# Intersections of dual $\operatorname{SL}_{3}$-webs

Linhui Shen  Department of Mathematics, Michigan State University, 619 Red
Cedar Road, 302 Wells Hall, East Lansing, Michigan 48824, United States
<EMAIL_ADDRESS>,  Zhe Sun  Key Laboratory of Wu Wen-Tsun Mathematics,
Chinese Academy of Sciences,
School of Mathematical Sciences, University of Science and Technology of
China, 96 Jinzhai Road, 230026 Hefei, Anhui, China<EMAIL_ADDRESS>and
Daping Weng  Department of Mathematics, University of California, Davis, One
Shields Avenue, Davis, CA 95616, United States<EMAIL_ADDRESS>
###### Abstract.

We introduce a topological intersection number for an ordered pair of
$\operatorname{SL}_{3}$-webs on a decorated surface. Using this intersection
pairing between reduced $(\operatorname{SL}_{3},\mathcal{A})$-webs and a
collection of $(\operatorname{SL}_{3},\mathcal{X})$-webs associated with the
Fock–Goncharov cluster coordinates, we provide a natural combinatorial
interpretation of the bijection from the set of reduced
$(\operatorname{SL}_{3},\mathcal{A})$-webs to the tropical set
$\mathcal{A}^{+}_{\operatorname{PGL}_{3},\hat{S}}(\mathbb{Z}^{t})$, as
established by Douglas and Sun in [DS20a, DS20b]. We provide a new proof of
the flip equivariance of the above bijection, which is crucial for proving the
Fock–Goncharov duality conjecture of higher Teichmüller spaces for
$\operatorname{SL}_{3}$.

###### Key words and phrases:

$\operatorname{SL}_{3}$-webs, intersection numbers, tropical points of
$\mathcal{A}$ moduli space, mapping class group equivariance.

###### Contents

  1. 1 Introduction
    1. 1.1 Background
    2. 1.2 Main results
    3. 1.3 Historical Comments
  2. 2 Definitions
    1. 2.1 Ideal triangulations
    2. 2.2 Webs
    3. 2.3 Intersections
  3. 3 Reduced Webs and Hives
    1. 3.1 Hives
    2. 3.2 Intersection metric
    3. 3.3 Dual surfacoids
    4. 3.4 From webs to hives
    5. 3.5 Compatibility
  4. 4 Fock-Goncharov Moduli Spaces
    1. 4.1 Definitions
    2. 4.2 Tropicalization.
    3. 4.3 Intersection pairings revisited

##  1\. Introduction

The ${\rm SL}_{3}$-webs, introduced by Kuperberg [Kup96], are oriented
trivalent graphs on a disk used to construct natural bases for the tensor
invariant spaces of irreducible representations of ${\rm SL}_{3}$. These webs
have been extended to surfaces and have been utilized in studying character
varieties and skein algebras associated with those surfaces. Sikora and
Westbury [SW07] developed a theory of confluence of graphs, and as an
application, constructed natural bases of ${\rm SL}_{3}$-skein algebras
associated with surfaces by using reduced ${\rm SL}_{3}$-webs.

Let $\hat{S}$ be a compact oriented topological surface with marked points on
its boundary and punctures inside. There are two versions of reduced webs on
$\hat{S}$, denoted by $\mathscr{W}_{\hat{S}}^{\mathcal{A}}$ the set of reduced
$\mathcal{A}$-webs and by $\mathscr{W}_{\hat{S}}^{\mathcal{X}}$ the set of
reduced $\mathcal{X}$-webs respectively, whose definition will be recalled in
Section 2.2. Recently, there are several works focus on the $\mathcal{A}$-webs
(cf. [DS20a, DS20b, FS22, Kim20, NY22]), while other works focus on the
$\mathcal{X}$-webs (cf. [FP16, Fra22, FP23, IK22, IOS23, IY23]).

The present paper, for the first time, puts these two versions of webs
together and proposes a mapping class group equivariant intersection pairing

(1)
$\mathbb{I}:~{}~{}\mathscr{W}_{\hat{S}}^{\mathcal{A}}\times\mathscr{W}_{\hat{S}}^{\mathcal{X}}\longrightarrow\frac{1}{3}\mathbb{Z}.$

We place the pairing $\mathbb{I}$ in the context of the cluster duality of
higher Teichmüller spaces.

###  1.1. Background

The moduli spaces $\mathcal{X}_{{\rm G},\hat{S}}$ and $\mathcal{A}_{{\rm
G}^{L},\hat{S}}$, introduced by Fock and Goncharov in [FG06], are variations
of the character varieties of the fundamental group $\pi_{1}(S)$ into a pair
of Langlands dual groups ${\rm G}$ and ${\rm G}^{L}$. For ${\rm PGL_{2}}$,
these moduli spaces recover the enhanced Teichmüller space and the decorated
Teichmüller space of Penner in [Pen87]. Fock and Goncharov proposed a duality
conjecture ([FG06, Conjecture 12.3]) relating $\mathcal{X}_{{\rm G},\hat{S}}$
and $\mathcal{A}_{{\rm G}^{L},\hat{S}}$. This conjecture has found numerous
interactions with cluster algebras, representation theory, mirror symmetry,
low dimensional topology, and Poisson geometry.

To elaborate, when $\hat{S}$ has no marked points, the duality conjecture
asserts that the ring of regular functions on $\mathcal{X}_{{\rm G},\hat{S}}$
admits a canonical linear basis parameterized by the integer tropical points
of $\mathcal{A}_{{\rm G}^{L},\hat{S}}$, and the parametrization is equivariant
under the action of the mapping class group of the surface.111The original
conjecture also includes the reversed direction; namely, the ring of regular
functions on $\mathcal{A}_{{\rm G}^{L},\hat{S}}$ has a canonical linear basis
parameterized by the tropical points of $\mathcal{X}_{{\rm G}^{L},\hat{S}}$.
However, when $\hat{S}$ has punctures, the conjecture does not hold and
requires a slight modification. As explained in [GS15], when $\hat{S}$ has
marked points on its boundary, the space $\mathcal{X}_{{\rm G},\hat{S}}$
should be replaced by $\mathcal{P}_{{\rm G},\hat{S}}$, an enhancement of
$\mathcal{X}_{{\rm G},\hat{S}}$. It is further conjectured that when the
tropical points of $\mathcal{A}_{{\rm G}^{L},\hat{S}}$ are cut out by the
tropical potential function, the dual space should be changed to the character
variety of $\hat{S}$.

The duality conjecture is proven for ${\rm G}=\operatorname{PGL}_{2}$ in
[FG06, Theorem 12.3] by relating Thurston’s transversely integer measured
laminations [Thu22] to the tropical coordinates using the topological
intersection numbers with the ideal triangulation of $\hat{S}$, and to the
trace functions of the laminations. Motivated by the lamination approach, the
duality for ${\rm G}=\operatorname{PGL}_{3}$ has recently been investigated by
using webs. The first step is to provide a canonical bijection between the
tropical points and the reduced webs, which has been established by Douglas
and Sun [DS20a, DS20b]. When $\hat{S}$ is a punctured surface without marked
points, based on the bijection of Douglas and Sun, Kim [Kim20] showed that the
highest degrees of the trace functions of the reduced webs are equal to the
tropical coordinates of $\mathcal{A}_{{\rm SL}_{3},\hat{S}}(\mathbb{Z}^{t})$,
completing the proof of the duality conjecture for ${\rm PGL}_{3}$.

An alternative approach to the duality conjecture is through cluster theory.
Fock and Goncharov generalized the duality conjecture to the setting of
cluster ensembles ([FG09, Conjecture 4.1]). Using scattering diagrams and
broken lines, Gross, Hacking, Keel, and Kontsevich [GHKK18] proved the duality
conjecture under certain conditions, which follow from the existence of
cluster Donaldson-Thomas transformations (a.k.a. reddening sequences).
Goncharov and Shen [GS18, GS19] showed that the pair $(\mathcal{P}_{{\rm
G},\hat{S}},\mathcal{A}_{{\rm G}^{L},\hat{S}})$ forms a cluster ensemble, and
explicitly determine their cluster Donaldson-Thomas transformations when
$\hat{S}$ is not a once-punctured surface $S_{g,1}$. For the once punctured
surface $S_{g,1}$, by slightly adjusting Goncharov–Shen’s construction, Fraser
and Pylyavskyy [FP23] obtained the cluster Donaldson-Thomas transformation for
$\mathcal{A}_{\operatorname{SL}_{n},S_{g,1}}$ when $n>2$. Combining the above
results, the duality conjecture holds for these cases.

Comparing the two distinct approaches to the duality conjecture mentioned
above, one advantage of the lamination-web approach is the explicitness in its
construction,which makes it more convenient for concrete calculations. The
theta bases in [GHKK18], although applicable to all cluster ensembles with
reddening sequences, are challenging to compute explicitly. Recently, Mandel
and Qin [MQ23] proved that for $\operatorname{PGL}_{2}$, the theta bases and
the Fock–Goncharov’s lamination bases coincide. For $\operatorname{PGL}_{3}$,
the comparison between the theta bases and the web bases is an interesting
direction for future research. Moreover, the lamination-web approach aligns
more closely with the study of skein algebras and the quantum trace maps
[BW11, LY23], which establish connections between two different quantizations
of the rings of regular functions of the ${\rm SL}_{n}$ character varieties.

###  1.2. Main results

Following Knutson and Tao [KT99], a hive is an arrangement of numbers
satisfying a family of rhombus conditions. In Section 3.1, we recall the set
${\bf Hive}(\mathcal{T})$ of ${\rm SL}_{3}$ hives associated with an arbitrary
ideal triangulation $\mathcal{T}$ of $\hat{S}$. As in (7), the hives for
different ideal triangulations are related by a sequence
$\varphi_{\mathcal{T}^{\prime},\mathcal{T}}$ of octahedron relations

$\varphi_{\mathcal{T}^{\prime},\mathcal{T}}:~{}{\bf
Hive}(\mathcal{T})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\bf
Hive}(\mathcal{T}^{\prime}).$

In Section 4.2, we review the cone $\mathcal{A}^{+}_{{\rm
PGL}_{3},\hat{S}}(\mathbb{Z}^{t})$ of integer tropical points cut out by the
tropical potential function of Goncharov and Shen [GS15]. The paper loc.cit.
constructed a bijection $\alpha_{\mathcal{T}}$ between $\mathcal{A}^{+}_{{\rm
PGL}_{3},\hat{S}}(\mathbb{Z}^{t})$ and ${\bf Hive}(\mathcal{T})$, and obtained
the following commutative diagram of bijections

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In Section 2.3, we associate with each ideal triangulation $\mathcal{T}$ of
$\hat{S}$ a family $\\{[V_{i}]\\}$ of $\mathcal{X}$-webs, which consists of
the oriented ideal edges of $\mathcal{T}$ and the inward tripods inside ideal
triangles of $\mathcal{T}$. These webs correspond to the Fock-Goncharov
cluster $\mathcal{A}$-coordinates. Using the intersection pairing with
$[V_{i}]$, we define the following intersection map in Definition 2.12

${i}_{\mathcal{T}}:\mathscr{W}_{\hat{S}}^{\mathcal{A}}\longrightarrow\left(\frac{1}{3}\mathbb{Z}\right)^{n}.$

The following theorem is the main result of this paper.

######  Theorem 1.1.

The map $i_{\mathcal{T}}$ gives rise to a bijection between
$\mathscr{W}_{\hat{S}}^{\mathcal{A}}$ and ${\bf Hive}(\mathcal{T})$. For any
two ideal triangulations $\mathcal{T}$ and $\mathcal{T}^{\prime}$ of
$\hat{S}$, we have the following commutative diagram of bijections

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Comparing with (2), we get a canonical bijection between
$\mathscr{W}_{\hat{S}}^{\mathcal{A}}$ and $\mathcal{A}_{{\rm
PGL}_{3},\hat{S}}^{+}(\mathbb{Z}^{t})$.

A further tropicalization of the duality between $\mathcal{A}_{{\rm
G}^{L},\hat{S}}$ and $\mathcal{P}_{{\rm G},\hat{S}}$ yields a mysterious
mapping class group equivariant pairing

(3) $\mathcal{I}:~{}\mathcal{A}_{{\rm
G}^{L},\hat{S}}(\mathbb{R}^{t})\times\mathcal{P}_{{\rm
G},\hat{S}}(\mathbb{R}^{t})\longrightarrow\mathbb{R}.$

A better understanding of the pairing is important to the study of the moduli
spaces $\mathcal{A}_{{\rm G}^{L},\hat{S}}$ and $\mathcal{P}_{{\rm
G},\hat{S}}$. We conjecture a natural bijection between
$\mathscr{W}_{\hat{S}}^{\mathcal{X}}$ and the cone $\mathcal{P}^{+}_{{\rm
PGL}_{3},\hat{S}}(\mathbb{Z}^{t})$. See [IK22] for recent results in this
direction. We expect that our intersection paring (1) provides a geometric-
combinatorial way to realize the pairing $\mathcal{I}$. We wish to apply the
paring (1) to investigate the compactification of the higher Teichmüller
spaces in the future. See [FG16] for related results for ${\rm PGL}_{2}$.

###  1.3. Historical Comments

We include a list of results related to this paper.

Douglas and Sun [DS20a, DS20b] constructed a mapping class group equivariant
bijection

$\Phi:~{}\mathscr{W}_{\hat{S}}^{\mathcal{A}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathcal{A}_{{\rm
PGL}_{3},\hat{S}}^{+}(\mathbb{Z}^{t}).$

Frohman and Sikora [FS22] provided a different topological integer coordinate
system for the reduced $\operatorname{SL}_{3}$-webs depending on the ideal
triangulation chosen. Theorem 1.1 provides an alternative interpretation of
the bijection $\Phi$ in terms of the intersections.

For the disk case and for $\operatorname{SL}_{3}$, Fontaine, Kamnitzer, and
Kuperberg [FKK13] showed that reduced $\mathcal{A}$-webs are dual to CAT(0)
triangulated diskoids in the affine building, and that each reduced web with
minuscule boundary $\vec{\lambda}$ corresponds to a component of the Satake
fiber $F(\vec{\lambda})$. See Theorem 1.4 of loc. cit. In [Akh20], Akhmejanov
showed that each diskoid is the intersection of min-convex set and max-convex
hulls of a generic point of a component of $F(\vec{\lambda})$. Goncharov and
Shen [GS15] give a bijection between the top dimensional components of the
Satake fibers and the tropical points. Thus, the above works implicitly imply
a bijection between reduced webs and tropical points. In this paper, we
crucially use the dual graphs of reduced webs on surfaces, which we propose to
call surfacoids, as a slight extension of the diskoids.

Several approaches in the literature aim to understand the pairing (3). In
[Le21], Le utilized the length-minimal weighted networks in the affine
buildings to interpret tropical Fock-Goncharov coordinates on
$\mathcal{A}_{{\rm SL}_{n},\hat{S}}$ for disks. However, as explained in
Section 4.4 of loc.cit., crucial ingredients are still missing to fully
realize the pairing $\mathcal{I}$. In some special cases, a representation-
theoretic understanding of the pairing $\mathcal{I}$ can be found in [Fei23].
It is worth mentioning that our intersection number pairing (1) is more
suitable for topological visualization and can be broadly defined for any
ordered pairs of webs on marked surfaces, not limited to disks.

In the ongoing joint work [ISY] for the $\operatorname{Sp}_{4}$ case,
Ishibashi, Yuasa, and Sun constructed the intersection number coordinates for
the crossroad webs [IY22].

Acknowledgements. We are grateful to Alexander Goncharov for his encouragement
and enlightening discussions. We thank Ian Le for the conversations on the
interactions with affine buildings. L. Shen is supported by the NSF grant
DMS-2200738.

##  2\. Definitions

This section recalls the definitions of $\mathcal{A}$\- and $\mathcal{X}$\-
webs of type $A_{2}$ over decorated surfaces. We introduce a canonical
intersection pairing between them.

###  2.1. Ideal triangulations

A decorated surface $\hat{S}$ is a pair $(S,m_{b})$ where $S$ is a connected
oriented compact surface with boundary and $m_{b}$ is a finite (possibly
empty) set of marked points on its boundary considered up to isotopy. Note
that $\partial S\backslash m_{b}$ is a union of circles and open intervals,
which are called boundary circles and boundary intervals respectively. The
boundary circles are considered as punctures. We denote the set of punctures
by $m_{p}$.

An ideal triangulation $\mathcal{T}$ of $\hat{S}$ is a maximal collection of
arcs joining the marked points and the punctures of $\hat{S}$ such that these
arcs are pairwise disjoint on the interior of $S$ and non-homotopic. The edges
(respectively triangles) of the ideal triangulations are called ideal edges
(respectively ideal triangles). In this paper, we focus on decorated surfaces
that admit ideal triangulations. Suppose the underlying surface of $\hat{S}$
is of genus $g$ and of $c$ many boundary components. Let $m$ be the total
number of marked points on $\hat{S}$. A direct calculation shows that every
ideal triangulation of $\hat{S}$ has $f$ many ideal triangles and $e$ many
ideal edges, where

$f=2c+m+4g-4,\qquad e=3c+2m+6g-6.$

We place two vertices on each ideal edge, and place one vertex on the center
of each ideal triangle. As illustrated by Figure 1, we draw nine arrows within
each ideal triangle, obtaining a quiver $Q_{\mathcal{T}}$. Denote by
$\Theta_{\mathcal{T}}$ the set of vertices of $Q_{\mathcal{T}}$. By
definition, the cardinality of $\Theta_{\mathcal{T}}$ is

$2e+f=8c+5m+16g-16.$

Figure 1. The quiver $Q_{\mathcal{T}}$ associated with an ideal triangulation
$\mathcal{T}$

###  2.2. Webs

######  Definition 2.1.

A $(\operatorname{SL}_{3},\mathcal{A})$-web is an oriented graph embedded in
$\hat{S}$ with finitely many connected components, where each component is one
of the following graphs

  * •

a simple oriented loop,

  * •

an oriented arc connecting the boundary intervals of $\hat{S}$,

  * •

an oriented graph such that every interior vertex is a $3$-valent sink or a
$3$-valent source and the rest vertices are 1-valent vertices lying on the
boundary intervals of ${\hat{S}}$.

A $k$-gon of a web $W$ is a contractible component of $\hat{S}\backslash W$
with $k$ sides. We say a $k$-gon is internal if all of its vertices are
interior vertices of $W$. A web is non-elliptic if it contains no contractible
loops and all of its internal $k$-gons have at least $6$ sides. A non-elliptic
web is reduced if each $k$-gon with only one side along $\partial\hat{S}$ must
satisfy $k\geq 5$.

Denote by $\mathscr{W}_{\hat{S}}^{\mathcal{A}}$ be the space of reduced
$(\operatorname{SL}_{3},\mathcal{A})$-webs up to homotopy equivalence on
$\hat{S}\times[0,1]$.

######  Example 2.2.

Let $\hat{S}$ be a once-punctured disk with two marked points on its boundary.
In Figure 2, we represent the puncture and the marked points by red circles.
We denote the sinks of the webs by black dots and the sources by white dots.
The left web is not reduced because the shaded region is a square with one
side along $\partial\hat{S}$. The right web is elliptic because the shaded
region is an internal square.

Figure 2. $(\operatorname{SL}_{3},\mathcal{A})$-webs on a once-punctured disk

######  Example 2.3.

A bigon is a disk with two marked points on its boundary. As shown on Figure
3(2), we consider a collection of arcs going between the two boundaries of a
bigon such that

  * •

every pair of arcs intersect with each other at most once,

  * •

if two arcs intersect each other, then they point to different boundaries of
the bigon.

We replace each crossing of the arcs with an “I”, obtaining a web as in Figure
3(1). The webs obtained this way are called minimal ladders. It is easy to see
that every minimal ladder is uniquely determined by its intersection with the
bigon.

Figure 3. (1) A minimal ladder on a bigon. (2) Its schematic diagram.

Moreover, by choosing one of the two marked points of the bigon, we can label
the endpoints of the arcs along each boundary by $1,2,3,\dots,n$ in the order
according to their distance from the chosen marked point (there are
necessarily the same number of endpoints along each boundary by construction).
Along a chosen boundary $\alpha$ of the bigon, we say an arc is _ascending_ if
its endpoint $i$ at $\alpha$ is smaller than its other endpoint $i^{\prime}$
at the other boundary. Now for each ascending arc along a boundary, we define
its _staircase_ to be the collection of “I”s in the minimal ladder that
correspond to crossings between this particular ascending arc and other arcs.
For example, if we choose the top marked point in Figure 3(2), then the left
boundary has three ascending arcs and the right boundary has one ascending
arc; correspondingly, each “I” in Figure 3(1) is a staircase for an ascending
arc along the left boundary, and the three “I”s together form a staircase for
the ascending arc along the right boundary.

######  Example 2.4.

Let $\Delta$ be a disk with three marked points. A reduced $({\rm
SL}_{3},\mathcal{A})$-web on $\Delta$ is shown on Figure 4(1). In general,
every reduced web on $\Delta$ consists of an oriented honeycomb in the middle
and oriented arcs in the corners. Let $\mathbb{N}$ be the set of non-negative
integers. Each reduced web on $\Delta$ corresponds to a tuple

$(x,y,z,t,u,v,w)\in\mathbb{Z}\times\mathbb{N}^{6}.$

Here $|x|$ denotes the number of the arcs of the center oriented honeycomb
intersecting with each side of $\Delta$. The sign of $x$ determines the
orientation of the honeycomb: when $x$ is positive, then the arrows of the
honeycomb are coming out from the center; otherwise, they are getting into the
center. The numbers of oriented corner arcs around each marked points are
determined by the nonnegative integers $y,z,t,u,v,w$. Hence, we obtain a
bijection

$\mathscr{W}_{\Delta}^{\mathcal{A}}\stackrel{{\scriptstyle\sim}}{{=}}\mathbb{Z}\times\mathbb{N}^{6}$

Figure 4. A reduced web on $\Delta$. Here $x=3$, $y=2$, $z=t=u=v=w=1$.

Below we recall the good position property of reduced webs in Proposition 2.5.
It shows every reduced web can be obtained by gluing minimal ladders and
reduced webs on ideal triangles. The good position property is originally due
to Kuperberg [Kup96] on the disk cases and can be easily generalized to
surfaces (see e.g. [DS20a],[FS22]).

Let $\mathcal{T}$ be an ideal triangulation of $\hat{S}$. The split ideal
triangulation $\mathcal{T}_{2}$ associated with $\mathcal{T}$ is constructed
by splitting each internal ideal edge of $\mathcal{T}$ into two disjoint
homotopic ideal edges. As a result, $\hat{S}$ is cut into bigons and triangles
as in Figure 5.

Figure 5. Split ideal triangulation.

######  Proposition 2.5.

Every reduced $(\operatorname{SL}_{3},\mathcal{A})$-web $W$ on $\hat{S}$ can
be placed in a good position with respect to $\mathcal{T}_{2}$ such that

  1. (1)

$W$ intersects transversely with every edge of $\mathcal{T}_{2}$;

  2. (2)

the restriction of $W$ to every bigon of $\mathcal{T}_{2}$ is a minimal ladder
as in Example 2.3;

  3. (3)

the restriction of $W$ to every triangle of $\mathcal{T}_{2}$ is a reduced web
as in Example 2.4.

In contrast to the $(\operatorname{SL}_{3},\mathcal{A})$-webs, we introduce
their dual counterparts, the $(\operatorname{SL}_{3},\mathcal{X})$-webs, which
have their endpoints placed at the marked points and the punctures of
$\hat{S}$.

######  Definition 2.6.

A $(\operatorname{SL}_{3},\mathcal{X})$-web is an oriented graph embedded in
$\hat{S}$ with finitely many connected components, where each component is one
of the following graphs

  * •

a simple oriented loop,

  * •

an oriented arc connecting points in $m_{b}\cup m_{p}$,

  * •

a connected oriented graph on $S$ such that each interior vertex is a
$3$-valent sink or a $3$-valent source and the rest vertices are in $m_{b}\cup
m_{p}$.

A $(\operatorname{SL}_{3},\mathcal{X})$-web is reduced if

  * •

it contains no peripheral loops, a.k.a., the loops surrounding a puncture;

  * •

each of its internal $k$-gons has at least 6 sides;

  * •

each of its $k$-gons with only one vertex in $m_{b}\cup m_{p}$ must satisfy
$k\geq 4$.

Denote by $\mathscr{W}_{\hat{S}}^{\mathcal{X}}$ the space of reduced
$(\operatorname{SL}_{3},\mathcal{X})$-webs up to homotopy equivalence on
$\hat{S}\times[0,1]$.

######  Example 2.7.

Figure 6 is a reduced $(\operatorname{SL}_{3},\mathcal{X})$-web on a once-
punctured disk with two marked points. Note that, unlike $\mathcal{A}$-webs,
$\mathcal{X}$-webs can enter into marked points and punctures.

Figure 6. A reduced $(\operatorname{SL}_{3},\mathcal{X})$-web.

From simplicity, we will call the aforementioned webs $\mathcal{A}$\- and
$\mathcal{X}$-webs from now on, respectively, without mentioning
$\operatorname{SL}_{3}$.

###  2.3. Intersections

Let $W$ be an $\mathcal{A}$-web and let $V$ be a $\mathcal{X}$-web on
$\hat{S}$ such that they intersect transversely. For each intersection point
$p\in W\cap V$, the intersection number of the ordered pair $(W,V)$ at $p$ is
defined as

$\epsilon_{p}(W,V):=\left\\{\begin{aligned} &\frac{1}{3}\;\;\;\text{if $W$
crosses $V$ through $p$ towards the right side of $V$};\\\ &\\\
&\frac{2}{3}\;\;\;\text{if $W$ crosses $V$ through $p$ towards the left side
of $V$}.\end{aligned}\right.$

The intersection number of the pair $(W,V)$ is defined as

(4) $i(W,V):=\sum_{p\in\,W\cap V}\epsilon_{p}(W,V).$

######  Example 2.8.

Figure 7 shows the intersection numbers for several pairs of webs on a
triangle.

Figure 7. Intersection numbers for ordered pairs of webs on a triangle.

######  Definition 2.9.

We define the intersection paring

(5)
$\mathbb{I}:~{}\mathscr{W}_{\hat{S}}^{\mathcal{A}}\times\mathscr{W}_{\hat{S}}^{\mathcal{X}}\longrightarrow\frac{1}{3}\mathbb{Z},\qquad([W],[V])\,\longmapsto\,\min_{W\in[W],V\in[V]}\left\\{i(W,V)\right\\}$

where $W$ and $V$ are in the homotopy classes $[W]$ and $[V]$ respectively
such that they intersect transversely.

######  Remark 2.10.

Note that the intersection number between $[W]$ and $[V]$ only depends on
their relative position. As a result, in order to compute
$\mathbb{I}([W],[V])$, one can fix the web $W$ and vary $V$ within its
homotopy class.

######  Remark 2.11.

Let $\mathfrak{g}$ be a simple Lie algebra and let $\mathfrak{g}^{\vee}$ be
its Langlands dual. We expect the existences of the notions of
$(\mathfrak{g},\mathcal{A})$ and $(\mathfrak{g}^{\vee},\mathcal{X})$ , whose
oriented edges are labelled by the dominant weights of $\mathfrak{g}$ and
$\mathfrak{g}^{\vee}$ respectively. The change of orientations of an edge is
equivalent to changing the label $\lambda$ to $-w_{0}(\lambda)$, where $w_{0}$
is the longest Weyl group element.

$\lambda$$-w_{0}(\lambda)$$=$

Let $X$ and $X^{\vee}$ be the weight lattices of $\mathfrak{g}$ and
$\mathfrak{g}^{\vee}$ respectively. There is a canonical pairing

$\langle~{},~{}\rangle:~{}X\times
X^{\vee}\longrightarrow\frac{1}{n}\mathbb{Z},$

where $n$ is the determinant of the Cartan matrix of $\mathfrak{g}$. Given a
$(\mathfrak{g},\mathcal{A})$ web $W$ and a $(\mathfrak{g},\mathcal{X})$ web
$V$, intersecting transversely at $p$ as illustrated by the figure

$p$$W$$\lambda$$V$$\mu$

We shall define the intersection number of $W$ and $V$ at $p$ as

$\epsilon_{p}(W,V)=\langle\lambda,\mu\rangle.$

Now fix an ideal triangulation $\mathcal{T}$ of $\hat{S}$. Recall the quiver
$Q_{\mathcal{T}}$ with the vertex set $\Theta_{\mathcal{T}}$. We assign a
reduced $\mathcal{X}$ web $[V_{i}]$ to each vertex $i\in\Theta_{\mathcal{T}}$.
The vertex at the center of each ideal triangle corresponds to a tripod inside
the same triangle, as depicted in Figure 8(8). Every vertex at each ideal edge
$e$ corresponds to an oriented arc that is homotopy to $e$, with the
orientation determined by the position of the vertex on $e$.

Figure 8. The correspondence between
$(\operatorname{SL}_{3},\mathcal{X})$-webs and vertices of $Q_{\mathcal{T}}$.

By using the above collection of reduced $\mathcal{X}$ webs associated with an
ideal triangulation, we arrive at the definition of the intersection map.

######  Definition 2.12.

For each ideal triangulation $\mathcal{T}$ of $\hat{S}$, we define the
_intersection map_

(6)
$i_{\mathcal{T}}:\mathscr{W}_{\hat{S}}^{\mathcal{A}}\longrightarrow\left(\frac{1}{3}\mathbb{Z}\right)^{\Theta_{\mathcal{T}}}$

by

$i_{\mathcal{T}}([W]):=\left\\{\mathbb{I}([W],[V_{i}])\right\\}_{i\in\Theta_{\mathcal{T}}}.$

##  3\. Reduced Webs and Hives

In this section, we show that the intersection map (6) establishes a natural
one-to-one correspondence between reduced $\mathcal{A}$-webs and Knutson-Tao
hives. For different ideal triangulations of $\hat{S}$, the corresponding
Knutson-Tao hives are related by octahedron relations.

###  3.1. Hives

Following Knutson and Tao [KT99], a hive is an arrangement of numbers
satisfying a family of rhombus conditions. Below, we include the definition of
hives for ${\rm SL}_{3}$.

Let $\Delta$ be a disk with three marked points. Let $a_{1},\ldots,a_{7}$ be
real numbers associated with the vertices of $Q_{\Delta}$ as follows

$a_{6}$$a_{7}$$a_{3}$$a_{1}$$a_{5}$$a_{2}$$a_{4}$ Figure 9. Coordinates
associated with vertices of $Q_{\Delta}$.

We say $(a_{1},\ldots,a_{7})$ is a hive if it satisfies the rhombus condition,
that is, the following

$a_{1}+a_{2}-a_{4},\qquad a_{3}+a_{4}-a_{1}-a_{6},\qquad
a_{4}+a_{5}-a_{2}-a_{7},$ $a_{5}+a_{7}-a_{4},\qquad
a_{2}+a_{4}-a_{1}-a_{5},\qquad a_{4}+a_{6}-a_{3}-a_{7},$
$a_{3}+a_{6}-a_{4},\qquad a_{4}+a_{7}-a_{5}-a_{6},\qquad
a_{1}+a_{4}-a_{2}-a_{3}$

are non-negative integers.

Note that every arrow of $Q_{\Delta}$ is the short diagonal of a unit rhombus.
The rhombus condition can be interpreted as saying that for any unit rhombus,
the sum across the short diagonal minus the sum across the long diagonal is a
non-negative integer.

Let $\mathcal{T}$ be an ideal triangulation of a decorated surface $\hat{S}$.
A hive of $\mathcal{T}$ is an assignment of numbers to the vertices of
$Q_{\mathcal{T}}$ such that they satisfy the rhombus condition within each
ideal triangle of $\mathcal{T}$. Denote by ${\bf Hive}(\mathcal{T})$ the set
of hives of $\mathcal{T}$.

$a_{6}$$a_{2}$$a_{11}$$a_{12}$$a_{10}$$a_{9}$$a_{3}$$a_{8}$$a_{7}$$a_{5}$$a_{1}$$a_{4}$$b_{6}$$b_{2}$$a_{11}$$a_{12}$$a_{10}$$a_{9}$$a_{3}$$a_{8}$$a_{7}$$a_{5}$$a_{1}$$a_{4}$$b_{6}$$b_{2}$$a_{11}$$a_{12}$$a_{10}$$a_{9}$$a_{3}$$a_{8}$$b_{7}$$b_{5}$$a_{1}$$a_{4}$
Figure 10. Hives for different ideal triangulations

Let $\\{a_{\bullet}\\}\in{\bf Hive}(\mathcal{T})$ be a hive whose restriction
to a quadrilateral is illustrated by the left graph of Figure 10. Flipping the
diagonal of the quadrilateral, we obtain a new ideal triangulation
$\mathcal{T}^{\prime}$. We impose the octahedron relation

(7)
$\begin{array}[]{lll}a_{2}+b_{2}=\max\\{a_{1}+a_{7},\,a_{5}+a_{3}\\},&\hskip
71.13188pt&a_{6}+b_{6}=\max\\{a_{5}+a_{11},\,a_{7}+a_{10}\\};\\\
b_{5}+a_{5}=\max\\{a_{4}+b_{6},\,a_{9}+b_{2}\\},&&b_{7}+a_{7}=\max\\{b_{2}+a_{12},\,a_{8}+b_{6}\\}.\end{array}$

It gives rise to the numbers $b_{2},b_{5},b_{6},b_{7}$ to the four new
vertices of $Q_{\mathcal{T}^{\prime}}$ as depicted on the right graph of
Figure 10. We keep the rest $a_{\bullet}$ invariant. The newly obtained
assignment satisfies the rhombus condition. In this way, we obtain a hive for
$\mathcal{T}^{\prime}$.

The above procedure can be reversed. Any two ideal triangulations
$\mathcal{T}$ and $\mathcal{T}^{\prime}$ can be related by a sequence of flips
of diagonals. By applying the relations (7) recursively, we obtain a bijection

(8) $\varphi_{\mathcal{T}^{\prime},\mathcal{T}}:\,{\bf
Hive}(\mathcal{T})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\bf
Hive}(\mathcal{T}^{\prime}).$

Note that there might be different sequences of flips relating $\mathcal{T}$
and $\mathcal{T}^{\prime}$. The paper [GS15] shows that the hives can be
viewed as a tropicalization of the Fock-Goncharov moduli space
$\mathcal{A}_{{\rm PGL}_{3},\hat{S}}$ (See Section 4). As a result, the
bijection (8) is canonical and does not depend on the sequence of flips used.

In the rest of this section, we prove the following main theorem of this
paper.

######  Theorem 3.1.

The intersection map $i_{\mathcal{T}}$ in (6) gives rise to a bijection

$i_{\mathcal{T}}:~{}\mathscr{W}_{\hat{S}}^{\mathcal{A}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\bf
Hive}(\mathcal{T}).$

For any two ideal triangulations $\mathcal{T}$ and $\mathcal{T}^{\prime}$ of
$\hat{S}$, the bijections $i_{\mathcal{T}}$ and $i_{\mathcal{T}^{\prime}}$ are
compatible with the transition map
$\varphi_{\mathcal{T}^{\prime},\mathcal{T}}$:

$i_{\mathcal{T}^{\prime}}=\varphi_{\mathcal{T}^{\prime},\mathcal{T}}\circ
i_{\mathcal{T}}.$

###  3.2. Intersection metric

Let $G=(V,E)$ be a connected oriented graph with the set of vertices $V$ and
the set of oriented edges $E$. In preparation for proving Theorem 3.1, we
introduce the following intersection metric on $G$.

######  Definition 3.2.

A path $\gamma$ in an oriented graph $G$ is an alternating sequence of
vertices and edges $(a_{0},e_{1},a_{1},e_{2},a_{2},...,e_{n},a_{n})$ such that
$a_{i-1}$ and $a_{i}$ are endpoints of $e_{i}$ for each $i$. We define the
length of $\gamma$ as

$|\gamma|:=\sum_{i=1}^{l}d_{e_{i}}(a_{i-1},a_{i}),$

where

$d_{e_{i}}(a_{i-1},a_{i}):=\left\\{\begin{aligned} &\frac{1}{3}\;\;\;\text{if
$e_{i}$ goes from $a_{i-1}$ to $a_{i}$};\\\ &\frac{2}{3}\;\;\;\text{if $e_{i}$
goes from $a_{i}$ to $a_{i-1}$}.\end{aligned}\right.$

Let $a,b$ be a pair of vertices of $G$. Let $P(a,b)$ denote the set of paths
from $a$ to $b$. We define the intersection metric on $G$ by

(9) $d(a,b):=\min_{\gamma\in P(a,b)}|\gamma|.$

A path from $a$ to $b$ attaining the minimum value is called a _geodesic_.

######  Remark 3.3.

It is easy to see that the intersection metric satisfies the following
properties:

  * •

for any $a,b$, we have $d(a,b)\geq 0$ and the equality attains if and only if
$a=b$;

  * •

for any $a,b,c$, we have $d(a,b)+d(b,c)\geq d(a,c)$.

However, it is asymmetric, i.e., $d(a,b)\neq d(b,a)$ in general.

$\bullet$$O$

$A$$B$$C$$\Omega$

Figure 11. Left: the graph $\Gamma$ associated with the lattice
$\mathbb{Z}^{2}$. Right: the region $\Omega$ defined by a triple of points
$(A,B,C)$.

Below we consider a particular oriented graph $\Gamma$ with the integer
lattice $\mathbb{Z}^{2}$ as its vertex set and with an edge from $(x,y)$ to
each of $(x+1,y)$, $(x,y+1)$, and $(x-1,y-1)$ for every vertex $(x,y)$. See
the left picture in Figure 11.

######  Lemma 3.4.

Let $O$ be the origin and let $A=(x,y)\in\mathbb{Z}^{2}$. Then

$d(O,A)=\frac{\max\\{x+y,\,y-2x,\,x-2y\\}}{3}=\left\\{\begin{array}[]{ll}\frac{x+y}{3}&\mbox{if
$x\geq 0$ and $y\geq 0$;}\\\ \frac{y-2x}{3}&\mbox{if $x\leq 0$ and $y\geq
x$;}\\\ \frac{x-2y}{3}&\mbox{if $y\leq 0$ and $x\geq y$.}\\\
\end{array}\right.$

###### Proof.

We prove the case when $x\geq 0$ and $y\geq 0$. The proof for the rest two
cases goes along the same line. Consider the vectors

$\vec{u}=(1,0),\qquad\vec{v}=(0,1),\qquad\vec{w}=(-1,-1).$

Let $\gamma$ be a path from $O$ to $A$. Let $k,p,q,r,s,t$ be the numbers of
the oriented edges on $\gamma$ that are equal to $\vec{u}$, $-\vec{u}$,
$\vec{v}$, $-\vec{v}$, $\vec{w}$, and $-\vec{w}$, respectively. We have

$(x,y)=(k-p+t-s,q-r+t-s).$

Since $k,p,q,r,s,t$ are non-negative, we get

$|\gamma|=\frac{(k+q+s)+2(p+r+t)}{3}\geq\frac{(k+q+2t)-(p+r+2s)}{3}=\frac{x+y}{3}.$

Meanwhile, if $\gamma$ is a path that first goes horizontally from $O$ to
$(x,0)$ and then vertically to $A=(x,y)$, we have $|\gamma|=\frac{x+y}{3}$,
which concludes the proof of the Lemma. ∎

######  Lemma 3.5.

Let $A=(0,p)$ and $B=(q,0)$ with $pq\geq 0$. The path $\gamma$ going
vertically from $A$ to $O$ and then horizontally from $O$ to $B$ is a
geodesic.

###### Proof.

Without loss of generality we may assume $p,q\geq 0$. By translating the plane
by $(0,-p)$, we move $A$ to $O$ and $B$ to $B^{\prime}=(q,-p)$. Note that due
to the translational symmetry of $\Gamma$, $d(A,B)=d(O,B^{\prime})$. By Lemma
3.4, $d(O,B^{\prime})=\frac{q+2p}{3}$. But this is exactly $|\gamma|$. ∎

######  Lemma 3.6.

Let $A=(a_{1},a_{2}),B=(b_{1},b_{2}),C=(c_{1},c_{2})$ be vertices on $\Gamma$
such that the region

$\Omega:=\left\\{(x,y)\in\mathbb{Z}^{2}~{}\middle|~{}a_{1}\leq x\leq
b_{1},~{}~{}a_{2}\leq y\leq c_{2},~{}~{}b_{2}-b_{1}\leq y-x\leq
c_{2}-c_{1}\right\\}$

is nonempty (the right picture in Figure 11). Let $X$ be an arbitrary vertex
of $\Gamma$. We have

$d(A,X)+d(B,X)+d(C,X)\geq\frac{-a_{1}-a_{2}+2b_{1}-b_{2}-c_{1}+2c_{2}}{3}.$

It attains the minimum when $X\in\Omega$.

###### Proof.

Let $X=(x,y)$. By Lemma 3.4, we have

$\displaystyle d(A,X)+d(B,X)+d(C,X)$
$\displaystyle\geq\frac{(x-a_{1})+(y-a_{2})}{3}+\frac{(y-b_{2})-2(x-b_{1})}{3}+\frac{(x-c_{1})-2(y-c_{2})}{3}$
$\displaystyle=\frac{-a_{1}-a_{2}-b_{2}+2b_{1}-c_{1}+2c_{2}}{3}.$

It attains equality when and only when $X\in\Omega$. ∎

###  3.3. Dual surfacoids

The dual diskoids, introduced in [FKK13, Fon12], are dual graphs of webs on
disks. We now extend this notion from disks to decorated surfaces.

######  Definition 3.7.

Given an $\mathcal{A}$-web $W$ on $\hat{S}$, its dual surfacoid $W^{*}$ is
obtained as follows.

  1. (1)

Place a vertex in each connected component of $\hat{S}\setminus W$;

  2. (2)

For every oriented edge $e$ of $W$ separating two connected components of
$\hat{S}\setminus W$, we draw an oriented edge $e^{*}$ connecting the
corresponding vertices of $W^{*}$ such that $i(e,e^{*})=\frac{1}{3}$.

As in Figure 12, each trivalent sink in $W$ corresponds to a clockwise
oriented triangle in $W^{*}$, and each trivalent source in $W$ corresponds to
a counterclockwise oriented triangle in $W^{*}$.

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$

Figure 12. Local picture of an $\mathcal{A}$-web $W$ and its dual surfacoid
$W^{*}$.

######  Example 3.8.

Continuing as in Example 2.3, let $W$ be a minimal ladder in a disk with two
marked points. The dual surfacoid $W^{*}$ is as in the left picture of Figure
13. Note that each “I” in $W$ corresponds to a rhombus formed by two adjacent
triangles in $W^{*}$. Moreover, by choosing a boundary marked point and a
boundary $\alpha$ of the bigon, the “I”s in the staircase of each ascending
arc along $\alpha$ give rise to a parallelogram formed by a row of rhombi; we
call such a parallelogram a _pendant_. Moreover, we can joint each pendant
with two chains of edges that go counterclockwise around all other pendants
and go to the two vertices near the marked points of the bigon. We call the
union of a pendant and its two chains of edges a _necklace_. We call each of
the two chains of edges a _string_ of the necklace. For each necklace, we can
travel from the region near one marked point of the bigon, along a string, and
then either clockwise or counterclockwise around the pendant, and then along
the other string, to get to the region near the other marked point of the
bigon; we call these two paths of edges the two _boundary paths_ of the
necklace. Note that necklaces can only intersect along their boundary paths,
and the union of all necklaces is the whole of $W^{*}$.

Figure 13. Left: a bigon region and an example of a dual surfacoid $W^{*}$.
Right: the three necklaces in the surfacoid $W^{*}$ defined by the three
ascending arcs along the left boundary of the bigon with respect to the top
marked point.

######  Example 3.9.

Continuing as in Example 2.4, let $W$ be a reduced $\mathcal{A}$ web on a disk
$\Delta$ with three marked points. The dual surfacoid $W^{*}$ is as in Figure
14. It is the union of a triangle consisting of oriented $3$-cycles and three
possible oriented curves going from each corner of the triangle to a marked
point. We call dual surfacoid $W^{*}$ of this form a _net_ ; in particular, we
refer to the subgraph spanned by oriented $3$-cycles as the _mesh_ of the net
$W^{*}$ and the three curves as the _strings_ of the net $W^{*}$. Let $\alpha$
be a boundary interval of $\Delta$. Tracing the edges of $W$ intersected by
$\alpha$ results in a path in $W^{*}$ with two strings and one side of the
mesh, which we continue to call $\alpha$ and refer to it as a _boundary path_
of the net $W^{*}$.

Figure 14. A triangle region

######  Proposition 3.10.

The dual surfacoid $W^{*}$ for a reduced web $W$ can be obtained by gluing
nets and necklaces along their boundary paths.

###### Proof.

It follows from Proposition 2.5. ∎

######  Definition 3.11.

Let $\alpha$ be a boundary path of a net or a necklace and let $A$ and $B$ be
vertices on $\alpha$. The path $\gamma_{AB}$ on $\alpha$ that goes directly
from $A$ to $B$ is called the straight path from $A$ to $B$.

######  Lemma 3.12.

Let $W^{*}$ be a net or a necklace and let $A$ and $B$ be vertices on a
boundary path $\alpha$ of $W^{*}$. Then the straight path $\gamma_{AB}$ is a
geodesic on $W^{*}$.

###### Proof.

Let $\gamma$ be an arbitrary path from $A$ to $B$. It is equivalent to show
that $|\gamma|\geq|\gamma_{AB}|$.

First, we claim that if $W^{*}$ is a net, then we may assume $\gamma$ never
travels along any edges inside the string $\beta$ that is not part of
$\alpha$. To see this, let $C$ be the vertex jointing $\beta$ to the mesh. If
$\gamma$ ever travels along an edge in $\beta$, then there must be a subpath
$\gamma^{\prime}$ of $\gamma$ that goes from $C$ into $\beta$ and then back
out to $C$; removing this subpath $\gamma^{\prime}$ from $\gamma$ shortens
$\gamma$.

Under the assumption above, we can find vertices
$A=A_{0},A_{1},A_{2},\dots,A_{n}=B$ in $\gamma\cap\alpha$ to split $\gamma$
into consecutive paths $\gamma_{1},\ldots,\gamma_{n}$ (with each $\gamma_{i}$
goes from $A_{i-1}$ to $A_{i}$) such that

  * (a)

$\gamma_{i}$ is contained in the mesh (when $W^{*}$ is a net);

  * (b)

$\gamma_{i}$ is contained in the pendant (when $W^{*}$ is a necklace);

  * (c)

$\gamma_{i}$ is contained in $\alpha$;

$O$$I$$II$$III$$IV$ Figure 15. Subgraphs of $\Gamma$.

As in Figure 15, we place the mesh in the second or the fourth quadrant and
place a pendant in the first or the third quadrant, realizing them as
subgraphs of $\Gamma$. Let $\gamma_{A_{i-1}A_{i}}$ be the straight path from
$A_{i-1}$ to $A_{i}$. By Lemma 3.5, we have

$|\gamma_{i}|\geq|\gamma_{A_{i-1}A_{i}}|.$

Therefore

$|\gamma|=\sum_{i}|\gamma_{i}|\geq\sum_{i}|\gamma_{A_{i-1}A_{i}}|\geq|\gamma_{AB}|.\qed$

######  Lemma 3.13.

Let $W^{*}$ be a net or a necklace and let $A_{1},A_{2},A_{3}$ be vertices on
a boundary path $\alpha$ of $W^{*}$. Let $m$ be the minimum of

$d(A_{1},X)+d(A_{2},X)+d(A_{3},X)$

as $X$ varies over all vertices of $W^{*}$. Then there exists a vertex $Y$ on
$\alpha$ such that $d(A_{1},Y)+d(A_{2},Y)+d(A_{3},Y)=m$.

###### Proof.

For $i\in\\{1,2,3\\}$, let $\gamma_{i}$ be a path from $A_{i}$ to $X$. It
suffices to show that there exists a vertex $Y$ on $\alpha$ such that

(10)
$d(A_{1},Y)+d(A_{2},Y)+d(A_{3},Y)\leq|\gamma_{1}|+|\gamma_{2}|+|\gamma_{3}|.$

If $X$ is on $\alpha$, then (10) follows directly.

If $W^{*}$ is a net and $X$ is on the string $\beta$ that is not part of
$\alpha$, then we may replace $X$ by the vertex $C$ where $\beta$ joints the
mesh, and this replacement shortens all three paths $\gamma_{i}$. Thus,
without loss of generality, we assume that $X$ is either contained in the mesh
(when $W^{*}$ is a net) or in the pendant (when $W^{*}$ is a necklace).

For each path $\gamma_{i}$, let us assume that $A_{i}$ is the only vertex
contained in $\alpha$. In this case, the paths $\gamma_{1}$, $\gamma_{2}$, and
$\gamma_{3}$ are all contained in the mesh (when $W^{*}$ is a net) or in the
pendant (when $W^{*}$ is a necklace). As in Figure 15, we may realize them as
subgraphs on $\Gamma$ such that the vertices $A_{1}$, $A_{2}$, and $A_{3}$ are
on the axes of $\mathbb{Z}^{2}$. It is then reduced into the following three
cases

  * •

$A_{1}=(a,0)$, $A_{2}=(b,0)$, $A_{3}=(c,0)$, where $c\leq a\leq b$;

  * •

$A_{1}=(0,a)$, $A_{2}=(b,0)$, $A_{3}=(0,c)$, where $b\geq 0$, and $c\geq a\geq
0$;

  * •

$A_{1}=(0,a)$, $A_{2}=(0,b)$, $A_{3}=(c,0)$, where $b\leq a\leq 0$, and $c\leq
0$.

Then the Lemma follows directly from Lemmas 3.5 and 3.6.

Now for a general path $\gamma_{i}$, let $B_{i}$ be the last vertex on
$\gamma_{i}$ such that $B_{i}\in\alpha$. It splits $\gamma_{i}$ into two
paths: the path $\gamma_{i}^{\prime}$ from $A_{i}$ to $B_{i}$, and the path
$\gamma_{i}^{\prime\prime}$ from $B_{i}$ to $X$. Then

(11) $|\gamma_{i}|=|\gamma_{i}^{\prime}|+|\gamma_{i}^{\prime\prime}|.$

By the above discussion, there exists a $Y\in\alpha$, such that

(12)
$d(B_{1},Y)+d(B_{2},Y)+d(B_{3},Y)\leq|\gamma_{1}^{\prime\prime}|+|\gamma_{2}^{\prime\prime}|+|\gamma_{3}^{\prime\prime}|.$

Meanwhile

(13) $d(A_{i},Y)\leq|\gamma_{i}^{\prime}|+d(B_{i},Y).$

Combining (11)-(13), we obtain the inequality (10). ∎

###  3.4. From webs to hives

We first prove Theorem 3.1 when $\hat{S}=\Delta$ is a triangle.

######  Lemma 3.14.

Let $W\in\mathscr{W}_{\Delta}^{\mathcal{A}}$ be a reduced web parametrized by
$(x,y,z,t,u,v,w)\in\mathbb{Z}\times\mathbb{N}^{6}$ as in Example 2.4. Let
$(a_{1},\ldots,a_{7})$ be the coordinates of $i_{\Delta}(W)$ associated with
the vertices of $Q_{\Delta}$ as in Figure 9. Then

$\displaystyle a_{1}$ $\displaystyle=\frac{2t+u+2w+v+\max\\{2x,-x\\}}{3},$
$\displaystyle a_{2}$ $\displaystyle=\frac{2w+v+2z+y+\max\\{x,-2x\\}}{3},$
$\displaystyle a_{3}$ $\displaystyle=\frac{2v+w+2u+t+\max\\{x,-2x\\}}{3},$
$\displaystyle a_{4}$
$\displaystyle=\frac{2v+w+2t+u+2z+y+\max\\{3x,-3x\\}}{3},$ $\displaystyle
a_{5}$ $\displaystyle=\frac{2v+w+2y+z+\max\\{2x,-x\\}}{3},$ $\displaystyle
a_{6}$ $\displaystyle=\frac{2z+y+2u+t+\max\\{2x,-x\\}}{3},$ $\displaystyle
a_{7}$ $\displaystyle=\frac{2t+u+2y+z+\max\\{x,-2x\\}}{3}.$

###### Proof.

Recall that the dual surfacoid $W^{*}$, which is a net consisting of a mesh
with three strings attached. We denote the endpoints of the strings by
$A,A^{\prime},B,B^{\prime},C,C^{\prime}$, as in Figure 16.

$\bullet$$A$$\bullet$$\bullet$$A^{\prime}$$\bullet$$B$$\bullet$$\bullet$$B^{\prime}$$\bullet$$C^{\prime}$$\bullet$$\bullet$$C$
Figure 16. The dual surfacoid $W^{*}$ on a triangle

Let $V$ be an inward tripod as in Figure 8. Let $X$ be an arbitrary vertex on
$W^{*}$. By definition, we have

$a_{4}=\mathbb{I}([W],[V])=\min_{X\in W^{*}}\\{d(A,X)+d(B,X)+d(C,X)\\}.$

Note that if $X$ is placed on one of the strings, say $AA^{\prime}$, then we
can move $X$ to the vertex $A^{\prime}$ without increasing the value of the
sum $d(A,X)+d(B,X)+d(C,X)$. Thus, we may assume that $X$ is inside the mesh.
On the one hand, by Lemma (3.6), we get

$d(A^{\prime},X)+d(B^{\prime},X)+d(C^{\prime},X)=|x|.$

On the other hand, it is not hard to see that
$d(A,X)=d(A,A^{\prime})+d(A^{\prime},X)$,
$d(B,X)=d(B,B^{\prime})+d(B^{\prime},X)$, and
$d(C,X)=d(C,A^{\prime})+d(C^{\prime},X)$. Therefore

$\displaystyle a_{4}=$ $\displaystyle
d(A,A^{\prime})+d(A^{\prime},X)+d(B,B^{\prime})+d(B^{\prime},X)+d(C,C^{\prime})+d(C^{\prime},X)$
$\displaystyle=$ $\displaystyle\frac{2v+w+2t+u+2z+y+\max\\{3x,-3x\\}}{3}$

If $V$ is a boundary arc as in Figure 8, then by Lemma 3.12, the intersection
$i(W,V)$ attains the minimum when $V$ is a straight path on $W^{*}$. In this
way, we obtain the values of the rest $a_{i}$’s as desired. ∎

######  Proposition 3.15.

The map $i_{\Delta}$ is a bijection from $\mathscr{W}_{\Delta}^{\mathcal{A}}$
to ${\bf Hive}(\Delta)$.

###### Proof.

First, we show that the data $i_{\Delta}(W)$ satisfies the rhombus condition.
Indeed, by Lemma 3.14,

$a_{1}+a_{2}-a_{4}=w;\qquad a_{3}+a_{4}-a_{1}-a_{6}=v+\max\\{0,-x\\};\qquad
a_{4}+a_{5}-a_{2}-a_{7}=v+\max\\{x,0\\}$

are all non-negative integers. The same hold for the rest six values.

Conversely, let $(a_{1},\ldots,a_{7})$ be a hive, then we get
$(x,y,z,u,v,w,t)\in\mathbb{Z}\times\mathbb{N}^{6}$, where

$\displaystyle x$ $\displaystyle=a_{1}+a_{5}+a_{6}-a_{2}-a_{3}-a_{7},$
$\displaystyle w$ $\displaystyle=a_{1}+a_{2}-a_{4},\quad\quad
u=a_{3}+a_{6}-a_{4},\quad\quad y=a_{5}+a_{7}-a_{4},$ $\displaystyle v$
$\displaystyle=\min\\{a_{3}+a_{4}-a_{6}-a_{1},~{}a_{4}+a_{5}-a_{2}-a_{7}\\},$
$\displaystyle t$
$\displaystyle=\min\\{a_{1}+a_{4}-a_{3}-a_{2},~{}a_{4}+a_{7}-a_{6}-a_{5}\\},$
$\displaystyle z$
$\displaystyle=\min\\{a_{2}+a_{4}-a_{1}-a_{5},~{}a_{4}+a_{6}-a_{3}-a_{7}\\}.$

Therefore, the map $i_{\Delta}$ is a bijection. ∎

Now let $\hat{S}$ be an arbitrary decorated surface. Fix a reduced web
$W\in\mathscr{W}_{\hat{S}}^{\mathcal{A}}$. Let $\gamma$ be an oriented path on
$\hat{S}$ that connects two components of $\hat{S}\backslash W$. We assume
that $\gamma$ intersects with $W$ transversely and consider the intersection
number $i(W,\gamma)$ via (4). Note that $\gamma$ corresponds to a path on the
dual surfacoid $W^{*}$, and $i(W,\gamma)$ is equal to the intersection metric
length of that path. Abusing notation, we define the length of $\gamma$ as

$|\gamma|:=i(W,\gamma).$

Let $\mathcal{T}_{2}$ be a split ideal triangulation of $\hat{S}$. Following
Propositions 2.5 and 3.10, we can split the bigons in $\mathcal{T}_{2}$ into
smaller bigons such that the restriction of the dual surfacoids $W^{*}$ to
each smaller bigon is a necklace, and to each triangle is a net. Denote by
resulted decomposition of $\hat{S}$ by ${\mathcal{T}}_{2}^{\prime}$. After
necessarily perturbing the edges of ${\mathcal{T}}_{2}^{\prime}$, we assume
that

  * •

the path $\gamma$ intersects transitively with the edges of
${\mathcal{T}}_{2}^{\prime}$;

  * •

the endpoints of $\gamma$ are not on the edges of
${\mathcal{T}}_{2}^{\prime}$.

We define the _crossing sequence_ of $\gamma$ to be the sequence of edges
$(\alpha_{1},\dots,\alpha_{n})$ of ${\mathcal{T}}_{2}^{\prime}$ that $\gamma$
crosses. Let $(B_{1},\ldots,B_{n})$ be the corresponding intersecting points
of $\gamma$ with the edges $\alpha_{i}$’s. They divide $\gamma$ into paths
$\gamma_{0}$, …, $\gamma_{n}$. When $\alpha_{i}=\alpha_{i+1}=\alpha$, let us
replace the subpath $\gamma_{i}$ from $B_{i}$ and $B_{i+1}$ by the path along
$\alpha$, and then move it slightly to the other side of $\alpha$. In this
way, we obtain a new path $\gamma^{\prime}$ as in Figure 17. Note that
$\gamma^{\prime}$ is homotopy equivalent to $\gamma$.

$\gamma$$B_{i}$$B_{i+1}$

$B_{i}$$B_{i+1}$$\gamma^{\prime}$

Figure 17. Local modification of $\gamma$.

######  Lemma 3.16.

We have $|\gamma^{\prime}|\leq|\gamma|$.

###### Proof.

It is a direct consequence of Lemma 3.12. ∎

######  Remark 3.17.

As a consequence of Lemma 3.16, when we try to minimize the intersection
metric length of a path $\gamma$, we may assume without loss of generality
that the crossing sequence of $\gamma$ does not have two identical ideal edges
next to each other.

######  Lemma 3.18.

Let $W$ and ${\mathcal{T}}_{2}$ be as above. Let $\alpha$ be an oriented ideal
edge of $\mathcal{T}_{2}$ and let $\gamma$ be an arbitrary path that is
homotopic to $\alpha$. Then $|\alpha|\leq|\gamma|.$

###### Proof.

Let $\Delta$ be the ideal triangle adjacent to $\alpha$. Let us perturb the
endpoints of $\gamma$ and $\alpha$ slightly, obtaining two paths
$\gamma^{\prime}$ and $\alpha^{\prime}$, such that

  * •

$|\alpha|=|\alpha^{\prime}|$ and $|\gamma|=|\gamma^{\prime}|$,

  * •

$\alpha^{\prime}$ is contained inside $\Delta$,

  * •

$\gamma^{\prime}$ is homotopic to $\alpha^{\prime}$.

Let $\tilde{S}$ be the universal cover of $\hat{S}$. We consider the further
decomposition $\mathcal{T}^{\prime}_{2}$ as above. Let us lift
$\mathcal{T}^{\prime}_{2}$ to a decomposition
$\tilde{\mathcal{T}}_{2}^{\prime}$ of $\tilde{S}$, and lift $\alpha^{\prime}$
and $\gamma^{\prime}$ to one of their representatives, denoted by
$\tilde{\alpha}^{\prime}$ and $\tilde{\gamma}^{\prime}$ respectively.

Let $(\alpha_{1},\alpha_{2},\cdots,\alpha_{n})$ be the crossing sequence of
$\tilde{\gamma}^{\prime}$ with $\tilde{\mathcal{T}}_{2}^{\prime}$. According
to Remark 3.17, we may assume that $\alpha_{i}\neq\alpha_{i+1}$ for any $i$.
If the crossing sequence of $\tilde{\gamma}^{\prime}$ is not empty, then
$\tilde{\gamma}^{\prime}$ will cross a sequence of distinct triangles and
bigons in $\tilde{\mathcal{T}}_{2}^{\prime}$ (Figure 18). In particular, it
will never return back to the ideal triangle that it starts with, which
contradicts with the assumption that $\alpha^{\prime}$ and $\gamma^{\prime}$
are isotopic. Thus, the crossing sequence of $\tilde{\gamma}^{\prime}$ must be
empty, i.e., $\tilde{\gamma}^{\prime}$ stays within the same ideal triangle as
$\alpha^{\prime}$ resides. By Lemma 3.12, we have

$|\alpha|=|\alpha^{\prime}|=|\tilde{\alpha}^{\prime}|\leq|\tilde{\gamma}^{\prime}|=|\gamma^{\prime}|=|\gamma|.$

This proves our lemma. ∎

$\bullet$$\bullet$$\tilde{\alpha}^{\prime}$$\alpha_{1}$$\alpha_{2}$$\alpha_{3}$$\alpha_{4}$$\tilde{\gamma}^{\prime}$
Figure 18. A lift of $\gamma^{\prime}$ with a non-empty crossing sequence to
the universal cover $\tilde{S}$.

######  Remark 3.19.

Lemma 3.18 has been proved by Kuperburg [Kup96, Lemma 6.5] using a discrete
version of the Gauss-Bonnet theorem. The advantage of our strategy is that it
could be used to prove the following key lemma dealing with the tripod.

Let $W$ and $\mathcal{T}_{2}$ be as above. Let $\Delta$ be an ideal triangle
of $\mathcal{T}_{2}$ with vertices $a,b,c$. Let $Y$ be an inward tripod going
from $a,b,c$ to the center of $\Delta$, as depicted in Figure 8(8).

######  Lemma 3.20.

Let $V$ be a reduced $\mathcal{X}$ web that is homotopic to $Y$. The
intersection number $i(W,V)$ attains the minimum when $V$ is contained in the
triangle $\Delta$.

###### Proof.

Let us perturb the three vertices of $V$ such that they are all contained
inside the triangle $\Delta$. The web $V$ is homotopic equivalent to the web
$Y$ as in the picture.

$\bullet$$\bullet$$\bullet$$Y$$V$${\gamma}_{1}$${\gamma}_{2}$${\gamma}_{3}$
Figure 19. A web $V$ that is homotopic to an inward tripod $Y$.

Without loss of generality, let us assume that $\hat{S}$ is a disk. Otherwise,
we may lift everything to the universal cover of $\hat{S}$ as in the proof of
Lemma 3.18. Denote the three legs of $V$ by $\gamma_{1}$, $\gamma_{2}$,
$\gamma_{3}$ respectively. By Remark 3.17, we assume that their crossing
sequences with the decomposition $\tilde{\mathcal{T}}_{2}^{\prime}$ do not
have two identical arcs next to each other. Since $\gamma_{1}$, $\gamma_{2}$,
$\gamma_{3}$ connects the triangle $\Delta$ to the same vertex, their crossing
sequences must be the same. In other words, they cross the same sequences of
triangles and bigons. The cross sequence separates $\gamma_{i}$ into several
paths. As illustrated by Figure 20, we replace the last part of each
$\gamma_{i}$ by a straight path along the ideal arc, and move them slightly
back to the other side of the arc. The newly obtained tripod $V^{\prime}$ is
homotopic to $V$. By Lemma 3.13, we can find $V^{\prime}$ such that

$i(W,V^{\prime})\leq i(W,V).$

Let us repeat the same procedure. Eventually, we obtain a tripod inside the
triangle $\Delta$, which concludes the proof of the Lemma. ∎

$V$

$V^{\prime}$

Figure 20. Local modification of $V$.

We are now ready to prove the following theorem, which is the first part of
Theorem 3.1.

######  Theorem 3.21.

For any ideal triangulation $\mathcal{T}$ of the marked surface $\hat{S}$, the
map $i_{\mathcal{T}}$ is a bijection from
$\mathscr{W}_{\hat{S}}^{\mathcal{A}}$ to ${\bf Hive}(\mathcal{T})$.

###### Proof.

Let $W\in\mathscr{W}_{\hat{S}}^{\mathcal{A}}$ be in a good position with
respect to $\mathcal{T}_{2}$. By Lemma 3.16 and Lemma 3.20, we may restrict
the ideal edges and the tripods within their corresponding ideal triangles. By
Proposition 3.15, we see that $i_{\mathcal{T}}(W)$ is a hive in ${\bf
Hive}(\mathcal{T})$.

It remains to show that for any hive $\\{a_{\bullet}\\}$ in ${\bf
Hive}(\mathcal{T})$, there is a unique reduced web $W$ such that
$i_{\mathcal{T}}(W)=\\{a_{\bullet}\\}$. By restricting the data
$\\{a_{\bullet}\\}$ to each ideal triangle in $\mathcal{T}$ and applying
Proposition 3.15, we obtain a reduced web $W_{t}$ for each ideal triangle $t$
of $\mathcal{T}$. Meanwhile, by a direct calculation, if the hive
$\\{a_{\bullet}\\}$ on one side of $t$ are $a_{1}$ and $a_{2}$, then the
number of the oriented arcs of $W_{t}$ intersecting with the same side of $t$
are $2a_{1}-a_{2}$ and $2a_{2}-a_{1}$ respectively.

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$a_{1}$$a_{2}$$2a_{1}-a_{2}$$2a_{2}-a_{1}$
Figure 21. Oriented arcs that intersect a side of a triangle.

As a result, we see that for any neighbored ideal triangles $t$ and
$t^{\prime}$, the webs $W_{t}$ and $W_{t^{\prime}}$ can be connected by a
minimal ladder in the bigon separating them. In this way, we obtain a web $W$
in a good position with respect to $\mathcal{T}_{2}$. Following the procedure
in Section 6.3 of [DS20a], one may reorder the position of the corner arcs in
each ideal triangle, and resolve the “I” bars in the bigons, obtaining a
reduced web $W^{\prime}.$ Meanwhile, by Lemma 57 of loc.cit., the obtained
reduced web $W^{\prime}$ is unique upto homotopy equivalence on
$\hat{S}\times[0,1]$. Hence, the map $i_{\mathcal{T}}$ is a bijection. ∎

###  3.5. Compatibility

In this subsection, we prove the second part of Theorem 3.1, that is, the
images of intersection maps for different ideal triangulations are related by
octahedron relations.

$A$

$B$

$C$

$D$

Figure 22. Exchanging branches

######  Lemma 3.22.

Let $W$ be a reduced $\mathcal{A}$ web on $\hat{S}$ and let $A$, $B$, $C$, and
$D$ be reduced $\mathcal{X}$ webs in Figure 22. We have

$\mathbb{I}([W],[A])+\mathbb{I}([W],[B])=\max\\{\mathbb{I}([W],[C]),\mathbb{I}([W],[D])\\}.$

###### Proof.

By the local properties of the intersection maps for tripods and ideal edges
(Lemma 3.18 and Lemma 3.20), it suffices to consider the case when $\hat{S}$
is a disk with four marked points. Let $W^{*}$ be the dual surfacoid of $W$ as
in Figure 12. We compute the desired intersection numbers by making the four
$\mathcal{X}$-webs travel along $W^{*}$.

With respect to the top-bottom triangulations of $\hat{S}$, $W^{*}$ is
separated into three subgraphs:

  * •

a net $T(W^{*})$ in the top triangle,

  * •

a union of necklaces $H(W^{*})$ in the middle bigon,

  * •

a net $B(W^{*})$ in the bottom triangle.

The pattern of each region is illustrated by Figure 14. By Lemma 3.20, there
exists a representative $A$ of $[A]$ attaining the value
$\mathbb{I}([W],[A])$, with the trivalent vertex lying inside the mesh of
$T(W^{*})$. In particular, by Lemma 3.18, we can fix the trajectory of the
tripod $A$ along $W^{*}$ so that its two lower legs travel along the boundary
of $T(W^{*})$, meeting at the trivalent vertex located at the top corner $x$
of the mesh of $T(W^{*})$.

$W$$W^{*}$corner arcs$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$x$

$W^{*}$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$

Figure 23. Left: local picture of $W^{*}$ when $W$ is in good position with
respect to the top-bottom triangulation. Right: trajectory of the tripod $A$
along $W^{*}$.

With respect to the left-right triangulation of $\hat{S}$, $W^{*}$ is
seperated into three subgraphs:

  * •

a net $L(W^{*})$ in the left triangle;

  * •

a union of necklaces $V(W^{*})$ in the middle triangle;

  * •

a net $R(W^{*})$ in the right triangle.

The vertex $x$ of $W^{*}$ now belongs to the bigon $V(W^{*})$ in the middle.
By Lemma 3.18, $i(W,B)$ attains its minimum when $B$ travels along a straight
path that is a boundary of the bigon region. Note that this straight path must
pass through the vertex $x$. By exchanging a leg of $A$ and part of $B$ at the
vertex $x$, we get a representative of $[C]$ and a representative of $[D]$ as
in Figure 24. This shows that

(14)
$\mathbb{I}([W],[A])+\mathbb{I}([W],[B])\geq\max\\{\mathbb{I}([W],[C]),\mathbb{I}([W],[D])\\}.$

$\rightarrow$ $\rightarrow$

Figure 24. Exchanging legs

$r_{1}$$l_{1}$$r_{2}$$l_{2}$

$r$$l$

Figure 25. Corner arcs with respect to different triangulations

As illustrated by Figure 25, let $r_{1}$ and $l_{1}$ be the numbers of
oriented corner arcs of $W$ restricted to the left triangle. Let $r_{2}$ and
$l_{2}$ be the numbers of corner arcs restricted to the right triangle. Let
$r$ and $l$ be the numbers of corner arcs restricted to the top triangle.
Following the procedure of gluing ideal triangles, we have

$r=\min\\{r_{1},r_{2}\\};\qquad l=\min\\{l_{1},l_{2}\\}$

To show the equality of (14), we split the proof into two cases.

Case 1: $l_{1}\leq l_{2}$. Let $y$ be the vertex on the top corner of the mesh
of $L(W^{*})$. Recall the vertex $x$ in Figure 23. After exchanging the legs
as in the right picture of Figure 24, the resulted representative $D$ has two
$\mathcal{X}$-web components: a straight path that is parallel to the upper
right side of the quadrilateral, and a tripod on the left with trivalent
vertex at $x$. Note that one of the branches of the tripod is still traveling
along the left boundary of the bigon region until it reaches $x$. By Lemma
3.18, the intersection number between $W$ and the straight path component
already attains its minimum.

If $r_{1}\leq r_{2}$, then $x$ coincides with $y$. Hence $x$ resides on the
mesh of $L(W^{*})$. By Lemma 3.20, the intersection number between $W$ and the
left tripod reaches its minimum as well, and hence we can conclude that

$\mathbb{I}([W],[A])+\mathbb{I}([W],[B])=\mathbb{I}([W],[D]).$

If $r_{1}>r_{2}$, then there are $r_{1}-r_{2}$ many upward edges from $y$ to
$x$. By construction, two of the branches of the tripod travel along the
straight path from $y$ to $x$ (left picture in Figure 26). If we move the
trivalent vertex of the tripod from $x$ to $y$ along the straight path,
shrinking the two lower branches while extending the upper branch, the
intersection number between the tripod and $W$ remains unchanged. In the end,
we arrive at a configuration as in the right picture of Figure 26. Since the
trivalent vertex of the tripod is inside the triangle region now, we can again
apply Lemma 3.20 and conclude that

$\mathbb{I}([W],[A])+\mathbb{I}([W],[B])=\mathbb{I}([W],[D]).$

$\bullet$$x$$\bullet$$y$

$\bullet$$x$$\bullet$$y$

Figure 26. Local pictures between the vertices $x$ and $y$ in $W^{*}$.

Case 2: $l_{1}\geq l_{2}$. This is symmetric to the previous case and can be
handled analogously, yielding
$\mathbb{I}([W],[A])+\mathbb{I}([W],[B])=\mathbb{I}([W],[C])$ instead. ∎

Let $\mathcal{T}$ and $\mathcal{T}^{\prime}$ be two ideal triangulations
related by a flip of a diagonal. Following Figure (27), the four new webs
associated with $\mathcal{T}^{\prime}$ can be obtained in four steps of
changes. In each step, we only use the relation in Lemma 3.22. A direct
comparison shows that they coincide with the four octahedron relations in (7),
which concludes the proof of the second part of Theorem 3.1.

Figure 27. The flip $\mu=\mu_{4}\circ\mu_{3}\circ\mu_{2}\circ\mu_{1}$.

##  4\. Fock-Goncharov Moduli Spaces

In this section, we recall the Fock-Goncharov $\mathcal{A}$-moduli spaces and
their cluster structures. We further consider the tropicalization of the
$\mathcal{A}$-moduli spaces restricted by the potential function $P$ following
[GS15].

###  4.1. Definitions

A _flag_ $F$ in a 3-dimensional vector space $E$ is a filtration of subspaces

$\\{0\\}=F^{(0)}\subset F^{(1)}\subset F^{(2)}\subset
F^{(3)}=E,\qquad\mbox{where }{\dim}F^{(i)}=i.$

A _decorated flag_ is a flag $F$ with a pair of non-zero vectors
${f}^{i}\in\wedge^{i}F^{(i)}$ for $i=1,2$. Let $\mathcal{A}$ denote the moduli
space of decorated flags. The action of ${\rm SL}_{3}$ on $E$ induces a
transitive action of $\operatorname{SL}_{3}$ on $\mathcal{A}$. After fixing a
decorated flag, we obtain an isomorphism
$\mathcal{A}\stackrel{{\scriptstyle\sim}}{{=}}{\rm SL}_{3}/{\rm U}$, where
${\rm U}\subset{\rm SL}_{3}$ is the subgroup of unipotent upper triangular
matrices.

Let $\hat{S}$ be a decorated surface as in Section 2.2. Following [FG06,
Definition 2.4], we recall the Fock-Goncharov moduli space
$\mathcal{A}_{\operatorname{SL}_{3},\hat{S}}$.

######  Definition 4.1.

Let $\rho$ be a $\operatorname{SL}_{3}$-local system on $S$. A decoration on
$\rho$ is a flat section $\xi$ of the restriction of the associated decorated
flag bundle $\rho\underset{\operatorname{SL}_{3}}{\times}\mathcal{A}$ to
$m_{b}\cup m_{p}$. The moduli space
$\mathcal{A}_{\operatorname{SL}_{3},\hat{S}}$ parametrizes the pairs
$(\rho,\xi)$ up to the equivalence $(\rho,\xi)\sim(g\rho g^{-1},g\xi)$ for
$g\in{\rm SL}_{3}$.

######  Example 4.2.

Let $\hat{S}=\Delta$ be a disk with three marked points. Note that the
fundamental group $\pi_{1}(\Delta)$ is trivial. Therefore the moduli space
$\mathcal{A}_{{\rm SL}_{3},\Delta}$ parametrizes a triple of decorated flags
modulo the diagonal action of ${\rm SL}_{3}$:

$\mathcal{A}_{{\rm SL}_{3},\Delta}={\rm SL}_{3}\backslash\mathcal{A}^{3}.$

Recall the quiver $Q_{\Delta}$. Each vertex $v$ of $Q_{\Delta}$ corresponds to
a triple $(i,j,k)$ of non-negative integers with $i+j+k=3$.

$(0,2,1)$$(0,1,2)$$(1,2,0)$$(2,1,0)$$(1,0,2)$$(2,0,1)$$(1,1,1)$$(f_{1},f_{2})$$(g_{1},g_{2})$$(h_{1},h_{2})$
Figure 28. The quiver $Q_{\Delta}$.

Every marked point of $\Delta$ is associated with a decorated flag. Let

$\displaystyle(f_{1},f_{2}),\;(g_{1},g_{2}),\;(h_{1},h_{2})$

be their corresponding pairs of nonzero vectors. Let us fix a volume form
$\Omega$ on the 3-dimensional vector space $E$. Following [FG06, Section 9],
we set

(15) $A_{i,j,k}:=\Omega\left(f^{i}\wedge g^{j}\wedge h^{k}\right).$

In general, let $\mathcal{T}$ be an ideal triangulation of a decorated surface
$\hat{S}$. Recall the quiver $Q_{\mathcal{T}}$ with the vertex set
$\Theta_{\mathcal{T}}$. The restriction of $(\rho,\xi)$ to every ideal
triangle in $\mathcal{T}$ gives rise to a triple of decorated flags. By
applying (15), we obtain a collection $\\{A_{v}\\}_{v\in\Theta_{\mathcal{T}}}$
of Fock-Goncharov $\mathcal{A}$-coordinates for $\mathcal{A}_{{\rm
SL}_{3},\hat{S}}$.

$A_{6}$$A_{2}$$A_{11}$$A_{12}$$A_{10}$$A_{9}$$A_{3}$$A_{8}$$A_{7}$$A_{5}$$A_{1}$$A_{4}$$B_{6}$$B_{2}$$A_{11}$$A_{12}$$A_{10}$$A_{9}$$A_{3}$$A_{8}$$A_{7}$$A_{5}$$A_{1}$$A_{4}$$B_{6}$$B_{2}$$A_{11}$$A_{12}$$A_{10}$$A_{9}$$A_{3}$$A_{8}$$B_{7}$$B_{5}$$A_{1}$$A_{4}$
Figure 29. Fock-Goncharov coordinates related by a flip of a diagonal

The coordinate charts $\\{A_{v}\\}_{v\in\Theta_{\mathcal{T}}}$ corresponding
to different 3-triangulations of $\hat{S}$ are related by a sequence of
cluster mutations [FG06, Section 10]. In detail, suppose $\mathcal{T}$ and
$\mathcal{T}^{\prime}$ are related by a flip of a diagonal. Within the
quadrilateral containing the diagonal, the left quiver in Figure 29 consists
of the coordinates associated with $\mathcal{T}$, and the right quiver
consists of the coordinates associated with $\mathcal{T}^{\prime}$. They
related by four cluster mutations:

(16) $\begin{array}[]{l}A_{2}B_{2}=A_{1}A_{7}+A_{3}A_{5},\qquad
A_{6}B_{6}=A_{5}A_{11}+A_{7}A_{10}\\\ A_{5}B_{5}=A_{4}B_{6}+A_{9}B_{2},\qquad
A_{7}B_{7}=A_{12}B_{2}+A_{8}B_{6}.\\\ \end{array}$

The rest coordinates are kept invariant. Putting them together, we obtain a
canonical cluster $\mathcal{A}$ structure on the moduli space
$\mathcal{A}_{{\rm SL}_{3},\hat{S}}$.

The potential $P$, introduced in [GS15], is a function on the moduli space
$\mathcal{A}_{G,\hat{S}}$ for an arbitrary reductive group $G$. For the
purpose of the present paper, below we present an explicit expression of $P$
for $G={\rm SL}_{3}$ in terms of the Fock-Goncharov coordinates. Let $m$ be a
marked point or a puncture of $\hat{S}$. The monodromy $g_{m}$ of each
$(\rho,\xi)\in\mathcal{A}_{{\rm SL}_{3},\hat{S}}$ associted with $m$ is a
$3\times 3$ unipotent upper triangular matrix. Let $g_{ij}$ be the ${ij}$-th
entry of $g_{m}$. The potential associated with $m$ is defined as the sum of
the entries along the sub-diagonal $P_{m}:=g_{12}+g_{23}.$ The total potential
is

$P:=\sum_{m\in m_{b}\cup m_{p}}P_{m}.$

######  Example 4.3.

Continuing as in Example 4.2, let $a,b,c$ be the marked points of $\Delta$.
There are three unit rhombi whose short diagonals are parallel to the side
$bc$. As in Lemma 3.1 of [GS15], we define

(17)
$\alpha_{g}=\frac{A_{1,1,1}}{A_{2,1,0}A_{2,0,1}};\qquad\alpha_{b}=\frac{A_{2,1,0}A_{0,2,1}}{A_{1,2,0}A_{1,1,1}};\qquad\alpha_{r}=\frac{A_{0,1,2}A_{2,0,1}}{A_{1,1,1}A_{1,0,2}}.$

Here the $A$-coordinates on the right correspond to the vertices of each unit
rhombus, where the two vertices of the short diagonals are the denominators.
The potential associated with the vertex $a$ is the sum

$P(\Delta)_{a}:=\alpha_{g}+\alpha_{b}+\alpha_{r}.$

Similarly, by taking the unit rhombi with respect to $b$ and $c$, we get the
potential associated with the ideal triangle $\Delta$

$P(\Delta):=P(\Delta)_{a}+P(\Delta)_{b}+P(\Delta)_{c}.$

$a$$b$$c$ Figure 30. The green rhombus contains vertices that appear in the
definition of $\alpha_{g}$; the same goes for the blue rhombus with
$\alpha_{b}$ and the red rhombus with $\alpha_{r}$.

######  Remark 4.4.

Let $\mathcal{T}$ be an arbitrary ideal triangulation of $\hat{S}$. By the
scissor congruence invariance property of the potential in [GS15], the
potential $P$ is a function on $\mathcal{A}_{\operatorname{SL}_{3},\hat{S}}$
presented by

$P=\sum P(\Delta),$

where the sum is over all the ideal triangles $\Delta$ in $\mathcal{T}$. Note
that the original geometric definition of $P$ does not depend on the ideal
triangulation $\mathcal{T}$ chosen.

######  Remark 4.5.

Below we include a list of applications of the potential in the higher
Teichmüller theory.

  1. (1)

In [GS15], the potentials are understood as the mirror Landau–Ginzburg
potentials where they formulated a concrete homological mirror symmetry
between $(\mathcal{A}_{G,\hat{S}},P)$ and the generalized character variety
$\mathcal{L}_{G^{L},\hat{S}}$.

  2. (2)

When $\hat{S}$ is a disk with three marked points on the boundary and
$G=\operatorname{GL}_{n}$, the tropicalization of the potential $P$ recovers
the Knutson–Tao’s hive model [KT99].

  3. (3)

When $\hat{S}$ is a punctured surface, in [HS23], for each simple root and
each puncture, the partial potential is understood as generalized horocycle
length which provides a family of McShane-type identities.

  4. (4)

The partial potentials play important roles in the quantization of the moduli
spaces of $G$-local systems in [GS19] and the punctured skein relations [SSW].
In particular, when $\hat{S}$ is a punctured disk with two marked points, the
quantized partial potentials correspond to generators of the quantum group
$\mathcal{U}_{q}(\mathfrak{g})$. See [She22].

###  4.2. Tropicalization.

Every cluster variety admits a totally positive structure and can be further
tropicalized. Below we briefly recall the tropicalization of
$\mathcal{A}_{{\rm SL}_{3},\hat{S}}$.

Note that the transition maps (16) among different cluster charts of
$\mathcal{A}_{{\rm SL}_{3},\hat{S}}$ are subtraction-free. A positive rational
function is a nonzero function that can be presented as a ratio of two
polynomials with positive integer coefficients in one (and therefore every)
cluster chart of $\mathcal{A}_{{\rm SL}_{3},\hat{S}}$. Denote by
$\mathbb{Q}_{+}(\mathcal{A}_{{\rm SL}_{3},\hat{S}})$ the collection of all
positive functions.

A semifield is a set $F$ equipped with operations of addition and
multiplication, so that addition is commutative and associative,
multiplication makes $F$ into an abelian group, and they satisfy the usual
distributivity: $(a+b)c=ac+bc$ for $a,b,c\in F$. Note that
$\mathbb{Q}_{+}(\mathcal{A}_{{\rm SL}_{3},\hat{S}})$ is a semifield. For an
arbitrary semifield $F$, we define its set of $F$-points as

$\mathcal{A}_{{\rm SL}_{3},\hat{S}}(F):={\bf
Hom}_{\mbox{semifield}}\left(\mathbb{Q}_{+}(\mathcal{A}_{{\rm
SL}_{3},\hat{S}}),F\right).$

######  Example 4.6.

Let $F=\mathbb{R}_{>0}$ be the semifield of positive real numbers with usual
addition and multiplication. The $\mathbb{R}_{>0}$-points of
$\mathcal{A}_{{\rm SL}_{3},\hat{S}}$ gives rise to the total positive part of
$\mathcal{A}_{{\rm SL}_{3},\hat{S}}$

(18)
$\mathcal{A}_{\operatorname{SL}_{3},\hat{S}}(\mathbb{R}_{>0}):=\left\\{x\in\mathcal{A}_{\operatorname{SL}_{3},\hat{S}}~{}\middle|~{}A_{v}(x)>0~{}\mbox{for
every cluster coordinate $A_{v}$}\right\\}.$

Since the transition maps between any pair of cluster charts are given by
positive rational functions, it is enough to require that $A_{v}(x)>0$ in (18)
for $A_{v}$ associated with one cluster chart.

######  Example 4.7.

The tropical semifield $\mathbb{R}^{t}:=(\mathbb{R},+,\max)$ is the set
$\mathbb{R}$ with the usual addition as the multiplication and the $\max$ as
the addition. Let $f\in\mathbb{Q}_{+}(\mathcal{A}_{{\rm SL}_{3},\hat{S}})$ be
a positive rational function. Its tropicalization is a function defined
tautologically as

$f^{t}:\mathcal{A}_{\operatorname{SL}_{3},\hat{S}}(\mathbb{R}^{t})\longrightarrow\mathbb{R},\qquad
f^{t}(l):=l(f).$

The tropicalization $f^{t}$ is a piecewise linear function that replaces the
multiplication in $f$ by addition, and the addition by $\max$. For example, if
$f=\frac{A_{1}^{2}+A_{2}^{3}}{A_{1}+2}$, then

$f^{t}=\max\\{2A_{1}^{t},3A_{2}^{t}\\}-\max\\{A_{1}^{t},0\\}.$

Let $n=\dim\mathcal{A}_{{\rm SL}_{3},\hat{S}}$. The tropical set
$\mathcal{A}_{\operatorname{SL}_{3},\hat{S}}(\mathbb{R}^{t})$ has a piecewise-
linear structure, isomorphic to $\mathbb{R}^{n}$ in many different ways. In
detail, let $\alpha_{\mathcal{T}}:=\\{A_{1},\ldots,A_{n}\\}$ be the cluster
chart associated with an ideal triangulation $\mathcal{T}$ of $\hat{S}$. Its
tropicalization is a bijective map

$\alpha_{\mathcal{T}}^{t}:=(A_{1}^{t},\ldots,A_{n}^{t}):~{}\mathcal{A}_{\operatorname{SL}_{3},\hat{S}}\left(\mathbb{R}^{t}\right)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathbb{R}^{n}.$

For a different ideal triangulation $\mathcal{T}^{\prime}$, the transition map
$\alpha_{\mathcal{T}^{\prime}}^{t}\circ(\alpha_{\mathcal{T}}^{t})^{-1}$ is
given by the tropicalization of the positive birational map
$\alpha_{\mathcal{T}^{\prime}}\circ\alpha_{\mathcal{T}}^{-1}$. Similarly, we
may replace $\mathbb{R}$ by $\mathbb{Z}$ in the above construction, and define
the subsets of $\mathbb{Z}$\- tropical points

$\mathcal{A}_{\operatorname{SL}_{3},\hat{S}}\left(\mathbb{Z}^{t}\right)\subset\mathcal{A}_{\operatorname{SL}_{3},\hat{S}}\left(\mathbb{R}^{t}\right).$

By Equation (17), for each ideal triangulation $\mathcal{T}$, the
tropicalization of the potential $P$ is a piecewise linear function $P^{t}$ on
$\mathcal{A}_{{\rm SL}_{3},\hat{S}}(\mathbb{R}^{t})$ that can be presented as

(19) $P^{t}=\max\left\\{\alpha^{t}\right\\},$

where $\alpha$ are the Laurent polynomials as in (17) associated with unit
rhombus of $Q_{\mathcal{T}}$. By imposing the condition $P^{t}\leq 0$, we
obtain a convex cone

(20) $\mathcal{A}_{{\rm
SL}_{3},\hat{S}}^{+}\left(\mathbb{R}^{t}\right):=\left\\{x\in\mathcal{A}_{{\rm
SL}_{3},\hat{S}}\left(\mathbb{R}^{t}\right)~{}\middle|~{}P^{t}(x)\leq
0\right\\}.$

######  Remark 4.8.

The paper [GS15] uses the tropical semifield $(\mathbb{Z},+,\min)$, and define
the positive cone via requiring $P^{t}\geq 0$. The positive cone defined in
loc.cit. is isomorphic to (20) via the map taking $t\in\mathbb{Z}$ to $-t$.

By replacing the group ${\rm SL}_{3}$ by ${\rm PGL}_{3}$, we define the moduli
space $\mathcal{A}_{{\rm PGL}_{3},\hat{S}}$. Note that $\mathcal{A}_{{\rm
PGL}_{3},\hat{S}}$ may have several disconnected components. Meanwhile, there
is a natural finite-to-one map from $\mathcal{A}_{{\rm SL}_{3},\hat{S}}$ to
$\mathcal{A}_{{\rm PGL}_{3},\hat{S}}$, whose image is a component of
$\mathcal{A}_{{\rm PGL}_{3},\hat{S}}$ with a totally positive structure.
Similarly, we define the tropical points of $\mathcal{A}_{{\rm
PGL}_{3},\hat{S}}$ with a different lattice structure:

$\mathcal{A}_{\operatorname{SL}_{3},\hat{S}}(\mathbb{Z}^{t})\subset\mathcal{A}_{\operatorname{PGL}_{3},\hat{S}}(\mathbb{Z}^{t})\subset\mathcal{A}_{\operatorname{PGL}_{3},\hat{S}}\left(\mathbb{R}^{t}\right)=\mathcal{A}_{\operatorname{SL}_{3},\hat{S}}\left(\mathbb{R}^{t}\right).$

By imposing the condition $P^{t}\leq 0$, we obtain the cone

$\mathcal{A}_{\operatorname{PGL}_{3},\hat{S}}^{+}(\mathbb{Z}^{t})=\mathcal{A}_{\operatorname{PGL}_{3},\hat{S}}(\mathbb{Z}^{t})\cap\mathcal{A}_{\operatorname{SL}_{3},\hat{S}}^{+}(\mathbb{R}^{t})$

More explicitly, following [GS15], we see that

(21)
$\mathcal{A}_{\operatorname{PGL}_{3},\hat{S}}^{+}(\mathbb{Z}^{t}):=\left\\{x\in\mathcal{A}_{\operatorname{SL}_{3},\hat{S}}\left(\mathbb{R}^{t}\right)\;\bigg{|}\;\text{for
any }\alpha\text{ of $P^{t}$ in
\eqref{trop.potential}},\alpha^{t}(x)\in\mathbb{Z}_{\leq 0}\right\\}.$

By definition, the map $\alpha^{t}_{\mathcal{T}}$ identifies the set
$\mathcal{A}_{{\rm PGL}_{3},\hat{S}}^{+}$ with ${\bf Hive}(\mathcal{T})$. The
transition maps relating different triangulations are related by the
tropicalization of the cluster mutations (16), which coincides with octahedron
relations (7). Following Theorem 3.1, we get a natural isomorphism

$\mathcal{A}_{{\rm
PGL}_{3},\hat{S}}^{+}(\mathbb{Z}^{t})\stackrel{{\scriptstyle\sim}}{{=}}\mathscr{W}_{\hat{S}}^{\mathcal{A}}.$

###  4.3. Intersection pairings revisited

In this subsection, we discuss several different perspectives on the
intersection pairings.

A. Geometric interpretation of reduced $\mathcal{X}$-webs. The paper [GS15]
introduces the moduli space $\mathcal{P}_{{\rm G},\hat{S}}$, as a refinement
of the Fock-Goncharov moduli space $\mathcal{X}_{{\rm G},\hat{S}}$ in [FG06].
Let $\mathcal{B}\stackrel{{\scriptstyle\sim}}{{=}}{\rm G}/{\rm B}$ be the flag
variety associated with ${\rm G}$.

######  Definition 4.9.

The moduli space $\mathcal{P}_{{\rm G},\hat{S}}$ parametrizes the ${\rm
G}$-orbits of the data $(\rho,\xi,\gamma)$, where

  * •

$\rho$ is a ${\rm G}$-local system on $S$;

  * •

$\xi$ is flat section of the associated bundle $\rho\times_{{\rm
G}}\mathcal{A}$ restricted to every boundary interval in
$\partial{S}\backslash m_{b}$;

  * •

$\gamma$ is a flat section of the associated bundle $\rho\times_{{\rm
G}}\mathcal{B}$ restricted to every boundary circle in $\partial{S}\backslash
m_{b}$.

When the group ${\rm G}$ is adjoint, e.g., ${\rm G}={\rm PGL}_{n}$, then the
moduli space carries a cluster Poisson structure [GS19]. For example, if ${\rm
G}={\rm PGL}_{3}$, recall the quiver $Q_{\mathcal{T}}$ with the vertex set
$\Theta_{\mathcal{T}}$. By consider cross ratios associated with flags, the
space $\mathcal{P}_{{\rm PGL}_{3},\hat{S}}$ is equipped with a collection
$\\{X_{v}\\}_{v\in\Theta_{\mathcal{T}}}$ of cluster $\mathcal{X}$-coordinates.
Locally, a cluster mutation yields the following change of coordinates.

$X$$X_{4}$$X_{3}$$X_{2}$$X_{1}$

$X^{-1}$$X_{4}(1+X^{-1})^{-1}$$X_{3}(1+X)$$X_{2}(1+X^{-1})^{-1}$$X_{1}(1+X)$

Note that the transition maps are subtraction-free. Therefore
$\mathcal{P}_{{\rm PGL}_{3},\hat{S}}$ admits a totally positive structure and
can be further tropicalized. Let us impose the potential condition (20) to
every boundary interval and further require that the monodromy surrounding
each puncture corresponds to a dominant coweight of ${\rm PGL}_{3}$. In this
way, we obtain a cone

$\mathcal{P}_{{\rm
PGL}_{3},\hat{S}}^{+}(\mathbb{R}^{t})\subset\mathcal{P}_{{\rm
PGL}_{3},\hat{S}}(\mathbb{R}^{t}).$

Furthermore, We expect the following conjecture to be true.

######  Conjecture 4.10.

There is a natural bijection $\mathcal{P}_{{\rm
PGL}_{3},\hat{S}}^{+}(\mathbb{Z}^{t})\stackrel{{\scriptstyle\sim}}{{=}}\mathscr{W}_{\hat{S}}^{\mathcal{X}}$.

######  Example 4.11.

The following tropical $\mathcal{X}$-coordinates corresponds to an outward
tripod.

$1$$0$$0$$1$$1$$0$$-1$

Figure 31. Cluster $\mathcal{X}$-coordinates corresponding to an outward
tripod.

The intersection pairing among reduced webs induces a pairing among tropical
points

(22) $\mathbb{I}:~{}\mathcal{A}_{{\rm
PGL}_{3},\hat{S}}^{+}(\mathbb{Z}^{t})\times\mathcal{P}_{{\rm
PGL}_{3},\hat{S}}^{+}(\mathbb{Z}^{t})\longrightarrow\frac{1}{3}\mathbb{Z}.$

B. The canonical pairing from the perspective of cluster duality. We expect
that the pairing (22) coincides with the canonical pairing in the setting of
cluster ensembles due to Fock and Goncharov [FG09, Conj. 4.3]. Following the
notation of loc.cit., let $\mathcal{A}$ be a cluster $\mathcal{A}$-variety and
let $\mathcal{X}^{\vee}$ be the cluster $\mathcal{X}$-variety of Langlands
dual type. For example, our main example $(\mathcal{A}_{{\rm
SL}_{3},\hat{S}},\mathcal{P}_{{\rm PGL}_{3},\hat{S}})$ is such a pair.

The Fock-Goncharov duality conjecture asserts that every tropical point
$l\in\mathcal{A}(\mathbb{Z}^{t})$ naturally corresponds to a positive regular
function $\theta_{l}$ on $\mathcal{X}^{\vee}$. Its tropicalization induces a
pairing

$\mathbb{I}_{FG}:\mathcal{A}(\mathbb{Z}^{t})\times\mathcal{X}^{\vee}(\mathbb{Z}^{t})\longrightarrow\mathbb{Z},\qquad(l,m)\longmapsto\theta_{l}^{t}(m).$

Meanwhile, every tropical point $m\in\mathcal{X}^{\vee}(\mathbb{Z}^{t})$
corresponds to a function $\vartheta_{m}$ on $\mathcal{A}$. We expect that it
will give rise to the same pairing $\theta_{l}^{t}(m)=\vartheta_{m}^{t}(l)$.

Each cluster seed ${\bf i}$ provides a coordinate coordinate system
$\alpha_{\bf i}$ for $\mathcal{A}$ and a cluster coordinate system $\chi_{\bf
i}$ for $\mathcal{X}^{\vee}$. Let $(a_{1},\ldots,a_{n})$ be the coordinate of
$l\in\mathcal{A}(\mathbb{Z}^{t})$ under $\alpha_{\bf i}$ and let
$(x_{1},\ldots,x_{n})$ be the coordinates of
$m\in\mathcal{X}^{\vee}(\mathbb{Z}^{t})$ under $\chi_{\bf i}$. By the cluster
duality, we shall have

$\langle l,m\rangle_{\bf i}:=\sum_{j}a_{j}x_{j}\leq\mathbb{I}_{FG}(l,m).$

######  Conjecture 4.12.

We conjecture that

$\mathbb{I}_{FG}(l,m)=\max_{\bf i}~{}\langle l,m\rangle_{\bf i}.$

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