IPPP/14/75
DCPT/14/150
CALT-TH-2014-152
Anatomy of the Amplituhedron
Sebasti ́an Franco,
1
Daniele Galloni,
1
Alberto Mariotti,
1
Jaroslav Trnka
2
1
Institute for Particle Physics Phenomenology, Department of Physics
Durham University, Durham DH1 3LE, United Kingdom
2
Walter Burke Institute for Theoretical Physics
California Institute of Technology, Pasadena, CA 91125, USA
E-mail:
sebastian.franco@durham.ac.uk,daniele.galloni@durham.ac.uk,
alberto.mariotti@durham.ac.uk,trnka@caltech.edu
Abstract:
We initiate a comprehensive investigation of the geometry of the ampli-
tuhedron, a recently found geometric object whose volume calculates the integrand of
scattering amplitudes in planar
N
= 4 SYM theory. We do so by introducing and
studying its stratification, focusing on four-point amplitudes. The new stratification
exhibits interesting combinatorial properties and positivity is neatly captured by per-
mutations. As explicit examples, we find all boundaries for the two and three loop
amplitudes and related geometries. We recover the stratifications of some of these ge-
ometries from the singularities of the corresponding integrands, providing a non-trivial
test of the amplituhedron/scattering amplitude correspondence. We finally introduce
a deformation of the stratification with remarkably simple topological properties.
arXiv:1408.3410v1 [hep-th] 14 Aug 2014
Contents
1 Introduction
1
2 The Amplituhedron
3
2.1 Tree-Level Amplituhedron
3
2.2 Loop Geometry
4
2.3 The Full Amplituhedron
5
2.4 The Scattering Amplitude
5
3 Stratification of the Amplituhedron: Loop Geometry
6
3.1 The Degrees of Freedom of
C
7
3.2 Extended Positivity and Boundaries
7
3.3 Mini Stratification
9
3.4 Full Stratification
10
3.5 Summary of the Method and Structure of the Stratification
12
4 Simple Examples: Basic Properties
15
4.1 Stratification of
G
+
(0
,n
; 1) =
G
+
(2
,n
)
15
4.2 Non-Minimal Minors
17
5 Combinatorial Stratification
19
5.1 Perfect Matchings and the Stratification of
G
+
(
k,n
)
19
5.2 Multi-Loop Geometry and Hyper Perfect Matchings
24
6 The Combinatorics of Extended Positivity
25
6.1 Further Thoughts on Extended positivity
25
6.2 Hyper Perfect Matchings: Good, Bad and Neutral
26
6.3 Extended Positivity and the Return of Permutations
28
7 Two Loops
30
7.1 Mini Stratification
31
7.1.1
The Amplituhedron
31
7.1.2
The Log of the Amplitude
33
7.1.3
Gluing the Amplitude to its Log
34
7.2 Full Stratification
36
– i –
8 Three loops
42
8.1 Mini Stratification
42
9 An Alternative Path to Stratification: Integrand Poles
47
9.1 The Amplitude
47
9.2 The Log of the Amplitude
49
10 The Deformed
G
+
(0
,n
;
L
)
50
10.1 Examples
53
10.1.1 1-loop
53
10.1.2 2-loops
53
10.1.3 3-loops
54
10.1.4 4-loops
54
11 Conclusions and Outlook
56
A Two-Loop Boundaries Before Extended Positivity
60
B Geometric Versus Integrand Stratification: Explicit Examples
61
1 Introduction
Formidable progress in our understanding of scattering amplitudes in gauge theory
has been achieved in the last two decades (see e.g.[1–7] and reviews [8–11]). The
progress is especially impressive for amplitudes in planar
N
= 4 super Yang-Mills the-
ory where explicit results have been obtained up to high loop order [12–19], and many
interesting connections and dualities have been found including twistor strings [20],
the amplitude/Wilson loop correspondence [21–23] and many others. Amazingly, this
theory enjoys an infinite-dimensional Yangian symmetry [24], which results from the
combination of superconformal and dual superconformal invariance [25, 26] making an
interesting connection to the integrability of the theory [27, 28]. This infinite symmetry
is obscured in the standard Feynman diagram approach while it is completely manifest
in the dual formulation of amplitudes in this theory using the positive Grassmannian
[29] (see also [15, 30–33] and recent work on a deformed version of the story [34–38])
and the
amplituhedron
[39, 40]. This is a new algebraic geometric object which gen-
eralizes the positive Grassmannian and encodes scattering amplitudes in a maximally
geometric way: they are simply given by its volume. The amplituhedron is the missing
– 1 –
link explaining how to combine Yangian invariant building blocks to give rise to the
amplitude. Different representations of the same amplitude are beautifully translated
into different triangulations of the amplituhedron. In this approach the standard pil-
lars of quantum field theory like locality or unitarity are derived properties from the
geometry of the amplituhedron. The existence of such a structure in planar
N
= 4
SYM suggests that there might be a very different formulation of the field theory which
does not use the standard Lagrangian description of physics.
The correspondence between scattering amplitudes and the amplituhedron has
passed numerous tests [39, 40], although it still remains conjectural and its study is at
its infancy. In this article, we introduce new tools analyzing the amplituhedron and
initiate the most comprehensive investigation of its geometry to date. A clear goal is to
achieve a systematic understanding similar to the one available for cells in the positive
Grassmannian [41]. Among other things, we expect our ideas to be instrumental for
triangulating the amplituhedron, and hence contribute to its practical use in construct-
ing scattering amplitudes. A beautiful interplay between experimental exploration of
examples, discovery of new structures and theoretical new ideas has been a constant
driving force for progress in the understanding of scattering amplitudes. It is reason-
able to expect that the examples we study in this paper, and the ones which will be
studied in the future with the help of the tools we introduce, will nicely fit into this
trend.
This paper is organized as follows.
§
2 provides a quick review of the basics of the
amplituhedron. In
§
3, we introduce a stratification for it, which captures all detailed
structures of the corresponding differential form and allows us to explore its geometry
in depth. We also introduce a reduced version of the stratification, which we call mini
stratification, which captures broader features of the geometry and is amenable to
a combinatorial implementation.
§
4 contains a first encounter with the stratification
through simple examples.
§
5 introduces a powerful combinatorial implementation of the
mini stratification in terms of graphs and a new class of objects we denote hyper perfect
matchings. The combinatorics of extended positivity is the subject of
§
6. Interestingly,
we find that positivity can be neatly discussed in terms of permutations.
§
7 puts our
techniques at work and investigates various geometries at 2 and 3-loops. In
§
9 we study
an alternative approach to stratification, based on the singularities of the integrand.
For four particles, we find exact agreement with the geometric stratification of the
amplitude and its log, providing new and significant evidence for the amplituhedron
conjecture. In
§
10 we introduce and investigate the deformed amplituhedron, which
seems to exhibit an outstandingly simple geometry. We conclude and present a vision
for future work in
§
11. We also include two appendices with supporting material.
– 2 –
2 The Amplituhedron
In this section we provide a brief introduction to the amplituhedron. We refer the
reader to [39, 40] for further details.
2.1 Tree-Level Amplituhedron
The amplituhedron is a generalization of the positive Grassmannian conjectured to give
all scattering amplitudes in planar
N
= 4 SYM theory when integrated over with an
appropriate volume form. The amplituhedron can be regarded as a generalization of
the interior of a set of
n
vertices
Z
I
of dimension (
k
+ 4), where (
k
+ 2) is the number
of negative-helicity gluons,
I
= 1
,
2
,...,k
+ 4, and
n
is the total number of external
gluons. In this notation,
k
= 0 corresponds to MHV amplitudes. These vertices can be
combined into matrix
Z
I
a
, where
a
= 1
,
2
,...,n
. In order to have a notion of interior
we need vertices to be ordered in a specific way. In the familiar 2-dimensional case
of polygons, vertices must be cyclically ordered to avoid the crossing of external edges
connecting consecutive vertices. The generalization of this cyclicity constraint takes
the form of a positivity condition on the matrix
Z
I
a
: all maximal minors of
Z
I
a
must be
positive, i.e.
Z
I
a
∈
M
+
(4 +
k,n
) where
M
+
(4 +
k,n
) is the space of positive (4 +
k
)
×
n
matrices.
External vertices form a polytope. For
k
= 1 we consider a point in the interior of
this polytope, which corresponds to a linear combination of the external vertices, where
the coefficients must be positive. Each of these points will be considered projectively,
and can thus be seen as 1-planes (or lines) in
k
+ 4 dimensions. For general
k
, we
consider a
k
-plane and impose positivity conditions on the matrix of coefficients of its
expansion in terms of external points. Explicitly, a
k
-plane
Y
in the interior of the
tree-level amplituhedron is given by
Y
=
C
·
Z ,
(2.1)
where
Z
is the (
k
+ 4)
×
n
matrix of external vertices,
C
is a
k
×
n
matrix in
G
+
(
k,n
),
and
Y
is the tree-level amplituhedron interior, given by a
k
×
(
k
+ 4) matrix.
1
We are
not imposing positivity on each of the
k
rows of the matrix
C
, but a condition on how
the rows of
C
interact with each other such that minors are positive. As a result, the
1
A warning to the reader: whenever we refer to the positive Grassmannian
G
+
(
k, n
), we mean the
totally non-negative Grassmannian. The boundaries of this space arise when the positive degrees of
freedom become zero. Similarly, we will use positive as a synonym of non-negative and emphasize
when a given quantity is not zero. This slight abuse of terminology will persist throughout; we hope
it will not cause any confusion.
– 3 –
amplituhedron is
not
simply given by
k
copies of “the interior of the vertices”, but it
is a more complicated geometric object. We can also think of the amplituhedron as a
map:
G
+
(
k,n
)
Z
−→
G
(
k,k
+ 4)
.
(2.2)
The
GL
(
k
) degree of freedom of the Grassmannian, which acts on
C
, must also apply
to
Y
, thus implying the matrix
Y
∈
G
(
k,k
+ 4).
2.2 Loop Geometry
Each point of the tree-level amplituhedron spans a
k
-plane in (
k
+ 4) dimensions;
the full amplituhedron spans all possible
k
-planes in (
k
+ 4) dimensions. For each
point, the transverse space is 4-dimensional and this is where the loop-level part of
the amplituhedron lives. The degrees of freedom of each loop span a 2-plane in this
transverse space. Let us start our discussion with the
k
= 0 case, which at tree-level
is given by the empty projective space
P
3
, since
Y
is 0-dimensional. At loop level, it
corresponds to what we call the pure
loop geometry
. In this case, every loop
L
(
i
)
is a
different linear combination of the external vertices, which lies in
P
3
:
L
(
i
)
=
D
(
i
)
·
Z ,
(2.3)
where the
Z
’s are 4-dimensional vectors,
D
(
i
)
∈
G
+
(2
,n
) maps the vertices in
Z
to the
transverse space, and so
L
(
i
)
∈
G
(2
,
4). Multiple loops are implemented by increasing
the number of matrices
D
(
i
)
:
L
(1)
L
(2)
.
.
.
L
(
L
)
=
D
(1)
D
(2)
.
.
.
D
(
L
)
·
Z .
(2.4)
The matrices
D
(
i
)
satisfy
extended positivity
conditions, i.e. for any subset of them
we define
D
(
ij
)
=
(
D
(
i
)
D
(
j
)
)
, D
(
ijk
)
=
D
(
i
)
D
(
j
)
D
(
k
)
,
etc.
(2.5)
and demand all maximal minors of each of these extended matrices to be positive,
namely
D
(
ij
)
∈
M
+
(4
,n
),
D
(
ijk
)
∈
M
+
(6
,n
), etc. In general,
D
(
a
1
...a
m
)
∈
M
+
(2
m,n
).
These conditions apply only for
m
≤
n/
2. In the special case of
n
= 4 and arbitrary
L
, the only surviving conditions are mutual positivities:
D
(
ij
)
∈
M
+
(4
,n
) for all pairs
of
i
and
j
.
– 4 –
2.3 The Full Amplituhedron
To obtain the full amplituhedron for any
n,k,L
, we combine the tree-level space and
the loop space into a larger matrix
L
(1)
L
(2)
.
.
.
L
(
L
)
Y
=
D
(1)
D
(2)
.
.
.
D
(
L
)
C
·
Z
(2.6)
or more neatly
Y
=
C·
Z ,
(2.7)
where
C
is the (
k
+ 2
L
)
×
n
matrix specifying the set of (
k
+ 2
L
) different linear
combinations of external vertices, and
Y
is the full amplituhedron interior. Here the
positivity condition for
C
is not the same as the one for
C
:
C 6∈
G
+
(
k
+ 2
L,n
) (in fact,
k
+ 2
L
maybe be much larger than
n
). As for the pure loop geometry, the positivity
condition is now an extended positivity. The requirements are that the combination
of
C
with any subset of the
D
(
i
)
matrices is positive, i.e. all their maximal minors are
positive, as long as the matrix has at least as many columns as rows, i.e. that
(
C
)
,
(
D
(1)
C
)
,
···
,
(
D
(
L
)
C
)
,
D
(1)
D
(2)
C
,
···
(2.8)
are all positive, where we stop stacking
D
(
i
)
’s onto
C
when the resulting matrix has
more rows than columns.
2
Note that there is no condition that only relates the various
D
(
i
)
’s to each other, except in the absence of
C
, i.e. for
k
= 0. This novel space
inhabited by
C
, characterized by the extended positivity, is denoted
G
+
(
k,n
;
L
).
2.4 The Scattering Amplitude
The scattering amplitude is obtained by integrating over all of the degrees of freedom
of the amplituhedron, with a specific form constrained to have
logarithmic singularities
on the boundaries of the space. This form is the amplitude integrand, and can in
principle be constructed using methods such as Feynman diagrams, unitary cuts or
BCFW recursion relations. For arbitrary numbers of particles and loops such methods
become very laborious, and it would be desirable to construct the integrand directly
2
It is possible to stack more matrices but the maximal minors would be insensitive to this.
– 5 –
from the definition of the amplituhedron. There are several strategies for doing this:
the first one is to try to triangulate the amplituhedron in terms of smaller elementary
spaces which have trivial dlog forms. Recursion relations via on-shell diagrams provide
examples of such triangulations, where the rules for triangulating are dictated by the
physics rather than the amplituhedron geometry.
3
Another strategy is to nail down the
integrand directly, by requiring that all spurious singularities (which do not correspond
to amplituhedron boundaries) cancel. In either approach, an understanding of the
boundary structure of the space will be crucial for systematically constructing the
integrand form.
3 Stratification of the Amplituhedron: Loop Geometry
In this section we develop tools for
stratifying
the amplituhedron, by which we mean
finding its boundary structure.
In this paper, we focus our attention on the
k
= 0 case, i.e. on the pure
loop
geometry
, and also restrict to
n
= 4. For
k
= 0, the matrix
C
disappears, and we are
only left with the
D
(
i
)
matrices:
C
=
D
(1)
D
(1)
.
.
.
D
(
L
)
.
(3.1)
The structure at loop level is rather non-trivial due to the extended positivity condition
imposed on matrices. Note that
C
is
not
an element of the positive Grassmannian,
except for
L
= 1.
For
n
=
k
+4, the positivity of external data, encoded in the matrix
Z
, is trivial and
the stratification of the amplituhedron corresponds to the stratification of
C
.
4
Even in
this simplified situation, the geometry of the amplituhedron will exhibit extraordinary
richness. For general
n
, the process we will discuss can be regarded as the stratification
of
G
+
(0
,n
;
L
) rather than the stratification of the amplituhedron. Independently of
its relation to the amplituhedron, the stratification of
G
+
(0
,n
;
L
) is an interesting
geometric question in its own right.
3
See [42] for alternative diagrammatic tools for addressing this problem and [43] for interesting new
ideas on the computation of volumes of polytopes associated to scattering amplitudes.
4
This follows directly from the fact that when
Z
is a square matrix we may choose a basis for which
Z
equals the unit matrix. Then from (2.7) we see that
Y
=
C·
Z
=
C
.
– 6 –
3.1 The Degrees of Freedom of
C
Each
D
(
i
)
∈
G
+
(2
,n
) has 2(
n
−
2) degrees of freedom, best parametrized by its 2
×
2
minors, known as Pl ̈ucker coordinates. There are
(
n
2
)
different Pl ̈ucker coordinates ∆
(
i
)
I
,
with
I
=
{
a,b
}
specifying which two columns
a
and
b
are involved in the minor. The
∆
(
i
)
I
’s are not all independent but are subject to relations, known as Pl ̈ucker relations.
C
gets a contribution from each
D
(
i
)
, giving a total of 2
L
(
n
−
2) degrees of freedom.
Note that extended positivity, despite imposing a condition on the degrees of free-
dom of different
D
(
i
)
, does not decrease the dimension, for the simple reason that it is
just an inequality and cannot determine any Pl ̈ucker coordinate in terms of the oth-
ers. This is akin to the fact that the restriction to the positive Grassmannian, i.e.
that ∆
(
i
)
I
>
0, does not create new relations between the coordinates ∆
(
i
)
I
, but simply
constrains them to be positive.
However, extended positivity can restrict the allowed domain of the ∆
(
i
)
I
further
than the simple ∆
(
i
)
I
>
0 condition. This additional restriction can in certain cases be
quite non-trivial, and may even split the domain into disjoint
regions
. Later in this
section, we will introduce a
mini stratification
of
C
which is insensitive to this subtlety,
and a
full stratification
which refines the mini stratification and fully accounts for it.
The full stratification in effect counts all domain regions of the amplituhedron.
Regardless of which stratification we are interested in, for the purposes of counting
dimensions we only count the number of independent equalities between various ∆
(
i
)
I
’s.
For example, when
C
is top-dimensional the only relations come from the Pl ̈ucker
relations which are independently present in each
D
(
i
)
, e.g. for
i
= 1 there is a Pl ̈ucker
relation between various ∆
(1)
I
’s, for
i
= 2 there is a separate Pl ̈ucker relation between
the ∆
(2)
I
’s, but we cannot write any ∆
(1)
I
in terms of ∆
(2)
J
’s.
3.2 Extended Positivity and Boundaries
For
k
= 0, extended positivity enforces the condition that all
D
(
i
)
are positive, as well
as all subsets of them when stacked onto each other (as long as the number of rows
does not exceed the number of columns; these larger matrices produce no additional
conditions), i.e. that
(
D
(
i
)
)
,
(
D
(
i
)
D
(
j
)
)
,
···
(3.2)
are all positive. This translates into various conditions on the Pl ̈ucker coordinates. To
unify the conditions it is convenient to define 2
m
×
2
m
minors ∆
(
i
1
,...,i
m
)
I
,
m
= 1
,...,L
,
– 7 –
which are all the maximal minors when stacking the matrices
D
i
1
,...D
i
m
.
5
First,
all ∆
(
i
)
I
must be positive. Extended positivity also requires the ∆
(
i
1
,...,
ı
m
)
I
’s, which are
polynomials of order
m
in the ∆
(
i
)
I
’s, to be positive. In order to emphasize the contrast
with Pl ̈ucker coordinates ∆
(
i
)
I
, we will often refer to the
m >
1 minors as
non-minimal
minors
.
For a given number of loops
L
, there are
(
L
m
)
ways of choosing
m
matrices
D
(
i
)
to
form a ∆
(
i
1
,...,i
m
)
J
. For each of these choices, there are
(
n
2
m
)
ways of choosing the set
J
of
2
m
columns out of all the
n
external nodes. Hence, the number of non-minimal minors
becomes
m
≤
n/
2
∑
m
=2
(
L
m
)(
n
2
m
)
.
(3.3)
These larger minors are not all independent, there are Pl ̈ucker-like relations among
them.
Boundaries of
C
are reached by killing degrees of freedom in it by setting minors
to zero. In other words, ∆
(
i
1
,...,i
m
)
I
≥
0 has its boundary when ∆
(
i
1
,...,i
m
)
I
= 0. The more
complicated inequalities arising from minors with
m >
1 give rise to relations between
∆
(
i
)
I
’s. Each independent relation of this form reduces the degrees of freedom by 1.
A more precision characterization of boundaries is given below, when we discuss the
stratification.
Labels.
To every boundary we can associate the corresponding list of vanishing
∆
(
i
1
,...,i
m
)
I
. In each list, all ∆
(
i
1
,...,i
m
)
I
, i.e. for both
m
= 1 and
m >
1, are treated
democratically
. We will refer to such lists of minors as
labels
. The minors which are
not in the label are not vanishing. Labels are very useful for characterizing boundaries
and other configurations of minors, although they do not fully specify them.
These labels will form the basis of the mini stratification described in
§
3.3, which
will only distinguish elements in the stratification by them. However, motivated by the
physical problem of using the amplituhedron to identify all possible singularities of the
integrand, we will refine this counting in
§
3.4 by noticing that there are several indepen-
dent domain regions for each label, or equivalently by identifying
independent
solutions
consistent with a given label.
6
It is thus important to emphasize that, generically,
labels
do not fully specify boundaries
.
5
This notation includes the 2
×
2 Pl ̈ucker coordinates. In order to maintain an economic notation,
we use a single subindex
I
to indicate the set of columns in the larger minors.
6
As it will become clearer in
§
3.4, and exemplified in
§
7.2, the definition automatically accounts for
the information about the sequence or path in which minors are turned off to reach a given boundary.
– 8 –
However, labels are still subject to interesting restrictions, since not every arbitrary
set of minors can be set to zero. There are two sources of hindrance:
•
Pl ̈ucker relations relate different ∆
(
i
)
I
’s and hence it is sometimes impossible to
kill a given Pl ̈ucker coordinate without some other coordinate also becoming zero.
The same is in fact true for all ∆
(
i
1
,...,i
m
)
I
’s: they are not all independent, since
there are Pl ̈ucker-like relations between them. As a result, it is not possible to
exclusively
set any arbitrary combination of ∆
(
i
1
,...,i
m
)
I
’s to zero.
•
Relations belonging to different levels of minors may be incompatible, i.e. the full
extended positivity can become impossible to satisfy, despite only being given
in terms of inequalities. This is because the relations arising from non-minimal
minors typically contain positive and negative terms, and the sum must be non-
negative. When all the Pl ̈ucker coordinates are turned on, extended positivity is
easily satisfied. On the contrary if, for example, we kill a subset such that only
the negative terms survive, we can no longer satisfy positivity. Similarly, setting
a ∆
(
i
1
,...,i
m
)
I
to zero becomes impossible if only positive terms in it are turned on.
We shall later see explicit examples of both of these occurrences.
From the above discussion we conclude that while Pl ̈ucker relations and their gen-
eralizations for
m >
1 may invalidate boundaries in an automatic way, extended pos-
itivity does so more aggressively: it imposes by hand an ulterior check to determine
whether a given boundary exists or not. This is analogous to what happens when im-
posing positivity on the Grassmannian:
G
(
k,n
)
→
G
+
(
k,n
) kills “by hand” a subset
of boundaries. In our case, we go from
G
(
k,n
;
L
)
→
G
+
(
k,n
;
L
). For the tree-level
case
G
+
(
k,n
; 0)
≡
G
+
(
k,n
), it is a beautiful result that certain potential boundaries
7
are removed in such a way so as to generate an Eulerian poset [44].
3.3 Mini Stratification
As mentioned above, the full stratification of the amplituhedron counts all independent
solutions for a given positivity-preserving label. At this point in our discussion, it is
natural to define an unrefined counting, which we call
mini stratification
, and serves as a
close proxy of the full stratification introduced in next section. The mini stratification
corresponds to only considering the labels of the boundaries. This counting can be
used to generate a “poor man’s” label stratification, in which multiple solutions for a
given label are collapsed into a single point, which is assigned the highest dimension
7
By this we mean configurations in which some minors vanish.
– 9 –
of all these solutions. In other words, the mini stratification combines boundaries into
equivalence classes determined by the labels. For brevity, we will simply refer to these
equivalence classes as the boundaries of the mini stratification.
While the mini stratification does not capture the full singularity structure of the
amplitude, it is valuable for various reasons. First, it provides a rather complete geo-
metric characterization of the amplituhedron. More importantly, as we discuss in
§
5
and
§
6, its value follows from the fact that it admits a very efficient combinatorial
implementation. We will present examples of the mini stratification in
§
7 and
§
8.
3.4 Full Stratification
As already discussed above, labels only include information on which minors are van-
ishing and which are non-vanishing. Their level of refinement is identical to that of the
matroid strata for
G
+
(
k,n
). It is often possible, however, that there are disjoint regions
of domain for the minimal minors ∆
(
i
)
I
which satisfy the equalities of a given label, i.e.
that there are multiple solutions to the set of equalities described by the label.
We are thus naturally led to the definition of a
region
, which is a set of equalities and
inequalities for the ∆
(
i
1
,...,i
m
)
I
,
m
= 1
,...,L
, which has a unique solution. In general, the
equalities and inequalities needed to describe a region are more than those specifying
a label: given the label, we must also specify which of the solutions the region refers
to. In the future, when we refer to a boundary of
G
+
(
k,n
;
L
) we will mean a region as
defined here. The
full stratification
is defined as the stratification which distinguishes
all such regions. This suggests a natural extension of the labels introduced in the last
section, to which we refer as
extended labels
. Extended labels correspond to specifying
not only the vanishing ∆
(
i
1
,...,i
m
)
I
’s but also all other relations between minors. Such an
extended label then fully specifies a given boundary. While the mini stratification is
based on labels, the full stratification uses extended labels.
For concreteness, let us focus on
n
= 4, for which all non-minimal minors are 4
×
4.
Consider one such minor which, without loss of generality, we can assume to be ∆
(1
,
2)
1234
.
8
When all ∆
(
i
)
I
are turned on, ∆
(1
,
2)
1234
can be expressed in terms of Pl ̈ucker coordinates as
follows:
∆
(1
,
2)
1234
= ∆
(1)
12
∆
(2)
34
+ ∆
(1)
23
∆
(2)
14
+ ∆
(1)
34
∆
(2)
12
+ ∆
(1)
14
∆
(2)
23
−
∆
(1)
13
∆
(2)
24
−
∆
(1)
24
∆
(2)
13
.
(3.4)
8
The simplest situation in which such a minor arises is for 2-loops, i.e.
G
+
(0
,
4; 2). In this case,
this is the only non-minimal minor.
– 10 –
After using the Pl ̈ucker relations ∆
(
i
)
12
∆
(
i
)
34
+ ∆
(
i
)
23
∆
(
i
)
14
= ∆
(
i
)
13
∆
(
i
)
24
for
i
= 1
,
2, this can be
turned into the convenient form
∆
(1
,
2)
1234
=
(
∆
(1)
12
∆
(2)
13
−
∆
(1)
13
∆
(2)
12
)(
∆
(1)
13
∆
(2)
34
−
∆
(1)
34
∆
(2)
13
)
∆
(1)
13
∆
(2)
13
+
(
∆
(1)
23
∆
(2)
13
−
∆
(1)
13
∆
(2)
23
)(
∆
(1)
13
∆
(2)
14
−
∆
(1)
14
∆
(2)
13
)
∆
(1)
13
∆
(2)
13
.
(3.5)
If we now turn off ∆
(1)
23
= ∆
(1)
14
= 0, we obtain
∆
(1
,
2)
1234
=
(
∆
(1)
12
∆
(2)
13
−
∆
(1)
13
∆
(2)
12
)(
∆
(1)
13
∆
(2)
34
−
∆
(1)
34
∆
(2)
13
)
∆
(1)
13
∆
(2)
13
−
∆
(1)
13
∆
(2)
23
∆
(2)
14
∆
(2)
13
(3.6)
The mini stratification label for this is simply
{
∆
(1)
14
,
∆
(1)
23
}
, which is the full set of
vanishing minors. All other ∆
(
i
)
I
’s are strictly positive. However, we notice that there
are two regions in which we may satisfy ∆
(1
,
2)
1234
>
0:
•
Region 1:
(
∆
(1)
12
∆
(2)
13
−
∆
(1)
13
∆
(2)
12
)
>
0 and
(
∆
(1)
13
∆
(2)
34
−
∆
(1)
34
∆
(2)
13
)
>
0
•
Region 2:
(
∆
(1)
12
∆
(2)
13
−
∆
(1)
13
∆
(2)
12
)
<
0 and
(
∆
(1)
13
∆
(2)
34
−
∆
(1)
34
∆
(2)
13
)
<
0
These two regions are very easy to understand: denoting
x
≡
(
∆
(1)
12
∆
(2)
13
−
∆
(1)
13
∆
(2)
12
)
,
y
≡
(
∆
(1)
13
∆
(2)
34
−
∆
(1)
34
∆
(2)
13
)
and
k
≡
∆
(1)
13
∆
(2)
23
∆
(2)
14
∆
(2)
13
, we have the simple condition that
∆
(1
,
2)
1234
≥
0
⇔
xy
≥
k
(
k >
0)
(3.7)
which on the
x
−
y
plane simply corresponds to two regions whose boundary is the
hyperbolic curve
xy
=
k
. Here we see that to specify the regions within this label,
all we need to do is additionally specify the sign of
x
and
y
. The relations specifying
regions 1 and 2 are explicit examples of the type of relations included in extended
labels.
In this example, if we go to a different label where we have also shut off ∆
(1
,
2)
1234
, i.e.
{
∆
(1)
14
,
∆
(1)
23
,
∆
(1
,
2)
1234
}
, we again have two regions:
xy
=
k
with
x,y >
0, and
xy
=
k
with
x,y <
0.
The full stratification contains all possible poles of the integrand. In fact, it is even
more refined than the integrand: while there are several different integrand poles that
correspond to the same label in the mini stratification, here it sometimes happens that
– 11 –
there are several regions contained within the same integrand pole. The example above
is an instance where this happens: as will be clear in subsequent sections, the pole of
the integrand when we set ∆
(1)
23
= ∆
(1)
14
= 0 is
〈
AB
34
〉〈
CD
12
〉
+
〈
AB
12
〉〈
CD
34
〉
〈
ABCD
〉〈
AB
12
〉〈
AB
34
〉〈
CD
12
〉〈
CD
14
〉〈
CD
23
〉〈
CD
34
〉
.
(3.8)
We have just shown that this object is composed of two disjoint regions. Provided the
amplituhedron proposal holds, identifying those regions in the full stratification which
correspond to the same integrand pole exactly reproduces the pole structure of the
integrand.
3.5 Summary of the Method and Structure of the Stratification
In this section we summarize the general procedure for stratifying
C ∈
G
+
(0
,n
;
L
). As
stated earlier, in this article we will almost exclusively focus on the case of
k
= 0,
n
= 4
and arbitrary
L
. This case is particularly simple owing the fact that for
n
= 4 the
Z
I
matrix can be chosen to be diagonal, and hence trivial, thus positivity of external data
becomes unimportant and the stratification of
G
+
(0
,
4;
L
) actually coincides with the
one for the loop amplituhedron.
9
As previously mentioned, every boundary of
G
+
(0
,n
;
L
) has an associated label,
i.e. a list of vanishing minors. For any given label, there is one boundary (or region)
for each independent solution giving rise to it, in general specified by some additional
inequalities.
All minors should be treated democratically. When implementing the stratifica-
tion, however, it is natural to give the Pl ̈ucker coordinates ∆
(
i
)
I
a special treatment.
The reasons for this choice include the facts that every minor ∆
(
i
1
,...,i
m
)
I
is an order
m
polynomial in ∆
(
i
)
I
’s and, as we will discuss in
§
5, the ∆
(
i
)
I
’s are related to certain collec-
tions of edges, denoted perfect matchings, of simply connected graphs. Moreover, the
Pl ̈ucker coordinates for each
D
(
i
)
scale with a common factor under the
GL
(2) acting
on
D
(
i
)
. The dimension of each boundary is given by the number of degrees of freedom
in the ∆
(
i
)
I
’s:
d
=
N
∆
I
−
N
rel
−
L,
(3.9)
where
N
∆
I
is the number of non-vanishing ∆
(
i
)
I
on the boundary and
N
rel
is the number
of independent equations relating the ∆
(
i
)
I
.
10
These equations may be Pl ̈ucker relations
9
The case of
k >
0 is further complicated by the fact that the minors of the
D
(
i
)
matrices do not
have a definite sign, and tuning these to zero does not constitute a boundary of the amplituhedron.
Boundaries are only obtained by shutting off degrees of freedom that have a definite sign.
10
The subtraction of
L
degrees of freedom follows from the fact that Pl ̈ucker coordinates are pro-
jectively defined.
– 12 –
or follow from non-minimal minors that have been independently set to zero on a
given boundary. In the mini stratification, each label is assigned the dimension of the
top-dimensional region associated to it.
In this way we split the positivity constraint on the matrix
C
in two:
•
∆
(
i
)
I
≥
0.
•
Larger minors ∆
(
i
1
,...,i
m
)
I
, expressed as sums of products of ∆
(
i
)
I
, also satisfy
∆
(
i
1
,...,i
m
)
I
≥
0.
The aforementioned distinction between Pl ̈ucker coordinates and non-minimal mi-
nors reflects into a natural separation of the stratification of
G
+
(0
,n
;
L
) into two stages.
First, we obtain all possible sets of vanishing Pl ̈ucker coordinates ∆
(
i
)
I
, subject to ex-
tended positivity conditions. At this step larger minors are not set to zero, unless they
trivially vanish as a result of the vanishing Pl ̈ucker coordinates. If we are considering
the full stratification, some of these configurations can be further divided in different
regions, specified by inequalities among the non-vanishing Pl ̈ucker coordinates. Next,
we introduce for each of these elements a further structure corresponding to the van-
ishing of non-minimal minors. This second stage reduces the dimension of boundaries
by imposing constraints on the non-vanishing ∆
(
i
)
I
’s. Depending on whether we are
interested in the mini or the full stratification, it is implemented slightly differently.
The first stage in the stratification thus corresponds to the following two steps:
1. Classify potential boundaries according only to the vanishing Pl ̈ucker coordinates.
This corresponds to independently performing the positroid stratification of each
D
(
i
)
, i.e. of each
G
+
(2
,n
).
2. Some of these collections violate the extended positivity of the larger minors
∆
(
i
1
,...,i
m
)
I
≥
0 and are thus removed. The surviving collections of ∆
(
i
)
I
represent
all the labels of
G
+
(0
,n
;
L
) for which non-minimal minors can be non-negative.
Step 1 produces the
L
th
power of the positroid stratification of
G
+
(2
,n
) and is inde-
pendent of what type of stratification we are considering. We will denote the numbers
of potential boundaries with dimension
d
obtained at this first step as
N
(
d
)
, where
d
is
determined using (3.9). Step 2 represents a further refinement of this decomposition,
removing some of the potential boundaries obtained at step 1 by demanding extended
positivity. We refer to the number of remaining boundaries as
N
(
d
)
. These boundaries
can be organized in a poset that we denote Γ
0
, where at the top element corresponds to
all minors non-vanishing. Every element in Γ
0
is associated to a set of vanishing ∆
(
i
)
I
’s.
In the case of the full stratification, this information might not uniquely fix the element
– 13 –
of Γ
0
, due to the multiplicity of regions. A combinatorial approach for constructing Γ
0
in the mini stratification will be introduced in
§
5.
Independently of whether we are constructing the mini or the full stratification, for
each element in Γ
0
there are, generally, multiple boundaries, which arise from setting
to zero non-minimal minors which are not automatically vanishing due to vanishing
Pl ̈ucker coordinates. The procedure for systematically constructing these boundaries
is:
3. For each element of Γ
0
and its collections of surviving ∆
(
i
)
I
, we first classify non-
minimal minors ∆
(
i
1
,...,i
m
)
I
≥
0,
m >
1, into three categories:
(i) Those that are trivially zero given the list of vanishing ∆
(
i
)
I
.
(ii) Those that are manifestly positive, because only positive terms are turned
on by the given collection of non-zero ∆
(
i
)
I
.
(iii) Those that have both positive and negative terms turned on.
4. Given the previous classification, for each element of Γ
0
the additional boundary
structure is obtained by turning off combinations of type (iii) ∆
(
i
1
,...,i
m
)
I
. Addi-
tionally, for the full stratification we may sometimes obtain additional boundaries
from type (i) non-minimal minors. The mini and the full stratifications differ in
the structure arising from this step.
This new set of boundaries can be nicely captured by additional posets Γ
1
emanat-
ing from every point in Γ
0
. It is important to emphasize that, in general, each point in
Γ
0
can have a different Γ
1
. In addition, the explicit form of Γ
0
and the Γ
1
’s generically
depends on whether we are considering the mini or full stratification. The top element
of each Γ
1
is characterized by having all non-minimal minors of types (ii) and (iii)
non-vanishing. Figure 1 shows a cartoon of the structure of the full stratification poset.
Note that the construction of the Γ
1
’s requires caution. First, not all type (iii)
minors can always be set to zero. Non-minimal minors are in general not indepen-
dent and it is necessary to explicitly check whether it is possible to shut them off
while preserving the positivity of the type (ii) and type (iii) larger minors and of the
Pl ̈ucker coordinates ∆
(
i
)
I
. This becomes particularly important when trying to turn off
combinations of them. Moreover, if considering the full stratification, for every label
we should consider all separate regions. Finally, the computation of the dimension of
the boundaries via equation (3.9) can be subtle. The vanishing of the larger minors
should be taken into account as extra relations among Pl ̈ucker coordinates, and hence
contribute to
N
rel
in (3.9), only if they are independent from the other conditions, i.e.
– 14 –
Γ
0
Γ
1
Figure 1
. A natural decomposition of the poset associated to the stratification. Γ
0
corre-
sponds to 2
×
2 minors and Γ
1
corresponds to non-minimal ones.
Pl ̈ucker relations plus the possible vanishing of other larger minors. Explicit examples
of all these issues are given in
§
7.
4 Simple Examples: Basic Properties
This section further illustrates some of the basic properties of positivity in terms of
simple examples.
4.1 Stratification of
G
+
(0
,n
; 1) =
G
+
(2
,n
)
Let us first consider the 1-loop geometry. A top-dimensional cell of
G
+
(0
,n,
1)
≡
G
+
(2
,n
) has all
(
n
2
)
=
1
2
n
(
n
−
1) Pl ̈ucker coordinates turned on. There are (
n
2
2
−
n
2
−
2
n
+ 3) independent Pl ̈ucker relations; together with the
GL
(2) invariance which
removes one extra degree of freedom by rescaling the coordinates, we get
1
2
n
(
n
−
1)
−
(
n
2
2
−
n
2
−
2
n
+ 3)
−
1 = 2(
n
−
2)
(4.1)
degrees of freedom. Boundaries are obtained by setting some ∆
I
’s to zero in a way that
is compatible with the Pl ̈ucker relations and ∆
J
>
0. Since in this case there are no
non-minimal minors, there is no distinction between mini and full stratification. From
each boundary it is then possible to further set more ∆
I
to zero in a way compatible
with the Pl ̈ucker relations and ∆
J
>
0 to obtain all of the sub-boundaries. Iterating this
procedure until reaching the zero-dimensional boundaries produces the stratification of
G
+
(2
,n
). There are efficient combinatorial techniques that can be employed for doing
this in a quick and systematic way [45], which will be briefly reviewed in
§
5.1.
– 15 –
The boundaries can be conveniently organized into levels according to their dimen-
sions. Connecting with arrows each boundary to its sub-boundaries creates a poset.
An example is provided in Figure 2, where we illustrate the stratification of
G
+
(2
,
4).
11
In this example there are 6 Pl ̈ucker coordinates: ∆
12
, ∆
13
, ∆
14
, ∆
23
, ∆
24
, ∆
34
and one
Pl ̈ucker relation:
∆
12
∆
34
+ ∆
23
∆
14
= ∆
13
∆
24
.
(4.2)
Some remarks are already in order:
(
1
4
)
(
2
3
)
(
2
4
)
(
1
3
)
(
3
4
)
(
1
2
)
(
1
2
,
2
3
)
(
2
4
,
3
4
)
(
1
2
,
2
4
)
(
1
4
,
2
4
)
(
2
3
,
2
4
)
(
1
3
,
3
4
)
(
1
3
,
1
4
)
(
1
2
,
1
3
)
(
1
3
,
2
3
)
(
1
4
,
3
4
)
(
2
3
,
3
4
)
(
1
2
,
1
4
)
(
1
2
,
1
4
,
2
4
)
(
1
2
,
2
3
,
2
4
)
(
1
3
,
1
4
,
3
4
)
(
1
3
,
2
3
,
3
4
)
(
1
2
,
1
3
,
1
4
)
(
1
2
,
1
3
,
2
3
)
(
1
2
,
1
3
,
2
4
,
3
4
)
(
1
3
,
1
4
,
2
3
,
2
4
)
(
1
4
,
2
4
,
3
4
)
(
2
3
,
2
4
,
3
4
)
(
1
2
,
1
3
,
1
4
,
2
3
,
2
4
)
(
1
2
,
1
3
,
1
4
,
2
4
,
3
4
)
(
1
2
,
1
3
,
2
3
,
2
4
,
3
4
)
(
1
3
,
1
4
,
2
3
,
2
4
,
3
4
)
(
1
2
,
1
3
,
1
4
,
2
3
,
2
4
,
3
4
)
Figure 2
. Boundaries of
G
+
(2
,
4). The parentheses indicate which Pl ̈ucker coordinates are
turned on. The top level has all 6 coordinates turned on and has dimension 4, the bottom
level has only one coordinate turned on and has dimension 0.
•
At the first step, going to the 3-dimensional boundaries, we only turn off one
Pl ̈ucker coordinate. Since there are six Pl ̈ucker coordinates that can be turned
off, we would naively expect six different 3-dimensional boundaries. Instead, as
shown in Figure 2, there are only four of them. This is because once we restrict
the ∆
I
’s to be positive, two of these would-be boundaries are inconsistent with
the Pl ̈ucker relations. For example, killing ∆
13
gives
∆
12
∆
34
+ ∆
23
∆
14
= 0
,
(4.3)
11
This poset has already appeared in the literature, see e.g. [29, 45].
– 16 –
which can only be satisfied if we do not restrict ourselves to the strictly positive
domain. This is the first example of positivity killing boundaries “by hand”. This
phenomenon was already studied in [45] and emerged naturally from the methods
therein. We note that this is not imposing extended positivity yet, which imposes
compatibility of relations from different loops; this is positivity at a single loop
level.
•
For several 2-dimensional boundaries some extra ∆
I
had to be set to zero in
order to satisfy the Pl ̈ucker relation. For example, starting from the boundary
with non-vanishing (12
,
13
,
14
,
24
,
34), i.e. where we have turned off ∆
23
, it is not
possible to only kill ∆
12
, because the Pl ̈ucker relation would then become
∆
13
∆
24
= 0
,
(4.4)
which is not possible on
any
non-zero domain. Note here that positivity is not
the issue, it is the violation of the Pl ̈ucker relation.
•
As mentioned, the boundaries constructed in this way form a poset. Moreover,
this poset is Eulerian, i.e.
4
∑
d
=0
(
−
1)
d
N
(
d
)
= 1
,
(4.5)
where
N
(
d
)
is the number of boundaries of dimension
d
. We note that for this
simple example there is no distinction between mini and full stratification.
•
The full extent of extended positivity never comes into play in this example.
Having only one matrix, we never need to consider whether minors of different
matrices are compatible. This will however not be the case for the example of
G
+
(0
,n
;
L
= 2).
4.2 Non-Minimal Minors
Before developing a practical implementation for it in the coming section, it is illumi-
nating to consider a few explicit examples of the classification of non-minimal minors
introduced in
§
3.5.
Let us consider the simple case of
G
+
(0
,
4; 2), which has 12 Pl ̈ucker coordinates.
From Figure 2, we see that
G
+
(0
,
4; 1) has 33 boundaries. The square of this positroid
stratification then has 33
2
= 1 089 configurations, the top-dimensional one being that
with all 12 ∆
(
i
)
I
’s turned on, giving dimension 8. All these configurations automatically
– 17 –
satisfy the two Pl ̈ucker relations, both of the form (4.2), as well as the non-negativity
of all Pl ̈ucker coordinates.
Some of these configurations, however, do not satisfy the extended positivity ∆
(1
,
2)
1234
≥
0, with ∆
(1
,
2)
1234
given in terms of Pl ̈ucker coordinates in (3.4). One such configurations
corresponds to the set of vanishing Pl ̈ucker coordinates, i.e. label,
{
∆
(2)
12
,
∆
(2)
23
,
∆
(2)
14
,
∆
(2)
34
,
∆
(2)
24
}
. In this case, we have
∆
(1
,
2)
1234
= 0 + 0 + 0 + 0 + 0
−
∆
(1)
24
∆
(2)
13
,
(4.6)
which is explicitly negative. We hence conclude that this label does not correspond to
a boundary.
Let us now present examples of the three different types of behaviors identified in
§
3.5.
•
Type (i)
: for the label
{
∆
(1)
12
,
∆
(2)
12
,
∆
(1)
14
,
∆
(2)
14
,
∆
(1)
13
,
∆
(2)
13
}
, we automatically have
∆
(1
,
2)
1234
= 0
.
(4.7)
•
Type (ii)
: for the label
{
∆
(2)
12
,
∆
(2)
23
,
∆
(2)
14
,
∆
(2)
13
,
∆
(2)
24
}
, we have
∆
(1
,
2)
1234
= ∆
(1)
12
∆
(2)
34
+ 0 + 0 + 0
−
0
−
0
,
(4.8)
which is strictly positive. We then cannot reach new boundaries by only turning
off ∆
(1
,
2)
1234
.
•
Type (iii)
: for the label
{
∆
(1)
12
,
∆
(1)
34
}
, we obtain
∆
(1
,
2)
1234
= 0 + 0 + ∆
(1)
23
∆
(2)
14
+ ∆
(1)
14
∆
(2)
23
−
∆
(1)
13
∆
(2)
24
−
∆
(1)
24
∆
(2)
13
,
(4.9)
which has both positive and negative contributions. This type of non-minimal
minor can in principle be turned off without turning off Pl ̈ucker coordinates. This
is possible whenever there are no obstructions coming from relations with other
non-minimal minors, which in this particular case do not exist.
In the combinatorial approach we will introduce in the coming sections, the building
blocks naturally correspond to entire terms in the non-minimal minors rather than only
factors within them.
– 18 –
5 Combinatorial Stratification
There is a natural, combinatorial implementation of the mini stratification of the loop
geometry, to which we will refer to as
combinatorial stratification
, which generalizes
the graphical stratification first introduced by Postnikov for
G
+
(
k,n
) [41]. This ex-
tension includes the more general cases that appear in
G
+
(0
,n
;
L
), for which extended
positivity can be systematically incorporated as explained in
§
6. The language of this
stratification is not matroids, positroids, Pl ̈ucker coordinates, and permutations, but
is simply that of perfect matchings and perfect orientations. The combinatorial struc-
tures discussed in this section only depend on labels and hence correspond to the mini
stratification.
5.1 Perfect Matchings and the Stratification of
G
+
(
k,n
)
The stratification illustrated in Figure 2 can be achieved through a variety of meth-
ods, extensively discussed in [45]. Here we provide a brief summary of its graphical
implementation.
Following [41], every cell of the positive Grassmannian
G
+
(
k,n
) can be associ-
ated to a planar bicolored graph,
12
which in turn determines a specific set of totally
positive Pl ̈ucker coordinates. Furthermore, it is also possible, as we do in this paper,
to restrict to graphs which are not only bicolored but that are bipartite. Figure 3
shows the graphical representation of the top-dimensional cell of
G
+
(2
,
4) and its lower
dimensional boundaries.
Perfect matchings
are fundamental objects in the study of bipartite graphs. A
perfect matching is a sub-collection of edges such that every internal node is the end-
point of only one edge, while external nodes may or may not be contained in the
perfect matching.
13
As an example, the top-dimensional cell of
G
+
(2
,
4) has 7 perfect
matchings, which we present in Figure 4.
14
There exists a precise map between perfect matchings and Pl ̈ucker coordinates.
The map is based on perfect orientations, which are flows over the edges of the graph
constructed according to the following rules:
•
White nodes must have one incoming arrow and the rest outgoing.
12
To be precise, it is associated to an equivalence class of graphs, which differ by certain moves and
reductions.
13
External nodes are those that lie on the boundary. The objects we have just defined are, more
precisely, denoted
almost perfect matchings
in the literature. For brevity, we will simply refer to them
as perfect matchings. Similarly, we refer to edges as external or internal depending on whether they
terminate on external nodes or not.
14
There are powerful methods for obtaining the perfect matchings of a graph, see e.g. [46].
– 19 –
(
1
2
,
1
3
,
1
4
,
2
3
,
2
4
,
3
4
)
(
1
2
,
1
3
,
1
4
,
2
3
,
2
4
)
(
1
2
,
1
3
,
1
4
,
2
4
,
3
4
)
(
1
2
,
1
3
,
2
3
,
2
4
,
3
4
)
(
1
3
,
1
4
,
2
3
,
2
4
,
3
4
)
(
1
2
,
1
4
,
2
4
)
(
1
2
,
2
3
,
2
4
)
(
1
3
,
1
4
,
3
4
)
(
1
3
,
2
3
,
3
4
)
(
1
2
,
1
3
,
1
4
)
(
1
2
,
1
3
,
2
3
)
(
1
2
,
1
3
,
2
4
,
3
4
)
(
1
3
,
1
4
,
2
3
,
2
4
)
(
1
4
,
2
4
,
3
4
)
(
2
3
,
2
4
,
3
4
)
(
1
2
,
2
3
)
(
2
4
,
3
4
)
(
1
2
,
2
4
)
(
1
4
,
2
4
)
(
2
3
,
2
4
)
(
1
3
,
3
4
)
(
1
3
,
1
4
)
(
1
2
,
1
3
)
(
1
3
,
2
3
)
(
1
4
,
3
4
)
(
2
3
,
3
4
)
(
1
2
,
1
4
)
(
1
4
)
(
2
3
)
(
2
4
)
(
1
3
)
(
3
4
)
(
1
2
)
Figure 3
. Boundary structure of
G
+
(2
,
4) and the graphs associated to each boundary. For
each graph we indicate the set of non-vanishing Pl ̈ucker coordinates.
•
Black nodes must have one outgoing arrow and the rest incoming.
Going from a perfect matching to a perfect orientation is a simple matter of drawing
an arrow pointing from black node to white node over those edges that the perfect
matching occupies, i.e. the red edges in Figure 4, and the rest of the arrows according
to the above rules. Given a perfect orientation,
its source set
is the set of external
nodes whose edges point into the graph. The label
I
of the source set of a perfect
orientation corresponds to the index of the associated Pl ̈ucker coordinate ∆
I
. Multiple
perfect matchings can share the same source set, which indicates that they represent
contributions to the same Pl ̈ucker coordinate. Such perfect matchings correspond to
– 20 –