# Positive Geometries, Corolla Polynomial and Gauge Theory Amplitudes

Alok Laddha  Amit Suthar

###### Abstract

Arkani-Hamed, Bai, He and Yan (ABHY) discovered a convex realisation of the
associahedron whose combinatorial and geometric structure generates tree-level
amplitudes in bi-adjoint scalar theory. ABHY associahedron of dimension $k$
determines a unique meromorphic $k$-form in the kinematic space of Mandelstam
invariants which is $k+3$ point tree-level amplitude in bi-adjoint $\phi^{3}$
theory. As ABHY further proved in [1], while the color-ordered amplitude is a
form in the kinematic space, the (double) _color-dressed_ amplitudes are
scalars obtained by the so called color-form duality. Color-form duality can
be implemented by contracting a d-log form with a multi-vector field (MVF)
whose coefficients are fixed by the color factors of the theory. We extend the
latter result to identify tree-level S-matrix of Yang Mills theory with a
scalar obtained by contracting the canonical form of ABHY associahedron with a
MVF in the kinematic space. Components of this MVF are also determined by the
combinatorial structures that underlie associahedron and Corolla polynomial
which was introduced by Kreimer, Sars and van Suijlekom (KSVS) in [2]. KSVS
used the Corolla polynomial to obtain (at all orders in the loop expansion)
the parametric representation of gauge theory Feynman integral from the
corresponding Feynman integral in $\phi^{3}$ theory. The map from latter
integral to the former was generated by the Corolla polynomial. Using the full
power of Corolla polynomial, we then extend these results to obtain Yang-Mills
one loop planar integrand by contracting the Corolla generated MVF with the
canonical form defined by $\hat{D}_{n}$ polytope discovered by Arkani-Hamed,
Frost, Plamondon, Salvatori, Thomas recently. We also demonstrate that KSVS
representation of Corolla graph differential in the parametric space can be
readily extended to “spin up" the curve integral formulae for
$\textrm{Tr}(\phi^{3})$ amplitude discovered in [3, 4] and give an explicit
construction of such formulae for tree-level and planar one loop gluon
amplitudes.

##  1 Introduction

One of the central questions in the analytic S matrix program is to identify
the class of meromorphic functions in the Poincare invariant space of
scattering data (also known as kinematic space in the literature) which
qualify as amplitudes of unitary and local quantum field theories. Over last
few decades, a striking answer to this question with an underlying
universality has begun to emerge for a class of quantum field theories.
Namely, in the kinematic space, there is a class of combinatorial geometries
that generate S matrix. The strongest evidence in the favour of this radical
proposal is ${\cal N}$ = 4 super Yang-Mills (SYM) theory in the planar limit.
The integrand of planar amplitudes in this theory is a canonical form in the
kinematic space parametrized by momentum twistors. This form is in fact
completely fixed by a convex geometry known as amplituhedron [5]. The
“amplituhedron program" has led to rather striking advances in our
understanding of planar amplitudes in ${\cal N}=4$ SYM theory. Seminal attempt
at going beyond planarity can be found in [6].

In the world of scalar quantum field theories, the kinematic space
corresponding to spin-less external particles is simply the Cartesian space of
independent Mandelstam invariants .111For a fixed space-time dimension $D$,
this space is in fact a algebraic variety as opposed to a Cartesian space and
is also parametrized by momentum twistors. However so long as number of
external particles $n\leq\,D$, the kinematic space can also be co-ordinatized
by the Mandelstam invariants modulo linear relations imposed by momentum
conservation. A number of seminal developments in physics and mathematics has
led to a discovery of rich class of convex polytopes in this kinematic space.
The star of the show is a convex polytope known as associahedron (Stasheff
polytope) whose rich combinatorial structure makes it amenable to a convex
realisation inside the kinematic space associated to tree-level
scattering,[7]. This realisation was discovered by Arkani-Hamed, Bai, He and
Yun (ABHY) and is known as ABHY associahedron,[1]. In fact associahedron is a
specific (and in many ways the most prominent!) member of a family of such
combinatorial polytopes known as accordiohedra [8, 9, 10, 11, 12, 13]. All the
accordiohedra are convexly realised in the kinematic space where the embedding
is completely fixed by underlying combinatorial structures of these polytopes.

Accordiohedra share a common property that it has boundaries of all co-
dimensions and that every vertex is adjacent to number of co-dimension one
faces which equals the dimension of the polytope. More generally, they belong
to a specific family of geometries known as positive geometry [14].

Any positive geometry which is realised in the kinematic space, defines a
meromorphic form that produce the tree-level S-matrices of a large class of
colored scalar theories. For a sampling of the papers where the (ever-
deepening) relationship between combinatorics, geometry and amplitudes is
analysed, we refer the reader to the following papers and references there
in,[1, 14, 15, 16, 17, 18, 10, 12, 19, 20, 21, 22]. The one loop integrands of
colored scalar theories also belong to this category where the geometries
reside in enlarged kinematic space including loop momenta and the geometries
arise from combinatorics of chords that dissect abtract polygon with genus
$>0$, [23, 24, 25, 26]. The geometry whose form generates one loop planar
integrands of bi-adjoint scalar theory is known as $\hat{D}_{n}$ polytope
which has much the same combinatorial skeleton as Associahedron.222At higher
loop orders, the corresponding polytopes are known as surfacohedra and are
subject of an upcoming work by Arkani-Hamed, Frost, Plamondon, Salvatori,
Thomas, [24]. $\hat{D}_{n}$ polytope is also a positive geometry with boundary
of all codimensions with a corresponding unique meromorphic form. Analysing
emergence of S-matrix via geometric and combinatorial structures is a radical
attempt to penetrate the S-matrix program as some of the hard postulates such
as unitarity and locality emerge thanks to the boundary structure of the
positive geometries.

Although the class of amplitudes (or more precisely amplitude integrands at
loop level) that are meromorphic forms associated to positive geometries is
ever increasing, (see e.g. [27, 28]), various garden variety QFTs such as QED,
Yang-Mills (YM) theory has been resistant to positive geometries program
untill recently.

During past couple of years however, the real potential of positive geometry
program is being realised. The ABHY associahedron (or $\hat{D}_{n}$ polytope
e.g.) are a convex polytopes whose family of normals (called the ${\bf
g}$-vectos) to the co-dimension one facets form a fan in ${\bf R}^{|{\cal
K}_{n}|}$. This fan uniquely determines a set of piece-wise linear forms which
are dual to the ${\bf g}$-vectors which are known as headlight functions, [3,
8].333In representation theory literature, headlight functions are known as
$c$-vectors, see e.g. [8].

In [3, 4] it was shown that the headlight functions along with co-ordinates of
the kinematic space can be used to calculate the $\textrm{Tr}(\phi^{3})$
amplitude at arbitrary order in the loop expansion and the genus
($\frac{1}{N}$ for $SU(N)$) expansion in the parametric space of Schwinger
parameters. This representation of amplitude is known as the curve integral
formula and once we view the relationship between positive geometries and
scalar amplitudes through the lens of parametric space, then a large class of
amplitudes of theories such as non-linear sigma model (NLSM) theory and Yang-
Mills theory arise as certain deformations (obtained via shift of kinematic
variables) of the $\textrm{Tr}(\phi^{3})$ amplitudes,[21, 29]. In the seminal
paper [29], the authors proved that a specific deformation of the (stringy)
$\textrm{Tr}(\phi^{3})$ curve integral formula generates YM amplitudes at all
orders in perturbative expansion !

As striking as these results are, it is worth asking if there is a map from
the canonical form defined by positive geometries in the kinematic space and
S-matrix of gauge theories. _This is the question we analyse in the current
paper._ There have been several tantalising hints in the recent literature
that there should exist a geometric operation that maps the meromorphic form
in the kinematic space to the Yang-Mills amplitude. The most prominent among
those is the well known color kinematics duality which has been explicitly
established for tree-level and one loop amplitudes [30, 31, 32]. For tree-
level amplitudes another evidence came from the beautiful paper of Cheung,
Shen and Wen [33] in which an explicit map was constructed that maps the Yang-
Mills amplitude to bi-adjoint scalar amplitude. In [34], authors derived a
direct relationship between the canonical form of the ABHY associahedron and
the canonical form of the (tree-level) momentum amplituhedron. But perhaps the
strongest hint from our perspective, that such a map between associahedron
form and Yang-mills amplitude may exist in fact resides in one of the many
results which were obtained in [1] for tree-level amplitudes.

As reviewed above, the ABHY associahedron generates color ordered bi-adjoint
scalar amplitude as the canonical form in the kinematic space. However, a
perhaps less appreciated result of ABHY in [1] was their interpretation of the
color dressed amplitude in bi-adjoint scalar theory. It was shown that the
color dressed amplitude is in fact a scalar (as opposed to a meromorphic form)
which is defined over an extension of the physical kinematic space, that ABHY
termed big Kinematic space. Intuitively, the big kinematic space is the space
of all (including the non-planar) generalised Mandelstam invariants generated
by the on-shell momenta and momentum conservation. We will review it’s
definition in section (2.1). ABHY proved that whereas the color-ordered
amplitude in the bi-adjoint scalar theory is the (pullback of the) canonical
form fixed by the associahedron, the color-dressed amplitude is a scalar in
the big Kinematic space. Geometrically, this scalar can be obtained by
considering the contraction of a certain multi-vector field (MVF) with the
d-log form in the big Kinematic space. The multi-vector field is in fact a sum
over all the vertices of the associahedron (which is in bijection with the set
of all the tri-valent graphs) such that each term in the sum is determined by
the orientation at each vertex of the graph, and structure constants
associated to each of these tri-valent vertices.

Although the re-writing of the color-dressed amplitude as a contraction of the
form with an contravariant tensor in the kinematic space is rather a trivial
observation, it turns out to have some intriguing graph theoretic properties.
These properties are also shared by an interesting mathematical object known
as Corolla polynomial which was defined in [2] by Kreimer, Sars and van
Suijlekom. At it’s heart Corolla polynomial is a sum over monomials where each
monomial is determined by the variables that label half edges of the graph and
co-efficient of each monomial is either 0 or 1, [35].

In their seminal work, KSVS used the Corolla polynomial to construct a third
graph polynomial $\hat{C}_{\gamma}$ (along with First and Second Symanzik
graph polynomials) in the parametric space of Schwinger parameters.
$\hat{C}_{\gamma}$ was a differential operator valued meromorphic function in
the parametric space which was used to spin up the parametric space integrand
to obtain gauge theory integrand. KSVS proved the result at all loop orders as
the Corolla differential naturally maps the $\phi^{3}$ vertex to a three gluon
vertex and maps a tri-valent vertex adjacent to a closed cycle to a “gauge-
invariant" three-gluon vertex such that the unphysical modes are removed, (see
section 4 for details). As the differential is simply a representation of a
polynomial, it has been used to construct parametric space representation of
large class of gauge theory amplitudes on a fixed graph.

This motivates us to consider contraction of the associahedron form (or in
general canonical forms associated to positive geometries such as
$\hat{D}_{n}$ polytope) with multi-vector field which is defined via Corolla
polynomial. As we show in this paper, Kreimer’s Corolla graph differential
rather naturally leads to a map from each vertex of the positive geometry $v$
to a Corolla differential in the kinematic space $\hat{C}_{v}$. This leads to
a Multi-vector field in the kinematic space whose contraction with the d log
form generates color ordered YM amplitude at tree-level and the planar
integrand at one loop. As we prove in section 6, the following diagram depicts
the complete picture that combine our results with the seminal results
obtained in [33].

(1.1)

Our representation of Corolla differential is in the kinematic space as
opposed to the Corolla graph differential defined by KSVS which was in the
parametric space. However “fusion of the curve integral formula" with
definition of Corolla graph differential follows immediately thanks to the
existence of headlight functions and hence an off-shoot of our analysis is the
curve integral formula for YM amplitude. We explicitly derive the formula for
the integrand in the parametric space for both the tree-level amplitude and
(planar) one loop amplitudes.

This paper is organised as follows. In sections 2 and 3, we review the
definition of kinematic space, positive geometries and d-log forms that these
geometries define in the kinematic space. In section 4, we review the
definition of Corolla polynomial. In section 5 we contract the Corolla induced
MVF with the scattering form defined by the associahedron and show that it
generates color-ordered YM amplitude at tree-level. In section 6, we present
the inverse of the MVF, the differential form that _transmutes_ the YM
amplitude into the canonical form $\Omega_{n}$ for associahedron. In section 7
we extend this analysis to $\hat{D}_{n}$ polytopes and use the Corolla
polynomial associated to graphs with one closed cycle to define the MVF. Once
again the planar one loop integrand for YM amplitude is realised as a
contracted scalar. In section 8, we derive curve integral formula for gluon
amplitudes. Although the power of curve integral formula resides in their
universality with respect to loop expansion as well as genus expansion, we
give explicit formula only for tree-level and planar one loop amplitudes. But
as the Corolla polynomial is defined for a tri-valent graph with arbitrary
number of cycles, the generalisation of our results to obtain curve integral
formula for gluon at arbitrary order in perturbation theory including non
planar contributions should be immediate. We finally end with a discussion of
our results and several open issues.

##  2 Kinematic Space for Tree-level and One loop amplitudes

In this section we review the construction of the kinematic space ${\cal
K}_{n}^{(I)}\,|\,I\,\in\,\\{0,1\\}$ when external states are scalars. The
super-script $(I)$ indicates that the kinematic space is co-ordinatized by the
kinematic variables whose zeros correspond to poles of tree-level amplitudes
or one loop integrands respectively. We will also define an ambient space
known as off-shell kinematic space which treats all the kinematic variables
associated to internal or external edges as independent variables.

###  2.1 Kinematic space for tree-level S matrix and it’s enlarged cousins

The space of $n$ massless momenta modulo action of the Poincare group is known
as the kinematic space ${\cal K}_{n}$. In $D\,>\,n$ dimensions, ${\cal K}_{n}$
is a $\frac{n(n-3)}{2}$ dimensional linear space and can be coordinatized in
terms of so-called planar kinematic variables.

$\displaystyle X_{ij}\,=\,(p_{i}\,+\,\dots\,+\,p_{j-1})^{2}$ (2.1)

In recent years, one has discovered that inspite of being a rather barren
space with no dynamical information, ${\cal K}_{n}$ is the home of an infinite
family of linear convex polytopes of a certain kind known as the positive
geometry. Discovery of these positive geometries realised in ${\cal K}_{n}$
has geometrized the notion of S-matrix as amplitudes are naturally realised as
canonical meromorphic forms on these positive geometries.

In [1], ABHY introduced a $2^{n-1}-n-1$ dimensional space ${\cal
K}_{n}^{\star}$ which contains ${\cal K}_{n}$ as a linear subspace. We review
it’s definition essentially following [1]. Consider a family of variables
$S_{I}$. Each $S_{I}$ is labelled by a subset
$I\,\subset\,\\{1,\,\dots\,n\\}$. These variables are abstract at this stage
and are subjected to two conditions

  1. 1.

$S_{I}=S_{\overline{I}}$

  2. 2.

$S_{I}=0~{},\quad|I|=\\{\,0,1,n-1,n\\}$

Thanks to these conditions, it can be seen that $S_{I}$ can be identified with
all Mandelstam variables without imposing relations arising from momentum
conservation at each vertex.444$S_{n-1}=S_{n}=0$ can be identified with
overall momentum conservation and on-shell constraints on external momenta. A
simple example illustrates this point.

$\displaystyle{\cal K}_{4}^{\star}\,=\,\\{\,S_{12},\,S_{23},\,S_{13}\,\\}$

It thus follows that ${\cal K}_{n}$ is a subspace of ${\cal K}_{n}^{\star}$ by
imposing “Seven term identity" [1]

$\displaystyle
S_{I_{1}I_{2}}\,+\,S_{I_{2}I_{3}}\,+\,S_{I_{1}I_{3}}\,=\,\sum_{i=1}^{4}\,S_{I_{i}}$
(2.2)

${\cal K}_{n}^{\star}$ is the kinematic space on which color dressed
amplitudes are well defined. We will now define a further enlargement of big
kinematic space and refer to it as “off-shell Kinematic space", ${\cal
K}_{n}^{\prime}$.

${\cal K}_{n}^{\prime}$ is defined by relaxing the second condition on $S_{I}$
variables, (condition (2)). That is, ${\cal K}_{n}^{\prime}$ is spanned by
$S_{I}$ which are subjected to the following condition.

  1. 1.

$S_{I}=S_{\overline{I}}$

  2. 2.

$S_{I}=0,|I|=\\{\,0,n\\}$

${\cal K}_{n}^{\prime}$ is $2^{n-1}-1$ dimensional. As an example, ${\cal
K}_{4}^{\prime}$ is co-ordinatized by,

$\displaystyle{\cal
K}_{4}^{\prime}\,=\,\\{\,S_{12},S_{23},S_{13},S_{1},S_{2},S_{3},S_{4}\,\\}~{}.$

For color ordered gauge theory amplitudes, the relevant space will in fact be
a subspace ${\cal K}_{n}^{\textrm{os}}$ of ${\cal K}_{n}^{\prime}$ which is
parametrized by treating all the chords and sides of an $n$-gon (whose
vertices are labelled clockwise from $1\,\dots,\,n$) wih independent
variables. As can be easily checked, ${\cal K}_{n}^{\textrm{os}}$ is
$\frac{n(n-1)}{2}$ dimensional.

The projection, $\pi\,:\,{\cal K}_{n}^{\prime}\,\rightarrow\,{\cal
K}_{n}^{\star}$ is via the following restriction.

$\displaystyle S_{I}\,=\,0\,\forall\,|I|=\\{1,n-1\\}$ (2.3)

As ${\cal K}_{n}^{\textrm{os}}\,:=\,{\bf R}^{\frac{n(n-1)}{2}}$, we can co-
ordinatize it with orthogonal co-ordinates $x_{i},x_{ij}$. Upon mapping to the
kinematic space of scattering data, these variables will become

$\displaystyle x_{i}$ $\displaystyle\rightarrow\,0~{},\qquad(1\leq i\leq n)$
(2.4) $\displaystyle x_{ij}$
$\displaystyle\rightarrow\,(p_{i}\,+\,\dots\,p_{j-1})^{2}~{},\qquad(1\leq
i<j-1<n)$

We note that the $d\ln$ form on ${\cal K}_{n}$ can be trivially pulled back on
${\cal K}_{n}^{\textrm{os}}$ and without loss of generality we denote the
pull-back as $\Omega_{n-3}$.

We also introduce a family of four vectors
$\\{\xi_{i}\,|1\,\leq\,i\,\leq\,n\,\\}\cup\,\\{\,\xi_{ij}\,|\,(1\leq
i<j-1<n)\\}$ such that

$\displaystyle x_{i}=\xi_{i}^{2}~{},\quad
x_{ij}\,=\,\xi_{ij}^{2}~{},\quad(\xi_{ij}=-\xi_{ji})~{}.$ (2.5)

###  2.2 Kinematic space with one loop planar variables

We will denote the one loop kinematic space as ${\cal K}_{n}^{1-\textrm{l}}$.
It can be co-ordinatized by planar kinematic variables which are in bijection
with set of all internal chords that can dissect a planar $n$-gon with a
puncture. These variables are defined as,

$\displaystyle X_{ij}$ $\displaystyle=\,(p_{i}\,+\,\dots\,p_{j-1})^{2}$ (2.6)
$\displaystyle X_{i0}$ $\displaystyle=\,(p_{1}\,+\,\dots\,p_{i-1}\,+\,l)^{2}$
$\displaystyle X_{ii}$
$\displaystyle=\,(p_{i+1}\,+\,\dots\,+\,p_{n}\,+\,p_{1}\,+\,\dots\,+p_{i-1})^{2}$

One could enlarge ${\cal K}_{n}^{1-\textrm{l}}$ to the off-shell kinematic
space ${\cal K}_{n}^{1-\textrm{l}\ \textrm{os}}$ exactly as for ${\cal
K}_{n}$. That is ${\cal K}_{n}^{1-\textrm{l}\,\textrm{os}}$ is a
$\frac{n(n-3)}{2}+3n$ dimensional linear space which is co-ordinatized by
chordal lengths of internal chords as well as sides of the $n$-gon. We denote
the cartersian co-ordinates of ${\cal K}_{n}^{1-\textrm{l}\,\textrm{os}}$ as
$\\{\,x_{ij},x_{i0},x_{ii},x_{i}\,\\}$ each of which can be written as a
square of a 4-vector associated to the chords or boundaries of an $n$-gon with
a puncture.

$\displaystyle x_{ij}=\xi_{ij}^{2},\quad
x_{ii}\,=\,\xi_{ii}^{2},\quad\xi_{i0}\,=\,\xi_{i0}^{2},\,x_{i}\,=\,\xi_{i}^{2}~{}.$
(2.7)

The projection from ${\cal
K}_{n}^{1-\textrm{l}\,\textrm{os}}\,\rightarrow\,{\cal K}_{n}^{1-\textrm{l}}$
is obtained by

$\displaystyle x_{i}$ $\displaystyle\rightarrow\,0$ $\displaystyle x_{ii}$
$\displaystyle\rightarrow\,(\sum_{j=1}^{n}p_{j})^{2}$ $\displaystyle
x_{i0}-x_{i-1,0}$ $\displaystyle=\,2p_{i}\cdot l+X_{1,i+1}-X_{1,i}$ (2.8)

One can pull-back the d$\ln$ form on ${\cal K}_{n}^{1-l}$ to the big kinematic
space. As we will now show, the contraction of this form with a multi-vector
field defined by Corolla representation over the pseudo-triangulation
generates color ordered one-loop Yang-Mills integrand.

##  3 Positive geometries for scalars: Associahedron and $\hat{D}_{n}$
polytope.

In this section, we briefly review the definition of the (combinatorial)
associahedron, $\hat{D}_{n}$ polytope and their convex realisation discovered
by ABHY and AHST respectively, [1, 23]. A more detailed discussion on these
geometries and their embeddings inside the kinematic space (as well as their
relationship with dissection quivers and so-called gentle algebras) can be
found in [1, 17, 18, 15, 16, 20]. We caution the reader that this section is
not a self-contained introduction to the rich subject of positive geometries
and their realisations via quiver representation theory. One of our goals in
this section is to introduce the concept of ${\bf g}$ vector fan which forms
the back-bone of many of the recent developments in emergence of scattering
amplitudes from the positive geometry.

Associahedron $A_{n-3}$ is a simple polytope whose any two adjacent vertices
are related by precisely one mutation.555A polytope of dimension $D$ is called
simple if every vertex is adjacent to $D$ co-dimension one faces. Although
various convex realisations of $A_{n-3}$, all of which are diffeomorphic to
the ABHY realisation generate $n$-point amplitudes of an infinite class of
scalar field theories, we will not explicitly rely on this realisation in the
kinematic space, but will only use certain data that $A_{n-3}$ generates.

The positive geometry associated to one loop integrand of bi-adjoint scalar
theory is known to be a close cousin of the cluster $D_{n}$ polytope which was
discovered by Arkani-Hamed, Frost, Plamondon, Salavatori, and Thomas in [24].
Their construction has been reviewed in [20].

###  3.1 Combinatorial polytopes

Here, we review the construction of the _combinatorial polytopes_ , in terms
of triangulations of the $n$-gons.

####  3.1.1 The associahedron

Associahedron (also known as Stasheff Polytope) is a a set of relations
between triangulations of an $n$-gon. More in detail, this set has subsets
indexed by $k\in\,\\{0,\dots,n-3\\}$ where $T_{k}$ is the set of all partial
k-triangulations of an $n$-gon. Thus ${\cal T}_{n-3}$ is the set of all
complete triangulations of an $n$-gon. We refer to a complete triangulation as
a “vertex" and similarly all the elements in ${\cal T}^{k}$ as “faces of co-
dimension k".

If

$\displaystyle T_{1}^{n-3},T_{2}^{n-3}\,\in\,{\cal T}_{n-3}~{},$ (3.1)

such that

$\displaystyle T_{1}^{n-3}\,\cap\,T_{2}^{n-3}\,=\,T_{12}^{n-2}~{},$ (3.2)

where $T_{12}^{n-2}\,\in\,{\cal T}^{n-2}$, then the two vertices are adjacent
to each other through a one dimensional face. This defines a (hierarchy) of
relations among partial triangulations which can be coherently put together as
a positive geometry of dimension $n-3$ whose set of co-dimension $k$-faces is
isomorphic to set of all $k$-partial triangulations. Refer to figure 1 for a
picture of $A_{1}$ and $A_{2}$, the one and two dimensional associahedra.

All the vertices of the associahedron (co-dimension $n$ facets) map to the
complete triangulation of the $(n+3)$-gon. The definition implies that the two
vertices are adjacent to each other, if and only if the two complete
triangulations differ only by the flip of a single chord, keeping $(n-1)$
chords intact. The operation of changing precisely one chord is known as the
mutation. Figure 1 show the simplest cases of associahedron: $\mathcal{A}_{1}$
and $\mathcal{A}_{2}$.

Let $(ik)~{},\ (1\leq i<k-1<n)$ be the chord of $(n+3)$-gon connecting $i$ and
$k$ vertices. Since the co-dimension 1 faces of $\mathcal{A}_{n}$ map to the
triangulations using only one chord, we can label the faces by the chord
present in partial triangulation bijective to the face: ($ik$). Counting the
chords, the number of co-dimension 1 faces of the associahedron is $n(n-3)/2$.
We denote the $k$ dimensional faces of $n$ dimensional associahedron as
$\mathcal{A}_{n}^{[k]}$ so that we have: $|\mathcal{A}_{n}^{[n-1]}|=n(n-3)/2$.
The number of vertices of the associahedron is as follows:

$\displaystyle|\mathcal{A}_{n}^{[0]}|=C_{n+1}=\frac{(2n+2)!}{(n+2)!\,(n+1)!}~{},$
(3.3)

$C_{n}$ being the Catalan number. For instance, we have
$|\mathcal{A}_{1}^{[0]}|=2~{},\quad|\mathcal{A}_{2}^{[0]}|=5~{},\quad|\mathcal{A}_{3}^{[0]}|=14~{},\quad\ldots$.

The chord $(ik)$, of the $(n+3)$-gon decomposes the $(n+3)$-gon into an
$(k-i+1)$-gon and another $(n-k+i+1)$-gon. So, we can conclude that the face
of the associahedron is a product of smaller associahedra:

$\displaystyle\text{For }i\in[1,\ldots,n+1]~{},\quad
k\in[2,\ldots,n+3-i]~{}:\qquad\mathcal{A}_{n}\xrightarrow{\partial_{(i,i+k)}}\mathcal{A}_{k+1}\times\mathcal{A}_{n+1-k}$
(3.4)

$\mathcal{A}_{1}$

$\mathcal{A}_{2}$

Figure 1: Combinatorial associahedron: $\mathcal{A}_{1}$ and
$\mathcal{A}_{2}$.

$\hat{D}_{1}$

$\hat{D}_{2}$

Figure 2: Combinatorial $\hat{D}_{1}$ and $\hat{D}_{2}$ polytopes.

##### Bracketing

A planar $n$-gon with clockwise orientation of vertices
$\\{\,1,\,\dots,\,n\,\\}$ a triangulation is a set of $(n-3)$ chords which can
only intersect in their end points $\in\,\\{1,\,\dots,\,n\,\\}$.

$\displaystyle T\,=\,\\{\,(i_{1},j_{1}),\,\dots,\,(i_{n-3},j_{n-3})\,\\}$
(3.5)

$T$ dissects the $n$-gon into $(n-2)$ triangles and there is a bijection
between the set of all triangulation and placing $n-2$ parenthesis on the
sequence $\\{\,1,\,\dots,\,n\,\\}$.

$\displaystyle\\{(13)\,,\,(14)\,,\ldots,\,(1,n-1)\\}\equiv\big{(}\ldots(((12)3)4)\ldots\big{)}n$
(3.6)

Mutation exchanges one of the brackets with a distinct one, keeping the rest
of the sequence intact. For instance,

$\displaystyle\big{(}\ldots(((12)3)4)\ldots\big{)}n\ \ \longleftrightarrow\ \
\big{(}\ldots((1(23))4)\ldots\big{)}n$ (3.7)

####  3.1.2 $\hat{D}_{n}$ Polytope

The $\hat{D}_{n}$ polytope, or the surfacehedron is the polytope that has a
bijection between its co-dimension $k$ facets and the pseudo-triangulations of
an $n$-gon with a puncture in its interior. In the presence of the puncture,
the chords dissecting the $n$-gon can originate or terminate either at the
vertices of the $n$-gon, or via spiralling around the puncture. As a result,
there are two mutually incompatible chords joining a vertex of the $n$-gon
with the puncture. To differentiate these two possible chords, we draw them
curved, as shown in figure 2. Let us label the two possible chords joining the
vertex $k$ of the $n$-gon with the puncture as $(k0)$ and $(k0^{\prime})$.
Note that the curve $(k0)$ is incompatible with $(i0^{\prime})$, for all $i$
and $k$, i.e., the two can not appear together in a triangulation.

Along with the chords $(k0)$ and $(k0^{\prime})$, and the chords connecting
the vertices of the $n$-gon, here are all the possible chords (equivalently
the faces of $\hat{D}_{n}$ polytope):

$\displaystyle\big{\\{}\,(i,i+k)\ \ \big{|}\ \ i\in[1,n]\,,\
k\in[2,n]\,\big{\\}}~{},\qquad\big{\\{}\big{(}k0\big{)}\ \ \big{|}\ \
k\in[1,n]\big{\\}}~{},\qquad\big{\\{}\big{(}k0^{\prime}\big{)}\ \ \big{|}\ \
k\in[1,n]\big{\\}}$ (3.8)

Note that the chords $(ik)$ and $(ki)$ are distinct, corresponding to two ways
of going around the puncture. We refer to the curve going _left_ around the
puncture, while going from $i$ to $(i+k)$, as $(i,i+k)$. For instance, look at
(3.11).

The $\hat{D}_{n}$ polytope has $n(n+1)$ faces. The faces are themselves
smaller $\hat{D}_{n}$ polytopes, or the associahedra. Let us consider the face
$(k0)$. With a $(k0)$ chord present, the $n$-gon with vertices $[1,\ldots,n]$
with a puncture effectively become a $(n+2)$-gon with vertices
$[1,\ldots,k-1,k,0,k,k+1,\ldots,n]$ without any puncture. So, we have

$\displaystyle\hat{D}_{n}\xrightarrow{\partial_{(k0)}}\mathcal{A}_{n-1}~{},\qquad\quad\hat{D}_{n}\xrightarrow{\partial_{(k0^{\prime})}}\mathcal{A}_{n-1}~{}.$
(3.9)

Let us consider the face $(i,i+k)$. This chord splits the $n$-gon into
$(k+1)$-gon without the puncture, and an $(n-k+1)$-gon with the puncture, so
we have:

$\displaystyle\hat{D}_{n}\xrightarrow{\partial_{(i,i+k)}}\mathcal{A}_{k-2}\times\hat{D}_{n-k+1}~{}.$
(3.10)

For example, let us look at the faces $(25)$ and $(52)$ of $\hat{D}_{5}$:

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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\small$(52)$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-8.00002pt}{14.39458pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\small$(25)$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\hskip
28.45274pt\mathcal{A}_{1}\times\hat{D}_{3}\xleftarrow{\partial_{(25)}}\hat{D}_{5}\xrightarrow{\partial_{(52)}}\mathcal{A}_{0}\times\hat{D}_{4}~{}.$
(3.11)

While discussing the convex realization of the $\hat{D}_{n}$ polytope, we
shall count the number of vertices. Here, we mention the result:

$\displaystyle|\hat{D}_{n}^{[0]}|=2(2n-1)C_{n-1}=\frac{2(2n-1)!}{n!\,(n-1)!}$
(3.12)

###  3.2 The g-vector fan, and the convex realizations

In this section, we review the computation of $g$-vectors which are uniquely
determined by the combinatorics of dissections of an $n$-gon. $g$-vectors are
one of the most crucial ingredients in linking combinatorial polytopes such as
accordiohedra with S-matrix in quantum field theory. We will only analyse the
$g$-vectors which are generated by combinatorics of triangulation of an
$n$-gon and pseudo-triangulation of $n$-gon with a puncture in the center,
[16, 20]. Recent advancements in the quiver representation theory lead to
following results.

Let $\\{{\cal D}^{I}_{ij}\\}$ be the set of all the dissections of an n-gon
with $I$ punctures where $I\,\in\,\\{0,1\\}$. Then,

$\displaystyle\big{\\{}{\cal D}_{ij}^{I}\big{\\}}\,\xleftrightarrow{\text{one
to one}}\,\left\\{{\bf g}_{ij}^{I}\,\in\,\left(\mathcal{E}\equiv{\bf
R}^{\frac{n(n-3)}{2}+I(\frac{n^{2}}{2}+\frac{3n}{2})}\right)\right\\}$ (3.13)

That is, each dissection corresponds to a unique vector in a cartesian space
of dimension $\frac{n(n-3)}{2}$ in the case of triangulation and $n^{2}$ in
the case of pseudo-triangulations. The map depends on a reference
(pseudo)-triangulation which fixes the labels of the axis of embedding space.
Set of all $g$ vectors in both of these cases form an essential and complete
fan, which means the following:

  * •

Given a set of dissections that triangulate an n-gon, or pseudo-triangulate an
$n$-gon with a puncture, the positive span of the corresponding set of ${\bf
g}$ vectors forms a cone such that set of all such cones only intersect in the
origin and their union span the entire ${\cal E}$. For example, look at
figures 7 and 9.

  * •

There is a unique convex realisation of the associahedron and $\hat{D}_{n}$
polytope such that the set of normal vectors to all the co-dimension one faces
is in bi-jection with the set of all the ${\bf g}$-vectors.

  * •

These realisations are precisely, the ABHY associahedron and convex
realisation of $\hat{D}_{n}$ polytope discovered by Arkani-Hamed, Frost,
Plamondon, Salvatori and Thomas.

We start with a reference triangulation or a pseudo-triangulation (with a
puncture in case of loop-level graphs) and drawing triangular paths in each of
the cells of $T_{0}$. Let us mark the triangular paths clockwise in each cell.
Refer to figures 3, 4, and 5 for examples. Now, we look for paths starting
from an external edge (particle) $i$. We associate a unique vector $g_{C}$ in
$\mathcal{E}$ to a path $C$. Note that there are clockwise ‘directions’ on
each segment of the path. We obtain the ‘mountainscape’ for a path, with the
chords of $T_{0}$ as intermediate nodes, by imagining a ‘forward’ (in the
direction of path) direction as going up, and a ‘backward’ direction as coming
down. Let $\\{l_{i}\\}_{i=[1,\ldots,\text{dim}(\mathcal{E})]}$ be the edges of
the graph dual to $T_{0}$. With a peak (pinch in directions) in the
mountainscape at the edge $l_{i}$, we include $+1$ in the $i$th entry of
$\vec{g}_{C}$ and a $-1$ for a dip in the mountainscape (both directions
pointing away from the chord):

$\displaystyle\big{(}\vec{g}_{C}\big{)}{}_{m}=\sum_{\\{k|\ \text{peaks at
}l_{k}\\}}\delta_{m,k}-\sum_{\\{k|\ \text{dips at }l_{k}\\}}\delta_{m,k}$
(3.14)

Another way to characterize all the $g$-vectors is due to a recent work [3],
where the authors choose a particular triangulation $T_{0}$ and draw all the
non-self-intersecting curves on the ribbon graph for $T_{0}$. The two methods
are completely equivalent, and the _curves_ are precisely the _paths_
described earlier. We shall treat _curves_ , _paths_ and _chords_ as synonymes
in this article. As an extra structure, one can associate the following
momentum with a path/curve $C$:

$\displaystyle P_{C}=p_{\text{initial}}+\sum_{\text{right
turns}}p_{\text{(incoming from left)}}~{}.$ (3.15)

Analogous to the pseudo-triangulation, we can obtain the ‘mountainscape’ for a
curve, with the edges of the graph dual to $T_{0}$ as intermediate nodes, by
imagining a left turn as going up, and a right turn as coming down. Once we
have the mountainscape, we can read off the $g$-vectors for curves.

####  3.2.1 g-vector fan for tree-level graphs

Let us choose the ray-like triangulation as the reference triangulation
$T_{0}$. We draw the paths on the triangulated $n$-gon or equivalently the
curves on the ribbon graph for $T_{0}$. Let us denote the paths joining the
leg $i$ to the leg $j$ by $C_{ij}$. These are precisely the chords (or faces)
$(ij)$ discussed in the section 3.1.1. There are $n(n-3)/2$ such paths. Note
that $C_{ij}$ and $C_{ji}$ are equivalent and have the momenta
$P_{C_{ij}}=\big{(}p_{i}+\ldots+p_{j-1}\big{)}$ associated to these. For
example, figure 3 depicts all the curves and equivalent paths for $n=5$.

Figure 3: Triangulated paths and curves on the ribbon graph for tree level
$n=5$.

For arbitrary $n$, the following are the general $g$-vectors:

$\displaystyle\text{For }k\in[3,\ldots,n-1]~{},$
$\displaystyle\qquad\big{(}\vec{g}_{1k}\big{)}{}_{m}=\delta_{(k-2)m}$ (3.16)
$\displaystyle\text{For }k\in[2,\ldots,n-2]~{},$
$\displaystyle\qquad\big{(}\vec{g}_{kn}\big{)}{}_{m}=-\delta_{(k-1)m}$
$\displaystyle\text{For }i\in[2,\ldots,n-3]~{},k\in[i+2,\ldots,n-1]~{},$
$\displaystyle\qquad\big{(}\vec{g}_{ik}\big{)}{}_{m}=-\delta_{(i-1)m}+\delta_{(k-2~{})m}$

The set of all these vectors gives us the _g-vector fan_.

####  3.2.2 g-vector fan for planar one-loop

Let us choose $\\{(k0)\ |\ k\in[1,n]\\}$ as the preferred triangulation
$T_{0}$, dual to the $n$-gon loop graph. Let $\\{l_{i}\\}_{1,\ldots,n}$ denote
the chords. We draw all the paths on the $n$-gon with a puncture with
$\\{(k0)\\}$ chords, and equivalently the paths on the corresponding ribbon
graph. Thanks to the cyclic symmetry of the problem, all the paths come in $n$
copies. In figure 4, we enlist all the _non-spiral_ paths, momenta associated
with them, and the corresponding $g$-vectors. These are bijective to the
$(ik)$ chords (or faces) discussed in the section 3.1.2.

$C_{13}:$1$l_{1}$$l_{2}$3$~{},\quad
P_{C_{13}}^{2}=X_{13}=\big{(}p_{1}+p_{2}\big{)}^{2}~{},\qquad\vec{g}_{C_{13}}=(1,-1,0,\ldots)~{};$

$C_{14}:$1$l_{1}$$l_{2}$$l_{3}$4$~{},\quad
P_{C_{14}}^{2}=X_{14}=\big{(}p_{1}+p_{2}+p_{3}\big{)}^{2}~{},\qquad\vec{g}_{C_{14}}=(1,,0,-1,0,\ldots)~{};$$\vdots$

$C_{11}$1$l_{1}$$\ddots$$l_{n}$n$~{},\quad
P_{C_{11}}^{2}=p_{\Sigma}^{2}=\left(\sum_{i}p_{i}\right)^{2}~{},\qquad\vec{g}_{C_{11}}=(1,,0,\ldots,0,-1)~{}.$

Figure 4: Paths (curves) with their g-vectors lying on the $\Delta$ plane.
Attributable to the cyclic symmetry, there are $n$ copies of such curves,
obtained by shifting the starting point $1\to 2\to\ldots\to n$.

Let us denote the curve joining $i$ to $j$ by $C_{ij}$, such that it takes
immediate left turn after $i$:

$\displaystyle\text{For }\quad i\in[1,n]~{},\quad j\in[i+2,i+n]~{},\qquad\quad
C_{ij}=i\,L\,l{i}\,R\,l_{i+1}\,R\ldots R\,t_{j-1}Lj~{},$ $\displaystyle
P^{2}_{C_{ij}}=X_{ij}=(p_{i}+p_{i+1}+\ldots+p_{j-1})^{2}~{},\qquad(\vec{g}_{ij}){}_{m}=\delta_{im}-\delta_{(j-1)m}$
(3.17)

A total of $n(n-1)$ number of such $C_{ij}$ curves exist. Given
$\\{t_{i}\\}_{i=1}^{n}$ being the coordinates on the $\mathcal{E}$, we can see
that the g-vectors corresponding to these $C_{ij}$ curves lie on a co-
dimension 1 hyperplane: $\sum_{i=1}^{n}t_{i}=0$, called the $\Delta$-plane.
One can check this by the explicit form of $g$-vectors from (3.17).

Along with the paths mentioned above, there are spiral paths depicted in
figure 5, which are termed as $S_{i}$ and $S_{i}^{\prime}$:

$\displaystyle\text{For }\quad i\in[1,n],\qquad\quad S_{i}$
$\displaystyle=iL\,\big{(}l_{i}R\,l_{i+1}R\,l_{i+2}R\ldots
R\,l_{i+n-1}\,R\big{)}^{\infty}~{},$ $\displaystyle\hskip
56.9055pt(\vec{g}_{S_{i}})_{m}=\delta_{im}~{},\qquad\quad
P_{S_{i}}=l_{i-1}~{}.$ (3.18) $\displaystyle\text{For }\quad
i\in[1,n],\qquad\quad S_{i}^{\prime}$
$\displaystyle=iR\,\big{(}l_{i-1}L\,l_{i-2}L\,l_{i-3}L\ldots
L\,l_{i-n}\,L\big{)}^{\infty}~{},$ $\displaystyle\hskip
56.9055pt(\vec{g}_{S_{i}^{\prime}})_{m}=-\delta_{(i-1)m}~{},\qquad\quad
P_{S_{i}^{\prime}}=-l_{i-1}~{}.$ (3.19)

These are bijective to the $(k0)$ and $(k0^{\prime})$ chords or faces
discussed in combinatorial polytope.

$S_{1}~{}:$1$\bigg{(}$$l_{1}$$\ddots$$l_{n}$$\bigg{)}^{\infty}$$~{},\quad
P_{S_{1}}=l_{n}~{},\qquad\vec{g}_{S_{1}}=\big{(}1,0,\ldots,0\big{)}~{};$

$S_{1}^{\prime}~{}:$1$\bigg{(}$$l_{n}$$\ldots$$l_{1}$$\bigg{)}^{\infty}$$~{},\quad
P_{S_{1}^{\prime}}=-l_{n}~{},\qquad\vec{g}_{S_{1}^{\prime}}=\big{(}0,\ldots,0,-1\big{)}~{}.$

Figure 5: The spiraling paths (curves): $S_{1}$ and $S_{1}^{\prime}$. There
are $n$ copies of such curves, obtained by shifting the starting point $1\to
2\to\ldots\to n$.

There is a bijection between the pseudo-triangulations of $n$-gon with a
puncture and the one-loop planar graphs. The two curves $(k0)$ and
$(k0^{\prime})$ joining vertex $k$ with the puncture correspond to two
directions of loop momenta. So, there exists a mutation that flips the loop
momentum. For instance, refer to figure 6 and 7.

$\hat{D}_{1}$ Figure 6: $\hat{D}_{1}$ polytope: a line segment.

$g_{S_{1}}\,(l_{2})$$g_{C_{11}}\,(p_{\Sigma})$$g_{C_{22}}\,(p_{\Sigma})$$g_{S_{2}^{\prime}}\,(-l_{1})$$g_{S_{2}}\,(l_{1})$$g_{S_{1}^{\prime}}\,(-l_{2})$

$\hat{D}_{2}$

Figure 7: The Feynman fan for planar one-loop $n=2$, and the $\hat{D}_{2}$
polytope: a hexagon.

Note that all the graphs are doubled in the $\hat{D}_{n}$ polytope, on top of
the doubling of the tadpoles due to the ‘switch’ mutation. The co-dimension 1
hypersurface called $\Delta$-plane, defined by $\sum t_{i}=0$ divides the
$\hat{D}_{n}$ polytope into two halves.

##### Revisiting the decomposition of $\hat{D}_{n}$ on its faces

Let us revisit how the $\hat{D}_{n}$ polytope decomposes into smaller
polytopes as we approach a boundary, thinking of trivalent graphs. Consider
the face labeled by $S_{i}$ ($S_{i}^{\prime}$), conjugate to the loop momenta
$l_{i-1}$. Approaching the face is equivalent to performing a single cut on
the loop, resulting in a ‘forward limit’ $n+2$ tree-level diagram. So, we
have:

$\displaystyle\hat{D}_{n}\xrightarrow{\partial_{S_{i}}}\mathcal{A}_{n-1}~{},\qquad\quad\hat{D}_{n}\xrightarrow{\partial_{S_{i}^{\prime}}}\mathcal{A}_{n-1}~{}.$
(3.20)

Approaching the $C_{ij}$ face, the graphs decompose into a tree graph and a
loop graph. Precisely, we have:

$\displaystyle\text{For }i\in[1,n]~{},\qquad
k\in[2,n]~{},\qquad\hat{D}_{n}\xrightarrow{\partial
C_{i,i+k}}\mathcal{A}_{k-2}\times\hat{D}_{n-k+1}$ (3.21)

For instance, let us rewrite (3.11) in terms of graphs:

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28.45274pt\mathcal{A}_{1}\times\hat{D}_{3}\xleftarrow{\partial_{C_{25}}}\hat{D}_{5}\xrightarrow{\partial_{C_{52}}}\mathcal{A}_{0}\times\hat{D}_{4}~{}.$
(3.22)

Let us consider the the boundary along $C_{ii}$:
$\hat{D}_{n}\xrightarrow{\partial
C_{i,i+n}}\mathcal{A}_{n-2}\times\hat{D}_{1}$ . This is ‘cutting’ the ‘zero’
momentum propagator $p_{\Sigma}$. There are $n$ such faces, $C_{ii}$, in the
$\hat{D}_{n}$ polytope, and all the tadpole graphs necessarily lie on these
faces. Thus all the tadpoles on one side of the $\Delta$-plane, are arranged
neatly into $n$ copies of associahedra $\mathcal{A}_{n-2}$. The remaining
$\hat{D}_{1}$ mutates the tadpoles across the $\Delta$-plane.

All the tadpole graphs belonging to a particular $\mathcal{A}_{n-2}$
associahedron share a common loop propagator. On the upper side of
$\Delta$-plane, $\sum t_{i}>0$, all the vertices on the face $C_{ii}$ contains
the the loop momenta $l_{i-1}$ corresponding to $S_{i}$. Figure 9 depicts this
for $n=3$. On the other side of the $\Delta$-plane, $\sum t_{i}<0$, the
tadpole cones dual to the face $C_{ii}$ have the loop momenta $-l_{i-1}$
corresponding to $S^{\prime}_{i}$.

#### Counting the number of vertices

On one side of the $\Delta$ plane, the number of graphs (top dimensional cones
in $g$-vector space, or the vertices of $\hat{D}_{n}$ polytope) is as follows:

$\displaystyle\text{Number of tadpole
graphs}:2(2n-3)|\mathcal{A}_{n-3}^{[0]}|$ $\displaystyle\text{Number of bubble
graphs}:(2n-3)|\mathcal{A}_{n-3}^{[0]}|$ $\displaystyle\text{Number of
triangle graphs}:(n-2)|\mathcal{A}_{n-3}^{[0]}|$ $\displaystyle\qquad\ldots$
$\displaystyle\text{Number of }(n-1)\text{ -gon graphs}:n$
$\displaystyle\text{Number of }n\text{-gon graphs}:1$

$|\mathcal{A}_{n-3}^{[0]}|$ is the number of vertices of the $n-3$ dimensional
associahedron (3.3).

Note that the number of $g$-vectors in $\Delta$-plane for $n$ particles is the
same as the total number of $g$-vectors for $n-1$ particles: $n(n-1)$. Thus,
the total number of top dimensional cones for $n-1$ particles on both sides of
the $\Delta$-plane is the same as the ($n-1$ dim) cones inside the
$\Delta$-plane for $n$ particles. The latter is identical to the number of
tadpole graphs on one side of the $\Delta$-plane for $n$ particles, which is
$2(2n-3)|\mathcal{A}_{n-3}^{[0]}|$. Thus, on both sides of the $\Delta$-plane,
the total number of vertices of the $\hat{D}_{n-1}$ polytope is
$2(2n-3)|\mathcal{A}_{n-3}^{[0]}|$. We write down the explicit counting for
some simple cases in table 1.

$(n=1)\qquad$ | $2\times\big{(}1\big{)}$ | $=2$
---|---|---
$(n=2)\qquad$ | $2\times\big{(}1+2\big{)}$ | $=6$
$(n=3)\qquad$ | $2\times\big{(}1+3+6\big{)}$ | $=20$
$(n=4)\qquad$ | $2\times\big{(}1+4+10+20\big{)}$ | $=70$
$(n=5)\qquad$ | $2\times\big{(}1+5+15+35+70\big{)}$ | $=252$
$(n=6)\qquad$ | $2\times\big{(}1+6+21+56+126+252\big{)}$ | $=924$
| $\qquad\vdots$ |
$n\qquad$ |  $2\times\big{(}|n$-gons$|+|(n-1)$gons$|+\ldots+|$bubbles$|+|$tadpoles$|$ $\big{)}$ | $=2(2n-1)|\mathcal{A}_{n-2}^{[0]}|$
Table 1: Number of vertices of the $\hat{D}_{n}$ polytope.

We have thus counted the total number of trivalent planar one-loop diagrams
(accounting for the doubling of tadpole graphs):

$\displaystyle\text{Total planar one-loop graphs for }n\text{
particles}=\frac{1}{2}|\hat{D}_{n}{}^{[0]}|=(2n-1)\,C_{n-1}=\frac{(2n-1)!}{n!\,(n-1)!}~{}.$
(3.23)

In the $\Delta$ plane, there are $|\mathcal{A}_{n-3}^{[0]}|\times 2(2n-3)$
number of $n-1$ dim cones. These lead to an equal number of top-dimensional
cones dual to the tadpole graphs on one side of the $\Delta$-plane, sharing a
co-dimension 1 space with it. For example, figure 8 and 9 depicts the
situation for $n=3$. The plane of the paper is the two-dimensional
$\Delta$-plane. The six cones in the first two diagrams in figure 9 correspond
to tadpoles, sharing a 2d face with $\Delta$ plane. The cones corresponding to
bubbles on the propagators share co-dimension 2 space with the $\Delta$ plane.
In figure 9, the three cones in the third diagram correspond to bubbles and
share a one-dimensional edge with the $\Delta$ plane. Note that the g-vector
with ‘zero ’momentum: $p_{\Sigma}\equiv\sum_{1}^{n}p_{i}$ is adjacent only to
the cones dual to tadpoles. The final cone in figure 9 depicts the ‘triangle’
graph.

$g_{C_{11}}\,(p_{\Sigma})$$g_{C_{22}}\,(p_{\Sigma})$$(p_{\Sigma})\,g_{C_{33}}$$g_{C_{21}}\,(p_{21})$$(p_{13})\,g_{C_{13}}$$(p_{32})\,g_{C_{32}}$$g_{S_{1}}$$(l_{3})$$g_{S_{2}}\,(l_{1})$$(l_{2})\,g_{S_{3}}$

$g_{S^{\prime}_{2}}$$(-l_{1})$$g_{S_{1}^{\prime}}\,(-l_{3})$$(-l_{2})\,g_{S_{3}^{\prime}}$

Figure 8: The $g$-vectors for $n=3$, planar one-loop. The solid (dashed) blue
vectors are pointing out (in) of the plane. The six black vectors lie on the
$\Delta$-plane: $\sum t_{i}=0$.

Figure 9: Cones for $n=3$ on the positive side $\sum t_{i}>0$ of
$\Delta$-plane. Two left ones show total of six tadpoles, then three bubbles,
and one triangle graph.

###  3.3 From ABHY associahedron in ${\cal K}_{n}$ to Scattering form in
${\cal K}_{n}^{\star}$

As a positive geometry that sits inside ${\cal K}_{n}$ (or more precisely, in
the positive octant of ${\cal K}_{n}$), it defines a unique $n-3$ form
$\Omega_{n-3}$ on ${\cal K}_{n}$. $\Omega_{n-3}$ has simple poles on all the
hyperplanes that contain any boundary of the ABHY associahedron such that the
residue on a hyper-plane $H$ containing a co-dimension $k$ boundary
$\partial_{k}A_{n-3}$ equals the canonical form on the (product) of
associahedron $\partial_{k}A_{n-3}$.

More explicitly, given _any_ vertex $v_{0}$ of $A_{n-3}$, let
$\sigma(v,v_{0})$ be the number of mutations required to flip the
triangulation associated to $v_{0}$ to the triangulation associated to $v$.
Then the planar scattering form is defined as,

$\displaystyle\Omega_{n-3}\,=\,\sum_{v\,\in\,A_{n-3}}\,(-1)^{\sigma(v,v_{0})}\,\bigwedge_{(i,j)\,\in\,v}\,d\ln
X_{ij}$ (3.24)

This form can be pulled back to ${\cal K}_{n}^{\star}$ as

$\displaystyle\pi^{\star}\Omega_{n-3}\,=\,\sum_{v\,\in\,A_{n-3}}\,(-1)^{\sigma(v,v_{0})}\,\bigwedge_{(i,j)\,\in\,v}\,d\ln
S_{i,j}$ (3.25)

where

$\displaystyle S_{i,j}=:S_{I}\,|I=\\{i,\dots,j-1\\}$ (3.26)

A remarkable and by now well known property of $\Omega_{n-3}$ is that the form
is projective under $X_{ij}\,\rightarrow\,f(\\{X_{mn}\\})\,X_{ij}$. In fact it
was proved in [1] that projective invariance uniquely fixes $\Omega_{n-3}$
(upto an overall sign which would change the S-matrix only by a phase $\pm
1$.)

It was proved in [1] that although the invariance under of projective
transformation uniquely fixes the planar scattering form in ${\cal
K}_{n}$,there exists a class of scattering forms in the big kinematic space
${\cal K}_{n}^{\star}$ which satisfy this property. Of course $\Omega_{n-3}$
can be pulled back to ${\cal K}_{n}^{\star},\,{\cal K}_{n}^{\textrm{os}}$ and
defines one such projective invariant $n-3$ form, but there are many other
interesting examples, [1] which are relevant for study of scattering
amplitudes. One such form leads to the color dressed amplitudes of biadjoint
scalar theory. We review this construction below as it will be central to us
going forward.

Let
$\alpha\,\in\,{\raisebox{1.99997pt}{$S_{n}$}\left/\raisebox{-1.99997pt}{$Z_{n}$}\right.}$
and fix a planar 3-valent graph
$\gamma_{\alpha}\,\forall\,\alpha\,\in\,{\raisebox{1.99997pt}{$S_{n}$}\left/\raisebox{-1.99997pt}{$Z_{n}$}\right.}$
which is compatible with the $\alpha$ ordering. Let the (internal) edges of
the graph be denoted as $e_{a}^{\alpha}\,|\,1\,\leq\,a\,\leq\,n-3$. Let the
variable of ${\cal K}_{n}^{\star}$ corresponding to $e_{a}^{\alpha}$ be
denoted as $x_{e_{a}^{\alpha}}$. We denote,

  1. 1.

The standard clock-wise ordering $(1,\,\dots,\,n)$ as ${\bf I}$ and the
corresponding chosen graph as $\gamma_{{\bf I}}$,

  2. 2.

The number of ( flips + mutations ) required to map a graph $\gamma_{{\bf I}}$
to $\gamma_{\alpha}$ as $\textrm{flip}(\alpha,{\bf I})$.

Consider the $n-3$ form in ${\cal K}_{n}^{\star}$,

$\displaystyle\omega_{n-3}\,=\,\sum_{\alpha\,\in\,{\raisebox{1.39998pt}{$S_{n}$}\left/\raisebox{-1.39998pt}{$Z_{n}$}\right.}}\,c(\gamma_{\alpha})\,w(\gamma_{\alpha})\,\prod_{a=1}^{n-3}\,\frac{1}{x_{e_{a}^{\alpha}}}$
(3.27)

where

$\displaystyle w(\gamma_{\alpha})\,=\,(-1)^{\textrm{flip}(\alpha,{\bf
I})}\,\bigwedge_{a=1}^{n-3}\,dx_{e_{a}^{\alpha}}$ (3.28)

Using the Seven term identity of eqn. (2.2) it was shown in [1] that
$\omega_{n-3}$ is projective and that $\omega_{n-3}$ can be used to generate a
scalar in ${\cal K}_{n}^{\star}$ which is the color-dressed amplitude of bi-
adjoint scalar theory. We recast the ABHY derivation in a slightly different
manner which will then be generalised to obtain tree-level _color ordered_
Yang-Mills amplitude from $\Omega_{n-3}$. Consider a three-regular graph
$\gamma$, where each vertex is equipped with a clockwise orientation.666For
graphs embedded in a plane, such an ordering can be inherited from the
orientation of the underlying manifold, but even for algebraic graphs, we can
simply choose an ordering at each vertex. We color each (internal as well as
external) edge of $\gamma$ with a color label. If the three edges
$e_{1}(v),e_{2}(v),e_{3}(v)$ are incident on $v$, we color them with
$a_{v},b_{v},c_{v}$ respectively. We can now associate a color tensor
$\tilde{c}(\gamma)$ to the graph as,

$\displaystyle\tilde{c}(\gamma)\,:=\,\prod_{v\,\in\,\gamma}\,f_{v}^{a_{v}b_{v}c_{v}}$
(3.29)

where $f^{abc}$ is the structure constant of a compact group and the ordering
of color indices at each vertex is determined by the orientation.

In ${\cal K}_{n}^{\star}$ we can now define a multi-vector field (MVF)777A
multi-vector field of rank $k\,\leq\,n$ on an ${\bf R}^{n}$ is a section of
the exterior algebra bundle $\Lambda^{k}\,TM$. Contraction of such a field
with a $k$ form is defined via
$\displaystyle\langle\,\partial_{x_{1}^{i_{1}}}\,\wedge\,\dots\,\wedge\,\partial_{x_{k}^{i_{k}}},\,dx^{j_{1}}\,\wedge\,\dots\,\wedge\,dx^{j_{k}}\,\rangle\,=\,\delta^{i_{1}\dots
i_{k}}_{[j_{1}\,\dots\,j_{k}]}$ (3.30)

$\displaystyle{\cal
P}^{\textrm{color}}_{n}\,=\,\sum_{\alpha\,\in\,{\raisebox{1.39998pt}{$S_{n}$}\left/\raisebox{-1.39998pt}{$Z_{n}$}\right.}}\,\tilde{c}(\gamma_{\alpha})\,(-1)^{\textrm{flip}(\alpha,{\bf
I})}\,\bigwedge_{a=1}^{n-3}\,\frac{\partial}{\partial s_{e_{a}^{\alpha}}}$
(3.31)

Then,

$\displaystyle\langle\,{\cal
P}^{\textrm{color}}_{n},\pi^{\star}\omega_{n}\,\rangle\,=\,\sum_{\alpha\,\in\,{\raisebox{1.39998pt}{$S_{n}$}\left/\raisebox{-1.39998pt}{$Z_{n}$}\right.}}\,\tilde{c}(\gamma_{\alpha})\,c(\gamma_{\alpha})\,\prod_{a=1}^{n-3}\,\frac{1}{s_{e^{\alpha}_{a}}}$
(3.32)

which is the color-dressed bi-adjoint amplitude $\sum_{\alpha}\,{\cal
M}_{n}(\alpha\,|\,\alpha)$ with color factors denoted as $c,\tilde{c}$
respectively. There are two important aspects of this interpretation of the
color-dressed bi-adjoint amplitude from our perspective.

  * •

As emphasized above, unlike the color ordered amplitude which is the volume of
the simplex dual to the ABHY associahedron in ${\cal K}_{n}$, the color
dressed amplitude is a scalar function in the big kinematic space.

  * •

This scalar can be understood as a contraction of the planar scattering form
with multi-vector field.

  * •

${\cal P}^{\textrm{color}}_{n}$ is a weighted sum over set of all graphs
obtained from $\gamma_{I}$ by mutation and flips. It’s valuation on each graph
is a polynomial (in color space) determined by the orientation at each vertex
of the graph and the set of half-edges incident at that vertex.888An half-edge
is simply a pair $(e,v)$ of an edge along with one of it’s end points. For a
closed loop incident on a single vertex, half edge is same as the edge itself.

##  4 A Brief Review of Corolla Polynomial

In [2], D. Kreimer, M. Sars and W. D. van Suijlekom defined a new graph
polynomial $C_{\gamma}$ for any 3-regular graph $\gamma$ which is uniquely
fixed by the

  1. 1.

The genus of the graph

  2. 2.

The set of half-edges of the graph and

  3. 3.

A clockwise orientation at each vertex of the graph.

As such Corolla polynomial is simply a counting polynomial (see [35]) which
enumerates the set ${\cal T}$ of all the subsets of the half-edges $T$ such
that $\frac{\gamma}{T}$ has no cycle,[35]. An intriguing mathematical object
in it’s own right, the real interest for us in the Corolla polynomial is due
to seminal results proved in [2], where a specific representation of the
Corolla polynomial (which we review below) was used to generate a large class
of gauge theory correlators from correlators in massless $\phi^{3}$ theory.

More in detail, it was shown in [2] that the Corolla polynomial can be
represented as a differential operator which can then “spin up" an
$n$-particle, $L$ loop Feynman diagram in $\phi^{3}$ theory in Schwinger
parametrisation, for all $n,\,L$. The result is a the Schwinger integrand of
Yang-Mills theory where the underlying graph is built out of (1) 3-gluon
vertex, (2) 4-gluon vertices and (3) closed ghost loops. In our opinion, this
is a rather striking result where the manifest gauge invariance and locality
arises from a remarkable fact that the action of the Corolla polynomial can
also be viewed as generating elements of the graph homology, [36, 37].
Although, the complete ramfications of the rich structure of the Corolla
polynomial still remains to be appreciated and explored, it has already been
used to define amplitude for a class of gauge theories including (1)
spontaneously broken gauge theories [38] and (2) gauge theories coupled to
charged matter fields such as QED [39].

We review this construction below. Many of the beautiful properties of the
Corolla polynomial have been derived in [35, 37]. Our review will only scratch
the surface of these results, as our primary goal is to define a new
representation of the Corolla polynomial as differential operators in ${\cal
K}_{n}^{\textrm{os}}$.999Kreimer’s introduction of the Corolla polynomial as a
third graph polynomial which along with First and Second symanzik polynomials
generate gauge theory amplitudes. We will review the Corolla graph
differential in the space of Schwinger parameters in section (8.1) and extend
it to define a “Corolla curve differential”. This will lead us to the curve
integral formulae for gluons as derivatives of the formulae discovered in [3].
Interested reader is encouraged to peruse the references cited above for more
details about Corolla Polynomial.

###  4.1 Half edges, Graph Orientation and Edge ordering

In this section we review the graph theoretic data required to define the
Corolla polynomial on tri-valent (cubic) graphs with a fixed ordering of
external particles. Throughout the paper, we will only encounter three-valent
graphs which can be embedded on a disc with $n$ maked points on the boundary
and at most one puncture.

Given an edge $e$ incident on a vertex $v$, by half edge we mean the pair
$(e,v)$. In a cubic graph, every vertex $v$ has three half edges associated to
it denoted by $h_{v}^{I}\,|\,I=\\{1,2,3\\}$. We denote the set of tri-valent
vertices of a graph $\gamma_{n}$ with $n$ external vertices as
$V(\gamma_{n})$.

######  Definition 4.1.

A corolla $C_{v}$ is a vertex $v\,\in\,V(\gamma_{n})$ along with the set of
all the half edges incident on $v$.

To define the Corolla polynomial, a bare graph is not enough but we need to
assign some additional data to it.

  1. 1.

All the external edges are outgoing.

  2. 2.

We assign a clockwise orientation to each vertex in $V(\gamma_{n})$ once and
for all. It generates a cyclic ordering on the half edges,
$h_{v}^{1}<h_{v}^{2}<h_{v}^{3}$.

  3. 3.

Each half edge $h_{e}$ is assigned a variable, $a_{h_{e}}$.

  4. 4.

Every internal edge is directed.

  5. 5.

To every vertex of the associahedron, we assign a unique “ordering label"
which orders the set of (internal) edges of the corresponding 3-regular planar
graph.

The last two points requires some deliberation. For a fixed graph, we can
assign a ordering to the set of vertices as well as set of internal edges once
and for all, but as $A_{n-3}$ “glues" all 3-regular planar graphs together,
the corresponding ordering on the set $E_{\textrm{int}}(\gamma_{1})$ and
$E_{\textrm{int}}(\gamma_{2})$ also has to commute with mutations. We claim
that indeed, such an ordering assignment exists.

Let us consider a given 3-regular graph with ordered external vertices as
(following J. Stasheff) parenthesizing $(1,2,\dots,n)$ with $n-3$ parenthesis
which are either non intersecting or nested, [7]. In the case of $n=4$ the two
possible triangulations can hence be represented as following parenthetical
expressions.

$\displaystyle T_{1}\,=\,(12),3,4$ $\displaystyle T_{2}\,=\,1,(23),4$ (4.1)

We note that given a parenthetical expression that corresponds to a fixed
triangulation, one can always add a redundant parenthesis without changing the
triangulation. We color the redundant parenthesis in blue.

$\displaystyle
T_{1}\,=\,(12),{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(}34{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1})}$
$\displaystyle
T_{2}=\,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(}1,(23),4\,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1})}$
(4.2)

We also note that in $n=4$ case, given a triangulation $T$ and the associated
parenthesised expression, every pair of parenthesis then corresponds to a
unique tri-valent vertex in $\gamma_{T}$.

$\displaystyle(12)\,\leftrightarrow\,v_{1,2,(12)}$
$\displaystyle(34)\,\leftrightarrow\,v_{3,4,(34)}$ (4.3)

We now provide a natural ordering on the set parenthetisized expressions via

$\displaystyle(i,j)<(k,l)\,\textrm{if}\,(i,j)\,\cap\,(k,l)=0\,\textrm{ and
}i<k$ $\displaystyle(i,j)<(k,l)\,\textrm{ if }\,(i,j)\,\subset\,(k,l)$ (4.4)

Such an ordering induces a canonical ordering on the sets of vertices
$V(\gamma_{T_{1}}),\,V(\gamma_{T_{2}})$.

$\displaystyle v_{1,2,(12)}<v_{3,4,(34)}\,\textrm{ in }\,\gamma_{T_{1}}$ (4.5)
$\displaystyle v_{2,3,(23)}<v_{4,1,(41)}\,\textrm{ in }\,\gamma_{T_{2}}$ (4.6)

We can now extend these definitions to arbitrary triangulations for all $n$.
But rather than giving a general set of conditions, we simply illustrate the
algorithm (to assign ordering on set of all vertices of a graph) through one
more example. The reader is encouraged to verify that a systematic algorithm
can be written down for any $n$.

Given a complete triangulation $T$ of the $n$-gon, we now argue that it’s
representation as parenthisizing $n$ letters generates a canonical complete
ordering on the set of all the vertices.

  1. 1.

Given the parenthetisized expression for $T$, any bracket that contains $k$
elements $(i_{1},\,\dots,i_{k})$ must also contain an additional $k-3$
parentheses nested inside the outermost parenthesis. We can then associate a
set $S_{(i_{1}\dots i_{k})}$ of vertices of $\gamma_{T}$ (with cardinality
$k-1$) to this expression.

  2. 2.

Between any two vertices in $S_{(i_{1},\dots,i_{k})}$ there is then a ordering
determined by criteria laid out in eqn.(4.1).

We then have a complete ordering on set of all the vertices in
$V(\gamma_{T})$. Let us denote the ordered vertices as
$(v_{1},\dots,\,v_{n-2})$. We can now order the set of all internal edges as
follows.

$\displaystyle e_{v_{i},v_{j}}\,<\,e_{v_{k},v_{l}}\quad\textrm{if}\quad
v_{i}<v_{k}$ (4.7)

We note that this algorithm makes all the internal edges directed and puts
ordering on the set $E_{\textrm{int}}$.

Let us take a specific example. Consider the following triangulation for
$n=6$.

$\displaystyle T_{1}=\\{\,(1,3),(3,5),(1,5)\,\\}$ (4.8)

$T_{1}$ can be mapped to the following bracketing of the sequence
$\\{1,\,\dots\,6\\}$.

$\displaystyle
T_{1}\,=\,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(}(12)\,(34)\,(56){\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1})}$
(4.9)

The bracket fixes a unique order on the set of all internal vertices.

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56.9055pt\begin{matrix}v_{1,2,(12)}<v_{3,4,(34)}<v_{5,6,(56)}<v_{(12)(34)(56)}\\\
\\\ e_{a}<e_{b}<e_{c}\end{matrix}$ (4.10) $\displaystyle e_{a}\equiv
e_{v_{1,2,(12)}\,v_{(12)(34)(56)}}~{},\quad e_{b}\equiv
e_{v_{(12)(34)(56)}\,v_{3,4,(34)}}~{},\quad e_{c}\equiv
e_{v_{(12)(34)(56)}\,v_{5,6,(56)}}$ (4.11)

This in turn fixes the orientation on the edges and a complete ordering on the
set of internal edges.

Hence we have placed additional data on any triangulation of the $n$-gon. That
is, with each vertex $T$ of the associahedron we associate,

$\displaystyle T_{\textrm{directed}}\,:=$ $\displaystyle
T\oplus\textrm{\big{(} Each dissection marked by an arrow}$ and a complete
ordering on the set of chords that constitute $T$. ) (4.12)

As there is a canonical map from $T\,\rightarrow\,T_{\textrm{directed}}$ that
is defined by the triangulation itself, we will continue to denote the marked
triangulation $T_{\textrm{directed}}$ as $T$.

###  4.2 Complete Ordering on a Pseudo-Triangulation

Similarly, given any pseudo-triangulation $PT$ of an $n$-gon with a puncture,
one can assign a complete ordering on the set of all the vertices and set of
all the chords of the dual graph $\gamma_{PT}$. The simplest way to understand
the “ordering map" is via the “forward-limit" construction as any
$\gamma_{PT}$ with $n$ external vertices can be obtained from a tree-level
graph with $n+2$ vertices by gluing the two additional edges. Rather than
elaborating on the algorithm for general $n$, we illustrate it with an
example. Interested reader can easily extend this discussion to a one loop
planar graph with arbitrary number of external legs.

We thus consider a pseudo-triangulation $PT_{0}$ of a punctured $n=4$-gon
whose dual graph $\gamma_{PT_{0}}=:\gamma_{1}^{13\,\textrm{t}}$ is as shown in
(4.13).101010The superscript is used to denote the fact that there is a
tadpole on the propagator labelled by the chord $(1,3)$. The corresponding
loop integrand can also be interpreted as a forward limit of a $6$ point tree-
level graph with the ordering $(1,2,0_{L},0_{R},3,4)$ as shown in (4.13). All
the external edges of this graph are outgoing with momenta on the two
additional edges being $l,-l$ as shown. We can now apply the algorithm devised
for 3-valent planar graphs and immediately deduce the total ordering on set of
vertices and map the set of edges to set of directed edges.

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e_{12}<e_{l}<e<e_{34}\end{matrix}$ (4.13)

We end this section with a couple of remarks on the relevance of the ordering
and orientation we have introduced, on the set of all vertices.

  * •

The ordering on set of all vertices is sufficient to turn all the internal
edges into directed edges. This is a necessary structure required to define
the Corolla differential in section 4.3, which will “spin up" bi-adjoint
$\phi^{3}$ amplitude to obtain Yang-Mills amplitude.

  * •

The action of Corolla differential on a 3-valent planar graph will result in a
set of graphs with marked edges. The ordering on $E_{\textrm{int}}$ is then
necessary to prove that the direct sum of all such graphs generate the graph
homology.111111It has been proved in [37] that the homology class itself is
invariant under a permutation of ordering. However the ordering is essential
to prove that any representative of the homology is anhilated by so called
graph differentials. These essential technicalities have been analysed in [2,
37] and we will not elaborate on them further.

  * •

We used a structure inherent to the associahedron to define a ordering on set
of all the vertices and set of all the edges. It is important to note for a
given graph $\gamma$ one can also define such an ordering by simply giving a
cyclic ordering on the set of all the corollas in $\gamma$, [36]. A detailed
comparison between the two prescriptions is outside the scope of this paper.

###  4.3 Review of a new graph polynomial and it’s Differential
Representation.

In this section, we review the definition of Corolla polynomial. Corolla
polynomial is defined for any tri-valent graph with arbitrary number of cycles
but we restrict ourself with either tree-level or one loop planar graphs with
tri-valent vertices. Although not nearly as well known some of the other graph
polynomials, (e.g. the Symanzik polynomials), Corolla polynomial has been
studied in the literature by physicists and mathematicians. For a study of
some of the fascinating properties of this polynomial, we refer the reader to
[35].

Let $\gamma_{n}$ be a three-regular graph with $n$ external edges and no
cycles.121212Recall from the previous section that $\gamma_{n}$ has additional
structures : ordering on the set of internal edges and clock-wise orientation
at each vertex. Then the Corolla polynomial associated to $\gamma$ is defined
as,

$\displaystyle
C_{\gamma_{n}}\,=\,C^{(0)}_{\gamma_{n}}\,-\,\delta_{l,1}\,C^{(1)}_{\gamma_{n}}$
(4.14)

where $0\,\leq\,l\,\leq\,1$ is the number of cycles in $\gamma_{n}$. $C^{(0)}$
and $C^{(1)}$ are defined as follows.

$\displaystyle C^{(0)}_{\gamma_{n}}\,$
$\displaystyle=\,\prod_{v\,\in\,V(\gamma_{n})}\,{C}_{v}~{},$ (4.15)
$\displaystyle\text{where}\qquad{C}_{v}\,$
$\displaystyle=\,\sum_{I=1}^{3}\,a_{h_{v}^{I}}$ (4.16)

In order to define $C^{(1)}$ we introduce some notations first.

  1. 1.

If $L_{\gamma_{n}}$ is the unique cycle in $\gamma_{n}$ then the set vertices
adjacent to it will be denoted as $V(L_{\gamma_{n}})$.

  2. 2.

The half-edge incident on $v^{\prime}\,\in\,V(L_{\gamma_{n}})$ which is not
inside $L_{\gamma_{n}}$ will be denoted as $l_{v^{\prime}}$.

  3. 3.

Given the cyclic ordering at $v^{\prime}$, the corresponding edges that
precede and follow $l_{v^{\prime}}$ will be denoted as $l_{v^{\prime}}^{\pm}$.

  4. 4.

$\forall\,v^{\prime}\,\in\,V(L_{\gamma_{n}})$, we define

$\displaystyle
b_{v^{\prime}}\,:=\,a_{l_{v^{\prime}}^{+}}\,+\,a_{l_{v^{\prime}}^{-}}.$

In summary, The Corolla polynomial is then a difference of two polynomials for
all cubic graphs with number of closed cycles $\leq\,1$.

$\displaystyle
C_{\gamma_{n}}\,=\,C^{(0)}_{\gamma_{n}}\,-\,C^{(1)}_{\gamma_{n}}$ (4.17)
$\displaystyle C^{(0)}_{\gamma_{n}}\,$
$\displaystyle=\,\prod_{v\,\in\,V(\gamma_{n})}\,C_{v}$ $\displaystyle
C^{(1)}_{\gamma_{n}}\,$
$\displaystyle=\,\prod_{v\,\in\,V(\gamma_{n})-V(L_{\gamma_{n}})}\,C_{v}\,\prod_{v^{\prime}\
\in V(L_{\gamma_{n}})}\,b_{v^{\prime}}$ (4.18)

The geometric interpretation of $-\,C^{(1)}_{\gamma_{n}}$ is that it maps the
gluon loop integrand to the ghost loop integrand. In other words it maps the
three-gluon vertex adjacent to the cycle to a gluon-ghost-ghost vertex.

We end this section by assigning 4-vectors to all the edges of $\gamma_{n}$
which will be used in defining a differential operator representation for the
Corolla polynomial.

  1. 1.

Every external edge $e\,\in\,E_{\textrm{ext}}$ will be labelled by a space-
time vector $\xi_{i}^{\mu}$.

  2. 2.

If $\gamma_{n}$ has no cycles and if $T_{\gamma_{n}}$ is the corresponding
triangulation, then for an internal edge $e^{\prime}$ which is dual to a chord
$(i,j)\,\in\,T_{\gamma_{n}}$, we assign a label $\xi^{\mu}_{i,j}$.

  3. 3.

If $\gamma_{n}$ contains exactly one cycle and if $PT_{\gamma_{n}}$ is the
corresponding (pseudo)-triangulation, then an internal edge $e^{\prime}$ which
is dual to any of the chords $(I,J):=(i,j),\,(i,0),(i,i)\in\,PT_{\gamma_{n}}$,
we assign a label $\xi^{\mu}_{I,J}$.131313Recall that in this case the set of
chords also include $(i,i+1)$ which are not homotopic to any of the boundaries
of the $n$-gon.

The scalar products of these vector labels generate the off-shell kinematic
space ${\cal K}_{n}^{\textrm{os}}$ and projection to the kinematic space is
obtained by simply using the following replacement rules.

$\displaystyle\xi_{i}^{\mu}\rightarrow\,p_{i}^{\mu}~{},\qquad\xi^{\mu}_{i,j}\rightarrow\,p_{i}^{\mu}\,+\,\dots\,+p_{j-1}^{\mu}$
(4.19)

##  5 From triangulations and flips to $n-3$ ranked Multi-vector field on
${\cal K}_{n}$

As every vertex $v_{T}$ of $A_{n-3}$ is labelled by a complete triangulation
$T$ of the $n$-gon, one can associate a unique Corolla polynomial $C_{T}$ to
each vertex of the associahedron. This assignment provides us with the
following $(n-3)$-rank multi-vector field (MVF) which we denote as ${\cal
P}_{n}$. We use the letter $c$ to denote a generic chord that dissects an
$n$-gon.

$\displaystyle{\cal
P}_{n-3}\,:=\,\prod_{i=1}^{n}x_{i}\,\sum_{T}\,(-1)^{T}\,C^{(0)}_{T}\,\prod_{c\,\in\,T}\,\prod_{j=1}^{n}\frac{1}{x_{j}}\,\frac{\partial}{\partial
x_{c}}$ (5.1)

Consider a specific triangulation

$\displaystyle T_{0}\,=\,\\{\,(1,3),\,(1,4),\,\dots,\,(1,n-1)\,\\}.$ (5.2)

The set of vertices of the graph $\gamma_{T_{0}}$ can be labelled as,

$\displaystyle
V(\gamma_{T_{0}})\,:=\,\\{\,(1,2,(12)),\,((12),3,((12),3)),\,\dots,\,((1\;n-2),n-1,(1\;n-1))\,\\}$
(5.3)

One can use this presentation of the vertices of $\gamma_{T_{0}}$ to express
Corolla polynomial as a product of monomials

$\displaystyle C^{(0)}_{T}\,=\,\prod_{v\,\in\,V(\gamma_{T})}\,C_{v}$ (5.4)

so that,

$\displaystyle
C^{(0)}_{T_{0}}\,=\,C^{(0)}_{(1,2,(12))}\,C^{(0)}_{((12),3,((12),3))}\,\dots\,C^{(0)}_{((1\;n-2),n-1,(1\;n-1))}$
(5.5)

We will now show that tree-level YM color-ordered amplitude is a scalar in
${\cal K}_{n}^{\textrm{os}}$ which is obtained by a contraction of a multi-
vector field with the pull-back of the planar scattering form $\Omega_{n}$
defined in ${\cal K}_{n}$, [1]. Restriction of the scalar to ${\cal
K}_{n}\,\subset\,{\cal K}_{n}^{\textrm{os}}\,\dots$.

###  5.1 From d-log form to Tree-level Amplitude

In [2] Corolla polynomial was represented as a map from a meromorphic function
in the space of Schwinger parameter to a differential operator of degree $n-2$
on ${\cal K}_{n}^{\textrm{os}}$.

Directly inspired by the construction of [2], we propose a new representation
of the Corolla polynomial which will appear inside ${\cal P}_{n-3}$ defined in
eqn.(5.1). We then show that contraction of ${\cal P}_{n-3}$ with the d-log
form $\Omega_{n-3}$ on ${\cal K}_{n}^{\textrm{os}}$ produces the tree-level
color ordered YM amplitude.141414Strictly speaking we have to pull back the
scalar obtained via contraction from ${\cal K}_{n}^{\textrm{os}}$ to ${\cal
K}_{n}$. Contraction of a d-log form by a multi-vector field determined by the
half-edges of the set of graphs is reminiscent of the dervation of color-
dressed amplitude of bi-adjoint $\phi^{3}$ theory from $\Omega_{n-3}$, (See
section 3.3). One technical difference of course is that whereas, ${\cal
P}_{n-3}^{\textrm{color}}$ is a multi-vector field in the big kinematic space
${\cal K}_{n}^{\star}$, we now define a multi-vector field in ${\cal
K}_{n}^{\textrm{os}}$.

Tthe contraction of ${\cal P}_{n-3}$ with the $\Omega_{n-3}$ is,

$\displaystyle\langle\,{\cal
P}_{n-},\Omega_{n-3}\,\rangle\,=\,\prod_{i=1}^{n}x_{i}\sum_{T\,\in\,V(A_{n-3})}\,C^{(0)}_{T}\,\prod_{i=1}^{n}\,\frac{1}{x_{i}}\,\prod_{C\,\in\,v}\,\frac{1}{x_{C}}$
(5.6)

Each vertex $T=\triangle_{1}^{T}\,\cup\,\dots\,\triangle_{n-2}^{T}$ of the
associahedron corresponds to a unique set of tri-valent vertices that we
denote as $\\{v_{1}^{T},\,\dots\,v_{n-2}^{T}\\}$. As reviewed in section 4.3,
one can then associate to $T$, a corolla polynomial $P_{T}$

$\displaystyle C^{(0)}_{T}\,:=\,\prod_{i=1}^{n-2}\,C_{v_{i}^{T}}$ (5.7)

If the vertex $v_{i}^{T}$ has the three incident half-edges
$h_{i,v}^{1}<h_{i,v}^{2}<h_{i,v}^{3}$ than the corolla polynomial associated
to $v_{i}^{T}$ is,

$\displaystyle\sum_{I=1}^{3}a_{h_{i,v}^{I}}$ (5.8)

We can in fact interpret the multi-vector field defined in eqn.(3.31) as a
specific presentation of Corolla polynomial.

  1. 1.

The multi-vector field is parametrized by the color factors
$\tilde{c}(\gamma_{\alpha})$ which is a polynomial in the color space and can
be written as

$\displaystyle\tilde{c}(\gamma)\,=\,\prod_{v\,\in\,V(\gamma)}\,\textrm{Tr}(\,T_{a_{v}}\,[\,T_{b_{v}},\,T_{c_{v}}\,]\,)$
(5.9)

  2. 2.

Each tensor at the vertex is associated to the half-edges and there is a
contraction over repeated color indices corresponding to gluing of half-edges.

  3. 3.

The color-dressed bi-adjoint ampliude is a scalar in ${\cal
K}_{n}^{\textrm{os}}$ as opposed to color ordered amplitude which is a form in
${\cal K}_{n}$.

  4. 4.

The polynomial at each vertex which is dependent on the (half)-edges incident
at $v$ is a specific representation of Corolla polynomial. Given a fixed
orientation at every vertex and a fixed ordering on the set of edges, Corolla
polynomial is determined by assigning two variables $(b_{e},a_{e})$ to each
edge $e$. Given a cyclic ordering of $e_{1},e_{2},e_{3}$ at $v$ which is
$e_{1}<e_{2}<e_{3}$ Corolla polynomial is

$\displaystyle\tilde{c}_{v}:=f^{a_{e_{1}}a_{e_{2}}a_{e_{3}}}$ (5.10)

We now define another representation of Corolla Polynomial as a multi-vector
field in ${\cal K}_{n}$. In this representation, Corolla polynomial $C_{T}$
associated to a given triangulation $T$ is not a polynomial in color space but
a differential operator which acts on meromorphic functions on ${\cal K}_{n}$.

The starting point is a representation for Corolla monomial associated to a a
cell $\triangle\,\in\,T$. Let, the three 4-vector labels associated to
boundary of $\triangle$ be $\\{\xi_{1},\xi_{2},\xi_{12}\,\\}$. $\triangle$ is
dual to a 3 point vertex depicted in (5.11).

(5.11) $\displaystyle C_{\triangle}$
$\displaystyle:=\eta^{\mu_{1}\mu_{2}}\left(\frac{\xi_{1}^{2}}{2}\,\partial_{\xi_{1}}-\frac{\xi_{2}^{2}}{2}\,\partial_{\xi_{2}}\right)^{\mu_{12}}+\eta^{\mu_{2}\mu_{12}}\left(\frac{\xi_{2}^{2}}{2}\,\partial_{\xi_{2}}-\frac{\xi_{12}^{2}}{2}\,\partial_{\xi_{12}}\right)^{\mu_{1}}+\eta^{\mu_{12}\mu_{1}}\left(\frac{\xi_{12}^{2}}{2}\,\partial_{\xi_{12}}-\frac{\xi_{1}^{2}}{2}\,\partial_{\xi_{1}}\right)^{\mu_{2}}$
(5.12)

One can use eqns. (5.5, 5.12) to write the Corolla polynomials any complete
triangulation $T$.

#### Corolla contraction with the one form

Let us consider the simplest example of $n=4$ and the corresponding 1
dimensional associahedron $A_{1}$. Vertices of $A_{1}$ are the two
triangulations $T_{1}=(1,3)$, $T_{2}=(2,4)$. The Corolla differential
associated to $T_{1}$ on ${\cal K}_{4}$ can be written down as a product of
two differentials $C_{v_{1}=(1,2,(12))},C_{v_{2}=((12),3,4)}$.

Following notation will be useful for writing the Corolla differential in a
compact form.

$\displaystyle(\eta\,\wedge\,\eta)^{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}}\,:=\,\eta^{\alpha_{1}\alpha_{2}}\,\eta^{\alpha_{3}\alpha_{4}}\,-\,\eta^{\alpha_{1}\alpha_{4}}\,\eta^{\alpha_{2}\alpha_{3}}$
(5.13)

We can then use eqn.(5.12) and after some algebra, both the Corolla
differentials at the two vertices of the graph $\gamma_{T_{1}}$ can be written
as,

$\displaystyle C_{v_{1}}\,$
$\displaystyle=\,-\,\frac{1}{2}\big{(}[\,\xi_{1}^{2}\,(\eta\,\wedge\,\eta)^{\mu_{1}\mu_{2}\alpha\mu}\frac{\partial}{\partial\xi_{1}^{\alpha}}+\xi_{2}^{2}\,(\eta\,\wedge\,\eta)^{\mu_{2}\mu\alpha\mu_{1}}\,\frac{\partial}{\partial\xi_{2}^{\alpha}}+\xi_{12}^{2}\,(\eta\,\wedge\,\eta)^{\mu\mu_{2}\alpha\mu_{1}}\,\frac{\partial}{\partial\xi_{12}^{\alpha}}\big{)}$
(5.14) $\displaystyle C_{v_{2}}\,$
$\displaystyle=\,-\,\frac{1}{2}\big{(}[\,\xi_{3}^{2}\,(\eta\,\wedge\,\eta)^{\mu_{3}\mu_{4}\alpha\mu}\,\frac{\partial}{\partial\xi_{3}^{\alpha}}+\xi_{4}^{2}\,(\eta\,\wedge\,\eta)^{\mu_{4}\mu\alpha\mu_{3}}\,\frac{\partial}{\partial\xi_{4}^{\alpha}}+\xi_{12}^{2}\,(\eta\,\wedge\,\eta)^{\mu\mu_{3}\alpha\mu_{4}}\,\frac{\partial}{\partial\xi_{12}^{\alpha}}\big{)}$
(5.15)

Similarly, if we denote the vertices of the graph $\gamma_{T_{2}}$ as

$\displaystyle v_{1}^{\prime}=(2,3,(23)),\,v_{2}^{\prime}=(23,4,1)$ (5.16)

One can obtain the $C_{v_{1}^{\prime}}$ and $C_{v_{1}^{\prime}}$ by performing
the following shift in (5.14): $i\to i+1~{},\forall i$.

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{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-22.76228pt}\pgfsys@lineto{0.0pt}{-21.04669pt}\pgfsys@stroke\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
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{}{}{}{}{}{}{{}}\pgfsys@moveto{-56.90552pt}{-14.22638pt}\pgfsys@lineto{-55.7757pt}{-13.09656pt}\pgfsys@stroke\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\hbox{{$(\mu_{2})~{}~{}~{}\xi_{2}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{$(\mu_{3})~{}~{}~{}\xi_{3}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\xi_{1}~{}~{}~{}(\mu_{1})$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-79.48001pt}{-28.1073pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}\hbox{{$\xi_{4}~{}~{}~{}(\mu_{4})$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ (5.17)
$\displaystyle{\cal
P}_{4}\,=\,\prod_{i=1}^{4}x_{i}C_{v_{1}}C_{v_{2}}\prod_{j=1}^{4}\frac{1}{x_{i}}\,\partial_{x_{13}}\,-\,\prod_{i=1}^{4}x_{i}C_{v^{\prime}_{1}}C_{v^{\prime}_{2}}\prod_{j=1}^{4}\frac{1}{x_{i}}\,\partial_{x_{24}}$
(5.18)

A moment of meditation reveals that inside either of the Corolla differential
$\hat{C}_{T_{1}}$ or $C_{T_{2}}$ the differential
$\xi_{i}^{2}\,\partial_{\xi_{i}}$ with respect to the 4-vector label
associated to any of the external edges appears only in one unique monomial
$C_{v}$ where $v\,\in\,V(\gamma_{T_{i}})|i=\\{1,2\\}$. Thus, we can make the
following replacement across all the vertices:
$\xi_{i}^{2}\,\partial_{\xi_{i}}\to-2\xi_{i}$. We also note that this
observation remains valid for any genus $0$ tri-valent graph $\gamma$.

As a result, a rather direct computation gives,

$\displaystyle\langle\,{\cal P}_{1},\Omega_{1}\,\rangle\,=$
$\displaystyle\hskip
14.22636pt=\left[\eta^{\mu_{1}\mu_{2}}\left({\xi_{1}}-{\xi_{2}}\right)^{\nu}+\eta^{\mu_{2}\nu}\left({\xi_{2}}-{\xi_{12}}\right)^{\mu_{1}}+\eta^{\nu\mu_{1}}\left({\xi_{12}}-{\xi_{1}}\right)^{\mu_{2}}\right]$
$\displaystyle\hskip
113.81102pt\times\left[\eta_{\nu}{}^{\mu_{4}}\left({\xi_{12}}+{\xi_{4}}\right)^{\mu_{3}}+\eta^{\mu_{4}\mu_{3}}\left(-{\xi_{4}}+{\xi_{3}}\right)_{\nu}+\eta^{\mu_{3}}{}_{\nu}\left(-{\xi_{3}}-{\xi_{12}}\right)^{\mu_{4}}\right]\frac{1}{\xi_{12}^{2}}$
$\displaystyle\hskip
19.91684pt-(\eta\,\wedge\,\eta)^{\mu_{1}\mu_{4}\mu_{2}\mu_{3}}$
$\displaystyle\hskip
19.91684pt+\left[\eta^{\mu_{1}}{}_{\nu}\left({\xi_{1}}+{\xi_{14}}\right)^{\mu_{4}}+\eta_{\nu}{}^{\mu_{4}}\left(-{\xi_{14}}-{\xi_{4}}\right)^{\mu_{1}}+\eta^{\mu_{4}\mu_{1}}\left({\xi_{4}}-{\xi_{1}}\right)_{\nu}\right]$
$\displaystyle\hskip
113.81102pt\times\left[\eta^{\nu\mu_{2}}\left({\xi_{14}}-{\xi_{2}}\right)^{\mu_{3}}+\eta^{\mu_{2}\mu_{3}}\left({\xi_{2}}-{\xi_{3}}\right)^{\nu}+\eta^{\mu_{3}\nu}\left({\xi_{3}}-{\xi_{14}}\right)^{\mu_{2}}\right]\frac{1}{\xi_{14}^{2}}$
$\displaystyle\hskip
19.91684pt+(\,\eta\wedge\eta\,)^{\mu_{1}\mu_{3}\mu_{2}\mu_{4}}$ (5.19)

The function $\langle\,{\cal P}_{1},\Omega_{1}\,\rangle$ can now be restricted
to ${\cal K}_{4}$ and equals the color ordered Four point tree-level gluon
amplitude, which we denote as ${\cal M}_{4}^{\mu_{1}\dots\mu_{4}}$. The
uncontracted Lorentz indices are simply substitutes for polarisation vectors
of the external gluons.

One can compute $n$ point tree-level gluon amplitude in a similar fashion.
Rather than going over the entire computation for arbitrary $n$, we now argue
as to why the result of the contraction between ${\cal P}_{n-3},\Omega_{n-3}$
will always generate tree-level gluon amplitude.

### Computing the scalar $\langle\,{\cal P}_{n-3},\Omega_{n-3}\,\rangle$

Let us start with the fixed triangulation of an $n$-gon

$\displaystyle T\,$
$\displaystyle=\,\\{\,(1,3),\,(1,4),\,\dots,\,(1,n-1)\,\\}$ (5.20)

which is dual to the graph $\gamma_{T}$ as shown in (5.21).

$\displaystyle\gamma_{T}=\ \ \leavevmode\hbox to131.67pt{\vbox
to43.91pt{\pgfpicture\makeatletter\hbox{\hskip 33.5745pt\lower-7.83301pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{}
{}{}{}\pgfsys@moveto{-21.33957pt}{0.0pt}\pgfsys@lineto{32.00935pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{21.33957pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{21.33957pt}{0.0pt}\pgfsys@lineto{21.33957pt}{21.33957pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}{}\pgfsys@moveto{53.34892pt}{0.0pt}\pgfsys@lineto{85.35828pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{64.0187pt}{0.0pt}\pgfsys@lineto{64.0187pt}{21.33957pt}\pgfsys@stroke\pgfsys@invoke{
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