
+through the corners $1$ and $2$ in some order. If we parametrize the
+matrices as
+\begin{equation}
+\left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & y_i & 1 & z_i
+\end{array} \right) \, \left(\begin{array}{cccc} 0 & 1 & 0 & 0 \\ -
+\alpha_I & 0 & \beta_I & 1 \end{array} \right)
+\end{equation}
+then the positivity conditions are just $y_{L_1} > \cdots > y_1,
+\beta_{L_2}> \cdots > \beta_1$, with the mutual positivity condition
+$z_i \beta_I > 1$, which just means $z_i > 1/\beta_1$ for all $i$.
+Then the form is trivially
+\begin{equation}
+\prod_I \frac{1}{\alpha_I} \times [y_{L_1}, \cdots, y_1] [\beta_{L_2}, \cdots, \beta_1] \prod_i
+\frac{1}{z_i - \beta_1^{-1}}
+\end{equation}
+
+This result generalizes trivially to the case with $L_1$ lines
+cutting $(41)$, $L_2$ lines cutting $(12)$, $L_3$ lines cutting
+$(23)$ and $L_4$ lines cutting $(34)$, then taken to pass through
+$1,2,3,4$.
+$$
+\includegraphics[scale=.75]{pic20.pdf}
+$$
+
+These results are very simple and arise from a single local term.
+Much more interesting are the mixed corner cuts, where we have the
+two different types of lines passing through different corners. One
+case is still extremely simple, where the two different lines pass
+through consecutive corners. Suppose we have $L_1$ lines passing
+through $1$, and $L_2$ lines lying in $(123)$. It is trivial to see
+that
+\begin{equation}
+\langle D_{{\rm through \, 1}} D_{{\rm lying \, in \, (123)}}
+\rangle = (14)_{{\rm through \, 1}} (23)_{{\rm lying \, in (123)}} >
+0
+\end{equation}
+and so the mutual positivity between these two sets is automatically
+satisfied. The form is then just the product of the form for the
+$L_1$ lines and the $L_2$ lines separately.
+
+The non-trivial case is
+when the corner cuts are different lines in opposite corners.
+Suppose we have $L_1$ lines cutting $(41)$ that were then sent to
+pass through $1$ in order $(L_1, \cdots, 1)$, and $L_3$ lines
+cutting $(23)$ that were made to pass through $(234)$ in order
+$(1,\cdots,L_3)$.
+$$
+\includegraphics[scale=.75]{pic21.pdf}
+$$
+For notational convenience we'll parametrize the $D$ matrices using different
+variable names in this case:
+\begin{equation}
+D_{{\rm through \, 1}} = \left(\begin{array}{cccc} 1 & 0 & 0 & 0
+\\ 0 & x_i & 1 & y_i
+\end{array} \right),\quad D_{{\rm lying \, in (234)}} =
+\left(\begin{array}{cccc} 0 & 1 & a_I^{-1} & 0 \\ 0 & 0 & b_I^{-1} &
+1 \end{array} \right)
+\end{equation}
+Then the positivity conditions are
+\begin{equation}
+x_1 < \cdots < x_{L_1}, \, a_1 <
+\cdots < a_{L_3}, \, \, {\rm and} \, \, \frac{x_i}{a_I} +
+\frac{y_i}{b_I} > 1
+\end{equation}
+
+It is quite straightforward to triangulate this space; let us work
+out the case $L_3 = 2$ explicitly. Here the geometry is very similar
+to the last of our warmup exercises. Suppose first that $b_1 < b_2$.
+Then the inequalities are just $x_i/a_2 + y_i/b_2 > 1$ together with
+the restriction $x_1 < \cdots < x_{L_1}$. We simply order the $x_i$
+relative to $a_2$. If $x_i > a_2$, then we just have $y_i > 0$ and
+the form is $1/y_i$, while if $x_i < a_2$, we have $Y_{i,2} > 0$ and
+the form is $1/Y_{i,2}$. Here we have defined
+\begin{equation}
+Y_{i,1} = y_i + \frac{b_1 x_i}{a_1} - b_1, Y_{i,2} = y_i + \frac{b_2
+x_i}{a_2} - b_2.
+\end{equation}
+Thus, for $b_1 < b_2$, the form is just
+\begin{equation}
+\frac{1}{a_1 (a_2 - a_1)} \frac{1}{b_1(b_2 - b_1)} \sum_m [\cdots,
+x_m, \underline{a_2},x_{m+1} \cdots ] \prod_k
+\left\{\begin{array}{c} Y_{k,2}^{-1} \, \, k \leq m \\ y_k^{-1} \,
+\, k > m \end{array} \right\}
+\end{equation}
+
+If instead $b_2 < b_1$, then we have to break $x$ space up into the
+three regions between $0,a_{12},a_2$ where $a_{12} = \frac{a_1 a_2
+(b_1 - b_2)}{a_2 b_1 - a_1 b_2}$. We have to sum over all the
+orderings of the $x$'s relative to $a_{12},a_2$'; for all the $x_i >
+a_2$, the form in $y$ space is just $1/y_i$, for the $x_i$ in the
+range $a_2 > x_i > a_{12}$ the $y$ form is just $1/Y_{i,2}$, while
+for $a_{12} > x_i > 0$ the $y$ form is $1/Y_{i,1}$. Thus in this
+case the form is
+\begin{equation}
+\frac{1}{a_1 (a_2 - a_1)} \frac{1}{b_1(b_2 - b_1)} \sum_{m \leq l}
+[\cdots, x_m,\underline{a_{12}},x_{m+1}, \cdots, x_l, \underline{a_2},x_{l+1}, \cdots]
+ \prod_k \left\{\begin{array}{c} Y_{k,1}^{-1}, \, \, k \leq m \\
+Y_{k,2}^{-1}, \, \, m < k \leq l \\ y_k^{-1} \, \, k > l \end{array}
+\right\}
+\end{equation}
+
+The full form is just the sum of these two pieces. While this result
+is completely straightforward from triangulation, it gives rise to
+highly non-trival local expressions even at comparatively low loop
+order. In the first really interesting case at 5 loops, with $L_1 =
+3$ and $L_3 = 2$, 19 local terms contribute to this cut, and when they are all combined under a common
+denominator, the numerator has 325 terms.
+
+There is another interesting feature of these cuts, that is not
+evident from any traditional point of view but is obvious from the
+positive geometry. We have seen that fixing the order in which the
+lines are brought to pass through $1$, imposes the constraint $x_1 <
+\cdots < x_{L_1}$. But, if we sum over all the different orderings,
+we simply remove these ordering constraints! We then expect that the
+form simplifies greatly. Indeed, if we stick with the case $L_3 =
+2$, then we just get several copies of the problem $x/a_1 + y/b_1 >
+1, x/a_2 + y/b_2 > 1$, which we analyzed in our warmup section.
+Thus, the sum over all the ways to start with $L_1$ lines on $(41)$
+which are then sent through $1$ (while sending $L_3 = 2$ lines to
+lie in (234) in the usual fixed order), is
+\begin{align}
+\frac{1}{a_1 (a_2 - a_1)}&\left[ \frac{1}{b_1 (b_2 - b_1)} \prod_i
+\left([x_i, \underline{a_2}] \frac{1}{Y_{i,2}} +
+[\underline{a_2},x_i] \frac{1}{y_i}\right)\right.\\
+&\hspace{0.3cm}\left.+ \frac{1}{b_2 (b_1 - b_2)} \prod_i \left([x_i,
+\underline{a_{12}}] \frac{1}{Y_{i,1}} + [\underline{a_{12}}, x_i,
+\underline{a_2}] \frac{1}{Y_{i,2}} + [\underline{a_2},x_i]
+\frac{1}{y_i}\right) \right]\nonumber
+\end{align}
+
+
+\subsection*{Internal Cuts}
+
+It is interesting that up to 4 loop order, every loop in the local
+expansion of the amplitude touches the external lines, but this
+behavior is obviously not generic. Starting at 5 loops, we have
+diagrams with purely internal loops, such as
+$$
+\includegraphics[scale=.8]{pic16.pdf}
+$$
+and it is interesting to probe these from positivity. Let us look at
+a particularly simple set of cuts that exposes the structure in a
+nice way. Suppose we take 4 lines $(AB)_1, \cdots, (AB)_4$, and take
+them to pass through $1,2,3,4$ respectively. But additionally, we
+take the cut where $\langle AB_1 AB_2 \rangle \to 0$, $\langle AB_2
+AB_3 \rangle \to 0$, $\langle AB_3 AB_4 \rangle \to 0$, $\langle
+AB_4 AB_1\rangle \to 0$, i.e. the lines are taken to one intersect
+the next. The $D$ matrices
+are simply
+\begin{align}
+D_{(1)} = \left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\
+0 & \alpha^{-1} & 1 & \beta \end{array} \right), \quad D_{(2)} =
+\left(\begin{array}{cccc} 0 & 1 & 0 & 0 \\ \gamma & 0 & \beta^{-1} &
+1 \end{array} \right),\nonumber\\ D_{(3)} =
+\left(\begin{array}{cccc} 0 & 0 & 1 & 0 \\ -1 & -\sigma & 0 &
+\gamma^{-1} \end{array} \right),\quad D_{(4)} =
+\left(\begin{array}{cccc} 0 & 0 & 0 & 1 \\ - \sigma^{-1} & -1 & -
+\alpha & 0 \end{array} \right)
+\end{align}
+
+Note that the mutual positivity between $D_{(1)} D_{(3)}$ and
+$D_{(2)} D_{(4)}$ is automatic. The geometry of the lines is
+$$
+\includegraphics[scale=.7]{pic17.pdf}
+$$
+and so we can think of the lines as $(AB)_1 = \hat{1} \hat{2},
+(AB)_2 = \hat{2} \hat{3}, (AB)_3 = \hat{3} \hat{4}, (AB)_4 = \hat{1}
+\hat{4}$, where
+\begin{align}
+\hat{1} = 1 + \sigma(2 + \alpha(3 + \beta 4)), \quad \hat{2} = 2 +
+\alpha(3 + \beta(4 - \gamma 1)) \nonumber\\ \hat{3} = 3 + \beta(4 -
+\gamma(1 + \sigma 2)),\quad \hat{4} = 4 - \gamma(1 + \sigma(2 +
+\alpha 3))
+\end{align}
+
+Now, it is easy to see that the remaining mutual positivity
+conditions between $D_{(1)}, \cdots, D_{(4)}$ and the other
+$D_{(i)}$ are just satisfied by the lower-loop shifted amplitude;
+thus we conclude that on this cut the form is
+\begin{equation}
+\frac{d \alpha}{\alpha} \frac{d \beta}{\beta} \frac{d
+\gamma}{\gamma} \frac{d \sigma}{\sigma} \times M^{L-4}(\hat{1},
+\hat{2}, \hat{3}, \hat{4})
+\end{equation}
+
+\section{Four Particle Outlook}
+
+We have only scratched the surface of the rich amplituhedron
+geometry controlling four-particle scattering in planar ${\cal N} =
+4$ SYM at all loop order. There is obviously much more to be done
+just along the elementary lines of this note,  minimally in further continuing a
+systematic exploration of other facets of the geometry,
+corresponding to different classes of cuts of physical interest. But
+we close with a few comments about some different avenues of
+exploration.
+
+In this note we have approached the determination of the integrand
+for four-particle scattering  by directly ``triangulating" the
+amplituhedron geometry. The $L-$ loop geometry is defined in a
+self-contained way, as a subspace living inside $L$ copies of
+space-time realized as $G(2,4)$. In particular, no-where do we need
+to refer to lower-loop, higher-$k$ amplitudes, as in necessary in
+the BCFW recursion approach \cite{BCFW2} to loop integrands
+\cite{BCFWloop}. Nonetheless, it is likely that some natural
+connection exists with the full problem, and perhaps a broader view
+of the bigger amplituhedron geometry in which the four-particle
+problem sits will be important for systematically determining the
+all-loop integrand. Certainly, experience with the positive Grassmannian
+\cite{alex, positive} strongly suggests that different faces  can't be properly  understood in
+isolation.
+
+As we have seen, the approach to computing the integrand by
+triangulating the amplituhedron does not give us the familiar
+expansions that are manifestly local. This is of course not
+surprising; however, what {\it is} surprising is that some special
+local expansions expose yet {\it another} aspect of positivity, that
+we are not making apparent in the triangulation approach. As also
+mentioned in \cite{P1}, we are still clearly missing a picture of
+the form $\Omega$ which is analogous to one available for convex
+polygons, determined by a literal volume of the {\it dual} polygon.
+We don't yet have a notion of a ``dual amplituhedron", but there is
+a powerful indication that such a formulation must exist: the form
+$\Omega$ is itself positive, inside the amplituhedron! More
+specifically, we can write the $L-$loop integrand as
+\begin{equation}
+\Omega_L(AB_i)  = \prod_{i=1}^L \langle AB_i d^2 A_i \rangle \langle
+AB_i d^2 B_i \rangle M_L(AB_i)
+\end{equation}
+Then, we claim that when the $(AB)_i$ are taken to lie inside
+the amplituhedron,
+\begin{equation}
+M_L(AB_i) > 0
+\end{equation}
+We will return to exploring this fact at greater in length in
+\cite{withandrew}. We stress that this property is {\it not}
+manifest term-by-term in the amplituhedron triangulation expansion
+of the integrand. Random forms of the local expansion also don't
+make this remarkable property manifest term-by-term, but there are
+particularly nice forms of local expansion that do make this
+manifest. As we will discuss in \cite{withandrew}, we suspect that
+this surprising positivity property of the integrand is pointing the
+way to a more direct and intrinsic, triangulation-independent
+definition for the canonical form $\Omega$ associated with the
+amplituhedron.
+
+From a mathematical point of view, it is interesting that the study
+of amplitudes leads to stratifications of various collections of
+objects in projective space. If we consider a collection of $n$
+vectors in $k$ dimensions, together with a cyclic structure on this
+data, we are led to the beautiful stratification of the space given
+by the positive Grassmannian. Even just with the the four-particle
+amplituhedron, we see something new, not needing a cyclic structure
+on the objects:  given a collection of $L$ 2-planes in 4 dimensions,
+the positivity conditions are fully permutation invariant between
+the $L$ lines. Just as with the positive Grassmannian, it is natural to expect the cell structure of
+the amplituhedron to be determined in a fundamentally combinatorial
+way. The fascinating path-dependence of the forms associated with
+the cuts, together with the combinatorics that arise just in the
+simple discussion of the multi-collinear limit, are perhaps
+indications of an underlying combinatorial structure.
+
+The four-particle amplitude is a truly remarkable object. At the
+level of the integrand, at multi-loop order it contains non-trivial
+information about all the more complicated multi-particle amplitudes
+in the theory. At the level of the final integrated expression, we
+have a function that smoothly interpolates between a picture of
+``interacting gluons" at weak coupling to ``minimal area surface in
+AdS space"\cite{DCI2} at strong coupling. We have explored a reformulation of
+this physics in terms of a simple to define, yet rich
+and intricate geometry. We hope that this will lead us a more direct
+understanding of how the picture of ``gluons"  and ``strings" arise
+as different limits of a single object. As a small step in this
+direction, it is encouraging to find a natural understanding,
+intrinsic to the geometry, of the behavior of the amplitude in the
+multi-collinear region, and an associated intrinsic-to-the-geometry
+rationale for taking the log of the amplitude. Trying to more
+completely determine the IR singular behavior of the integrand of
+the amplitude is an ideal laboratory to connect our approach to the
+loop integrand with the final integrated expressions, and especially
+to ideas related to integrability. Indeed the coefficient of the
+log$^2$ infrared divergence of the log of the  amplitude is given by
+the cusp anomalous dimension, which was brilliantly determined using
+integrability in \cite{BS, Beisert:2006ez,Eden:2006rx}. It is
+notable that this approach makes crucial use of a spectral
+parameter, something which is absent in our present discussion  of
+the amplituhedron. Given the spectral deformation of
+on-shell diagrams given in \cite{Ferro:2013dga,Ferro:2012xw}, it is
+natural to ask whether a similar deformation can be found directly at the
+level of the amplituhedron.
+
+\section*{Acknowledgements}
+
+We thank Jake Bourjaily, Freddy Cachazo, Simon Caron-Huot, Johannes
+Henn, Andrew Hodges, Jan Plefka, Dave Skinner and Matthias
+Staudacher for stimulating discussions. N.~A.-H. is supported by the
+Department of Energy under grant number DE-FG02-91ER40654. J.~T. is
+supported in part by the David and Ellen Lee Postdoctoral
+Scholarship and by DOE grant DE-FG03-92-ER40701 and also by NSF
+grant PHY-0756966.
+
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+
+\bibitem{BCFW2}
+  R.~Britto, F.~Cachazo, B.~Feng and E.~Witten,
+  %``Direct proof of tree-level recursion relation in Yang-Mills theory,''
+  Phys.\ Rev.\ Lett.\  {\bf 94}, 181602 (2005)
+  [hep-th/0501052].
+
+
+\bibitem{withandrew}
+ N.~Arkani-Hamed, A.~Hodges and  J.~Trnka, to appear.
+
+\bibitem{DCI2}
+  L.~F.~Alday and J.~M.~Maldacena,
+  %``Gluon scattering amplitudes at strong coupling,''
+  JHEP {\bf 0706}, 064 (2007)
+  [arXiv:0705.0303 [hep-th]].
+
+\bibitem{BS}
+  N.~Beisert and M.~Staudacher,
+  %``The N=4 SYM integrable super spin chain,''
+  Nucl.\ Phys.\ B {\bf 670}, 439 (2003)
+  [hep-th/0307042].
+
+\bibitem{Beisert:2006ez}
+  N.~Beisert, B.~Eden and M.~Staudacher,
+  %``Transcendentality and Crossing,''
+  J.\ Stat.\ Mech.\  {\bf 0701}, P01021 (2007)
+  [hep-th/0610251].
+
+\bibitem{Eden:2006rx}
+  B.~Eden and M.~Staudacher,
+  %``Integrability and transcendentality,''
+  J.\ Stat.\ Mech.\  {\bf 0611}, P11014 (2006)
+  [hep-th/0603157].
+
+
+\bibitem{Ferro:2013dga}
+  L.~Ferro, T.~Lukowski, C.~Meneghelli, J.~Plefka and M.~Staudacher,
+  %``Spectral Parameters for Scattering Amplitudes in N=4 Super Yang-Mills Theory,''
+  arXiv:1308.3494 [hep-th].
+
+\bibitem{Ferro:2012xw}
+  L.~Ferro, T.~Lukowski, C.~Meneghelli, J.~Plefka and M.~Staudacher,
+  %``Harmonic R-matrices for Scattering Amplitudes and Spectral Regularization,''
+  Phys.\ Rev.\ Lett.\  {\bf 110}, no. 12, 121602 (2013)
+  [arXiv:1212.0850 [hep-th]].
+
+\end{thebibliography}
+
+\end{document}
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+\documentclass[12pt,nohyper,twosided]{JHEP3}
+
+\usepackage{epsfig}
+\usepackage{float}
+\usepackage{amsmath}
+%\usepackage{multirow}
+%\usepackage{mathrsfs}
+\usepackage{color}
+\usepackage{array}
+\let\normalcolor\relax
+%\usepackage{graphicx}
+\usepackage{wick}
+\usepackage{cite}
+\usepackage{enumerate}
+\usepackage{arydshln}
+
+
+\newcommand{\nn}{\nonumber}
+\newcommand{\Mp}{M_{pl}}
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+\renewcommand{\L}{\mathcal{L}}
+\renewcommand{\H}{\mathcal{H}}
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+\newcommand{\N}{\mathcal{N}}
+\newcommand{\M}{\mathcal{M}}
+\newcommand{\e}{\epsilon}
+\renewcommand{\d}{\delta}
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+\newcommand{\Th}{\Theta}
+\renewcommand{\th}{\theta}
+\newcommand{\om}{\omega}
+\renewcommand{\d}{\partial}
+\newcommand{\be}{\begin{eqnarray}}
+\newcommand{\ee}{\end{eqnarray}}
+\newcommand{\en}{\ee}
+%\newcommand{\smallminus}{-}
+%\newcommand{\smallplus}{+}
+\newcommand{\smallminus}{{\rm\rule[2.4pt]{6pt}{0.65pt}}}
+\newcommand{\smallplus}{\hspace{0.5pt}\text{{\small+}}\hspace{-0.5pt}}
+\newcommand{\mi}{\smallminus}
+\newcommand{\pl}{\smallplus}
+\newcommand{\ab}[1]{\langle #1\rangle}
+%\newcommand{\h}{\hbox{-}}
+\newcommand{\A}{{\cal A}}   \newcommand{\la}{\langle}
+\newcommand{\ra}{\rangle}
+
+
+\title{\hspace{-0.0cm}{\LARGE The Amplituhedron}}
+\author{\vspace{-.5cm}Nima Arkani-Hamed$^{a}$ and Jaroslav Trnka$^{b}$\\
+{\footnotesize{\it $^{a}$ School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA}\\
+{\it $^{b}$ California Institute of Technology, Pasadena, CA 91125,
+USA}}\vspace{-.5cm}} \preprint{2013}
+
+\abstract{Perturbative scattering amplitudes in gauge theories have
+remarkable simplicity and hidden infinite dimensional symmetries
+that are completely obscured in the conventional formulation of
+field theory using Feynman diagrams. This suggests the existence of
+a new understanding for scattering amplitudes where
+locality and unitarity do not play a central role but are derived
+consequences from a different starting point. In this note  we
+provide such an understanding for ${\cal N} = 4$ SYM scattering
+amplitudes  in the planar limit, which we identify as  ``the volume"
+of a new mathematical object--the Amplituhedron--generalizing the
+positive Grassmannian.  Locality and unitarity emerge hand-in-hand
+from positive geometry.}
+
+\preprint{CALT-68-2872}
+
+\begin{document}
+
+\newpage
+
+\section{Scattering Without Space-Time}
+
+Scattering amplitudes in gauge theories are amongst the most
+fundamental observables in physics. The textbook approach to
+computing these amplitudes in perturbation theory, using Feynman
+diagrams, makes locality and unitarity as manifest as possible, at
+the expense of introducing large amounts of gauge redundancy into
+our description of the physics, leading to an explosion of apparent
+complexity for the computation of amplitudes for all but the very
+simplest processes. Over the last quarter-century it has become
+clear that this complexity is a defect of the Feynman diagram
+approach to this physics, and is not present in the final amplitudes
+themselves, which are astonishingly simpler than indicated from the
+diagrammatic expansion \cite{PT,Z1,Z2,Witten:2003nn, CSW,
+BCFW1,BCFW2}.
+
+This has been best understood  for maximally supersymmetric gauge
+theories in the planar limit. Planar ${\cal N} = 4$ SYM has been
+used as a toy model for real physics in many guises, but as toy
+models go, its application to scattering amplitudes is closer to the
+real world than any other. For instance the leading tree
+approximation to scattering amplitudes is identical to ordinary
+gluon scattering, and the most complicated part of loop amplitudes,
+involving virtual gluons, is also the same in ${\cal N} = 4$ SYM as
+in the real world.
+
+Planar ${\cal N} = 4$ SYM amplitudes turn out to be especially
+simple and beautiful, enjoying the hidden symmetry of dual superconformal
+invariance\cite{DCI1,DCI2}, associated with a dual interpretation of
+scattering amplitudes as a supersymmetric Wilson loop
+\cite{WL1,WL2,Alday:2010zy}. Dual superconformal symmetry combines
+with the ordinary conformal symmetry to generate an infinite
+dimension ``Yangian" symmetry \cite{Yangian}. Feynman diagrams
+conceal this marvelous structure precisely as a consequence of
+making locality and unitarity manifest. For instance, the Yangian
+symmetry is obscured in either one of the standard physical
+descriptions either as a``scattering amplitude" in one space-time or
+a ``Wilson-loop" in its dual.
+
+This suggests that there must be a different formulation of the
+physics, where locality and unitarity do not play a central role,
+but emerge as derived features from a different starting point. A
+program to find a reformulation along these lines was initiated in
+\cite{N1,N2}, and  in the context of a planar ${\cal N} = 4$ SYM was
+pursued in \cite{N3,N4,N5}, leading to a new physical and
+mathematical understanding of scattering amplitudes \cite{N6}. This
+picture builds on BCFW recursion relations for tree \cite{BCFW1,
+BCFW2} and loop \cite{N5,Rutger} amplitudes, and represents the
+amplitude as a sum over basic building blocks, which can be
+physically described as arising from gluing together the elementary
+three-particle amplitudes to build more complicated on-shell
+processes. These ``on-shell diagrams" (which are essentially the
+same as the ``twistor diagrams" of \cite{TD1,TD2,N3}) are remarkably
+connected with ``cells" of a beautiful new structure in algebraic
+geometry, that has been studied by mathematicians over the past
+number of years, known as the positive Grassmannian \cite{alex, N6}.
+The on-shell building blocks can not be associated with  local
+space-time processes. Instead, they enjoy all the symmetries of the
+theory, as made manifest by their connection with the
+Grassmannian--indeed, the infinite dimensional Yangian symmetry is
+easily seen to follow from ``positive" diffeomorphisms \cite{N6}.
+
+While these developments give a complete understanding for the
+on-shell building blocks of the amplitude, they do not go further to
+explain {\it why} the building blocks have to be combined in a
+particular way to determine the full amplitude itself. Indeed, the
+particular combination of on-shell diagrams is dictated by {\it
+imposing} that the final result is local and unitary--locality and
+unitarity specify the singularity structure of the amplitude, and
+this information is {\it used} to determine the full integrand. This
+is unsatisfying, since we want to see locality and unitarity emerge
+from more primitive ideas, not merely use them to obtain the
+amplitude.
+
+An important clue \cite{N4,A1,N6} pointing towards a deeper understanding is that
+the on-shell representation of scattering
+amplitudes is not unique: the recursion relations can be solved in
+many different ways, and so the final amplitude can be expressed as
+a sum of on-shell processes in different ways as well. The on-shell
+diagrams satisfy remarkable identities--now  most easily understood
+from their association with cells of the positive Grassmannian--that
+can be used to establish these equivalences. This observation led
+Hodges \cite{A1} to a remarkable observation for the simplest case of
+``NMHV" tree amplitudes, further developed  in \cite{N7}: the amplitude
+can be thought of as the volume of a certain polytope in momentum
+twistor space. However there was no a priori
+understanding of the origin of this polytope, and the picture
+resisted a direct generalization to more general trees or to loop amplitudes.
+Nonetheless, the polytope idea motivated a continuing search for a geometric representation of the amplitude
+ as ``the volume" of ``some canonical region" in
+``some space", somehow related to the positive Grassmannian, with
+different ``triangulations" of the space corresponding to different
+natural decompositions of the amplitude into building blocks.
+
+In this note we finally realize this picture. We will introduce a
+new mathematical object whose ``volume" directly computes the
+scattering amplitude. We call this object the ``Amplituhedron", to
+denote its connection both to scattering amplitudes and positive
+geometry. The amplituhedron can be given a self-contained definition
+in a few lines as done below in section 9. We will motivate its
+definition from elementary considerations,  and show how scattering
+amplitudes are extracted from it.
+
+Everything flows from generalizing the notion of the ``inside of a
+triangle in a plane". The first obvious generalization is
+to the inside of a simplex in projective space, which further
+extends to the positive Grassmannian. The second generalization
+is to move from triangles to convex polygons, and then extend this
+into the Grassmannian. This gives us the amplituhedron for tree
+amplitudes, generalizing the positive Grassmannian by extending the
+notion of positivity to include external kinematical data. The full
+amplituhedron at all loop order further generalizes the notion of
+positivity in a way motivated by the natural idea of ``hiding
+particles".
+
+Another familiar  notion associated with triangles and polygons is
+their area. This is more naturally described in a projective way by
+a canonical 2-form with logarithmic singularities on the boundaries
+of the polygon. This form also generalizes to the full
+amplituhedron, and determines the (integrand of) the scattering
+amplitude. The geometry of the amplituhedron is completely bosonic,
+so the extraction of the superamplitude from this canonical form
+involves a novel treatment of supersymmetry, directly motivated by
+the Grassmannian structure.
+
+The connection between the amplituhedron and scattering amplitudes
+is a conjecture which has  passed a large number of non-trivial
+checks, including an understanding of how locality and unitarity
+arise as consequences of positivity. Our purpose in this note is to
+motivate and give the complete definition of the  amplituhedron and
+its connection to the superamplitude in planar ${\cal N}=4$ SYM. The
+discussion will be otherwise telegraphic and few details or examples
+will be given. In two accompanying notes \cite{Into, Threeviews}, we
+will initiate a systematic exploration of various aspects of the
+associated geometry and physics. A much more thorough exposition of
+these ideas, together with many examples worked out in detail, will
+be presented in \cite{Long}.
+
+\subsection*{Notation}
+
+The external data for massless $n$ particle scattering amplitudes
+(for an excellent review see \cite{review}) are labeled as
+$|\lambda_a,\tilde \lambda_a, \tilde \eta_a \rangle$ for $a=1,
+\dots, n$. Here $\lambda_a, \tilde \lambda_a$ are the
+spinor-helicity variables, determining null momenta $p_a^{A \dot{A}}
+= \lambda_a^A \tilde \lambda_a^{\dot{A}}$. The $\tilde \eta_a$ are
+(four) grassmann variables for  on-shell superspace. The component
+of the color-stripped superamplitude with weight $4(k+2)$ in the $\tilde \eta$'s is
+$M_{n,k}$. We can write \be M_{n,k}(\lambda_a, \tilde \lambda_a,
+\tilde \eta_a) = \frac{\delta^4(\sum_a \lambda_a \tilde \lambda_a)
+\delta^8(\sum_a \lambda_a \tilde \eta_a)}{\langle 1 2 \rangle \dots
+\langle n 1 \rangle} \times {\cal M}_{n,k}( z_a, \eta_a) \ee where
+$(z_a, \eta_a)$ are the (super) ``momentum-twistor" variables
+\cite{A1}, with $ z_a = \left(\begin{array}{c} \lambda_a \\ \mu_a
+\end{array} \right)$. The $z_a, \eta_a$ are unconstrained, and
+determine the $\lambda_a, \tilde \lambda_a$ as
+\begin{eqnarray}
+\tilde \lambda_a &=& \frac{\langle a\mi1 \, a \rangle \mu_{a\pl1} + \langle a\pl1 \, a\mi1
+\rangle \mu_a + \langle a \, a\pl1 \rangle \mu_{a\mi1}}{\langle a\mi1 \, a \rangle \langle a \, a\pl1 \rangle}, \nonumber \\
+\tilde \eta_a &=& \frac{\langle a\mi1 \, a \rangle \eta_{a\pl1} +
+\langle a\pl1 \, a\mi1 \rangle \eta_a + \langle a \, a\pl1 \rangle
+\eta_{a\mi1}}{\langle a\mi1 \, a \rangle \langle a \, a\pl1 \rangle}
+\end{eqnarray}
+where throughout
+this paper, the angle brackets $\langle \dots \rangle$ denotes
+totally antisymmetric contraction with an $\epsilon$ tensor.
+${\cal M}_{n,k}$ is cyclically invariant. It is also invariant under the little group action $(z_a, \eta_a)
+\to t_a (z_a, \eta_a)$, so $(z_a, \eta_a)$ can be taken to live in
+$\mathbb{P}^{3|4}$.
+
+At loop level, there is a well-defined notion of ``the integrand"
+for scattering amplitudes, which at $L$ loops is a $4L$ form. The
+loop integration variables are points in the (dual) spacetime $x^\mu_i$,
+which in turn can be associated with $L$ lines in momentum-twistor
+space that we denote as ${\cal L}_{(i)}$ for $i = 1, \cdots, L$. The
+$4L$ form is \cite{Andrewloop, LDbox, LocalIntegrand} \be {\cal
+M}(z_a, \eta_a; {\cal L}_{(i)}) \ee We can specify the line by
+giving two points ${\cal L}_{1 (i)},{\cal L}_{2 (i)}$ on it, which
+we can collect as ${\cal L}_{\gamma(i)}$ for $\gamma = 1,2$. ${\cal
+L}$ can also be thought of as a $2$ plane in 4 dimensions. In
+previous work, we have often referred to the two points on the line
+${\cal L}_1, {\cal L}_2$ as ``$AB$", and we will use this notation here
+as well.
+
+Dual superconformal symmetry says that ${\cal M}_{n,k}$ is invariant
+under the $SL(4|4)$ symmetry acting on $(z_a,\eta_a)$ as
+(super)linear transformations. The full symmetry of the theory is the Yangian of
+$SL(4|4)$.
+
+\section{Triangles $\to$ Positive Grassmannian}
+
+To begin with, let us start with the simplest and most familiar
+geometric object of all, a triangle in two dimensions. Thinking
+projectively, the vertices are $Z_1^I,Z_2^I,Z_3^I$ where $I=1,\dots,
+3$. The interior of the triangle is a collection of points of the
+form \be Y^I = c_1 Z_1^I + c_2 Z_2^I + c_3 Z_3^I \ee
+where we span over all $c_a$ with
+\be
+c_a > 0
+\ee
+$$
+\includegraphics[scale=.75]{pix12.pdf}
+$$
+More precisely, the interior of a triangle is associated with a
+triplet $(c_1,c_2,c_3)/GL(1)$, with all ratios $c_a/c_b > 0$, so
+that the $c_a$ are either all positive or all negative, but here and
+in the generalizations that follow,  we  will abbreviate this by
+calling them all positive.  Including the closure of the triangle
+replaces ``positivity"  with ``non-negativity", but we will continue
+to refer to this as ``positivity" for brevity.
+
+One obvious generalization
+of the triangle is to an $(n-1)$ dimensional simplex in a general
+projective space, a collection $(c_1, \dots, c_n)/GL(1)$, with $c_a
+> 0$. The $n$-tuple $(c_1, \dots, c_n)/GL(1)$ specifies a line in $n$
+dimensions, or a point in $\mathbb{P}^{n-1}$. We can generalize this
+to the space of $k$-planes in $n$ dimensions--the Grassmannian
+$G(k,n)$--which we can take to be a collection of $n$
+$k-$dimensional vectors modulo $GL(k)$ transformations, grouped into
+a $k \times n$ matrix \be C =   \left( \begin{array}{ccc} & & \\ c_1
+&  \dots &  c_n \ \\ & & \end{array} \right)/GL(k) \ee
+
+Projective space is the special case of $G(1,n)$. The notion of
+positivity giving us the ``inside of a simplex" in projective space
+can be generalized to the Grassmannian. The only possible $GL(k)$
+invariant notion of positivity for the matrix $C$ requires us to fix
+a particular ordering of the columns, and demand that all minors in
+this ordering are positive: \be \langle c_{a_1} \dots c_{a_k}
+\rangle > 0 \, \, {\rm for} \, \, a_1<\dots<a_k \ee We can also talk
+about the very closely related space  of positive matrices
+$M_+(k,n)$, which are just $k \times n$ matrices with all positive
+ordered minors. The only difference with the positive Grassmannian
+is that in $M_+(k,n)$ we are not moding out by $GL(k)$.
+
+Note that while  both $M_+(k,n)$ and $G_+(k,n)$ are defined with a
+given ordering for the columns of the matrices, they have a natural
+cyclic structure. Indeed, if $(c_1, \dots, c_n)$ give a positive
+matrix, then positivity is preserved under the  (twisted) cyclic
+action  $c_1 \to c_2, \dots, c_n \to (-1)^{k-1} c_1$.
+
+\section{Polygons $\to$ (Tree) Amplituhedron ${\cal A}_{n,k}(Z)$}
+
+Another natural generalization of a triangle is to a more
+general polygon with $n$ vertices $Z^I_1, \dots, Z^I_n$. Once again
+we would like to discuss the interior of this region. However in
+general this is not canonically defined--if the points $Z_a$ are
+distributed randomly, they don't obviously enclose a region where all
+the $Z_a$ are all vertices. Only if the $Z_a$ form a {\it convex}
+polygon do we have a canonical ``interior" to talk about:
+
+$$
+\includegraphics[scale=.65]{pix21.pdf}
+$$
+
+
+Now,  convexity for the $Z_a$ is a special case of positivity in the
+sense of the positive matrices we have just defined. The points
+$Z_a$ form a closed polygon only if the $3 \times n$ matrix with
+columns $Z_a$ has all positive (ordered) minors: \be \langle Z_{a_1}
+Z_{a_2} Z_{a_3} \rangle > 0 \quad {\rm for} \quad a_1 < a_2 < a_3
+\ee Having arranged for this, the interior of the polygon is given
+by points of the form \be Y^I = c_1 Z_1^I + c_2 Z_2^I + \dots c_n
+Z_n^I \quad {\rm with}\quad  c_a > 0 \ee Note that this can be
+thought of as an interesting pairing of two different positive
+spaces. We have \be (c_1, \dots, c_n) \subset G_+(1,n),\quad
+\left(Z_1, \dots, Z_n \right) \subset M_+(3,n) \ee If we jam them
+together to produce \be Y^I = c_a Z_a^I \ee for fixed $Z_a$, ranging
+over all $c_a$ gives  us all the points on the inside of the
+polygon, living in $G(1,3)$.
+
+This object has a natural generalization to higher projective
+spaces; we can consider $n$ points $Z_a^I$ in $G(1,1+m)$, with $I =
+1, \dots, 1+m$, which are positive \be \langle Z_{a_1} \dots Z_{a_{1
++ m}} \rangle > 0 \ee Then, the analog of the ``inside of the
+polygon" are points of the form \be Y^I = c_a Z_a^I, \quad {\rm
+with} \quad c_a > 0 \ee This space is very closely related to the
+``cyclic polytope" \cite{cyclic}, which is the convex hull of $n$
+ordered points on the moment curve in $\mathbb{P}^{m}$, $Z_a =
+(1,t_a,t_a^2, \dots, t_a^{m})$, with $t_1 < t_2 \dots < t_n$. From
+our point of view, forcing the points to lie on the moment curve is
+overly restrictive; this is just one way of ensuring the positivity
+of the $Z_a$.
+
+We can further generalize this structure into the Grassmannian. We
+take positive  external data as $(k + m)$ dimensional vectors
+$Z_a^I$ for $I =1, \dots, k+m$.  It is natural to restrict $n \geq
+(k+m)$, so that the external $Z_a$ fill out the entire $(k+m)$
+dimensional space. Consider the space of $k$-planes in this $(k+m)$
+dimensional space, $Y \subset G(k,k+m)$, with co-ordinates \be
+Y_\alpha^I, \, \alpha = 1, \dots k, \, I = 1, \dots, k+m \ee We then
+consider a subspace of $G(k,k+m)$ determined by taking all possible
+``positive" linear combinations of the external data, \be Y = C
+\cdot Z \ee or more explicitly \be Y_\alpha^I = C_{\alpha a} Z_a^I
+\ee where \be C_{\alpha a} \subset G_+(k,n), Z_a^I \subset
+M_+(k+m,n) \ee
+It is trivial to see that this space is cyclically invariant if $m$ is even: under the twisted cyclic symmetry,
+$Z_n \to (-1)^{k+m-1} Z_1$ and $c_n \to (-1)^{k-1} c_1$, and the product is invariant for even $m$.
+
+We call this space the generalized tree amplituhedron ${\cal A}_{n,k,m}(Z)$. The polygon is the simplest case with $k=1,m=2$. Another special case is $n = (k+m)$, where we can  use $GL(k+m)$
+transformations to set the external data to the identity matrix
+$Z_a^I = \delta_a^I$. In this case ${\cal A}_{k+m,k,m}$ is identical to the usual
+positive Grassmannian $G_+(k,k+m)$.
+
+The case of immediate relevance to physics is $m=4$, and we will refer to this as the tree amplituhedron
+${\cal A}_{n,k}(Z)$. The tree amplituhedron lives in a
+$4k$ dimensional space and is not trivially visualizable. For $k=1$, it
+is  a polytope, with inequalities determined by linear
+equations, while for $k>1$, it is not a polytope and is
+more ``curvy". Just to have a picture,
+below we sketch a 3-dimensional face of the 4 dimensional
+amplituhedron for $n=8$, which turns out to be the space $Y = c_1 Z_1 +\dots c_7
+Z_7$ for $Z_a$ positive external data  in $\mathbb{P}^3$:
+$$
+\includegraphics[scale=.6]{pix11.pdf}
+$$
+
+\section{Why Positivity?}
+
+We have motivated the structure of the amplituhedron by mimicking
+the geometric idea of the ``inside" of a convex polygon.
+However there is a simpler and deeper origin of  the need for
+positivity. We can  attempt to define $Y = C \cdot Z$ with no
+positive  restrictions on $C$ or $Z$. But in general, this will not
+be projectively meaningful, and this expression  won't allow us to
+define a region in $G(k,k+m)$. The reason is that  for $n > k+m$,
+there is always some linear combination of the $Z_a$ which sum to
+zero! We have to take care to avoid this happening, and in order to
+avoid ``0" on the left hand side, we obviously need positivity
+properties on both the $Z$'s and the $C$'s.
+
+It is simple and instructive to see why positivity ensures that the
+$Y = C \cdot Z$ map is projectively well-defined. We will see this
+as a by-product of locating  the co-dimension one boundaries of the
+generalized tree amplituhedron. Let us illustrate the idea already
+for the simplest case of the polygon with $k=1, m=2$, with $Y = c_1
+Z_1 + \dots c_n Z_n$.  In order to look at the boundaries of the
+space, let us compute $\langle Y Z_i Z_j \rangle$ for some $i,j$. If
+as we sweep through all the allowed $c$'s, $\langle Y Z_i Z_j
+\rangle$ changes sign from being positive to negative, then
+somewhere $\langle Y Z_i Z_j \rangle \to 0$ and $Y$ lies on the line
+$(Z_i Z_j)$ in the interior of the space, thus $(Z_i Z_j)$ should
+not be a boundary of the polygon. On the other hand, if $\langle Y
+Z_i Z_j \rangle$ everywhere has a uniform sign, then $(Z_i Z_j)$ is
+a boundary of the polygon:
+
+$$
+\includegraphics[scale=.6]{pix22.pdf}
+$$
+
+Of course for the polygon it is  trivial to directly see that the
+co-dimension one boundaries are just the lines $(Z_i Z_{i+1})$, but
+we wish to see this more algebraically, in a way that will
+generalize to the amplituhedron where ``seeing" is harder.
+So, we compute \be \langle Y Z_i Z_j \rangle =\sum_a c_a \langle Z_a
+Z_i Z_j \rangle \ee We can see why there is some hope for the
+positivity of this sum, since the $c_a > 0$, and also ordered minors
+of the $Z's$ are positive. It is however obvious that if $i,j$ are
+not consecutive, some of the terms in this sum will be positive, but
+some (where $a$ is stuck between $i,j$) will be negative. But
+precisely when $i,j$ are consecutive, we get a manifestly
+positive sum: \be \langle Y Z_i Z_{i+1} \rangle = \sum_a c_a \langle
+Z_a Z_i Z_{i+1} \rangle > 0 \ee Since $\langle Z_a Z_i Z_{i+1}
+\rangle > 0$ for $a \neq i, i+1$, this is manifestly positive. Thus
+the boundaries are lines $(Z_i Z_{i+1})$ as expected.
+
+This also tells us that the map  $Y = C \cdot Z$ is projectively well-defined.
+There is no way to get $Y \to 0$, since this would make the left hand side
+identically zero, which is impossible without making all the $c_a$
+vanish, which is not permitted as we we mod out by
+$GL(1)$ on the $c_a$.
+
+We can extend this logic to higher $k,m$. Let's look at the case
+$m=4$ already for $k=1$.
+We can investigate whether the plane $(Z_i Z_j Z_k Z_l)$ is a boundary by computing
+\be \langle Y Z_i Z_j Z_k Z_l \rangle = \sum_a c_a  \langle Z_a Z_i
+Z_j Z_k Z_l \rangle \ee Again, this is not in general positive. Only for
+$(i,j,k,l)$ of the form $(i,i+1,j, j+1)$, we have \be \langle Y Z_i
+Z_{i+1} Z_j Z_{j+1} \rangle = \sum_a c_a  \langle Z_a Z_i Z_{ i+1}
+Z_j Z_{j+1} \rangle > 0 \ee For general even the  $m$, the
+boundaries are when $Y$ is on the plane\\ $(Z_{i} Z_{i+1}
+\dots Z_{i_{m/2 - 1}} Z_{i_{m/2}})$. This again shows that the $Y =
+C \cdot Z$ is projectively well-defined. The result extends
+trivially to general $k$, provided the positivity of $C$ is
+respected. For $m = 4$ the boundaries are again when the $k$-plane $(Y_1 \cdots Y_k)$ is on  $(Z_i Z_{i+1} Z_j
+Z_{j+1})$, as follows from \be \langle Y_1 \dots Y_k Z_i Z_{i+1} Z_j
+Z_{j+1} \rangle = \sum_{a_1< \dots < a_k} \langle c_{a_1} \dots
+c_{a_k} \rangle  \langle Z_{a_1} \dots Z_{a_k} Z_i Z_{i+1} Z_j
+Z_{j+1} \rangle > 0 \ee which also shows that $Y$ is always a full rank
+$k$-plane in $k+4$ dimensions.
+
+The emergence of boundaries on the plane $(Z_{i} Z_{i+1} Z_j
+Z_{j+1})$ is a simple and striking consequence of positivity. We will
+shortly understand that the location of these boundaries are the
+``positive origin" of locality from the geometry of the
+amplituhedron.
+
+\section{Cell Decomposition}
+
+The tree amplituhedron can be thought of as the image of the
+top-cell of the the positive Grassmannian $G_+(k,n)$ under the map
+$Y = C \cdot Z$. Since ${\rm dim}\, G(k,k+m) = m k \leq\,{\rm
+dim}\,G(k,n)= k (n -k)$ for $n \geq k+m$, this is in general a
+highly redundant map. We can already see this in the simplest case
+of the polygon, which lives in 2 dimensions, while the $c_a$ span an
+$(n-1)$ dimensional space. The non-redundant maps into $G(k,k+m)$
+can only come from the $m \times k$ dimensional ``cells" of
+$G_+(k,n)$. For the polygon, these are the cells we can label as
+$(i,j,k)$, where all but $(c_i,c_j,c_k)$ are non-vanishing. The
+image of these cells in the $Y$-space are of course just the
+triangles with vertices at $Z_i,Z_j,Z_k$, which lie inside the
+polygon.
+
+The  union of all these triangles covers the inside of the polygon.
+However, we can also cover the inside of the polyon more nicely with
+non-overlapping triangles, giving a triangulation. Said in a
+heavy-handed way, we find a collection of 2 dimensional cells of
+$G_+(1,n)$, so that their images in $Y$ space are non-overlapping
+except on boundaries, and collectively cover the entire polygon. Of
+course these collections of cells are not unique--there are many
+different triangulations of the polygon. A particularly simple one
+is
+$$
+\includegraphics[scale=.6]{pix23.pdf}
+$$
+which we can write as \be \sum_i (1\,i\,i\pl1) \ee Sticking with
+$k=1$ but moving to $m=4$, the four-dimensional cells of $G_+(1,n)$
+are labeled by five non-vanishing $c$'s  $(c_i,c_j,c_k,c_l,c_m)$.
+While it is harder to visualize, one can easily show algebraically
+that the above simple triangulation of the polygon generalizes to
+\be \sum_{i<j} (1\,i\,i\pl1\,j\,j\pl1) \ee
+
+This expression is immediately recognizable  to physicists familiar
+with scattering amplitudes in ${\cal N} = 4$ SYM. If  the
+$(i,j,k,l,m)$ are interpreted as ``R-invariants", this expression is
+one of the canonical BCFW representations of the $k=1$ ``NMHV" tree
+amplitudes. In the positive Grassmannian representation for
+amplitudes \cite{N4,N6},  R-invariants are precisely associated with
+the corresponding four-dimensional cells of $G(1,n)$.
+
+For general $k$, $m$ any $m \times k$ dimensional cell of $G_+(k,n)$
+maps under $Y = C \cdot Z$ into some region or cell in $G(k,k+m)$.
+Said more explicitly, consider an $m \times k$ dimensional cell
+$\Gamma$ of the $G_+(k,n)$, with ``positive co-ordinates"
+$C^\Gamma(\alpha^\Gamma_1, \dots, \alpha^\Gamma_{m \times k})$ \cite{N6}.
+Putting $Y = C(\alpha) \cdot Z$ and scanning over all positive
+$\alpha$'s, this carves out a region in $G(k,k+m)$ which is a
+corresponding cell $\Gamma$ of the tree amplituhedron. A cell
+decomposition is a collection $T$ of non-overlapping cells $\Gamma$
+which cover the entire amplituhedron.
+
+The case of immediate relevance for physics is $m=4$. For any $k$,
+the BCFW decomposition of tree amplitudes gives us a collection of
+$4 \times k$ dimensional cells of the positive Grassmannian. We have
+performed extensive checks for high $k$ and $n$, that for positive
+external $Z$, under $Y = C \cdot Z$ these cells are beautifully
+mapped into non-overlapping regions of $G(k,k+4)$ that together
+cover the entire tree amplituhedron. As we have stressed, other than
+the desire to make the final result local and unitary, we did not
+previously have a rational for thinking about this particular
+collection of cells of $G_+(k,n)$. Now we know what natural question
+this collection of cells are answering: taken together they
+``cellulate" the tree amplituhedron.  We will shortly see how to
+directly associate the amplitude itself directly with the geometry
+of the amplituhedron.
+
+
+\section{``Volume" as Canonical Form}
+
+Before discussing how to determine the (super)amplitude from the
+geometry, let us define the notion of a ``volume" associated with
+the amplituhedron. As should by now be expected, we will merely generalize a simple
+existing idea from the world of triangles and polygons.
+
+The usual notion of ``area" has units and is obviously not projectively
+meaningful. However there is a closely related idea that is. For the
+triangle, we can consider a rational 2-form in $Y$-space,  which has
+logarithmic singularities on the boundaries of the triangle. This is
+naturally associated with positive co-ordinates for the triangle, if
+we expand $Y = Z_3 + \alpha_1 Z_1 + \alpha_2 Z_2$, then the form is \be
+\Omega_{123} = \frac{d\alpha_1}{\alpha_1} \frac{d\alpha_2}{\alpha_2} \ee which can be
+re-written more invariantly as \be \Omega_{123} = \frac{\langle Y d
+Y d Y \rangle \langle 1 2 3 \rangle^2}{\langle Y 12 \rangle \langle
+Y 2 3 \rangle \langle Y 3 1 \rangle} \ee We can extend this to a
+form $\Omega_P$ for the convex polygon $P$. The defining property of
+$\Omega_P$ is that
+
+\begin{center}
+$\Omega_P$ has logarithmic singularities on all the boundaries of
+$P$.
+\end{center}
+$\Omega_P$ can be obtained by first triangulating the polygon in
+some way, then summing the elementary two-form for each triangle,
+for instance as \be \Omega_P = \sum_i \Omega_{1\,i\,i\pl1}. \ee Each
+term in this sum has singularities corresponding to $Y$ hitting the
+boundaries of the corresponding triangle. Most of these singularities, associated
+with the internal edges of the triangulation, are spurious, and
+cancel in the sum. Of course the full form $\Omega_P$ is independent
+of the particular triangulation.
+
+This form is closely related to an area, not directly of the polygon
+$P$, but its dual $\tilde{P}$, which is also a convex polygon
+\cite{N7}. If we dualize so that points are mapped to lines and
+lines to points, then a point $Y$ {\it inside} $P$ is mapped to a
+line $Y$ {\it outside} $\tilde{P}$. If we write $\Omega_P = \langle
+Y d^2 Y \rangle V(Y)$, then $V(Y)$ is the area of $\tilde{P}$ living
+in the euclidean space defined by $Y$ as the line at infinity.
+
+This form can be generalized to the tree amplituhedron in an obvious
+way. We define a rational form $\Omega_{n,k}(Y;Z)$ with the property
+that
+
+\begin{center}
+$\Omega_{n,k}(Y;Z)$ has logarithmic singularities on all the
+boundaries of ${\cal A}_{n,k}(Z)$.
+\end{center}
+
+Just as for the polygon, one concrete way of computing  $\Omega$ is
+to give a cell decomposition of the amplituhedron. For any cell
+$\Gamma$ associated with positive co-ordinate $(\alpha^\Gamma_1,
+\dots, \alpha^\Gamma_{4k})$, there is an associated form with
+logarithmic singularities on the boundaries of the cell \be
+\Omega_{n,k}^\Gamma(Y; Z) = \prod_{i = 1}^{4k} \frac{d
+\alpha_i^\Gamma}{\alpha_i^\Gamma} \ee For instance, consider 4
+dimensional cells for $k=1$, associated with cells in $G_+(1,n)$
+which are vanishing for all but columns $a_1, \dots, a_5$, with
+positive co-ordinates\\  $(\alpha_{a_1}, \dots, \alpha_{a4},
+\alpha_{a_5} =1)$.  Its image in $Y$ space is simply \be Y =
+\alpha_{a1} Z_{a_1} + \dots \alpha_{a_4} Z_{a_4}  + Z_{a_5} \ee and
+the form is \be \frac{d \alpha_{a1}}{\alpha_{a1}} \dots \frac{d
+\alpha_{a4}}{\alpha_{a4}} = \frac{\langle Y d^4 Y \rangle \langle
+Z_{a_1} Z_{a_2} Z_{a_3} Z_{a_4} Z_{a_5} \rangle^4}{\langle Y Z_{a_1}
+Z_{a_2} Z_{a_3} Z_{a_4} \rangle \dots \langle Y Z_{a_5} Z_{a_1}
+Z_{a_2} Z_{a_3} \rangle} \ee Now, given a collection of cells $T$
+that cover the full amplituhedron, $\Omega_{n,k}(Y;Z)$ is given by
+\be \Omega_{n,k}(Y;Z) = \sum_{\Gamma \subset T}
+\Omega_{n,k}^\Gamma(Y;Z) \ee As with the polygon, the form is
+independent of the particular cell decomposition.
+
+Note that the definition of the amplituhedron itself crucially
+depends on the positivity of the external data $Z$, and this
+geometry in turn determines the form $\Omega$. However, once this
+form is in hand, it can be analytically continued to any
+(complex!) $Y$ and $Z$.
+
+\section{The Superamplitude}
+
+We have already defined central objects in our story: the
+tree amplituhedron, together with the associated form $\Omega$
+that is loosely speaking its ``volume". The
+tree super-amplitude ${\cal M}_{n,k}$ can be
+directly extracted from $\Omega_{n,k}(Z)$. To see how, note that
+we we can always use a $GL(4 +k)$ transformation to send $Y \to Y_0$
+as \be Y_0 = \left(\begin{array}{ccc} & 0_{4 \times k} & \\
+\hdashline & 1_{k \times k} & \end{array} \right) \ee We can think
+of the 4 dimensional space complementary to $Y_0$, acted on by an
+unbroken $GL(4)$ symmetry, as the usual $\mathbb{P}^3$ of
+momentum-twistor space. Accordingly, we identify the top four
+components of the $Z_a$ with the usual bosonic momentum-twistor
+variables $z_a$: \be Z_a = \left(\begin{array}{c} z_a \\ *_1 \\
+\vdots \\ *_k
+\end{array}\right) \ee We still have to decide how to interpret the
+remaining $k$ entries of $Z_a$. Clearly, if they are normal bosonic
+variables, we have an infinite number of extra degrees of freedom.
+It is therefore natural to try and make the remaining components
+infinitesimal, by saying that some ${\cal N} + 1$'st power of them
+vanishes. This is equivalent to saying that each entry can be
+written as \be Z_a = \left(\begin{array}{c} z_a \\ \phi^A_1 \cdot
+\eta_{1 A} \\ \vdots \\ \phi^A_k \cdot \eta_{A k} \end{array}\right)
+\ee where $\phi_{1,\dots,k}$ and $\eta_a$ are Grassmann parameters,
+and $A = 1, \dots, {\cal N}$.
+
+Now there is a unique way to extract the amplitude. We simply
+localize the form $\Omega_{n,k}(Y;Z)$ to $Y_0$, and integrate over the
+$\phi$'s: \be {\cal M}_{n,k}(z_a,\eta_a) = \int d^{\cal N} \phi_1 \dots
+d^{\cal N} \phi_k \int \Omega_{n,k}(Y;Z)  \delta^{4k}(Y;Y_0)
+\label{super}\ee Here $\delta^{4k}(Y;Y_0)$ is a projective $\delta$
+function \be \delta^{4k}(Y;Y_0) = \int d^{k \times k}
+\rho_\alpha^\beta \, {\rm det}(\rho)^4 \, \delta^{k \times
+(k+4)}(Y_\alpha^I - \rho_{\alpha}^\beta Y_{0 \beta}^I) \ee Note that
+there is really no integral to perform in the second step; the delta
+functions fully fix $Y$. Any form on $G(k, k+4)$ is of the form
+\begin{equation}
+\Omega =  \langle Y_1 \dots Y_k d^4 Y_1 \rangle \dots \langle Y_1
+\dots Y_k d^4 Y_k \rangle \times \omega_{n,k}(Y;Z)
+\end{equation}
+and our expression just says that \be {\cal M}_{n,k}(z_a,\eta_a) = \int
+d^{\cal N} \phi_1 \dots d^{\cal N} \phi_k  \omega_{n,k}(Y_0;Z_a) \ee
+Note that we can define this operation for any ${\cal N}$, however,
+only for ${\cal N} = 4$ does ${\cal M}_{n,k}$ further have weight zero under the
+rescaling $(z_a,\eta_a)$.
+
+This connection between the form and the super-amplitude also allows
+us to directly exhibit the relation between our super-amplitude
+expressions and the Grassmannian formulae of \cite{N4,N6}. Consider
+the form in $Y$-space associated with a given $4k$ dimensional cell
+$\Gamma$ of $G_+(k,n)$. Then, if $C^\Gamma_{\alpha
+a}(\alpha_1,\dots, \alpha_{4k})$ are positive co-ordinates for the
+cell, and $\Omega^\Gamma = \frac{d\alpha^\Gamma_1}{\alpha^\Gamma_1}
+\dots \frac{d\alpha^\Gamma_{4k}}{\alpha^\Gamma_{4k}}$ is the
+associated form in $Y$ space, then it is easy to show that \be \int
+d^4 \phi_1 \dots d^4 \phi_k \int \Omega^\Gamma \delta^{4k}(Y;Y_0) =
+\int \frac{d\alpha^\Gamma_1}{\alpha^\Gamma_1} \dots
+\frac{d\alpha^\Gamma_{4k}}{\alpha^\Gamma_{4k}}
+\delta^{4k|4k}(C_{\alpha a}(z) {\cal Z}_a) \ee where ${\cal Z}_a =
+(z_a|\eta_a)$ are the super momentum-twistor variabes. This is
+precisely the formula for computing on-shell diagrams (in
+momentum-twistor space) as described in \cite{N4,LD1,N6}. Thus,
+while the amplituhedron geometry and the associated form $\Omega$
+are purely bosonic, we have extracted from them super-amplitudes
+which are manifestly supersymmetric. Indeed, the connection to the
+Grassmannian shows much more--the superamplitude obtained for each
+cell is manifestly Yangian invariant \cite{N6}.
+
+\section{Hiding Particles $\to$ Loop Positivity in $G_+(k,n;L)$}
+The direct generalization of ``convex polygons" into the
+Grassmannian $G(k,k+4)$ has given us the tree amplituhedron. We will
+now ask a simple question: can we ``hide particles" in a natural
+way? This will lead to an extended notion of positivity giving us
+loop amplitudes.
+
+It is trivial to imagine what we might mean by hiding a single
+particle, but as we will see momentarily, the idea of hiding
+particles is only natural if we hide {\it pairs} of {\it adjacent}
+particles. To pick a concrete example, suppose we have some positive
+matrix $C$ with columns we'll label $(A_1,B_1, 1, 2, \dots, m, A_2, B_2,
+m+1, \dots n)$. We can always gauge-fix the $A_1,B_1$ and $A_2,B_2$ columns
+so that the matrix takes the form
+
+$$\bordermatrix{\text{}&A_1&B_1&1&2&\ldots&m&A_2&B_2&m+1&\ldots&n\cr
+               &1&0&\ast&\ast&\ldots&\ast&0&0&\ast&\ldots & \ast\cr
+               &0&1&\ast&\ast&\ldots&\ast&0&0&\ast&\ldots & \ast\cr
+               &0&0&\ast&\ast&\ldots&\ast&1&0&\ast&\ldots & \ast\cr
+               &0&0&\ast&\ast&\ldots&\ast&0&1&\ast&\ldots & \ast\cr
+               &0&0&\ast&\ast&\ldots&\ast&0&0&\ast&\ldots & \ast\cr
+               &\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\cr
+               &0&0&\ast&\ast&\ldots&\ast&0&0&\ast&\ldots & \ast\cr}$$
+We would now like to ``hide" the particles $A_1,B_1,A_2,B_2$. We do
+this simply by chopping out the corresponding columns. The remaining
+matrix can be grouped into the form \be \left(\begin{array}{ccc}&
+D_{(1)} & \\ \hdashline  & D_{(2)} & \\ \hdashline & C & \end{array}
+\right) \ee But the ``hidden" particles leave an echo in the
+positivity properties of this matrix. The positivity of the minors
+involving all of $(A_1,B_1,A_2,B_2)$, $(A_2,B_2)$ and $(A_1,B_1)$
+individually, as well those not involving $A_1,B_1,A_2,B_2$ at all
+enforce that the ordered maximal minors of the following matrices
+\be \left(\begin{array}{ccc} & C & \end{array} \right), \left(
+\begin{array}{ccc}& D_{(1)}& \\ \hdashline & C & \end{array} \right),
+\left( \begin{array}{ccc} &D_{(2)}& \\ \hdashline & C  & \end{array}
+\right), \left(\begin{array}{ccc} &  D_{(1)}&  \\ \hdashline &
+D_{(2)}&
+\\ \hdashline &C& \end{array} \right) \ee are all positive.
+
+We can now see why particles are most naturally hidden in pairs. If
+we had instead hidden single particles as $A_1, A_2, A_3, \dots$ in
+separate columns, the remaining minors would be positive or negative
+depending on the orderings of $A_1,A_2,A_3, \dots$, which is
+additional structure over and above the cyclic ordering of the
+external data. In order to avoid this arbitrariness, we should hide
+particles in even numbers, with pairs the minimal case. In order to
+ensure that only minors involving the pairs $(A_i B_i)$ are taken into
+account,  we mod out by the $GL(2)$ action rotating the $(A_i,B_i)$ columns
+into each other.
+
+This ``hidden particle" picture has thus motivated an extended
+notion of positivity associated with the Grassmannian. We are used
+to considering a $k$-plane in $n$ dimensions $C$, with all ordered
+minors positive. But we can also imagine a collection of $L$
+2-planes $D_{(i)}$ in the $(n-k)$ dimensional complement of $C$,
+schematically
+$$
+\includegraphics[scale=.65]{pix3.pdf}
+$$
+We call this space $G(k,n;L)$, and we will denote the collection of
+$(D_{(i)}, C)$ as ${\cal C}$.  We can extend the notion of
+positivity to $G(k,n;L)$ by demanding that not only the ordered
+minors of $C$, but also of $C$ with any collection of the $D_{(i)}$,
+are positive. (All minors must include the matrix $C$, since the
+$D_{(i)}$ are defined to live in the complement of $C$). Note that
+this notion is completely permutation invariant in the $D_{(i)}$.
+
+Very interestingly, it turns out that while we motivated this notion
+of positivity by hiding particles from an underlying positive
+matrix, there are positive configurations of ${\cal C}$ that can not
+be obtained by hiding particles from a positive matrix in this way.
+
+Extending the map $Y = C . Z$ in the obvious way to include the
+$D$'s leads us to define the full amplituhedron.
+
+\section{The Amplituhedron ${\cal A}_{n,k,L}(Z)$}
+
+We can now give the full definition of the amplituhedron ${\cal
+A}_{n,k,L}(Z)$. First,  the external data for $n \geq k+4$
+particles is given by the vectors $Z_a^I$ living in a $(4 + k)$
+dimensional space; where $a= 1, \dots, n$ and $I = 1, \dots, 4 + k$.
+The data is positive \be \langle Z_{a_1} \dots Z_{a_{4 + k}} \rangle
+> 0 \quad {\rm for} \quad  a_1 < \dots < a_{4 + k} \ee The amplituhedron
+lives in $G(k,k+4;L)$: the space of $k$ planes $Y$ in $(k+4)$
+dimensions, together with $L$ 2-planes ${\cal L}_{(i)}$ in the 4
+dimensional complement of $Y$, schematically
+$$
+\includegraphics[scale=.65]{pix28.pdf}
+$$
+We will denote the collection of $({\cal L}_{(i)}, Y)$ as ${\cal
+Y}$.
+
+The amplituhedron ${\cal A}_{n,k,L}(Z)$ is the subspace of
+$G(k,k+4;L)$ consisting of all ${\cal Y}$'s which are a positive
+linear combination of the external data, \be {\cal Y} = {\cal C}
+\cdot Z \ee More explicitly in components, the $k$-plane is
+$Y_{\alpha}^I$, and the 2-planes are ${\cal L}_{\gamma (i)}^I$,
+where $\gamma = 1,2$ and  $i = 1, \dots, L$ . The amplituhedron is
+the space of all $Y, {\cal L}_{(i)}$ of the form \be Y_{\alpha}^I =
+C_{\alpha a} Z_a^I, \, \, {\cal L}_{\gamma (i)}^{I} = D_{\gamma a
+(i)} Z_a^I \ee where as in the previous section the $C_{\alpha a}$
+specifies a $k$-plane in $n$-dimensions, and the $D_{\gamma a (i)}$
+are $L$ 2-planes living in the $(n-k)$ dimensional complement of
+$C$, with the positivity property that for any $0 \leq l \leq L$,
+all the ordered $(k + 2l) \times (k+2l)$ minors of the $(k + 2
+l) \times n$ matrix \be \left(\begin{array}{ccc} & D_{(i_1)} & \\
+\hdashline & \vdots &  \\ \hdashline & D_{(i_l)} & \\ \hdashline & C
+\end{array} \right) \ee are positive.
+
+The notion of cells, cell decomposition and canonical form
+can be extended to the full amplituhedron. A cell $\Gamma$ is
+associated with a set of positive co-ordinates $\alpha^\Gamma =
+(\alpha^\Gamma_1, \dots, \alpha^\Gamma_{4(k + L)})$, rational in the ${\cal C}$, such that for $\alpha$'s positive,
+${\cal C}(\alpha) = (D_{(i)}(\alpha), C(\alpha))$ is in $G_+(k,n;L)$.  A
+cell decomposition is a collection $T$ of non-intersecting cells
+$\Gamma$ whose images under ${\cal Y} = {\cal C} \cdot Z$ cover the
+entire amplituhedron. The rational form $\Omega_{n,k,L}({\cal Y}; Z)$ is
+defined by having the property that
+
+\begin{center}
+$\Omega_{n,k,L}(Y;Z)$ has logarithmic singularities on all the
+boundaries of ${\cal A}_{n,k,L}(Z)$
+\end{center}
+
+A concrete formula follows from a cell decomposition as \be
+\Omega_{n,k,L}({\cal Y}; Z) = \sum_{\Gamma \subset T} \prod_{i =
+1}^{4(k + L)} \frac{d \alpha_i^\Gamma}{\alpha_i^\Gamma} \ee Of course any cell
+decomposition gives the same form $\Omega_{n,k,L}$.
+
+
+\section{The Loop Amplitude Form}
+
+We can extract the $4L$-form for the loop integrand from
+$\Omega_{n,k,L}$ in the obvious way. The 2-planes ${\cal L}_{(i)}$,
+being in the complement of $Y_0$, can be taken to be non-vanishing
+in the first 4 entries ${\cal L}^I_{ (i)}  = ({\cal L}_{(i) 2 \times
+4} | 0_{2 \times k})$. Each ${\cal L}_{\gamma (i)}$ gives us a line
+$({\cal L}_{\gamma =1} {\cal L}_{\gamma = 2})_{(i)}$ (which we have
+also been calling $(AB)_{(i)}$) in $\mathbb{P}^3$. These are the
+momentum-twistor representation of the loop integration variables.
+The analog of equation (\ref{super}) for the loop integrand form is
+\be {\cal M}_{n,k}(z_a,\eta_a; {\cal L}_{(\gamma (i)}) = \int d^{4}
+\phi_1 \dots d^4 \phi_k \int \Omega_{n,k,L}(Y,{\cal L}_{\gamma
+(i)};Z) \delta^{4k}(Y;Y_0) \ee Any form on $G(k,k+4k;L)$ can be
+written as
+\begin{equation}
+\Omega =  \langle Yd^4 Y_1 \rangle \dots \langle Yd^4 Y_k \rangle
+\prod_{i=1}^L \langle Y {\cal L}_{1(i)} {\cal L}_{2(i)} d^2 {\cal
+L}_{1(i)} \rangle \langle Y {\cal L}_{1(i)} {\cal L}_{2(i)} d^2
+{\cal L}_{2(i)} \rangle \times \omega_{n,k,L}(Y,{\cal L}_{(i)})(Z)
+\end{equation}
+where we denoted $Y=Y_1\dots Y_k$. So we have for the integrand of
+the all-loop amplitude
+\begin{align}
+{\cal M}_{n,k}(z_a,\eta_a,{\cal L}_{\gamma(i)}) &= \int d^4 \phi_1
+\dots d^4 \phi_k  \prod_{i=1}^L \langle {\cal L}_{1(i)} {\cal
+L}_{2(i)} d^2 {\cal L}_{1(i)} \rangle \langle  {\cal L}_{1(i)} {\cal
+L}_{2(i)} d^2 {\cal L}_{2(i)} \rangle \omega_{n,k}(Y_0,{\cal
+L}_{\gamma(i)};Z_a)
+\end{align}
+Already the simplest case $k=0$ of the amplituhedron is interesting
+at loop level. At 1-loop, we have a 2-plane in 4 dimensions $AB$,
+and  the $D$ matrix is just restricted to be in $G_+(2,n)$. It is
+easy to see that the 4 dimensional cells of $G_+(2,n)$ are labeled
+by a pair of triples $[a,b,c;x,y,z]$, where the top row of the
+matrix is non-zero in the columns $(a,b,c)$ and the bottom in
+columns $(x,y,z)$. A simple collection of these \be \sum _{i<j}
+[1\,i\,i\pl1;\,1\,j\,j\pl1] \ee beautifully covers the amplituhedron
+in this case. The map into $G(2,4)$ for each cell is \be A = Z_1 +
+\alpha_i Z_i + \alpha_{i+1} Z_{i+1}, \, B  = -Z_1 + \alpha_j Z_j +
+\alpha_{j+1} Z_{j+1} \ee and so the form associated with the cell is
+\begin{align} &\frac{d
+\alpha_i}{\alpha_i} \frac{d\alpha_{i+1}}{\alpha_{i+1}}
+\frac{\alpha_j}{\alpha_j} \frac{d \alpha_{j+1}}{\alpha_{j+1}} =
+\frac{\langle AB d^2 A \rangle \langle AB d^2 B \rangle \langle AB
+(1\,i\,i\pl1) \cap (1\,j\,j\pl1) \rangle^2}{\langle AB\,1\,i \rangle
+\langle AB\,1\,i\pl1 \rangle \langle AB\,i\,i\pl1 \rangle \langle
+AB\,1\,j\rangle \langle AB\,1\,j\pl1 \rangle \langle AB\,j\,j\pl1
+\rangle}
+\end{align}
+The form $\Omega$ gives exactly the ``Kermit" expansion for the MHV
+integrand given in \cite{N5}, now obtained without any reference to tree
+amplitudes, forward limits or recursion relations.
+
+In this simple case, direct triangulation of the space is straightforward. But we could also have worked backwards, starting with the BCFW formula, and  recognizing how each term in the ``Kermit" expansion is associated with positive co-ordinates for some cell of the amplituhedron. We could then observe that, remarkably,  these cells are non-overlapping, and together cover the full amplituhedron.
+
+In order to illustrate more of the structure of the loop
+amplituhedron, including the interplay between the $``C"$ and $``D"$
+matrices, let us consider the 1-loop $k=1$ amplitude for $n=6$.
+There are 16 terms in the BCFW recursion, which can all be mapped
+back to their $Y, AB$ space form, and in turn associated with
+positive co-ordinates in the amplituhedron. For instance, one of
+BCFW terms is
+
+$$
+\frac{\la YAB13\ra\la YAB(561)\cap(2345)\ra^4\la
+YAB(123)\cap(Y456)\ra^2}{\begin{array}{c} \la Y2345\ra\la
+YAB(561)\cap(Y345)\ra\la YAB(561)\cap(Y234)\ra\la
+YAB(561)\cap(Y235)\ra\la YAB56\ra \\ \la
+YAB(561)\cap(Y45(23)\cap(YAB1))\ra\la YAB12\ra\la YAB23\ra\la
+YAB13\ra\la YAB15\ra\la YAB16\ra\end{array}}
+$$
+While it may not be immediately apparently, this is nothing but the
+``dlog" canonical form associated with the following positive
+co-ordinates for the $(D,C)$ matrix
+$$
+\left(\begin{array}{ccc} & D & \\ \hdashline & C & \end{array}
+\right) = \left(
+  \begin{array}{cccccc}
+1 & x & y & 0 & 0 & 0 \\
+ -w & 0 & 0 & 0 & -1 & -z \\ \hdashline
+ w & xt_1 & t_2+t_1y & t_3 & 1+t_4 & z \\
+  \end{array}
+\right)
+$$
+This exercise can be repeated with all $16$ BCFW terms. The
+corresponding $(D,C)$ matrices are
+$$
+\hspace{-0.5cm}\includegraphics[scale=1]{six_nmhv.pdf}
+$$
+One can easily check that for all variables positive, the bottom row
+of these matrices is positive, and all the ordered  $3 \times 3$
+minors are also positive. For any cell, we can range over all the
+positive variables, which under the ${\cal Y} = {\cal C} \cdot Z$
+gives an image of the cell in $(Y,AB)$ space. Remarkably, we find
+that these cells are non-overlapping, and cover the entire space.
+This can be checked directly in a simple way. We begin by fixing
+positive external data $(Z_1, \cdots, Z_6)$. We then choose any
+positive matrix ${\cal C}$ at random, which gives an associated
+point ${\cal Y}$ inside the amplituhedron. We can ask whether or not
+this point is contained in one of the cells, by seeing whether
+${\cal Y}$ can be reproduced with positive values for all eight
+variables of the cell. Doing this we find that every point in the
+amplituhedron is contained in just one of these cells (except of
+course for points on the common boundaries of different cells). The
+cells taken together therefore give a cellulation of the
+amplituhedron.
+
+Note that the form shown above, associated with a BCFW term, has some
+physical poles (like $\langle Y AB 12 \rangle$), but also many
+unphysical poles. The unphysical poles are associated with
+boundaries of the cell that are ``inside" the amplituhedron, and not
+boundaries of the amplituhedron themselves. These boundaries are
+spurious, and so are the corresponding poles, which cancel in the
+sum over all BCFW terms.
+
+We have checked in many other examples, for higher $k$ and also at
+higher loops, that $(a)$ BCFW terms can be expressed as canonical
+forms associated with cells of the amplituhedron and $(b)$ these
+collection of cells do cover the amplituhedron.
+
+It is satisfying to have a definition of the loop amplituhedron that
+lives directly in the space relevant for loop amplitudes. This is in
+contrast with the approach to computing the loop integrand using
+recursion relations, which ultimately traces back to higher $k$ and
+$n$ tree amplitudes. Consider the simple case of the 2-loop
+4-particle amplitude. We are after a form in the space of two
+2-planes $(AB)_1, (AB)_2$ in four dimensions. The BCFW approach begins with
+the $k=2, n=8$ tree amplitudes, and arrives at the form we are
+interested in after taking two ``forward limits". But the
+amplituhedron lives directly in the $(AB)_1, (AB)_2$ space, and we can find
+a cell decomposition for it directly, yielding the form without
+having to refer to any tree amplitudes.
+
+We have understood how to directly ``cellulate" the amplituhedron in
+a number of other examples, and strongly suspect that there will be
+a general understanding for how to do this. The BCFW
+decomposition of tree amplitudes seems to be associated with
+particularly nice, canonical cellulations of the tree
+amplituhedron. Loop level BCFW also gives a cell decomposition. The ``direct" cellulations we have
+found in many cases are however simpler, without an obvious connection to the BCFW expansion.
+
+
+\section{Locality and Unitarity from Positivity}
+
+Locality and unitarity are encoded in the positive geometry of the
+amplituhedron in a beautiful way.  As is well-known, locality and
+unitarity are directly reflected in the singularity structure of the
+integrand for scattering amplitudes. In momentum-twistor language,
+the only allowed singularities at tree-level should occur when
+$\langle Z_i Z_{i+1} Z_j Z_{j+1} \rangle \to 0$; in the loop-level
+integrand, we can also have poles of the form $\langle AB\,i\,i\pl1
+\rangle \to 0$, and $\langle AB_{(i)} AB_{(j)} \rangle \to 0$.
+Unitarity is reflected in what happens as  poles are approached,
+schematically we have \cite{N6}
+$$
+\includegraphics[scale=.8]{pix30.pdf}
+$$
+
+Given the connection between the form $\Omega_{n,k,L}$ and the
+amplitude, it is obvious that the first (co-dimension one) poles of
+the amplitude are associated with the co-dimension one ``faces" of
+the amplituhedron. For trees, we have already seen that, remarkably,
+positivity forces these faces to be precisely where $\langle Y_1
+\dots Y_k Z_i Z_{i+1} Z_j Z_{j+1} \rangle \to 0$, exactly as needed
+for locality. The analog statement for the full loop amplituhedron
+also obviously includes $\langle Y_1 \cdots Y_k AB\,i\,i\pl1 \rangle
+\to 0$.
+
+The factorization properties of the amplitude also follow directly
+as a consequence of positivity. For instance, let us consider the
+boundary of the tree amplituhedron where the $k$ plane $(Y_1 \dots
+Y_k)$ is on the plane $(Z_i Z_{i+1} Z_j Z_{j+1})$. We can e.g.
+assume that $Y_1$ is a linear combination of $(Z_i, Z_{i+1}, Z_j,
+Z_{j+1})$, and thus that the top row of the $C$ matrix is only
+non-zero in these columns. But then, positivity remarkably forces
+the $C$ matrix to  ``factorize" in the form
+$$
+\includegraphics[scale=1]{matrix.pdf}
+$$
+for all possible $k_L, k_R$ such that $k_L + k_R = k - 1$. This
+factorized form of the $C$ matrix in turn implies that on this
+boundary, the amplituhedron does ``split" into lower-dimensional
+amplituhedra in exactly the way required for the factorization of
+the amplitude.
+
+We can similarly understand the single-cut of the loop integrand.
+Consider for concreteness the simplest case of the $n$ particle
+one-loop MHV amplitude. On the boundary where $\langle AB\,n1
+\rangle \to 0$, the $D$ matrix has the form
+
+$$\bordermatrix{\text{}&1&2& \dots &n\cr
+               &1&0& \dots & -x_n\cr
+               &y_1&y_2&\dots&y_n \cr}$$
+
+The connection of this $D$ matrix to the forward limit
+\cite{CaronHuot:2010zt} of the NMHV tree amplitude is simple. In the
+language of \cite{N5}, the forward limit in momentum-twistor space
+is represented as
+$$
+\includegraphics[scale=.55]{pix25.pdf}
+$$
+we start with the tree NMHV amplitude, associated with the positive
+$1 \times n$ matrix \be (y_A \, y_B \, y_1 \, y_2 \, \dots \, y_n)
+\ee and first we ``add" particle $n+1$ between $n$ and $A$, which
+adds three degrees of freedom $x_n, x_A, \alpha$
+$$\bordermatrix{\text{}&A&B&1&2& \dots &n & n+1\cr
+               &x_A& \alpha x_A & 0 & 0 & \dots & -x_n & -1\cr
+               &y_A&y_B + \alpha y_A& y_1 & y_2 & \dots&y_n & 0\cr}$$
+and we finally ``merge" $n+1,1$, which means shifting column 1 as
+$c_1 \to c_ 1 - c_{n+1}$ and removing column $(n+1)$. This gives us
+the matrix
+
+$$\bordermatrix{\text{}&A&B&1&2& \dots &n\cr
+               &x_A& \alpha x_A & 1 & 0 & \dots & -x_n \cr
+               &y_A&y_B + \alpha y_A& y_1 & y_2 & \dots&y_n\cr}$$
+note that the the $A,B$ columns have precisely four degrees of
+freedom $x_A,\alpha, y_A, y_B$ which we can remove by $GL(2)$ acting
+on the $A,B$ columns. Chopping off $A,B$ we are then left precisely
+with the $D$ matrix on the single cut. This shows that the single cut of the loop integrand is the forward limit
+of the tree amplitude, exactly as required by unitarity.
+
+\section{Four Particles at All Loops}
+
+Let us briefly describe the  simplest example illustrating the novelties of positivity at
+loop level, for four-particle scattering at $L$
+loops. We can parametrize each $D_{(i)}$ as \be D_{(i)} =
+\left(\begin{array}{cccc} 1 & x_i & 0 & -w_i \\ 0 & y_i & 1 & z_i
+\end{array} \right) \ee In this simple case the positivity
+constraints are just that all the $2 \times 2$ minors of $D_{(i)}$
+and the $4 \times 4$ minors \be {\rm det} \left(\begin{array}{ccc} &
+D_{(i)} & \\ \hdashline & D_{(j)} & \end{array} \right) \ee are
+positive. This translates to \be x_i, y_i, z_i, w_i > 0, \quad (x_i
+- x_j)(z_i - z_j) + (y_i - y_j)(w_i - w_j) < 0 \ee We can rephrase
+this problem in a  simple, purely geometrical way by  defining
+two dimensional vectors $\vec{a}_i = (x_i,y_i), \vec{b}_i = (z_i,
+w_i)$. The points are in the upper quadrant of the plane. The mutual
+positivity condition is just $(\vec{a}_i - \vec{a}_j) \cdot
+(\vec{b}_i - \vec{b}_j) < 0$. Geometrically this just means that the
+$\vec{a}, \vec{b}$ must be arranged so that for every pair $i,j$,
+the line directed from $\vec{a}_i \to \vec{a}_j$ is pointed in the
+opposite direction as the one directed from $\vec{b}_i \to
+\vec{b}_j$. An example of an allowed configuration of such points
+for $L=3$ is
+$$
+\includegraphics[scale=.6]{pix26.pdf}
+$$
+Finding a cell decomposition of this $4L$ dimensional space directly
+gives us the integrand for the four-particle amplitude at $L$-loops.
+
+Now, we know that the final form can be expressed as a sum over
+local, planar diagrams. This makes it all the more remarkable that
+no-where in the definition of our geometry problem do we
+reference to diagrams of any sort, planar or not!
+Nonetheless, this property is one of many that emerges from
+positivity.
+
+As we will describe at greater length in \cite{Into}, it is easy to
+find a cell decomposition for the full space ``manually" at low-loop
+orders. We suspect there is a more systematic approach to
+understanding the geometry that might crack the problem at all loop
+order. As an interesting  warmup to the full problem, we can
+investigate lower-dimensional ``faces" of the four-particle amplituhedron. Cellulations
+of these faces corresponds to computing certain
+cuts of the integrand, at all loop orders. We will discuss many of
+these faces and cuts systematically in \cite{Into}. Here we will
+content ourselves by presenting some especially simple but not
+completely trivial examples.
+
+Let us start by considering an extremely simple boundary of the space,
+where all the $w_i \to 0$. This corresponds to having all the lines
+intersect $(Z_1 Z_2)$. The positivity conditions then simply become
+\be (x_i - x_j)(z_i - z_j) < 0 \ee  which is trivial to
+triangulate. Whatever configuration of $x$'s we have are ordered in
+some way, say $x_1 < \dots < x_L$.  Then we must have $z_1 > \dots
+> z_L$. The $y_i$ just have to be positive. The associated form is
+then trivially (we omit the measure $\prod_i dx_i dz_i d y_i$): \be
+\frac{1}{y_1} \dots \frac{1}{y_L}  \frac{1}{x_1} \frac{1}{x_2 - x_1}
+\dots \frac{1}{x_L - x_{L-1}} \frac{1}{z_L} \frac{1}{z_{L-1} - z_L}
+\dots \frac{1}{z_1 - z_2} + {\rm perm.} \ee Now, this cut is
+particularly simple to understand from the point of view of the
+familiar ``local" expansions of the integrand--there is only  only
+local diagram that can possibly contribute to this cut: the
+``ladder" diagram. The corresponding cut is precisely what we
+have above from positivity.
+$$
+\includegraphics[scale=.8]{pix19.pdf}
+$$
+
+We can continue along these lines to explore  faces of the
+amplituhedron which determine cuts to all loop orders that are
+difficult (if not impossible) to derive in any other way. For
+instance, suppose that some of the lines  intersect $(Z_1
+Z_2)$, so that the $w_i \to 0$ for $i = 1, \dots, L_1$ and others
+intersects $(Z_3 Z_4)$, so that $y_I \to 0$ for $I =L_1+1, \dots,
+L$. To pick a concrete interesting example, let choose $L-2$ lines
+to intersect $(12)$ and 2 lines to intersect $(34)$. We can further
+specialize the geometry and take more cuts by making the $L$'th line
+pass through the point 3 -- this corresponds to sending $z_{L} \to
+0$. Let us also take the $(L-1)$'st line to pass through the point 4
+-- this corresponds to sending $z_{L-1}, w_{L-1} \to \infty$ with
+$w_{L-1}/z_{L-1} \equiv W_{L-1}$ fixed.
+
+We can again label the $x_i; x_I$ so they are in increasing order;
+then the positivity conditions become \be x_1 < \dots < x_{L-2}, z_1
+> \dots > z_{L-2}; \, x_{L-1} < x_{L} \ee and \be W_{L-1} y_i >
+(x_{L-1} - x_i), \, \, w_L y_i > z_i (x_i - x_L) \ee This space is
+also trivial to triangulate, but the corresponding form is more
+interesting. The ordering for the $z$'s is associated with the form
+$$
+\frac{1}{z_{L\mi2}(z_{L\mi3}-z_{L\mi2})(z_{L\mi4}-z_{L\mi3})\dots
+(z_1-z_2)}
+$$
+The interesting part of the space involves $x_i,y_i$. Note that if $x_i
+< x_{L-1}$, the second inequality on $y_i$ is trivially satisfied
+for positive $y_i$, and the only constraint on $y_i$ is just $y_i >
+(x_{L-1} - x_i)/W_{L-1}$. If $x_{L-1} < x_i < x_{L}$, then both
+inequalities are satisfied and we just have $y_i > 0$. Finally if
+$x_i > x_L$, the first inequality is trivially satisfied and we just
+have $y_i > z_i (x_i - x_L)/w_L$. Thus, given any ordering for all
+the $x's$, there is an associated set of inequalities on the $y$'s,
+and the corresponding form in $x,y$ space is trivially obtained. For
+instance, consider the case $L=5$, and an ordering for the $x$'s
+where $x_1 < x_4 < x_2 < x_5 < x_3$. The corresponding form in
+$(x,y)$ space is just
+\begin{align}
+\frac{1}{x_1 (x_4 - x_1) (x_2 - x_4) (x_5 - x_2)(x_3 - x_5)}
+\frac{1}{y_1 - (x_4 - x_1)/W_4} \frac{1}{y_2} \frac{1}{y_3 - z_3(x_3
+- x_5)/w_5}
+\end{align}
+By summing over all the possible orderings $x$'s, we get the final
+form. For general $L$, we can simply express the result (again
+omitting the measure) as a sum over permutations $\sigma$:
+
+\begin{align}
+&\prod_{l=1}^{L-2}\frac{1}{(z_l -
+z_{l+1})}\quad\times\hspace{-0.7cm}\sum_{\sigma; \sigma_1 < \dots <
+\sigma_{L-2}; \sigma_{L-1} <
+\sigma_L} \frac{1}{w_L W_{L-1}} \prod_{l=1}^{L} \frac{1}{(x_{\sigma^{-1}_l} - x_{\sigma^{-1}_{l-1}})}\\
+&\hspace{4cm}\times\prod_{i=1}^{L-2} \left\{ \begin{array}{cc}
+(y_i - (x_{L-1} - x_i)/W_{L-1})^{-1} &  \sigma_i < \sigma_{L-1} \\
+y_i^{-1} &  \sigma_{L-1} < \sigma_i < \sigma_{L} \\ (y_i - (x_i -
+x_L) z_i/w_L)^{-1} &  \sigma_L < \sigma_i  \end{array}\right\}
+\nonumber
+\end{align}
+where we define for convenience $z_{L-1} = x_{\sigma^{-1}_0} = 0$.
+
+This gives us non-trivial all-loop order information about
+the four-particle integrand. The expression has a feature familiar
+from BCFW recursion relation expressions for tree and loop level
+amplitudes. Each term has certain ``spurious" poles, which cancel in
+the sum. This result can be checked against the cuts of the
+corresponding amplitudes that are available in ``local form". The
+diagrams that contribute are of the type
+$$
+\includegraphics[scale=.7]{pix20.pdf}
+$$
+but now there are non-trivial numerator factors that don't trivially
+follow from the structure of propagators. The full integrand is
+available through to seven loops in the literature
+\cite{Bern:2005iz,Bern:2006ew,Bern:2007ct,Bourjaily:2011hi,Eden:2012tu}.
+The inspection of the available  local expansions on this cut does
+not indicate an obvious all-loop generalization, nor does it betray
+any hint that that the final result can be expressed in the one-line
+form given above. For instance just at 5 loops, the local form of
+the cut is given as a sum over diagrams,
+$$
+\includegraphics[scale=.8]{pix29.pdf}
+$$
+with intricate numerator factors. If all terms are combined with a
+common denominator of all physical propagators, the numerator has
+347 terms. Needless to say, the complicated expression obtained in
+this way perfectly matches the amplituhedron computation of the cut.
+
+
+\section{Master Amplituhedron}
+We have defined the amplituhedron ${\cal A}_{n,k,L}$ separately for
+every $n,k$ and loop order $L$. However, a trivial feature of the
+geometry is that ${\cal A}_{n,k,L}$ is contained in the ``faces" of
+${\cal A}_{n^\prime, k^\prime, L^\prime}$, for $n^\prime >n,
+k^\prime > k, L^\prime > L$. The objects needed to compute
+scattering amplitudes for any number of particles to all loop orders
+are thus contained in a ``master amplituhedron" with $n, k , L \to
+\infty$.
+
+In this vein it may also be worth considering natural
+mathematical generalizations of the amplituhedron. We have already seen
+that the generalized tree amplituhedron ${\cal A}_{n,k,m}$ lives in $G(k,k+m)$ and can be
+defined for any even $m$. It is obvious that the amplituhedron with
+$m=4$, of relevance to physics, is contained amongst the faces of
+the object defined for higher $m$.
+
+If we consider general even $m$, we can also generalize the notion of
+``hiding particles" in an obvious way: adjacent particles can be hidden
+in even numbers. This leads us to a bigger space in which
+to embed the generalized loop amplituhedron. Instead of just considering
+$G(k,k+4;L)$ of ($k-$ planes) $Y$ together with $L$ ($2-$planes) in
+$m=4$ dimensional complement of $Y$, we can consider a space
+$G(k,k+m;L_2,L_4, \dots, L_{m-2})$, of $k$-planes $Y$ in $(k+m)$
+dimensions, together with $L_2$ (2-planes), $L_4$ (4-planes), $\dots
+L_{m-2}$ ($(m-2)$-planes) in the $m$ dimensional complement of $Y$,
+schematically:
+$$
+\includegraphics[scale=.6]{pix10.pdf}
+$$
+Once again we have ${\cal Y} = {\cal C} \cdot Z$, with the obvious
+extension of the loop positivity conditions on ${\cal C}$ to
+$G(k,n;L_2, L_4, \dots, L_{m-2})$. We can call this space ${\cal
+A}_{n,k;m,L_2, \dots, L_{m-2}}(Z)$. The $m=4$ amplituhedron is again
+just a particular face of this object. It would be interesting to
+see whether this larger space has any interesting role to play in
+understanding the $m=4$ geometry relevant to physics.
+
+\newpage
+
+\section{Outlook}
+
+This paper has concerned itself with perturbative  scattering
+amplitudes in gauge theories. However the deeper motivations for
+studying this physics, articulated in \cite{N1,N2} have to do with
+some fundamental challenges of quantum gravity. We have long known
+that quantum mechanics and gravity together make it impossible to
+have local observables. Quantum mechanics forces us to divide the
+world in two pieces--an infinite measuring apparatus and a finite
+system being observed. However for any observations made in a finite
+region of space-time, gravity makes it impossible to make the
+apparatus arbitarily large, since it also becomes heavier, and
+collapses the observation region into a black hole. In some cases
+like asymptotically AdS or flat spaces, we still have precise
+quantum mechanical observables, that can be measured by infinitely
+large apparatuses pushed to the boundaries of space-time: boundary
+correlators for AdS space and the S-matrix for flat space. The fact
+that no precise observables can be associated with the inside of the
+space-time strongly suggests that there should be a way of computing
+these boundary observables without any reference to the interior
+space-time at all. For asymptotically AdS spaces, gauge-gravity
+duality \cite{Maldacena:1997re} gives us a wonderful description of
+the boundary correlators of this kind, and gives a first working
+example of emergent space and gravity. However, this duality is
+still an equivalence between ordinary physical systems described in
+standard physical language, with time running from infinite past to
+infinite future. This makes the duality inapplicable to our universe
+for cosmological questions. Heading back to the early universe,  an
+understanding of emergent time is likely necessary to make sense of
+the big-bang singularity. More disturbingly, even at late times, due
+to the accelerated expansion of our universe,  we only have access
+to a finite number of degrees of freedom, and thus the division of
+the world into ``infinite" and ``finite" systems, required by
+quantum mechanics to talk about precise observables, seems to be
+impossible \cite{Witten:2001kn}. This perhaps indicates the need for
+ an extension of quantum mechanics to deal with subtle cosmological questions.
+
+Understanding emergent space-time or possible cosmological
+extensions of quantum mechanics  will obviously be a tall order. The
+most obvious avenue for progress is directly attacking the
+quantum-gravitational questions where the relevant issues must be
+confronted. But there is another strategy that takes some
+inspiration from the similarly radical  step taken in the transition
+from classical to quantum mechanics, where classical determinism was
+lost. There is a powerful clue to the coming of quantum mechanics
+hidden in the structure of classical mechanics itself. While
+Newton's laws are manifestly deterministic, there is a completely
+different formulation of classical mechanics--in terms of the
+principle of least action--which is not manifestly deterministic.
+The existence of these very different starting points leading to the
+same physics was somewhat mysterious to classical physicists, but
+today we know why the least action formulation exists: the world is
+quantum-mechanical and not deterministic, and for this reason, the
+classical limit of quantum mechanics can't immediately land on
+Newton's laws, but must match to some formulation of classical
+physics where determinism is not a central but derived notion. The
+least action principle formulation is thus much closer to quantum
+mechanics than Newton's laws, and gives a better jumping off point
+for making the transition to quantum mechanics as a natural
+deformation, via the path integral.
+
+We may be in a similar situation today. If there is a more
+fundamental description of physics where space-time and perhaps even
+the usual formulation of quantum mechanics don't appear, then even
+in the limit where non-perturbative gravitational effects can be
+neglected and the physics reduces to perfectly local and unitary
+quantum field theory, this description is unlikely to directly
+reproduce the usual formulation of field theory, but must rather
+match on to some new formulation of the physics where locality and
+unitarity are derived notions. Finding such reformulations of
+standard physics might then better prepare us for the transition to
+the deeper underlying theory.
+
+In this paper, we have taken a baby first step in this direction,
+along the lines of the  program put forward in \cite{N1,N2} and
+pursued in \cite{N4,N5,N6}. We have given a formulation for planar
+${\cal N} = 4$ SYM scattering amplitudes with no reference to
+space-time or Hilbert space, no Hamiltonians, Lagrangians or gauge
+redundancies, no path integrals or Feynman diagrams, no mention of
+``cuts", ``factorization channels", or recursion relations. We have
+instead presented  a new geometric question, to which the scattering
+amplitudes are the answer. It is remarkable that such a  simple
+picture, merely moving from ``triangles" to ``polygons", suitably
+generalized to the Grassmannian, and with an extended notion of
+positivity reflecting ``hiding" particles, leads us to the
+amplituhedron ${\cal A}_{n,kL}$, whose ``volume" gives us the
+scattering amplitudes for a non-trivial interacting quantum field
+theory in four dimensions. It is also fascinating that while in  the
+usual formulation of field theory, locality and unitarity are in
+tension with each other, necessitating the introduction of the
+familiar redundancies to accommodate both, in the new picture they
+emerge together from positive geometry.
+
+A great deal remains to be done both to establish and more fully
+understand our conjecture. The usual  positive Grassmannian has a
+very rich cell structure. The task of understanding all possible
+ways to make ordered $k \times k$ minors of a $k \times n$ matrix
+positive seems daunting at first, but the key is to realize that the
+``big" Grassmannian can be obtained by gluing together
+(``amalgamating" \cite{FG}) ``little" $G(1,3)$'s and $G(2,3)$'s,
+building up larger positive matrices from smaller ones \cite{N6}.
+Remarkably, this extremely natural mathematical operation translates
+directly to the physical picture of building on-shell diagrams from
+gluing together elementary three-particle amplitudes.  This story of
+\cite{N6} is most naturally formulated in the original twistor space
+or momentum space, while the amplituhedron picture  is formulated in
+momentum-twistor space. At tree-level, there is a direct  connection
+between the cells of $G(k,n)$ that cellulate the amplituhedron, and
+those of $G(k+2,n)$, which give the corresponding on-shell diagram
+interpretation of the cell \cite{N6}.  In this way, the natural
+operation of decomposing the amplituhedron into pieces is ultimately
+turned into a vivid on-shell scattering picture in the original
+space-time.  Moving to loops, we don't have an analogous
+understanding of all possible cells of the extended positive space
+$G_+(k,n;L)$--we don't yet know how to systematically find positive
+co-ordinates, how to think about boundaries and so on, though
+certainly the on-shell representation of the loop integrand as
+``non-reduced" diagrams  in $G(k+2,n)$ \cite{N6} gives hope that the
+necessary understanding can be reached. Having control of  the cells
+and positive co-ordinates for  $G_+(k,n;L)$ will very likely be
+necessary to properly understand the cellulation ${\cal A}_{n,k;L}$.
+It would also clearly be very illuminating to find an analog of the
+amplituhedron, built around  positive external data in the original
+twistor variables.This might also shed light on the connection
+between these ideas and Witten's twistor-string theory
+\cite{Witten:2003nn,Roiban:2004ix}, along the lines of
+\cite{Dolan:2009wf, Nandan:2009cc, ArkaniHamed:2009dg,
+Bourjaily:2010kw}.
+
+While cell decompositions of the amplituhedron are geometrically
+interesting in their own right, from the point of view of physics,
+we need them only as a stepping-stone to determining the form
+$\Omega_{n,k,L}$. This form was motivated by the idea of the area of
+a (dual) polygon. For polygons, we have another definition of
+``area", as an integral, and this gives us a completely invariant
+definition for $\Omega$  free of the need for any triangulation. We
+do not yet have an analog of the notion of ``dual amplituhedron",
+and also  no integral representation for $\Omega_{n,k,L}$. However
+in  \cite{Threeviews}, we will give strong circumstantial evidence
+that such such an expression should exist. On a related note, while
+we have a simple geometric picture for the loop integrand at any
+fixed loop order, we still don't have a non-perturbative question to
+which  the full amplitude (rather than just the fixed-order loop
+integrand) is the answer.
+
+Note that the form $\Omega_{n,k,L}$ is given directly by
+construction as a sum of ``dlog" pieces. This is a highly
+non-trivial property of the integrand, made manifest (albeit less
+directly) in the on-shell diagram representation of the amplitude
+\cite{N6} (see also \cite{Lipstein:2012vs, Lipstein:2013xra}).
+ Optimistically, the great simplicity of this form should
+allow a new picture for carrying out the integrations and arriving
+at the final amplitudes. The crucial role that positive external
+data played in our story suggests that this positive structure must
+be reflected in the final amplitude in an important way. The
+striking appearance of ``cluster variables" for external data in
+\cite{posextamp} is an example of this.
+
+We also hope that with a complete geometric picture for the
+integrand of the amplitude in hand, we are now positioned to make direct
+contact with the explosion of progress in using ideas from
+integrability to determine the amplitude directly
+\cite{CaronHuot:2011kk,Basso:2013vsa,Basso:2013aha, Dixon:2013eka}. A particularly promising place
+to start forging this connection is with the four-particle amplitude
+at all loop orders. As we noted, the positive geometry problem in
+this case is especially simple, while the coefficient of the log$^2$
+infrared divergence of the (log of the)  amplitude gives the cusp
+anomalous dimension, famously determined using integrability
+techniques in \cite{BS, Beisert:2006ez,Eden:2006rx}. Another natural question is how the introduction of the spectral
+parameter in on-shell diagrams given in \cite{Ferro:2013dga,Ferro:2012xw} can be realized at the level of the amplituhedron.
+
+On-shell diagrams in ${\cal N} = 4$ SYM and the positive
+Grassmannian have a close analog with on-shell diagrams in ABJM
+theory and the positive null Grassmannian \cite{Huang:2013owa}, so
+it is natural to expect an analog of the amplituhedron for ABJM as
+well. Should we expect any of the ideas in this paper to extend to
+other field theories, with less or no supersymmetry, and beyond the
+planar limit? As explained in \cite{N6}, the connection between
+on-shell diagrams and the Grassmannian is valid for any theory in
+four dimensions, reflecting only the building-up of more complicated
+on-shell processes from gluing together the basic three-particle
+amplitudes. The connection with the positive Grassmannian in
+particular is universal for any planar theory: only the measure on
+the Grassmannian determining the on-shell form differs from theory
+to theory. Furthermore, on-shell BCFW representations of scattering
+amplitudes are also widely available--at loop level for planar gauge
+theories, and at the very least for gravitational tree amplitudes
+(where there has been much recent progress from other points of view
+\cite{Hodges:2012ym,Cachazo:2012da,Cachazo:2012kg,Skinner:2013xp,Cachazo:2013hca,Cachazo:2013iea}).
+As already mentioned, one of the crucial clues leading to the
+amplituhedron was the myriad of different BCFW representation of
+tree amplitudes, with equivalences guaranteed by remarkable rational
+function identities relating BCFW terms. We have finally come to
+understand these representations and identities as simple
+reflections of amplituhedron geometry. As we move beyond planar
+${\cal N} = 4$ SYM, we encounter even {\it more} identities with
+this character, such as the BCJ relations \cite{Bern:2008qj,
+Bern:2010ue}. Indeed even sticking to planar ${\cal N} = 4$ SYM,
+such identities, of a fundamentally non-planar origin, give rise to
+remarkable relations between amplitudes with different cyclic
+orderings of the external data. It is hard to believe that these
+on-shell objects and the identities they satisfy only have a
+geometric ``triangulation" interpretation in the planar case, while
+the even richer structure beyond the planar limit have no geometric
+interpretation at all. This provides a strong impetus to search for
+a geometry underlying more general theories.
+
+Planar ${\cal N} = 4$ SYM amplitudes are Yangian invariant, a fact
+that is invisible in the  conventional field-theoretic description
+in terms of amplitudes in one space or Wilson loops in the dual
+space. We have become accustomed to such striking facts  in string
+theory, which has a rich spectrum of $U$ dualities, that are
+impossible to make manifest simultaneously in conventional string
+perturbation theory. Indeed the Yangian symmetry of planar ${\cal N}
+= 4$ SYM is just fermionic $T$-duality \cite{Berkovits:2008ic}.
+The amplituhedron has now given us a new description of planar
+${\cal N} = 4$ SYM amplitudes which does not have a usual
+space-time/quantum mechanical description, but {\it does} make all
+the symmetries manifest. This is not a ``duality" in the usual sense, since we are not identifying an equivalence
+between existing theories with familiar physical interpretations. We
+are seeing something rather different: new mathematical
+structures for representing the physics without reference to
+standard physical ideas, but with all symmetries manifest. Might
+there be an analogous story for superstring scattering amplitudes?
+
+\newpage
+
+{\Large \bf Acknowledgements} \vskip .1in
+
+We thank Zvi Bern, Jake Bourjaily, Freddy Cachazo, Simon Caron-Huot,
+Clifford Cheung, Pierre Deligne, Lance Dixon, James Drummond, Sasha
+Goncharov, Song He, Johannes Henn, Andrew Hodges, Yu-tin Huang,
+Jared Kaplan, Gregory Korchemsky, David Kosower, Bob MacPherson,
+Juan Maldacena, Lionel Mason, David McGady, Jan Plefka, Alex
+Postnikov, Amit Sever, Dave Skinner, Mark Spradlin, Matthias
+Staudacher, Hugh Thomas, Pedro Vieira, Anastasia Volovich, Lauren
+Williams and Edward Witten for valuable discussions. Our research in
+this area over the past many years owes an enormous debt of
+gratitude to Edward Witten, Andrew Hodges, and especially Freddy
+Cachazo and Jake Bourjaily, without whom this work would not have
+been possible. N. A.-H. is supported by the Department of Energy
+under grant number DE-FG02-91ER40654. J.T. is supported in part by
+the David and Ellen Lee Postdoctoral Scholarship and by DOE grant
+DE-FG03-92-ER40701 and also by NSF grant PHY-0756966.
+
+\vskip .2in
+
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+  %``The N=4 SYM integrable super spin chain,''
+  Nucl.\ Phys.\ B {\bf 670}, 439 (2003)
+  [hep-th/0307042].
+
+\bibitem{Beisert:2006ez}
+  N.~Beisert, B.~Eden and M.~Staudacher,
+  %``Transcendentality and Crossing,''
+  J.\ Stat.\ Mech.\  {\bf 0701}, P01021 (2007)
+  [hep-th/0610251].
+
+\bibitem{Eden:2006rx}
+  B.~Eden and M.~Staudacher,
+  %``Integrability and transcendentality,''
+  J.\ Stat.\ Mech.\  {\bf 0611}, P11014 (2006)
+  [hep-th/0603157].
+
+\bibitem{Ferro:2013dga}
+  L.~Ferro, T.~Lukowski, C.~Meneghelli, J.~Plefka and M.~Staudacher,
+  %``Spectral Parameters for Scattering Amplitudes in N=4 Super Yang-Mills Theory,''
+  arXiv:1308.3494 [hep-th].
+
+\bibitem{Ferro:2012xw}
+  L.~Ferro, T.~Lukowski, C.~Meneghelli, J.~Plefka and M.~Staudacher,
+  %``Harmonic R-matrices for Scattering Amplitudes and Spectral Regularization,''
+  Phys.\ Rev.\ Lett.\  {\bf 110}, no. 12, 121602 (2013)
+  [arXiv:1212.0850 [hep-th]].
+
+\bibitem{Huang:2013owa}
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+  %``ABJM amplitudes and the positive orthogonal Grassmannian,''
+  arXiv:1309.3252 [hep-th].
+
+\bibitem{Hodges:2012ym}
+  A.~Hodges,
+  %``A simple formula for gravitational MHV amplitudes,''
+  arXiv:1204.1930 [hep-th].
+
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+  F.~Cachazo and Y.~Geyer,
+  %``A 'Twistor String' Inspired Formula For Tree-Level Scattering Amplitudes in N=8 SUGRA,''
+  arXiv:1206.6511 [hep-th].
+
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+  F.~Cachazo and D.~Skinner,
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+  [arXiv:1207.0741 [hep-th]].
+
+\bibitem{Skinner:2013xp}
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+  arXiv:1301.0868 [hep-th].
+
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+  F.~Cachazo, S.~He and E.~Y.~Yuan,
+  %``Scattering of Massless Particles in Arbitrary Dimension,''
+  arXiv:1307.2199 [hep-th].
+
+\bibitem{Cachazo:2013iea}
+  F.~Cachazo, S.~He and E.~Y.~Yuan,
+  %``Scattering of Massless Particles: Scalars, Gluons and Gravitons,''
+  arXiv:1309.0885 [hep-th].
+
+\bibitem{Bern:2008qj}
+  Z.~Bern, J.~J.~M.~Carrasco and H.~Johansson,
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+
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+
+
+\end{thebibliography}
+
+
+\end{document}
diff --git a/papers/references/positive_geometries_index.md b/papers/references/positive_geometries_index.md
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+# Reference Papers: Positive Geometries and Conscious Realism
+
+## The Core Physics (Amplituhedron)
+1. **The Amplituhedron**
+   - Authors: Nima Arkani-Hamed, Jaroslav Trnka
+   - ArXiv: `1312.2007`
+   - File: `Arkani_Hamed_The_Amplituhedron_1312.2007.pdf`
+   
+2. **Into the Amplituhedron**
+   - Authors: Nima Arkani-Hamed, Jaroslav Trnka
+   - ArXiv: `1312.7878`
+   - File: `Arkani_Hamed_Into_the_Amplituhedron_1312.7878.pdf`
+
+## The Hoffman Synthesis
+3. **Fusions of Consciousness**
+   - Authors: Donald D. Hoffman, Chetan Prakash, Robert Prentner
+   - Published: *Entropy* (2023), Vol. 25, Issue 1, 129
+   - Note: This is the seminal paper where Hoffman defines a mathematical map from the Markovian dynamics of conscious agents to "decorated permutations," which are the combinatorial objects that underpin positive geometries like the Amplituhedron.
+
+4. **Traces of Consciousness**
+   - Authors: Donald D. Hoffman, Chetan Prakash, et al.
+   - Note: Follow-up work further exploring how Conscious Agents project physical phenomena via positive geometries.