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WILSON LINES AND THEIR LAURENT POSITIVITY

TSUKASA ISHIBASHI AND HIRONORI OYA

Abstract. For a marked surface Σ and a semisimple algebraic group G of adjoint type,

we study the Wilson line morphism g[c] : PG,Σ → G associated with the homotopy class

of an arc c connecting boundary intervals of Σ. The matrix coefficients of the Wilson lines

give a generating set of the Betti algebra O(PG,Σ) when Σ has no punctures. The Wilson

lines have the multiplicative nature with respect to the gluing morphisms introduced by

Goncharov–Shen [GS19]. As a consequence, we obtain a decomposition formula for

Wilson lines with respect to a given ideal triangulation of Σ. Moreover we show that the

matrix coefficients cVf,v(g[c]) give Laurent polynomials with positive integral coefficients

in the Goncharov–Shen coordinate system associated with any decorated triangulation

of Σ, for suitable f and v.

Contents

1. Introduction 1

2. Configurations of pinnings 9

3. Wilson lines on the moduli space PG,Σ 18

4. Factorization coordinates and their relations 42

5. Coordinate expressions of Wilson lines and loops 57

6. Laurent positivity of Wilson lines and Wilson loops 63

Appendix A. Some maps related to the twist automorphism 71

Appendix B. Proof of Theorem 6.7 72

Appendix C. Cluster varieties, weighted quivers and their amalgamation 84

Appendix D. A short review on quotient stacks 90

References 92

1. Introduction

The moduli space of G-local systems on a topological surface is a classical object of

study, which has been investigated both from mathematical and physical viewpoints. Wil-

son loops give a class of important functions (or gauge-invariant observables), which are

obtained as the traces of the monodromies of G-local systems in some finite-dimensional

representations of G.

For a marked surface Σ, Fock–Goncharov [FG06] introduced two extensions AG,Σ and

XG,Σ of the moduli space of local systems, each of which admits a natural positive structure.

Here G is a simply-connected semisimple algebraic group, and G = G/Z(G) is its adjoint

Date: May 20, 2021.1

http://arxiv.org/abs/2011.14260v2

2 TSUKASA ISHIBASHI AND HIRONORI OYA

group. Such a structure in particular allows one to consider the semifield-valued points

of these moduli spaces, and in particular their positive real points give an analogue of

the Hitchin component of the moduli space of local systems on a closed surface, which

has been intensively studied as a higher-rank generalization of the Teichmuller space. See

[Wie18] for a comprehensive survey.

Moreover, the positive structures of these moduli spaces can be upgraded to cluster

structures, which are two kinds of collections of open embeddings of algebraic tori ac-

companied with weighted quivers, related by two kinds of cluster transformations. The

collection of weighted quivers is shared by AG,Σ and XG,Σ, and thus they form a cluster

ensemble in the sense of [FG09]. Such a cluster structure is first constructed by Fock–

Goncharov [FG06] when the gauge groups are of type An, by Le [Le16] for type Bn, Cn, Dn

(and further investigated in [IIO19]), and by Goncharov–Shen [GS19] for all semisimple

gauge groups, generalizing all the works mentioned above.

Strictly speaking, however, the moduli space XG,Σ misses the frozen coordinates, which

should be assigned to the vertices of the weighted quivers on the boundary when ∂Σ 6= ∅.Hence the dimension of XG,Σ is strictly smaller than AG,Σ, which breaks the symmetry. In

fact, one of the main innovations made by Goncharov–Shen in [GS19] is the introduction

of a new moduli space PG,Σ closely related to the moduli space XG,Σ, supplying the frozen

coordinates which have been missing in the latter so that the pair (AG,Σ,PG,Σ) forms

a cluster ensemble in the strict sense for any marked surface Σ. When ∂Σ = ∅, we

have PG,Σ = XG,Σ, and otherwise the former includes additional data called the pinnings

assigned to boundary intervals. Pinnings in particular allow one to define a nice gluing

morphism

qE1,E2 : PG,Σ → PG,Σ′

when Σ′ is obtained from Σ by gluing two boundary intervals E1 and E2 of Σ. The

supplement of frozen coordinates is also crucial in the connection to the Drinfel’d–Jimbo

quantum groups in their work [GS19, Section 11].

1.1. The Wilson lines. Pinnings also allow us to introduce a new class of G-valued

morphisms

g[c] : PG,Σ → G,

which we call the Wilson line along the homotopy classes [c] of a curve connecting two

boundary intervals called an arc class. Our aim in this paper is a detailed study of

these morphisms. Roughly speaking, the Wilson line g[c] is defined to be the comparison

element of the two pinnings assigned to the initial and boundary intervals under the

parallel-transport along the curve c.

The Wilson lines have the multiplicative nature for the gluing morphisms. If we have

two arc classes [c1] : E1 → E2 and [c2] : E′2 → E3 on Σ, then by gluing the boundary

intervals E2 and E ′2 we obtain another marked surface Σ′ equipped with an arc class

[c] := [c1]∗[c2], which is the concatenation of the two arcs. Then we will see that the Wilson

line g[c] is given by the product of the Wilson lines g[c1] and g[c2]. See Proposition 3.27

WILSON LINES AND THEIR LAURENT POSITIVITY 3

and Figure 7. In particular, the morphism

ρ|γ| : PG,Σ → [G/AdG]

given by the monodromy along a free loop |γ|, which we call the Wilson loop in this

paper1, can be computed from the Wilson line along the curve obtained by cutting the

loop γ along an edge. See Proposition 3.28 and Figure 8. In this sense, the Wilson lines

are “open analogues” of the Wilson loops.

A special kind of Wilson lines already appeared in the context of the multiplicative

canonical pairing (also known as the Fock–Goncharov duality map) [FG07] and a cer-

tain supersymmetric quantum field theory [GMN13], where they consider (certain matrix

coefficients of) the Wilson lines associated with collections of curves (or laminations)

satisfying a certain boundary condition rather than a single curve, so that they are well-

defined as functions on the moduli space XG,Σ. Our Wilson lines along any arc classes

are well-defined on the moduli space PG,Σ, whose product for a suitable lamination (a

collection of arc classes) descends to a function on XG,Σ.

The Wilson lines are morphisms from the moduli stack PG,Σ equipped with the natural

Betti structure (see Section 3.1.1), and hence their matrix coefficients give rise to regular

functions on PG,Σ. Moreover, we will see in Section 3.5 that when Σ has no punctures,

the Betti algebra O(PG,Σ) is generated by the matrix coefficients of the (twisted) Wilson

lines. Therefore Wilson lines can be considered as one of the main sources of regular

functions on the moduli space PG,Σ. On the other hand, Shen [She20] proved that the

algebra O(PG,Σ) of regular functions on this moduli stack is isomorphic to the cluster

Poisson algebra Ocl(PG,Σ), which is by definition the algebra of regular functions on

the corresponding cluster Poisson variety. Hence the matrix coefficients of Wilson lines

belong to Ocl(PG,Σ). In other words, they are universally Laurent polynomials, meaning

that they are expressed as Laurent polynomials in any cluster chart (including those not

coming from decorated triangulations). Our goal in this paper is a detailed study of

these coordinate expressions of matrix coefficients of Wilson lines, and moreover to prove

that certain matrix coefficients give rise to Laurent polynomials with non-negative integer

coefficients.

1.2. Decomposition formula for Wilson lines. We first study the decomposition of

Wilson lines with respect to a given ideal triangulation ∆ of Σ. It is obtained simply by

successively applying the multiplicative property mentioned above, if we consider an arc

class [c] which traverses each triangle of ∆ at most once. Let T1, . . . , TM be the triangles

that [c] traverses in this order. Then the Wilson line g[c] is decomposed into a product of

Wilson lines on PG,Tν for ν = 1, . . . ,M , which are morphisms gν : PG,Tν → B±∗ valued in

upper or lower triangular matrices depending on the patterns of the intersections c ∩ Tν(see Figure 4). The pattern of intersection with triangles is encoded in the turning pattern

τ∆([c]) = (τ1, . . . , τM ) ∈ {L,R}M . For instance, if τν = L for all ν, then the Wilson line

g[c] is upper-triangular.

1In literature, the composition of this function with the trace in a finite-dimensional representation of

G is called a Wilson line. We call them the trace functions in this paper.


When the surface is a polygon Π, every arc class satisfies the assumption. We call the

Wilson lines in this special case the side pairings in analogy with the side pairing elements

in the Fuchsian group for its fundamental polygon from the hyperbolic geometry. Indeed,

when we consider a fundamental polygon of a marked surface, every Wilson line lifts to a

side pairing. Assuming Π = T1 ∪ · · · ∪ TM without loss of generality, we get the following

simplest version of the decomposition formula:

Proposition 1 (Decomposition formula for side pairings, Proposition 3.23). We have

q∗∆gEin,Eout = µM ◦M∏

ν=1

gν ,

where µM denotes the multiplication of M elements in G, and q∆ :∏M

ν=1PG,Tν → PG,Πdenotes the gluing morphism with respect to the ideal triangulation ∆ of Π.

In general, an arc class [c] on Σ may traverse one triangle more than once. Let

T1, . . . , TM denote the triangles that [c] traverses in this order, some of which may be

identical. In this case, rather than introducing more triangular matrices corrsponding

to different patterns of intersection, it is better to adjust the expression by using the

cyclic shift automorphisms S3 in view of the coordinate expressions discussed later. Let

us briefly explain the ideas.

Note that the intersection of [c] with each triangle encircles a distinguished corner,

which we indicate by a dot near that corner. We call an ideal triangulation equipped

with one dot mT for each triangle T a dotted triangulation, which will be used to give

identifications fmT: PG,T → Conf3PG with the configuration space of triples of pinnings,

and define cluster coordinates on PG,Σ by “amalgamating” those on Conf3PG. The cyclicshift automorphisms on Conf3PG results in rotating locations of these dots, and induces

cluster transformations. The intersection pattern of [c] and ∆ insists a “canonical” choice

of dots as the corner encircled by the intersection of [c] with each triangle, and the

triangular matrices in proposition 1 above can be viewed as the composites

gν : PG,Tν∼−→ Conf3PG

bτν−−→ B±∗ .

Here the first isomorphism is given by the canonical dot, and bL : Conf3PG → B+∗ and

bR : Conf3PG → B−∗ are basic Wilson lines (Definition 2.12). When two of the triangles

T1, . . . , TM coincide, however, these canonical dots may disagree on some triangles and

therefore do not form a dotted triangulation. In turn, if we fix a dotted triangulation

over ∆, the discrepancy with the canonical dots for [c] can be expressed by Stν3 for some

tν ∈ {0, 1, 2}. As a consequence of a precise formulation of these observations, we get the

following general decomposition formula:

Theorem 2 (Decomposition formula for Wilson lines, Theorem 3.38). For an arc class

[c] : Ein → Eout in a marked surface Σ and a dotted triangulation ∆∗, we have

q∗∆g[c] = µM ◦M∏

ν=1

(bτν ◦ Stν3 ◦ fnν

).


Here µM is the multiplication of M elements of G, and fnν:∏

T PG,T → PG,Tν∼−→

Conf3PG is induced by the dots of ∆∗. See Section 3.6 for a detail. This formula is the

basis of our consideration on the Laurent expressions of Wilson lines in the cluster charts.

Note that a similar decomposition formula for Wilson loops (and hence for the trace

functions) can be obtained from this formula and Proposition 3.28.

1.3. Coordinate expressions of Wilson lines. Coordinate expressions of Wilson loops

(or the trace functions) have been studied by several authors. In the A1 case, a combina-

torial formula for the expressions of Wilson loops in terms of the cross ratio coordinates

is given by Fock [Fo94] (see also [Pen, FG07]). It expresses the Wilson loop along a free

loop |γ| as a product of the elementary matrices

L =

(1 1

0 1

), R =

(1 0

1 1

), H(x) =

(x1/2 0

0 x−1/2

)∈ PGL2,

which are multiplied according to the turning pattern after substituting the cross ratio

coordinates into x.

In the An case, Fock–Goncharov [FG06] introduced the cluster coordinates associated

with ideal triangulations (called the special coordinate systems), and gave a similar formula

for Wilson loops called the snake formula. In particular, the trace functions are positive

Laurent polynomials (with fractional powers) in any special coordinate systems. The

snake formula has been recognized as related to the spectral networks [GMN14] and

certain integrable systems [SS17].

In general, Goncharov–Shen [GS19] gave a uniform construction of coordinate systems

on PG,Σ associated with decorated triangulations ∆ = (∆∗, s∆). Here ∆∗ is a dotted

triangulation and s∆ is a choice of a reduced word of the longest element w0 ∈ W (G) for

each triangle. For type An case, thanks to the cyclic invariance of the coordinate system,

the choice of dots is irrelevant. Fock–Goncharov’s special coordinates are recovered when

s∆ is chosen to be the “standard” one (see (4.8)).

Locally, a natural generalization of the snake formula is given by the evaluation map

[FG06], which gives the coweight parametrizations of double Bruhat cells of G. We will

see that the basic Wilson lines bL, bR can be expressed using the evaluation maps, where

the coweight parameters are identified with some of the Goncharov–Shen coordinates (GS

coordinates for short) on Conf3PG. In particular, by substituting into the decomposition

formula given in Theorem 2, one can write the Wilson lines and loops as a product of eval-

uation maps. This is basially the same strategy as Fock–Goncharov [FG06], but manipu-

lations in the recently-innovated moduli space PG,Σ makes the computation much clearer,

thanks to the nice properties of the gluing morphism [GS19]. Moreover, the adjustment by

cyclic shifts in the decomposition formula results in the cluster transformations of the GS

coordinates, which gives the twisted chain of GS coordinates X[c] = (Xsi [ν])(i,s;ν)∈I1∗···∗IM

along [c] associated with a decorated triangulation ∆. Each Xsi [ν] is a positive ratonal

function of the relevant GS coordinates.

Theorem 3 (Theorem 5.2). Let ∆ be a decorated triangulation, [c] : Ein → Eout an arc

class, and τ∆([c]) = (τ1, . . . , τM) ∈ {L,R}M the associated turning pattern. Then the


Wilson line g[c] : PG,Σ → G is expressed as

(ψ∆)∗g[c] = evτ1,...,τMs1,...,sM

(X[c]), (1.1)

where ψ∆ : (C∗)I(∆) → PG,Σ denotes the Goncharov–Shen coordinate system associated

with ∆.

When G = PGLn+1 and the reduced words are chosen to be the standard one, our

formula recovers the snake formula of Fock–Goncharov.

Here a technicality is that, except for type An case, the adjustment by cyclic shifts,

which is included in the definition of the functions Xsi [ν], is quite complicated. In par-

ticular, each function Xsi [ν] is not a Laurent monomial in general. While the matrix

coefficients of g[c] are at least guaranteed to be Laurent polynomials by Corollary 4.23, it

is therefore non-trivial whether their coefficients are non-negative integers.

1.4. Positivity of Wilson lines. Based on the formula given in Theorem 3, we discuss

the positivity problem for the coefficients of the Laurent polynomials mentioned above.

Let us further clarify the problem which we will deal with. A rational function f on

PG,Σ is called a GS-universally positive Laurent polynomial if it is expressed as a Laurent

polynomial with non-negative integral coefficients in the GS coordinate system associated

with any decorated triangulation ∆. This is a straightforward generalization of special

good positive Laurent polynomials on XPGLn+1,Σ in [FG06]. Moreover, a rational G-valued

map F : PG,Σ → G is called a GS-universally positive G-valued Laurent polynomial if for

any decorated triangulation ∆ of Σ and any finite-dimensional representation V of G,

there exists a basis B of V such that

cVf,v ◦ F : PG,Σ → C

is a GS-universally positive Laurent polynomial for all v ∈ B and f ∈ F, where F is the

basis of V ∗ dual to B. Our result is the following:

Theorem 4 (Theorem 6.2). Let G be a semisimple algebraic group of adjoint type, and

assume that our marked surface Σ has non-empty boundary. Then, for any arc class

[c] : Ein → Eout, the Wilson line g[c] : PG,Σ → G is a regular GS-universally positive

G-valued Laurent polynomial.

Since the Wilson loops can be computed from the Wilson lines by Proposition 3.28, it

immediately implies the following:

Corollary 5 (Corollary 6.3). Let G be a semisimple algebraic group of adjoint type, and

|γ| ∈ π(Σ) a free loop. Then, for any finite dimensional representation V of G, the trace

function trV (ρ|γ|) := trV ◦ρ|γ| : PG,Σ → C is a regular GS-universally positive Laurent

polynomial.

Corollary 5 is a generalization of [FG06, Theorem 9.3, Corollary 9.2].

Here we briefly comment on the proof of Theorem 4. By the construction of the GS

coordinate system on PG,Σ associated with a decorated triangulation ∆, the Laurent

positivity of a regular function on PG,Σ can be deduced from the Laurent positivity of


its pull-back via the gluing morphism q∆ :∏

T∈t(∆) PG,T → PG,Σ associated with the

underlying ideal triangulation ∆. In other words, we can investigate the Laurent positivity

of a regular function on PG,Σ by a local argument on triangles. Indeed, a key to the proof

of Theorem 4 is a construction of a basis Fpos,T of O(PG,T ) consisting of GS-universally

positive Laurent polynomials, which is invariant under the cyclic shift and compatible

with certain matrix coefficients.

We show that such a nice basis is constructed whenever we have a nice basis Fpos of

the coordinate ring O(U+∗ ) of the unipotent cell U+

∗ of G. In particular, the invariance

of Fpos,T under the cyclic shift on PG,T comes from the invariance of Fpos under the

Berenstein-Fomin-Zelevinsky twist automorphism on U+∗ [BFZ96, BZ97]. An example of

a basis of O(U+∗ ) which satisfies the list of desired properties (Theorem 6.7) is obtained

from the theory of categorification of O(U+∗ ) via quiver Hecke algebras, which has been

investigated, for example, in [KL09, Rou08, KL11, Rou12, KK12, KKKO18, KKOP18,

KKOP19]. Based on their results, we show that the basis arising from this categorification

satisfies the desired properties. As an important step, we prove in Theorem B.22 that

the (quantum) Berenstein-Fomin-Zelevinsky twist automorphism on U+∗ is categorified

by using the left dualizing functor in a certain category Cw, which is constructed in

[KKOP19]. This in turn ensures the invariance of Fpos under the twist automorphism.

Notice that the GS-universally positive Laurent property is weaker than the universal

positive Laurent property [FG09], which requires a similar positive Laurent property for all

cluster charts. By replacing GS-universally positive Laurent polynomials with universal

positive Laurent polynomials, we can also define the notion of universally positive G-valued

Laurent polynomials. Then, it would be natural to expect the following:

Conjecture 6. For any arc class [c] : Ein → Eout, the Wilson line g[c] : PG,Σ → G is a

universally positive G-valued Laurent polynomial. Moreover, the trace function trV (ρ|γ|) :

PG,Σ → C is a universally positive Laurent polynomial.

Indeed, it is known that this conjecture on the trace functions holds true for type A1

case [FG06].

1.5. Future directions.

Poisson brackets of Wilson lines. The Poisson brackets of the trace functions trV (ρ|γ|) in

the natural representation with respect to the Atiyah–Bott–Goldman Poisson structure on

the moduli space of G-local systems on a punctured surface form the celebrated Goldman

algebra [Go86]. Even for the adjoint group G, the absolute values | trV (ρ|γ|)| of the tracesin the natural representation are smooth functions on the positive-real part PG,Σ(R>0) (or

XG,Σ(R>0)) and their cluster Poisson brackets make sense. In the type An case, Chekhov–

Shapiro [CS20] proved that the cluster Poisson brackets of these functions reproduce the

Goldman brackets. Their argument is local in nature and seems to be applicable also to

Wilson lines, and it can be expected that (absolute values of) certain matrix coefficients

of the Wilson lines form an open analogue of the Goldman algebra.


Quantum lifts of Wilson lines. Any cluster Poisson variety X admits a canonical quanti-

zation, namely a one-parameter deformation Oq(X ) of the cluster Poisson algebra O(X )

and its representation on a certain Hilbert space as self-adjoint operators [FG08]. It will

be an extremely interesting problem to consider a quantum analogue of the matrix coeffi-

cients of the Wilson lines, which belong to Oq(PG,Σ) and recovers cVf,v(g[c]) in the classical

limit q → 1. Indeed, it has been known that certain matrix coefficients of the Wilson

line along an arc class encircling exactly one special point, which is valued in B+ (see

Lemma 3.30), admit such quantum lifts and generate the quantum group Uq(b+) inside

the quantum cluster Poisson algebra [GS19, Section 11]. In the type An case, a quantum

lift of the evaluation map has been studied by Douglas [Dou21] in relation with the quan-

tized coordinate ring Oq(G). The work of Chekhov–Shapiro [CS20] mentioned above in

fact investigated the quantization of trace functions of type An using the network descrip-

tion, computing their commutation relations in terms of the R-matrix and reproducing

the Goldman bracket in the classical limit.

A comparison with the quantization of the moduli stacks in terms of the factorization

homology studied by [JLSS21] will also be an important problem.

Relation to the skein theory and bases of the cluster Poisson algebras. For a marked sur-

face without punctures, relations between a quantum upper cluster algebra Oq(AG,Σ) and

the skein theory has been studied [Mul16, FP16, IY21] among others. It is expected that

the skein theory in particular provides a natural basis of Oq(AG,Σ) with certain positiv-

ity called the graphical basis, which has been intensively studied by Musiker–Schiffler–

Williams [MSW13] and Thurston [Thu14] for type A1, and studied in [IY21] for type

A2.

In the absence of punctures, the ensemble map induces an injective homomorphism

p∗ : Oq(PG,Σ) → Oq(AG,Σ) (for the corresponding choice of the compatibility matrix as in

[GS19, Section 13]) which is an isomorphism over Q, and therefore one can compare the

functions on these moduli spaces. It can be expected that certain matrix coefficients of

the quantum Wilson lines and the traces of Wilson loops reproduce the graphical basis.

In the forthcoming paper [Ish], we will see that it is indeed true for the type A1, classical

case. In general cases a fruitful interaction between the study of Wilson lines and the

skein theory is expected.

Organization of the paper. In Section 2, after recalling the notions of flags and pin-

nings, we introduce certain parametrizations of configurations of pinnings which we call

the standard configurations, and in particular introduce the basic Wilson lines. The con-

tents in this section give the basis of our local computations in this paper.

In Section 3, we introduce the Wilson line morphisms and study their basic properties.

We also give a detailed description of the Betti structure of the moduli space PG,Σ for later

use. Some basic facts on the quotient stacks are summarized in Appendix D. In Section 3.5,

we prove that the Betti algebra O(PG,Σ) is generated by the matrix coefficients of the

(twisted) Wilson lines when Σ has no punctures. We give the decomposition formulae for

the Wilson lines in Section 3.6.


In Section 4, we recall the Goncharov–Shen coordinates on the moduli space PG,Σ as a

preparation for the study of the coordinate expressions of the Wilson lines. Some basic

notions on the cluster varieties, weighted quivers and their amalgamation procedure are

recollected in Appendix C. In Section 5, we study the coordinate expressions of the Wilson

lines. First we compute them on the fundamental polygon of the universal cover as the

side pairings where no adjustment by cyclic shifts are needed, and then investigate the

effects of cyclic shifts.

In Section 6, we show that the Wilson lines are regular GS-universally positive G-

valued Laurent polynomials on the moduli space PG,Σ, and the trace functions in a finite

dimensional representation of G are regular GS-universally positive Laurent polynomials.

In the course of the proof, we construct a basis of O(PG,T ) for a triangle T , which consists

of GS-universally positive Laurent polynomials and is invariant under the cyclic shift.

Acknowledgements. The authors’ deep gratitude goes to Linhui Shen for his insightful

comments on this paper at several stages and explaining his works with Alexander Gon-

charov. They are grateful to Tatsuki Kuwagaki and Takuma Hayashi for explaining some

basic notions and backgrounds on Artin stacks, and giving valuable comments on a draft

of this paper. They also wish to thank Ryo Fujita for helpful discussions on quiver Hecke

algebras.

T. I. would like to express his gratitude to his former supervisor Nariya Kawazumi for

his continuous guidance and encouragement in the earlier stage of this work.

T. I. is partially supported by JSPS KAKENHI Grant Numbers 18J13304 and 20K22304,

and the Program for Leading Graduate Schools, MEXT, Japan. H. O. is supported by

Grant-in-Aid for Young Scientists (No. 19K14515).

2. Configurations of pinnings

2.1. Notations from Lie theory. In this subsection, we briefly recall basic terminolo-

gies in Lie theory. See [Jan] for the details.

Let g be a complex finite dimensional semisimple Lie algebra associated with a Cartan

matrix C(g) = (Cst)s,t∈S. Namely, g is isomorphic to the complex Lie algebra generated

by {es, fs, α∨s | s ∈ S} with the following relations:

(i) [α∨s , α

∨t ] = 0,

(ii) [α∨s , et] = Cstet, [α

∨s , ft] = −Cstft,

(iii) [es, ft] = δstα∨s ,

(iv) (ades)1−Cst(et) = 0 and (adfs)

1−Cst(ft) = 0 for s 6= t. Here, (adx)(y) := [x, y] for

x, y ∈ g.

Set h :=∑

s∈S Cα∨s , and define αs ∈ h∗ by [η, es] = 〈η, αs〉es for η ∈ h and s ∈ S. Then g

has the following root space decomposition

g = h⊕⊕

β∈Φ

gβ, gβ := {x ∈ g | [η, x] = 〈η, β〉x for η ∈ h}, dim gβ = 1.

We will also use another basis of h, coweights (∨s )s∈S, which is determined by α∨

s =∑u∈S Csu

∨u . Then we have the following commutation relations:


(i′) [∨s ,

∨t ] = 0,

(ii′) [∨s , et] = δstet, [

∨s , ft] = −δstft,

(iii′) [es, ft] = δst∑

u∈S Csu∨u .

Notations for algebraic groups. For an algebraic torus T over C, let X∗(T ) :=

Hom(T,C∗) denote the lattice of characters.

Let G be the simply-connected connected algebraic group over C whose Lie algebra is

g, and take a maximal torus H of G whose Lie algebra is h. Then each gβ is a weight

space (=simultaneous eigenspace of H) for the adjoint action of G on g, whose weight

(=simultaneous eigenvalue) is again denoted by β ∈ X∗(H). The lattice X∗(H) of rank

|S| is called the weight lattice.

For β =∑

s∈S csαs ∈ Φ, either cs ≥ 0 for all s ∈ S (write β > 0) or cs ≤ 0 for all s ∈ S

(write β < 0) holds. We have the corresponding decomposition Φ = Φ+ ⊔ Φ−, where

Φ± := {β ∈ Φ | ±β > 0}.

Lemma 2.1. For β ∈ Φ+, there exist one-parameter subgroups xβ, yβ : C → G such that

hxβ(t)h−1 = xβ(h

βt), dxβ : C∼−→ gβ,

hyβ(t)h−1 = yβ(h

−βt), dyβ : C∼−→ g−β

for h ∈ H and t ∈ C. Here dxβ and dyβ are tangent maps of xβ and yβ, respectively.

Define U+ and U− as the closed subgroups of G generated by {xβ(t) | β ∈ Φ+, t ∈ C}and {yβ(t) | β ∈ Φ+, t ∈ C}, respectively.

The group G = G/Z(G) is called the adjoint group, where Z(G) denotes the center

of G. Note that the above mentioned one-parameter subgroups descend to the adjoint

group, and U± can also be regarded as subgroups of G. The subgroups B± := HU± ⊂ G

are called Borel subgroups, where H := H/Z(G) is the Cartan subgroup of G. Let

G0 := U−HU+ ⊂ G be the open subvariety of triangular-decomposable elements. In the

following, we write xs := xαsand ys := yαs

, and normalize them so that dxs(1) = es and

dys(1) = fs.

Definition 2.2. In the adjoint group, define Es := xs(1) and Fs := ys(1) for each s ∈ S.

Let Hs : C∗ → H be the one-parameter subgroup such that dHs(1) = ∨s ∈ h.

Returning to the simply-connected group, we have a homomorphism ϕs : SL2(C) → G

such that(1 t

0 1

)7→ xs(t),

(1 0

t 1

)7→ ys(t)

for s ∈ S. For a ∈ C∗, write aα∨s := ϕs

((a 0

0 a−1

)). Since G is simply-connected, we

have an isomorphism

(C∗)S∼−→ H, (as)s∈S 7→

∏

s∈S

aα∨ss .


It induces an isomorphism of lattices

{µ ∈ h∗ | 〈α∨s , µ〉 ∈ Z for s ∈ S}

∼−→ X∗(H), µ 7→

(∏

s∈S

aα∨ss 7→

∏

s∈S

a〈α∨s ,µ〉

s

). (2.1)

Henceforth we identify the both sides of (2.1), since it will cause no confusion. Note that

this identification is compatible with the previous identification between β ∈ Φ and its

weight β ∈ X∗(H). For µ ∈ X∗(H), the image of η ∈ h under µ is denoted by 〈η, µ〉,

and that of h ∈ H is written as hµ. For s ∈ S, define the s-th fundamental weight

s ∈ X∗(H) by 〈α∨t , s〉 = δst. Obviously, we have X∗(H) =

∑s∈S Zs. The sub-lattice

X∗(H) ⊂ X∗(H) is generated by αs for s ∈ S and called the root lattice.

Weyl groups. Let W (G) := NG(H)/H denote the Weyl group of G. Here NG(H) is the

normalizer subgroup of H in G. It is known that the Weyl group is a Coxeter group,

which is described as follows. Consider the group

W (g) := 〈rs (s ∈ S) | (rsrt)mst = 1 (s, t ∈ S)〉,

where mst ∈ Z is given as follows:

CstCts : 0 1 2 3

mst : 2 3 4 6

For s ∈ S, we set rs := ϕs

((0 −1

1 0

))∈ NG(H). Then we have a group isomorphism

W (g)∼−→ W (G) extending rs 7→ rsH for s ∈ S. For a reduced word s = (s1, . . . , sℓ) of

w ∈ W (g), let us write w := rs1 . . . rsℓ ∈ NG(H), which does not depend on the choice

of the reduced word. We have a left action of W (G) on X∗(H) induced from the (right)

conjugation action of NG(H) on H. Then via the identification (2.1), we have w.µ = w.µ

for µ ∈ X∗(H), where the right-hand side denotes the action of W (g) on h∗ defined by

rs.µ := µ− 〈α∨s , µ〉αs

for s ∈ S.

For w ∈ W (g), write the length of w as l(w). Let w0 ∈ W (g) be the longest element of

W (g), and set sG := w02 ∈ NG(H). It turns out that sG ∈ Z(G), and s2G = 1 (cf. [FG06,

§2]). We define an involution S → S, s 7→ s∗ by

αs∗ = −w0αs.

We note that the Weyl group W (G) := NG(H)/H is naturally isomorphic to the Weyl

group W (G) of G, and we will frequently regard w as an element of NG(H) by abuse of

notation. Remark that sG = w02 = 1 in G.

Irreducible modules and matrix coefficients. Set X∗(H)+ :=∑

s∈S Z≥0s ⊂ X∗(H)

and X∗(H)+ := X∗(H)∩X∗(H)+. For λ ∈ X∗(H)+, let V (λ) be the (rational) irreducible

G-module of highest weight λ. A fixed highest weight vector of V (λ) is denoted by vλ.


Set

vwλ := w.vλ

for w ∈ W (G). A G-module V carries a natural g-module structure. For s ∈ S and

v ∈ V , we have

xs(t).v =∞∑

k=0

tk

k!eks .v, ys(t).v =

∞∑

k=0

tk

k!fks .v. (2.2)

There exists an anti-involution T : G → G, g 7→ gT of the algebraic group G given by

xs(t)T = ys(t) and h

T = h for s ∈ S, t ∈ C, h ∈ H . This is called the transpose in G.

Proposition 2.3. Let λ ∈ X∗(H)+. Then there exists a unique non-degenerate symmetric

C-bilinear form ( , )λ on V (λ) such that

(vλ, vλ)λ = 1, (g.v, v′)λ = (v, gT.v′)λ

for v, v′ ∈ V (λ) and g ∈ G.

For v ∈ V (λ), we set

v∨ := (v′ 7→ (v, v′)λ) ∈ V (λ)∗, fwλ := v∨wλ. (2.3)

Note that (vwλ, vwλ)λ = 1 for all w ∈ W (G).

For a G-module V , the dual space V ∗ is considered as a (left) G-module by

〈g.f, v〉 := 〈f, gT.v〉

for g ∈ G, f ∈ V ∗ and v ∈ V . Note that, under this convention, the correspondence

v 7→ v∨ for v ∈ V (λ) gives a G-module isomorphism V (λ) → V (λ)∗ for λ ∈ X∗(H)+. For

f ∈ V ∗ and v ∈ V , define the element cVf,v ∈ O(G) by

g 7→ 〈f, g.v〉 (2.4)

for g ∈ G. An element of this form is called a matrix coefficient. For λ ∈ X∗(H)+, we

simply write cλf,v := cV (λ)f,u . Moreover, for w,w′ ∈ W (G), the matrix coefficient

∆wλ,w′λ := cλfwν ,vw′λ. (2.5)

is called a generalized minor.

The ∗-involutions. We conclude this subsection by recalling an involution on G associ-

ated with a certain Dynkin diagram automorphism (cf. [GS18, (2)]).

Lemma 2.4. Let ∗ : G→ G, g 7→ g∗ be a group automorphism defined by

g 7→ w0(g−1)Tw−1

0 .

Then (g∗)∗ = g for all g ∈ G, and xs(t)∗ = xs∗(t), ys(t)

∗ = ys∗(t) for s ∈ S.

For a proof, see [IIO19, Lemma 5.3].


2.2. The configuration space ConfkPG. Let G be an adjoint group. Here we introduce

the configuration space ConfkPG based on [GS19], which models the moduli space PG,Πfor a k-gon Π.

Definition 2.5. The homogeneous spaces AG := G/U+ and BG := G/B+ are called the

principal affine space and the flag variety, respectively. An element of AG (resp. BG) iscalled a decorated flag (resp. flag). We have a canonical projection π : AG → BG.

The principal affine space can be identified with the moduli space of pairs (U, ψ), where

U ⊂ G is a maximal unipotent subgroup and ψ : U → C is a non-degenerate character.

See [GS15, Section 1.1.1] for a detailed discussion. The basepoint of AG is denoted by

[U+]. The flag variety BG will be identified with the set of connected maximal solvable

subgroups of G via g.B+ 7→ gB+g−1.

The Cartan subgroup H acts on AG from the right by g.[U+].h := gh.[U+] for g ∈ G

and h ∈ H , which makes the projection π : AG → BG a principal H-bundle.

For k ∈ Z≥2, the configuration spaces are defined to be

ConfkAG := G\

k times︷︸︸︷AG × · · · × AG, and ConfkBG := G\

k times︷︸︸︷BG × · · · × BG,

where we consider the diagonal left action of G. These configuration spaces are elementary

building blocks for the moduli spaces AG,Σ and XG,Σ, respectively [FG06].

Notation 2.6. For some left G-spaces X1, . . . , Xk and elements xi ∈ Xi for i = 1, . . . , k,

the G-orbit of the tuple (x1, . . . , xk) is denoted by the square bracket [x1, . . . , xk].

A pair (B1, B2) of flags is said to be generic if there exists g ∈ G such that g.(B1, B2) =

(B+, B−).

Using the Bruhat decomposition G =⋃w∈W (G)U

+HwU+, the configuration space

Conf2AG is parametrized as

α2 :∐

w∈W (G)

H∼−→ Conf2AG, (h, w) 7→

[h.[U+], w.[U+]

].

We write the inverse map as α−12 (A1, A2) =: (h(A1, A2), w(A1, A2)). The parameters

h(A1, A2) and w(A1, A2) are called the h-invariant and the w-distance of (A1, A2), re-

spectively. Note that the w-distance only depends on the underlying pair (π(A1), π(A2))

of flags, and the pair is generic if and only if w(A1, A2) = w0. The following lemma justifies

the name “w-distance” and provides us a fundamental technique to define Goncharov–

Shen coordinates.

Lemma 2.7 ([GS19, Lemma 2.3]). Let u, v ∈ W (G) be two elements such that l(uv) =

l(u) + l(v). Then the followings hold.

(1) If a pair (B1, B2) of flags satisfies w(B1, B2) = uv, then there exists a unique flag

B′ such that

w(B1, B′) = u, w(B′, B2) = v.

(2) Conversely, if we have w(B1, B′) = u and w(B′, B2) = v, then w(B1, B2) = uv.


Corollary 2.8. Let (Bl, Br) be a pair of flags with w(Bl, Br) = w. Every reduced word

s = (s1, . . . , sp) of w gives rise to a unique chain of flags Bl = B0, B1, . . . , Bp = Br such

that w(Bk−1, Bk) = rsk .

Next we define an enhanced configuration space by adding extra data called pinnings.

Definition 2.9 (pinnings). A pinning is a pair p = (B1, B2) ∈ AG×BG of a decorated flag

and a flag such that the underlying pair (B1, B2) ∈ BG×BG is generic, where B1 := π(B1).

We say that p is a pinning over (B1, B2).

An important feature is that the set PG of pinnings is a principal G-space, and in

particular PG is an affine variety. In this paper, we fix the basepoint to be pstd :=

([U+], B−), so that any pinning can be writen as g.pstd for a unique g ∈ G. The right

H-action of AG induces a right H-action on PG, which is given by (g.pstd).h = gh.pstd for

g ∈ G and h ∈ H . Each fiber of the projection

(π+, π−) : PG → BG × BG, p = (B1, B2) 7→ (B1, B2) (2.6)

is a principal H-space.

For p = g.pstd, we define the opposite pinning to be p∗ := gw0.pstd. We have (g.pstd.h)∗ =

g.p∗std.w0(h) for g ∈ G and h ∈ H .

Remark 2.10. We have the following equivalent descriptions of a pinning. See [GS19]

for details.

(1) A pair p = (B1, B2) ∈ AG×AG of decorated flags such that h(B1, B2) = e and the

underlying pair of flags is generic (i.e., w(B1, B2) = w0). The opposite pinning is

given by p∗ = (B2, B1).

(2) A data p = (B,Bop; (ξ+s (t))s∈S, (ξ−s (t))s∈S), where (B,Bop) is a pair of opposite

Borel subgroups of G and (ξ+s (t))s, (ξ−s (t))s are one-parameter subgroups deter-

mined by a fundamental system for the root data with respect to the maximal torus

B∩Bop. The opposite pinning is given by p∗ = (Bop, B; (ξ−s (−t))s∈S, (ξ+s (−t))s∈S).

For k ∈ Z≥2, we consider the configuration space

ConfkPG := [G\{(B1, . . . , Bk; p12, . . . , pk−1,k, pk,1)}],

where Bi ∈ BG, and pi,i+1 is a pinning over (Bi, Bi+1) for cyclic indices i ∈ Zk. Here we

use the notation for a quotient stack. See Appendix D. By Lemma D.4, ConfkPG is in

fact an algebraic variety. We have a dominant morphism ConfkPG → ConfkBG forgetting

the pinnings, which is a principal Hk-bundle over its image.

We will sometimes write an element of ConfkPG (i.e. a G-orbit) as [p12, . . . , pk−1,k, pk,1],

since the remaining data of flags can be read off from it via projections. However, the

reader is reminded that the tuples of pinnings must satisfy the constraints π−(pi−1,i) =

π+(pi,i+1) for i ∈ Zk.

2.3. Standard configurations and basic Wilson lines. We are going to give a cer-

tain explicit representative of an element of Conf3PG or Conf4PG. We also introduce

certain functions on these spaces called the basic Wilson lines, which will be the local


building blocks for the general Wilson line morphisms. The standard configuration makes

it apparent that the values of the basic Wilson lines are upper or lower triangular.

2.3.1. The standard configuration for Conf3PG. Let U±∗ := U±∩B∓w0B

∓ denote the open

unipotent cell, which is an open subvariety of U±. For the structure of its coordinate ring,

see Appendix B.1. Let φ′ : U+∗

∼−→ U−

∗ be the unique isomorphism such that φ′(u+).B+ =

u+.B−. See Appendix A for details.

Lemma 2.11 (cf. [FG06, (8.7)]). We have an isomorphism

C3 : H ×H × U+∗

∼−→ Conf3PG,

(h1, h2, u+) 7→ [B+, B−, u+.B−; pstd, φ

′(u+)h1w0.pstd, u+h2w0.pstd]

of varieties.

We call the parametrization C3 the standard configuration.

Proof. Since C3 is clearly a morphism of varieties, it suffices to prove that it is bijective.

Let (B1, B2, B3; p12, p23, p31) be an arbitrary configuration. Using the genericity condition

for the pairs (B1, B2), (B2, B3) and (B3, B1), we can write [B1, B2, B3] = [B+, B−, u+.B−]

for some u+ ∈ U+∗ . Using an element of B+ ∩ B− = H , we can further translate the

configuration so that p12 = pstd. Note that a representative of (B1, B2, B3; p12, p23, p31)

satisfying these conditions is unique.

Since p31 is now a pinning over the pair (u+.B−, B+), there exists h2 ∈ H such that

p31 = u+h2w0.pstd. Let us write p23 = g.pstd for some g ∈ G. Since p23 is a pinning

over the pair (B−, u+.B−), we have g.B+ = B− and hence g = b−w0 for some b− ∈ B−.

We also have g.B− = u+.B− = φ′(u+)w0.B

−, where the latter is the very definition of

the map φ′. It implies that b− = φ′(u+)h1 for some h1 ∈ H . Thus we get the desired

parametrization. �

Thus we get an induced isomophism

O(Conf3PG)∼−→ O(H)⊗2 ⊗O(U+

∗ )

of the rings of regular functions. Note that we can represent a configuration C ∈ Conf3PGin the following two ways:

• C = [pstd, p23, p31], where the first component is normalized,

• C = [p′12, p′23, pstd], where the last component is normalized.

Such representatives are unique since the set of pinnings is a principal G-space.

Definition 2.12. Define the elements bL = bL(C), bR = bR(C) ∈ G (“left”, “right”) by

the condition

p31 = (bL.pstd)∗, p′12 = (bR.pstd)

∗.

The resulting maps bL, bR : Conf3PG → G are called the basic Wilson lines.

Note that we have (bRw0)−1 = bLw0, since

C = [p′12, p′23, pstd] = [bRw0.pstd, p

′23, pstd] = [pstd, (bRw0)

−1.p′23, (bRw0)−1.pstd].


We remark here that these functions already appeared in [GS15, Section 6.2]. The fol-

lowing is a direct consequence of Lemma 2.11:

Corollary 2.13. We have bL(C) ∈ B+∗ and bR(C) ∈ B−

∗ for any configuration C ∈Conf3PG. The resulting maps

bL : Conf3PG → B+∗ , C 7→ bL(C), (2.7)

bR : Conf3PG → B−∗ , C 7→ bR(C) (2.8)

are morphisms of varieties, which are explicitly given by

bL(C3(h1, h2, u+)) = u+h2 ∈ B+∗ ,

bR(C3(h1, h2, u+)) = w0−1(u+h2)

−1w0 = ((u+h2)∗)T ∈ B−

∗ .

Remark 2.14. The definition of the basic Wilson lines bL and bR can be rephrased as

follows. Let us write a configuration as C = [g1.pstd, g2.pstd, g3.pstd] ∈ Conf3PG. Then we

have

bL(C) = g−11 g3w0 and bR(C) = g−1

3 g1w0,

and their regularity is also clear from this presentation.

The H3-action. Recall the rightH3-action on Conf3PG given by [p12, p23, p31].(k1, k2, k3) =

[p12.k1, p23.k2, p31.k3] for (k1, k2, k3) ∈ H3. It is expressed in the standard configuration by

C3(h1, h2, u+).(k1, k2, k3) = C3(k−11 h1w0(k2), k

−11 h2w0(k3),Ad

−1k1(u+)) (2.9)

for (h1, h2, u+) ∈ H ×H × U+∗ . By this action, the functions bL and bR are rescaled as

bL(C.(k1, k2, k3)) = k−11 bL(C)w0(k3), (2.10)

bR(C.(k1, k2, k3)) = k−13 bL(C)w0(k1). (2.11)

2.3.2. The standard configuration for Conf4PG. Following [FZ99], let us consider the open

subvariety

Zw0,w0 := {(u−, u+) ∈ U− × U+ | w0u−1− u+w0

−1 ∈ G0}.

Lemma 2.15. We have an isomorphism

C4 : Zw0,w0 ×H ×H ×H → Conf4PG,

((u−, u+), h1, h2, h3)

7→ [B+, B−, u−.B+, u+.B

−; pstd, u−h1w0.pstd, u−u′+h3.pstd, u+h2w0.pstd]

of varieties. Here u′+ := w0−1[w0u

−1− u+w0

−1]−w0 ∈ U+.

Proof. Since C4 is clearly a morphism of varieties, it suffices to show that it is bijective.

Write an arbitrary configuration as

[B1, B2, B3, B4; p12, p23, p34, p41] = [B+, B−, u−B+, u+B

−; pstd, g2.pstd, g3.pstd, g4.pstd]

for some u± ∈ U± and gi ∈ G. Note that (B+, B−), (B−, u−B+) and (u+B

−, B+) are

generic for any u± ∈ U±. The condition that B3 = u−B+ and B4 = u+B

− = u+w0B+


are in generic position is equivalent to u−1− u+w0 ∈ B+w0B

+, which is further rewritten as

w0−1u−1

− u+w0 ∈ B−B+ = G0.

Since g2.pstd is a pinning over the pair (B−, u−B+), we have g2 = u−h1w0 for some

h1 ∈ H . Similarly we have g4 = u+h2w0 for some h2 ∈ H .

From g3.B+ = u−B

+, we can write g3 = u−u′+h3 for some u′+h3 ∈ B+. Then from

g3.B− = u+B

−, we get u−u′+B

− = u+B−, equivalently,

w0u′+w0

−1B+ = w0u−1− u+w0

−1B+ = [w0u−1− u+w0

−1]−B+.

Here we have used the triangular decomposability. Therefore we get the desired statement.

�

Let Gw0,w0 := B+w0B+ ∩ B−w0B

− denote the open double Bruhat cell, which is an

open subvariety of G. By [FZ99, Proposition 3.1], we have an isomorphism

Zw0,w0 ×H∼−→ Gw0,w0, ((u−, u+), h) 7→ u−u

′+hw0,

here u′+ is the one defined in Lemma 2.15. See also Remark 2.18 below. It induces an

isomorphism

Gw0,w0 ×H ×H∼−→ (Zw0,w0 ×H)×H ×H

C4−→ Conf4PG,

where the component Zw0,w0 ×H is regarded as the space for the parameter ((u−, u+), h3)

in Lemma 2.15. Thus we get an induced isomorphism

O(Conf4PG)∼−→ O(Gw0,w0)⊗O(H)⊗2

of rings of regular functions. This description is a special case of the description of

ConfkPG for k ≥ 3 by Bott–Samelson varieties given in [SW19].

Now we are going to define basic Wilson lines on Conf4PG. Take a representative of a

configuration C ∈ Conf4PG of the form C = [pstd, p23, p34, p41] where the first component

is normalized.

Definition 2.16. Define three elements bL = bL(C), bS = bS(C), bR = bR(C) ∈ G (“left”,

“straight”, “right”) by the condition

p41 = (bL.pstd)∗, p34 = (bS.pstd)

∗, p23 = (bR.pstd)∗.

The resulting maps bL, bS, bR : Conf3PG → G are again called the basic Wilson lines.

From Lemma 2.15, we get the following:

Corollary 2.17. We have bL(C) ∈ B+, bR(C) ∈ B− and bS(C) ∈ Gw0,w0. The resulting

map

bX : Conf4PG → G, C 7→ bX(C) (2.12)

is a morphism of varieties for X ∈ {L, S,R}. Explicitly, we have

bL(C4((u−, u+), h1, h2, h3)) = u+h2,

bS(C4((u−, u+), h1, h2, h3)) = u−u′+h3w0,

bR(C4((u−, u+), h1, h2, h3)) = u−h1.


Remark 2.18. The isomorphism given in [FZ99, Proposition 3.1] is related to ours by an

adjustment, as follows. In [FZ99, Proposition 3.1], Fomin–Zelevinsky gave an isomorphism

Zw0,w0 ×H∼−→ Gw0,w0, ((u−, u+), h) 7→ u+w0[w0

−1u−u+w0]−1+ h

with Zw0,w0 := {(u−, u+) ∈ U− × U+ | w0u−u+w0−1 ∈ G0}. Our isomorphism above can

be obtained by composing it with the isomorphism

Zw0,w0 ×H∼−→ Zw0,w0 ×H, ((u−, u+), h) 7→ ((u−1

− , u+), w0(h)[w0−1u−1

− u+w0]−10 ).

Indeed,

u+w0[w0−1u−1

− u+w0]−1+ w0(h)[w0

−1u−1− u+w0]

−10

= u−w0(w0−1u−1

− u+w0)[w0−1u−1

− u+w0]−1+ w0(h)[w0

−1u−1− u+w0]

−10

= u−w0[w0−1u−1

− u+w0]−[w0−1u−1

− u+w0]0w0(h)[w0−1u−1

− u+w0]−10

= u−w0[w0−1u−1

− u+w0]−w0−1hw0

= u−u′+hw0.

3. Wilson lines on the moduli space PG,Σ

In this section, we first recall the definition of the moduli space PG,Σ for a marked surface

Σ. We give an explicit description of the structure of PG,Σ as a quotient stack (which

we call the Betti structure) as an algebraic basis for the arguments in the subsequent

sections. Then we introduce the Wilson line and Wilson loop functions on the stack PG,Σand study their basic properties. Finally we give their decomposition formula for a given

ideal triangulation (or an ideal cell decomposition) of Σ.

3.1. The moduli space PG,Σ. A marked surface Σ is a (possibly disconnected) compact

oriented surface with a fixed non-empty finite set M ⊂ Σ of marked points. A marked

point is called a puncture if it lies in the interior of Σ, and special point if it lies on the

boundary. Let P = P(Σ) (resp. S = S(Σ)) denote the set of punctures (resp. special

points), so that M = P ∪ S. Let Σ∗ := Σ \ P. We assume the following conditions:

(1) Each boundary component has at least one marked point.

(2) n(Σ) := −3χ(Σ∗) + 2|S| > 0, and Σ is not a disk with one or two special points.

These conditions ensure that the marked surface Σ has an ideal triangulation, which is

the isotopy class ∆ of a triangulation of Σ by a collection of mutually disjoint simple

arcs connecting marked points. The number n(Σ) is the number of edges of any ideal

triangulation. Denote the set of triangles of ∆ by t(∆), and the set of edges by e(∆).

Let eint(∆) ⊂ e(∆) be the subset of internal edges (i.e., those cannot homotoped into the

boundary).

In this paper, we only consider an ideal triangulation having no self-folded triangle

(i.e. a triangle one of its edges is a loop) for simplicity. Indeed, thanks to the condition

(2), our marked surface admits such an ideal triangulation. See, for instance, [FST08]2. More generally, one can consider an ideal cell decomposition: it is the isotopy class

2Note that the number n loc. sit. is the number of interior edges of an ideal triangulation.


m1

m2

m3

aI

m1

m2

m3

Ca

Figure 1. The marked surface Σ (left) and the surface Σ (right). A bound-

ary interval I is shown in red.

of a collection of mutually disjoint simple arcs connecting marked points such that each

complementary region is a polygon.

Recall that a G-local system on a manifoldM is a principal G-bundle over M equipped

with a flat connection. Let Σ be the compact oriented surface obtained from Σ by remov-

ing a small open disk around each puncture. We will use the surface Σ as a combinatorial

model where an ideal triangulation is drawn, while Σ is a geometric model on which we

consider local systems. See Figure 1. Let Ca denote the boundary component correspond-

ing to a puncture a. Let P ⊂ ∂Σ be the union of Ca for all a ∈ P. We call a connected

component of the set ∂Σ \ (S ∪ P) a boundary interval. Let B = B(Σ) denote the set of

boundary intervals. Each boundary interval belongs to any ideal triangulation of Σ. We

endow each boundary interval with the orientation induced from ∂Σ.

Let L be a G-local system on Σ. A framing of L is a flat section β of the associated

bundle LB := L ×G BG on (a small neighborhood of) S ∪ P.

Definition 3.1 (Fock-Goncharov [FG06]). Let XG,Σ denote the set of gauge-equivalence

classes of framed G-local systems (L, β).

For a description of XG,Σ as a quotient stack, see Section 3.1.1 soon below. We are

going to mainly deal with a variant of this moduli stack obtained by adding the data of

pinnings.

A framing β of L is said to be generic if for each boundary interval E = (m+E, m

−E)

with initial (resp. terminal) special point m+E (resp. m−

E), the associated pair (β+E , β

−E ) is

generic. Here β±E is the section defined near m±

E , and such a pair is said to be generic if

the pair of flags obtained as the value at any point on E is generic.

Let (L, β) be a G-local system equipped with a generic framing β. A pinning over

(L, β) is a section p of the associated bundle LP := L×G PG on the set ∂Σ \ (S∪ P) suchthat for each boundary interval E ∈ B, the corresponding section pE is a pinning over

(β+E , β

−E ). Here the last sentence means that pE is projected to the pair (β+

E , β−E ) via the

bundle map

LP |E(π+,π−)−−−−→ LB|E × LB|E → (LB)m+

E× (LB)m−

E,


where the former map is induced by the projection (2.6), and the latter is the evaluation

at the points m±E . Since LP is a principal G-bundle, a pinning of (L, β) determines a

trivialization of L near each boundary interval.

Definition 3.2 (Goncharov–Shen [GS19]). Let PG,Σ denote the set of the gauge-equivalence

classes [L, β; p] of the triples (L, β; p) as above.

If the marked surface Σ has empty boundary, we have PG,Σ ∼= XG,Σ. In general we have

a map PG,Σ → XG,Σ forgetting pinnings, which turns out to be a dominant morphism

with respect to their Betti structures. The image X 0G,Σ consists of the G-local systems

with generic framings. For each boundary interval E, we have a natural action αE :

PG,Σ × H → PG,Σ given by the rescaling of the pinning pE . Here recall that the set

of pinnings over a given pair of flags is a principal H-space. Thus the dominant map

PG,Σ → XG,Σ coincides with the quotient by these actions.

The following variant of the moduli space is also useful. Let Ξ ⊂ B be a subset. A

framed G-local system is said to be Ξ-generic if the pair of flags associated with any

boundary interval in Ξ is generic. Then we define the notion of Ξ-pinning over a Ξ-

generic framed G-local system, where we only assign pinnings to the boundary intervals

in Ξ.

Definition 3.3. Let PG,Σ;Ξ denote the set of gauge-equivalence classes of the triples

(L, β, p), where (L, β) is a Ξ-generic framed G-local system and p is a Ξ-pinning.

Obviously we have PG,Σ;∅ = XG,Σ and PG,Σ;B = PG,Σ.

3.1.1. The Betti structures of XG,Σ and PG,Σ. For simplicity, consider a connected marked

surface Σ. Fix a basepoint x ∈ Σ. A rigidified framed G-local system on a based marked

surface (Σ, x) consists of a framed G-local system (L, β) on Σ together with a choice of a

point s ∈ Lx.In order to obtain a concrete parametrization of rigidified framed G-local systems, let

us prepare some notations:

• For each puncture a ∈ P, let γa ∈ π1(Σ, x) denote a based loop freely homotopic

to the boundary component Ca.

• Enumerate the connected components of ∂Σ\ P as ∂1, . . . , ∂b, and let δk ∈ π1(Σ, x)

be a based loop freely homotopic to ∂k and following its orientation for k = 1, . . . , b.

• For k = 1, . . . , b, choose a distinguished marked point mk on the boundary com-

ponent ∂k. It turns the cyclic ordering on the boundary intervals on ∂k into a

linear ordering, and denote them by E(k)1 , . . . , E

(k)Nk

for this ordering. Here the

distinguished marked point mk is the initial marked point of E(k)1 .

• Take a path ǫ(k)1 = ǫ

E(k)1

from x to a point on the boundary interval E(k)1 for k =

1, . . . , b, and let ǫ(k)j = ǫ

E(k)j

be a path from x to E(k)j such that the concatenation

ǫ(k)j,j−1 := (ǫ

(k)j )−1∗ǫ

(k)j−1 is based homotopic to a boundary arc which contains exactly

one marked point, the initial vertex of E(k)j for j = 2, . . . , Nk.


In the pictures, the location of distinguished marked points is indicated by dashed lines.

See Figure 2. We will use the notation ǫE,E′ := ǫ−1E ∗ǫE′ for two boundary intervals E 6= E ′.

∂k Ca

x

ǫ(k)0

ǫ(k)1

ǫ(k)2

δk

γa

x

Figure 2. Some of the curves in the defining data of the atlases of XG,Σ and PG,Σ.

Notice that given a rigidified framed G-local system, the flat section of LB on Ca gives

an element of BG via the parallel-transport along the path from Ca to x surrounded by

the loop γa and the isomorphism BG∼−→ Lx, g.B 7→ s.g−1. Similarly, the flat section at

the initial marked point of a boundary interval E gives a flag via the parallel transport

along the path ǫE . Then we have the following:

Lemma 3.4 ([FG06, Definition 2.2], [AB18, Lemma 4.2]). There is a bijection between

the set of isomorphism classes of the rigidified framed G-local systems on Σ and the set

of points of the complex quasi-projective variety

X({mk})G,Σ :=

{(ρ, λ) ∈ Hom(π1(Σ, x), G)× (BG)

M∣∣∣ ρ(γa).λa = λa for all a ∈ P

}.

The group G acts on the isomorphism classes of rigidified framed G-local systems

(L, β; x, s) by fixing (L, β; x) and by s 7→ s.g for g ∈ G. This action is interpreted to an

action on the variety X({mk})G,Σ defined as (ρ, λ) 7→ (gρg−1, g.λ) for g ∈ G. Therefore we

define the moduli stack of framed G-local systems on Σ to be the quotient stack

XG,Σ := [X({mk})G,Σ /G].

We similarly introduce the moduli stack of framed G-local systems with pinnings, as

follows. A rigidified framed G-local system with pinnings consists of a triple (L, β; p)

together with a choice of s ∈ Lx. Let m(k)j ∈ M denote the initial marked point of the

boundary interval E(k)j . By convention, m

(k)1 = mk for k = 1, . . . , b. Then we have:

Lemma 3.5. There is a bijection between the set of isomorphism classes of the rigidified

framed G-local systems with pinnings on (Σ, x) and the set of points of the complex quasi-

projective variety P({mk})G,Σ consisting of triples (ρ, λ, φ) ∈ Hom(π1(Σ, x), G)×(BG)

M×(PG)B

which satisfy the following conditions:

• ρ(γa).λa = λa for all a ∈ P.


• π+(φE(k)j

) = λm

(k)j

and π−(φE(k)j

) = λm

(k)j+1

for k = 1, . . . , b and j = 1, . . . , Nk, where

we set λm

(k)Nk+1

:= ρ(δk).λm(k)1.

Moreover the correspondence is G-equivariant, where the group G acts on P({mk})G,Σ by

(ρ, λ, φ) 7→ (gρg−1, g.λ, g.φ) for g ∈ G.

Definition 3.6. The moduli stack of framed G-local systems with pinnings on Σ is defined

to be the quotient stack

PG,Σ := [P({mk})G,Σ /G]

over C.

Lemma 3.7. Suppose we replace the distinguished marked points as m′k := m

(k)2 by a shift

on a boundary component ∂k, and m′k′ := mk′ for k

′ 6= k. Then we have a G-equivariant

isomorphism

P({mk})G,Σ

∼−→ P

({m′k})

G,Σ

given by sending

(λ1, . . . , λNk) 7→ (λ2, . . . , λNk

, ρ(δk).λ1), (φ1, . . . , φNk) 7→ (φ2, . . . , φNk

, ρ(δk).φ1)

and keeping the other data intact. Here λj := λm

(k)j

and φj := φE

(k)j

for j = 1, . . . , Nk.

Proof. Follows from π+(ρ(δk).φ1) = ρ(δk).π+(φ1) = ρ(δk).λ1 and π−(ρ(δk).φ1) = ρ(δk).π−(φ1) =

ρ(δk)2.λ1 = ρ(δk).(ρ(δk).λ1). �

Hence the quotient stack PG,Σ is independent of the choice of distinguished marked

points. When no confusion can occur, we simply write PG,Σ = P({mk})G,Σ .

Suppose that the marked surface Σ has non-empty boundary. Since the group G freely

acts on the space PG of pinnings and the variety PG,Σ has at least one PG-factor, theG-action on PG,Σ is free. Then we get the following from Lemma D.4:

Lemma 3.8. There exists a geometric quotient PG,Σ/G which represents the stack PG,Σ.

Therefore we can regard the moduli stack PG,Σ as an algebraic variety. When ∂Σ = ∅,the stack PG,Σ = XG,Σ is no more representable.

The G-equivariant dominant morphism PG,Σ → XG,Σ forgetting the PG-factors inducesa dominant morphism PG,Σ → XG,Σ, which is a principal

∏E∈BH-bundle over its image.

Partially generic case. For any subset Ξ ⊂ B, the moduli stack of Ξ-generic framed

G-local systems with Ξ-pinnings (recall Definition 3.3) is similarly defined as

PG,Σ;Ξ := [P({mk})G,Σ;Ξ /G], (3.1)

where the algebraic variety P({mk})G,Σ;Ξ is obtained from P

({mk})G,Σ by forgetting the PG-factors

corresponding to B \Ξ. Here some of the distinguished marked points may be redundant

to obtain the atlas. For Ξ′ ⊂ Ξ, we have an obvious dominant morphism PG,Σ;Ξ → PG,Σ;Ξ′.

When Ξ 6= ∅, the stack PG,Σ;Ξ is still representable.


Disconnected case. When the marked surface Σ has N connected components, we con-

sider a rigidification of a framed G-local system (with pinnings) on each connected com-

ponent. Then the atlases XG,Σ and PG,Σ are defined to be the direct products of those

for the connected components, on which GN acts. The moduli stacks XG,Σ and PG,Σ are

defined as the quotient stacks for this GN -action.

3.1.2. The moduli space for a triangle. Let T be a triangle, i.e., a disk with three special

points m1, m2, m3 in this counter-clockwise order. The choice m ∈ {m1, m2, m3} of a

distinguished marked point determines an atlas P(m)G,T of the moduli space PG,T . Note that

the representable stack [P(m)G,T /G] is nothing but the configuration space Conf3PG. In other

words, the moduli space PG,T can be identified with the configuration space Conf3PG in

three ways, depending on the choice of a distinguished marked point. Let us denote the

isomorphism by

fmi: PG,T

∼−→ Conf3PG (3.2)

for i = 1, 2, 3. For later use, it is useful to indicate the distinguished marked point by the

symbol ∗ on the corresponding corner in figures, which we call the dot. See Figures 3–5

for example.

In topological terms, the isomorphisms fmiare described as follows. Given [L, β; p] ∈

PG,T , the local system L is trivial. We have a section βj of LB defined near mj , and a

section pj,j+1 of LP defined on the boundary interval Ej for j = 1, 2, 3 mod 3. Then

fmi:PG,T

∼−→ Conf3PG,

[L, β; p] 7→ [βi(x), βi+1(x), βi+2(x); pi,i+1(x), pi+1,i+2(x), pi+2,i+3(x)].

Here we extend the domain of each section until a common point x ∈ T via the parallel

transport defined by L. The following is a special case of Lemma 3.7.

Lemma 3.9. The coordinate transformation fmi+1◦ f−1

miis given by the cyclic shift

S3 : Conf3PG∼−→ Conf3PG, [p12, p23, p31] 7→ [p23, p31, p12]

for i = 1, 2, 3, which is an isomorphism.

An explicit computation of the cyclic shift S3 in terms of the standard configuration is

given in Section 6. We obtain the following lemma from (2.9).

Lemma 3.10. For j ∈ {1, 2, 3}, let Ej denote the boundary interval of T connecting the

marked points mj and mj+1. Then via the isomorphism

C3,mi:= f−1

mi◦ C3 : H ×H × U+

∗∼−→ PG,T

for i ∈ {1, 2, 3}, the action PG,T ×H3 → PG,T defined by (αEi, αEi+1

, αEi+2) is given by

C3,mi(h1, h2, u+).(k1, k2, k3) = C3,mi

(k−11 h1w0(k2), k

−11 h2w0(k3),Ad

−1k1(u+))

for (h1, h2, u+) ∈ H ×H × U+∗ and (k1, k2, k3) ∈ H3.


3.1.3. The moduli space for a quadrilateral. Now let us proceed to the moduli space PG,Qfor a quadrilateral Q, which is a disk with four special points m1, m2, m3, m4 in this

counter-clockwise order. We have four isomorphisms

f (4)mi

: PG,Q∼−→ Conf4PG

for i = 1, 2, 3, 4, depending on the choice of a distinguished vertex of the quadrilateral Q.

The coordinate transformation fmi+1◦ f−1

miis given by the cyclic shift

S4 : Conf4PG∼−→ Conf4PG, [p12, p23, p34, p41] 7→ [p23, p34, p41, p12],

which is again an isomorphism. On the other hand, we can decompose the quadrilateral

into two triangles T1 and T2 by choosing a diagonal. Then we have the following simplest

version of the gluing morphism:

PG,T1 ×PG,T2 → PG,Q.

It is defined as follows. For i = 1, 2, label the marked points of Ti as m(i)1 , m

(i)2 , m

(i)3

in the clockwise order so that the shared edge corresponds to the boundary intervals

(m(1)3 , m

(1)1 ) and (m

(2)1 , m

(2)2 ). Given [L(i), β(i), p(i)] ∈ PG,Ti, represent each of them so that

we have p(1)31 = (p

(2)12 )

∗ and glue them. Forgetting the pinning data p(1)31 = (p

(2)12 )

∗, we get an

element of PG,Q. Note that the resulting element remain unchanged after a transformation

(p(1)31 , p

(2)12 ) 7→ (p

(1)31 .h, p

(2)12 .w0(h)) for some h ∈ H . The following is the simplest version of

[GS19, Lemma 2.12]:

Lemma 3.11. The map PG,T1 ×PG,T2 → PG,Q is a morphism of varieties, which induces

an open embedding PG,T1 ×H PG,T2 → PG,Q.

Here we give an explicit proof for completeness, based on the standard configurations.

Proof. Since the structures of algebraic variety on the relevant moduli spaces do not

depend on the choices of dots and a corner of a quadrilateral, we can choose them to

be near a common vertex. See Figure 3. Then via the corresponding isomorphisms

fm

(1)1

× fm

(2)1

and f(4)m1 , the gluing map induces a map Conf3PG × Conf3PG → Conf4PG.

Then we have the following commutative diagram:

(U+∗ ×H ×H)× (U+

∗ ×H ×H) Conf3PG × Conf3PG

Zw0,w0 ×H ×H ×H Conf4PG,

C3×C3

C4

where the left vertical map is explicitly given as follows:

((u+, h1, h2), (v+, k1, k2)) 7→ ((φ′(u+), u+Adh2(v+)), h1, h2k2, [u+w0]0w0(h2k1)).

This is clearly a regular map and we get the desired statement. �

The image of the gluing morphism consists of the configurations of flags and pinnings

such that the pair of flags associated with the chosen diagonal is generic.

Remark 3.12. For a presentation of the gluing morphism in the Betti atlas, see Sec-

tion 3.2 below.


m(1)3

m(1)2

m(1)1

m(2)2

m(2)1

m(2)3

φ′(u+)h1w0.pstd

pstd

u+h2w0.pstd

u+h2.pstd

u+h2v+k2w0.pstd

u+h2φ′(v+)k1w0.pstd

∗ ∗

T1 T2

m(1)3 = m

(2)2

m(1)2

m(1)1 = m

(2)1 =: m1

m(2)3

φ′(u+)h1w0.pstd

pstd u+h2v+k2w0.pstd

u+h2φ′(v+)k1w0.pstd

Q

Figure 3. The gluing morphism for a quadrilateral. The pinnings assigned

to the boundary intervals (with the councter-clockwise orientation) are

shown.

3.2. Gluing morphisms. An advantage of considering the moduli space PG,Σ, ratherthan XG,Σ, is its nice property under the gluing procedure of marked surfaces. Let us

first give the “topological” definition of the gluing morphism. An explicit description as

a morphism of stacks is given soon below.

Let Σ be a (possibly disconnected) marked surface which has two boundary intervals

E1 and E2. Identifying the intervals E1 and E2, we get a new marked surface Σ′. Let

E denote the common image of E1 and E2, which is a new interior edge. On the level

of moduli spaces, given (L, β; p), note that the pinning pEνassigned to the boundary

interval Eν determines a trivialization of L near Eν for ν = 1, 2, since PG is a principal

G-space. Then there is a unique isomorphism beween the restrictions of (L, β) on Σ to

neighborhoods of E1 and E2 which identify the pinnings pE1 and p∗E2. In this way we

get a framed G-local system with pinnings qE1,E2(L, β; p) on Σ′. Note that the result is

unchanged under the transformation αE1,Eop2(h) : (pE1 , pE2) 7→ (pE1.h, pE2.w0(h)) for some

h ∈ H . We get the gluing morphism [GS19, Lemma 2.12]

qE1,E2 : PG,Σ → PG,Σ′, (3.3)

which induces an open embedding qE1,E2: PG,Σ/H → PG,Σ′, where H acts on PG,Σ via

αE1,Eop2.

The gluing operation is clearly associative. In particular, given an ideal triangulation

∆ of Σ, we can decompose the moduli space PG,Σ into a product of the configuration

spaces Conf3PG as follows. Let H∆ denote the product of copies of Cartan subgroups H ,

one for each interior edge of ∆. It acts on the product space P∆G,Σ :=

∏T∈t(∆) PG,T from

the right via αE1,Eop2

for each glued pair (E1, E2) of edges.

Theorem 3.13 ([GS19, Theorem 2.13]). Let ∆ be an ideal triangulation of the marked

surface Σ. Then we have the gluing morphism

q∆ : P∆G,Σ → PG,Σ, (3.4)


which induces an open embedding q∆ : [P∆G,Σ/H

∆] → PG,Σ.

The image of q∆ is denoted by P∆G,Σ ⊂ PG,Σ, which consists of framed G-local systems

with pinnings such that the pair of flags associated with each interior edge of ∆ is generic.

Presentation of the gluing morphism. Let us give an explicit presentation qE1,E2 :

P({mk})G,Σ → P

({m′k})

G,Σ′ of the gluing morphism (3.3) for some atlases for later use. For sim-

plicity, we assume that the resulting marked surface Σ′ is connected. Then we distinguish

the two cases: (1) Σ has two connected components Σ1 and Σ2 containing E1 and E2, re-

spectively, or (2) Σ is also connected. For example, the gluing morphism in Theorem 3.13

is obtained by succesively applying the gluings of the first type.

(1) The disconnected case: In this case, we have the van Kampen isomorphism

π1(Σ′, x) ∼= π1(Σ1, x) ∗ π1(Σ2, x) by choosing their common basepoints on the

new edge E. We also use the identifications M(Σ′) = M(Σ1) ⊔ (M(Σ2) \ {m±E2})

and B(Σ′) = B(Σ) \ {E1, E2}. For simplicity, we assume that the distinguished

marked points on the boundary components containing E1 and E2 are identified

under the gluing. The other cases are then obtained by composing the coordinate

transformations given in Lemma 3.7.

Given (ρ1, λ1, φ1; ρ2, λ2, φ2) ∈ PG,Σ = PG,Σ1 ×PG,Σ2, let us write (φ1)E1 = g1.pstdand (φ2)E2 = g2.pstd. Define (ρ′, λ′, φ′) ∈ PG,Σ′ by

ρ′(γ) :=

{ρ1(γ) if γ ∈ π1(Σ1),

Adg1w0g−12(ρ2(γ)) if γ ∈ π1(Σ2),

λ′m :=

{(λ1)m if m ∈ M(Σ1),

g1w0g−12 .(λ2)m if m ∈ M(Σ2) \ {m

±E2},

φ′E :=

{(φ1)E if E ∈ B(Σ1),

g1w0g−12 .(φ2)E if E ∈ B(Σ2).

Here ρ′ is extended as a group homomorphism. In terms of the rigidified framed

G-local systems, we have chosen the rigidification data given on Σ1 as that for Σ.

(2) The connected case: In this case, we have an inclusion π1(Σ) = π1(Σ′ \ E) →

π1(Σ′) and rankπ1(Σ

′) = rankπ1(Σ) + 1. Choose the basepoint x on the new

edge E. Then π1(Σ′, x) is generated by the based loop α := ǫ−1

E1∗ ǫE2 and the

subgroup π1(Σ, x). We use the identifications M(Σ′) = M(Σ)\{m±E2} and B(Σ′) =

B(Σ) \ {E1, E2}. When E1 and E2 belong to distinct boundary components, we

assume that their distinguished marked points are identified under the gluing. The

other cases are then obtained by composing the coordinate transformations given

in Lemma 3.7.


Given (ρ, λ, φ) ∈ PG,Σ, let us write φE1 = g1.pstd and φE2 = g2.pstd. Define

(ρ′, λ′, φ′) ∈ PG,Σ′ by

ρ′(γ) :=

{ρ(γ) if γ ∈ π1(Σ),

g1w0g−12 if γ = α,

λ′ := λ|M(Σ′),

φ′ := φ|M(Σ′).

Here ρ is extended as a group homomorphism.

Lemma 3.14. The morphism qE1,E2 : P({mk})G,Σ → P

({m′k})

G,Σ′ given above descends to a mor-

phism PG,Σ → PG,Σ′, which agrees with the topological definition of the gluing morphism

(3.3).

Proof. The morphism qE1,E2 is clearly G-equivariant, and hence descends to a morphism

PG,Σ → PG,Σ′ . In order to see that it agrees with the topological definition, observe the

followings.

In the disconnected case, consider the action of the element g1w0g−12 on the triple

(ρ2, λ2, φ2) by rescaling the rigidification. After such rescaling, the pinning assigned to

the boundary interval E2 gives g1w0g−12 .(φ2)E2 = g1w0.pstd, which is the opposite of the

pinning (φ1)E1. Thus the gluing condition matches with the one explained in the beginning

of this subsection.

In the connected case, note that the monodromy ρ′(α) is the unique element such that

ρ′(α).(g2.pstd)∗ = g1.pstd, which is given by ρ′(α) = g1w0g

−12 . The remaining data is

unchanged, since the system of curves {δk, ǫE} on Σ is naturally inherited to Σ′. �

Substacks associated with decompositions of marked surfaces. For later use, we

recall certain substacks of PG,Σ associated with ideal cell decompositions of Σ, which has

been introduced by [She20, Theorem 1.1].

For an ideal triangulation ∆ of Σ and an interior edge E ∈ eint(∆), let P∆;EG,Σ ⊂ PG,Σ be

the open subvariety which consist of the rigidified framed G-local systems with pinnings

such that for each E ′ ∈ eint(∆) \ {E} the pair of flags associated with E ′ is generic. Let

P∆;EG,Σ := [P∆;E

G,Σ /G] ⊂ PG,Σ be the corresponding open substack. One can think of P∆;EG,Σ

as associated with the ideal cell decomposition obtained from ∆ by removing the edge E,

which is denoted by (∆;E).

Decompose the surface Σ into the disjoint union of the quadrilateral QE which has

the edge E as one of its diagonal and other triangles of ∆. Then P∆;EG,Σ is the image of

the gluing morphism q∆;E with respect this decomposition, and we have an isomorphism

q∆;E : [P∆;EG,Σ /H

∆;E]∼−→ P∆;E

G,Σ as a slight variant of Theorem 3.13. Here

P∆;EG,Σ := PG,QE

×∏

T : ∂T+E

PG,T


is an affine variety as described in Section 2.3, and H∆;E :=∏

eint(∆)\{E}H . In particular,

we get an isomorphism

O(P∆;EG,Σ ) = O(P∆;E

G,Σ )Gq∗∆;E−−→ O(P∆;E

G,Σ )H∆;E

from Lemma D.1. We have describedO(PG,T ) ∼= O(Conf3PG) andO(PG,QE) ∼= O(Conf4PG)

in Section 2.2, where an explicit isomorphism is obtained by choosing a dot for each tri-

angle and a quadrilateral of the ideal cell decomposition (∆;E). Such a data is referred

to as follows:

Definition 3.15. An ideal triangulation ∆ equipped with a dot for each triangle is called

a dotted triangulation, and denoted by ∆∗. Similarly we define a dotted cell decomposition

to be an ideal cell decomposition equipped with a dot for each polygon.

Recall from Lemma 3.8 that when Σ has non-empty boundary, the stack PG,Σ is repre-

sentable, and so is the open substack P∆;EG,Σ .

Proposition 3.16. Suppose that Σ has non-empty boundary. Then the H∆;E-action on

the affine variety P∆;EG,Σ is free. In particular, there exists a geometric quotient P∆;E

G,Σ /H∆;E

which represents the stack [P∆;EG,Σ /H

∆;E].

The proof is based on the following local statement:

Lemma 3.17. (1) If the right action of (k1, k2, k3) ∈ H3 on Conf3PG has a fixed

point, then k1 = k2 = k3 and k1 = w0(k1). In particular, the subgroup H2 ⊂ H3

obtained by setting one of ki’s to be 1 acts freely on Conf3PG.(2) If the right action of (k1, k2, k3, k4) ∈ H4 on Conf4PG has a fixed point, then

k1 = k3, k2 = k4, and k2 = w0(k1). In particular, the subgroup H3 ⊂ H4 obtained

by setting one of ki’s to be 1 acts freely on Conf4PG.

Proof. From the expression (2.9), the equations k−11 h1w0(k2) = h1 and k−1

1 h2w0(h3) = h2for some h1, h2 ∈ H lead to k2 = w0(k1) and k3 = w0(k1), respectively. If moreover

Ad−1k1(u+) = u+ for some u+ ∈ U+

∗ , then ∆λ,w0λ(k−11 u+k1) = ∆λ,w0λ(u+) 6= 0 for all

λ ∈ X∗(H)+. Since ∆λ,w0λ(k−11 u+hk1) = k−λ+w0λ

1 ∆λ,w0λ(u+), this leads to k−λ+w0λ1 = 1

for all λ ∈ X∗(H)+, and hence w0(k1) = k1. This proves the first part. One can similarly

prove the second part by using the standard configuration given in Lemma 2.15. �

Proof of Proposition 3.16. Suppose that a tuple (hE′)E′∈eint(∆),E′ 6=E ∈ H∆;E has a fixed

point in P∆;EG,Σ . By choosing an arbitary collection of dots on (∆;E), we may identify

P∆;EG,Σ with a product of copies of Conf3PG and one Conf4PG. Then by successively

applying the first part of Lemma 3.17 to PG,T ∼= Conf3PG for all triangles T in (∆;E),

we see that two elements hE1 and hE2 assigned to the edges E1, E2 6= E are equal to each

other if they belong to the same connected component of Σ \ intQE . It may happen that

the surface Σ\ intQE is disconnected, but the number of connected components is at most

two. Then combined with the second part of Lemma 3.17 to PG,QE∼= Conf4PG, we see

that all the elements hE′ for E ′ 6= E are equal to each other.


Now we claim that this common element h ∈ H must be trivial. Suppose first that

there exists a triangle T in (∆;E) one of whose edges belongs to the boundary of Σ.

Since the group H∆;E does not have components corresponding to the boundary intervals

of Σ, at most two H-components acts on PG,T ∼= Conf3PG. Hence the second statement of

Lemma 3.17 (1) for implies that the elements of theseH-components must be trivial. Thus

h = 1. If no such triangle exists, then the boundary intervals belong to the quadrilateral

QE and we get h = 1 similarly by the second statement of Lemma 3.17 (2). �

Remark 3.18. When ∂Σ = ∅, the stack [P∆;EG,Σ /H

∆;E] is no more representable due to

the non-free action of the diagonal subgroup H ⊂ H∆;E. This ill-behavior of the H-action

can be understood to be occuring in the “last step” of gluings. Namely, let us choose one

edge E ′ 6= E and consider the marked surface Σ′ obtained by cutting Σ along E ′. Then

the stack [P∆;EG,Σ /H

∆;E,E′] is representable by Proposition 3.16, where H∆;E,E′

⊂ H∆;E is

the subgroup obtained by setting hE′ := 1. Then the gluing map factors through

[P∆;EG,Σ /H

∆;E] ∼= [(P∆;EG,Σ /H

∆;E,E′

)/H ] → P∆;EG,Σ = [P∆;E

G,Σ /G],

which is an open embedding by Lemma D.2.

The Ptolemy algebra. Notice that since P∆;EG,Σ ⊂ PG,Σ is an open substack, the induced

algebra homomorphism

O(PG,Σ) → O(P∆;EG,Σ )

q∗∆;E−−→ O(P∆;E

G,Σ )H∆;E

(3.5)

is injective for any ideal cell decomposition (∆;E). Let us consider the algebra

OPt(PG,Σ) :=⋂

(∆;E)

O(P∆;EG,Σ )H

∆;E

,

where the intersection is taken over all ideal triangulations ∆ of Σ and all interior edges

E ∈ eint(∆). We call OPt(PG,Σ) the Ptolemy algebra after Claudius Ptolemy to distinguish

from the Betti algebra O(PG,Σ). Then the codimension 2 argument in the proof of [She20,

Theorem 1.1] implies that the inclusions (3.5) combine to give an isomorphism

O(PG,Σ)∼−→ OPt(PG,Σ). (3.6)

Therefore we have two descriptions of the function algebra of the moduli stack PG,Σ, bothbeing identified with the corresponding cluster Poisson algebra [She20] (see Theorem 4.22).

Note that P∆G,Σ ⊂ P∆;E

G,Σ is an open substack for any E ∈ eint(∆). Then the Ptolemy

algebra might be regarded as the ring of regular functions on the stack obtained as the

push-out of the moduli stacks P∆;EG,Σ along these open embeddings.

3.3. Side pairings. As a simple example of Wilson lines, we introduce the side pairings

for a configuration of pinnings on a polygon. When the polygon is given as (an appropriate

subpolygon of) a fundamental polygon of a marked surface, the side pairing plays a role

of the “universal” function from which we can compute the Wilson lines and loops defined

on the moduli space PG,Σ.


First we give a topological definition. Let Π be an oriented k-gon, which can be regarded

as a marked surface (i.e., a disk with k special points on the boundary). Choose two

boundary intervals (i.e., side edges) Ein, Eout of Π, which are endowed with the orientation

induced from the boundary as before. Since Π is contractible, any local system on Π is

trivial. For each point [L, β; p] ∈ PG,Π, choose a trivialization of L normalized so that the

section of LP defined on Ein corresponds to pstd. Let pEoutbe the pinning associated to

Eout under this normalized trivialization for Ein.

Definition 3.19. For a point C = [L, β; p] ∈ PG,Π, the side pairing for the pair (Ein, Eout)

is the element g = gEin,Eout(C) ∈ G uniquely determined by the equation g.p∗std = pEout.

Thus we get a map

gEin,Eout : PG,Π → G.

This can be defined as a morphism, as follows. Note that just similarly as the triangle

and quadrilateral, case, we have isomorphisms PG,Π ∼= ConfkPG depending on the choice of

a distinguished marked point. By Lemma 3.7, the transitions between these isomorphisms

are given by cyclic shifts of the PG-components.

Proposition 3.20. Define a G-invariant morphism gEin,Eout : PG,Π → G of varieties by

gEin,Eout(φE1, . . . , φEk) := g−1

EingEoutw0,

where we write φE = gE .pstd for a unique gE ∈ G for each boundary interval E. Then the

induced morphism gEin,Eout : PG,Π → G of varieties agrees with the topological definition

given above.

Proof. Recall that the points in the variety PG,Π is in bijection with rigidified framed G-

local systems with pinnings (L, β, p; s) on Π. Here the rigidification s determines a global

trivialization of L, and the value of the flat section p on a boundary interval E gives the

element φE = gE.pstd in this trivialization. Then the normalized trivialization for Ein is

obtained by multiplying g−1Ein

to s. In the latter trivialization, the value of p at Eout gives

g−1EingEout.pstd = g−1

EingEoutw0.p

∗std. Thus g

−1EingEoutw0 gives the side pairing. �

Remark 3.21. In order to define the side pairing gEin,Eout, we only need the pinnings

assigned to the boundary intervals Ein and Eout. Hence it is obvious from Proposition 3.20

that the side pairing can be defined as a morphism

gEin,Eout : PG,Π;{Ein,Eout} → G,

where recall the notation (3.1).

Let ∆ be an ideal triangulation of Π. Take a path c from the edge Ein to Eout so that

the intersection with ∆ is minimal. Let T1, . . . , TM be the triangles of ∆ which c traverses

in this order. Note that for each ν = 1, . . . ,M , the intersection c∩ Tν is either one of the

two patterns shown in Figure 4. The turning pattern of c with respect to ∆ is encoded

in the sequence τ∆(Ein, Eout) = (τν)Mν=1 ∈ {L,R}N , where τν = L (resp. τν = R) if c ∩ Tν

is the left (resp. right) pattern in Figure 4. For our purpose, it is enough to consider the

case T1 ∪ · · · ∪ TM = Π. An example for k = 6 is shown in Figure 5.


B(ν)1

∗

B(ν)2 B

(ν)3

c

τν = L

B(ν)1

∗

B(ν)2 B

(ν)3

c

τν = R

Figure 4. Two intersection patterns of c ∩ Tν

Ein

Eout

∗

∗

∗

∗c

Figure 5. A side pairing for k = 6. The turning pattern is τ∆(Ein, Eout) = (L,R, L,R).

Definition 3.22. Let ∆ be an ideal triangulation of Π. Let Ein, Eout be two side edges

of the polygon Π such that the associated path c traverses every triangle of ∆. We put

a dot mν on the triangle Tν so that the arc c ∪ Tν separates the corner with the dot mν

from the other two corners (see Figure 4). The resulting dotted triangulation is written as

∆∗ = ∆∗(Ein, Eout), and called the dotted triangulation associated with the pair (Ein, Eout).

Let us first consider the restriction of the side pairing to the substack P∆G,Π. Let q∆ :∏M

ν=1PG,Tν → P∆G,Π be the gluing morphism and define

gν :=

{bL ◦ fmν

: PG,Tν → B+∗ if τν = L,

bR ◦ fmν: PG,Tν → B−

∗ if τν = R

for ν = 1, . . . ,M . Then we have the following:

Proposition 3.23 (Decomposition formula for side pairings). We have

q∗∆gEin,Eout = µM ◦M∏

ν=1

gν ,

where µM denotes the multiplication of M elements in G.

Proof. We proceed by induction onM ≥ 1. The caseM = 1 follows from Proposition 3.20

and Remark 2.14. For M > 1, consider the polygon Π′ := T1 ∪ · · · ∪ TM−1 and the gluing

morphism

qM : PG,Π′⊔TM → PG,Π


where a boundary interval E of Π′ and that E ′ of TM are glued to give the common

edge of TM−1 and TM in Π. We claim that q∗MgEin,Eout = gEin,E · gE′,Eout. Since we know

gE′,Eout = gM , the induction step proceeds.

Recall from Proposition 3.20 that the relevant side pairings are given by the G-invariant

functions

gEin,Eout = g−1EingEoutw0, gEin,E = g−1

EingEw0, gE′,Eout = g−1

E′ gEoutw0.

From the presentation of the gluing morphism given in Section 3.2, we have

q∗M gEin,Eout = g−1Ein

(gEw0g−1E′ · gEout)w0

= g−1EingEw0 · g

−1E′ gEoutw0

= gEin,E · gE′,Eout

as desired. �

As a slight generalization, we can get a similar decomposition formula for q∗∆;EgEin,Eout

for any ideal cell decomposition (∆;E). To simplify the notation, let us consider the case

where E is the edge shared by the triangles T1 and T2.

Relabel the four marked points of the quadrilateral Q12 := T1∪T2 as m(12)k , k = 1, 2, 3, 4

in the counter-clockwise order so that the boundary interval (m(12)1 , m

(12)2 ) corresponds to

Ein. Choose the dot for Q12 to be m12 := m(12)1 , and consider the isomorphism f

(4)m12 :

PG,Q12

∼−→ Conf4PG. Define a function g12 : PG,Q12 → G by g12 := bX ◦ f

(4)m12 if τ12 = X ∈

{L, S,R}, where the turning pattern τ12 is defined as in Figure 6. Then similarly to the

argument in the proof of Proposition 3.23, one can verify that

q∗∆;EgEin,Eout = µM−1 ◦ (g12 × g3 × · · · × gM) : P∆;EG,Π → G,

which is an H∆;E-invariant morphism by Corollary 2.17.

m(12)3

m(12)2

m(12)1

m(12)4

Ein

∗

c

τ12 = L

m(12)3

m(12)2

m(12)1

m(12)4

Ein

∗

c

τ12 = S

m(12)3

m(12)2

m(12)1

m(12)4

Ein

∗

c

τ12 = R

Figure 6. The intersection patterns of c ∩Q12


3.4. Wilson lines and Wilson loops. Generalizing the side pairings, we introduce the

Wilson line morphisms (Wilson lines for short) on PG,Σ for a marked surface with non-

empty boundary.

Let Ein, Eout ∈ B be two boundary intervals, and c a path from Ein to Eout in Σ. After

applying an isotopy if necessary, the path c can be also viewed as a path in the surface

Σ: henceforth we tacitly use this identification. First we give a topological definition. For

a point [L, β; p] ∈ PG,Σ, choose a local trivialization of L on a vicinity of Ein so that the

flat section pEinof LP associated to Ein corresponds to pstd. This local trivialization can

be extended along the path c until it reaches Eout. Then the flat section pEout determines

a pinning under this trivialization, which is written as g.p∗std for a unique element g =

gc([L, β; p]) ∈ G. It depends only on the homotopy class [c] of c relative to Ein and Eout:

we call such a homotopy class [c] an arc class, and write [c] : Ein → Eout in the sequel.

Then we have a map

g[c] : PG,Σ → G,

which we call the Wilson line along the arc class [c] : Ein → Eout.

Example 3.24. Let Π be a polygon. Then for each pair (Ein, Eout) of its sides, there is a

unique arc class [c] : Ein → Eout. Then the associated Wilson line coincides with the side

pairing considered before: g[c] = gEin,Eout.

As in the case of side pairings, the Wilson lines can be defined as morphisms PG,Σ → G.

Fix a basepoint x ∈ Σ and the collection {mk} of distinguished marked points, and

consider the corresponding presentation PG,Σ = [P({mk})G,Σ /G]. Notice that any arc class

[c] : Ein → Eout can be decomposed as [c] = [ǫEin]−1 ∗ [γx] ∗ [ǫEout], where γx is a based

loop at x.

Proposition 3.25. Define a G-invariant morphism gEin,Eout : P({mk})G,Σ → Gof varieties by

gEin,Eout(ρ, λ, φ) := g−1Einρ(γx)gEoutw0,

where we write φE = gE .pstd for a unique element gE ∈ G for a boundary interval E.

Then the induced morphism gEin,Eout : PG,Σ → G of varieties agrees with the topological

definition given above.

Proof. The proof proceeds similarly to that of Proposition 3.20. Consider the rigidified

framed G-local system with pinnings (L, β, p; s) corresponding to a given point of PG,Σ.

The rigidification s determines a local trivialization of L near x, and the section pEin

(resp. pEout) gives the element gEin.pstd (resp. gEout.pstd) via the parallel-transport along

the path ǫEin(resp. ǫEout) under this local trivialization. Moreover, notice that the section

pEout gives ρ(γx)gEout.pstd via the parallel-transport along the path γx ∗ ǫEout . The local

trivialization of L near Ein so that pEincorresponds to pstd is given by the rigidification

s.g−1Ein

. The latter trivialization can be continued along the path ǫ−1Ein

∗ γx ∗ ǫEout, for which

the section pEout gives g−1Einρ(γx)gEout.pstd = g−1

Einρ(γx)gEoutw0.p

∗std as desired. �


Remark 3.26. The Wilson line g[c] along an arc class [c] : Ein → Eout can be defined as

a morphism

g[c] : PG,Σ;{Ein,Eout} → G.

Here recall (3.1).

The Wilson lines g[c] have the following multiplicative property with respect to the glu-

ing of marked surfaces. Let Σ be a (possibly disconnected) marked surface, and consider

two arc classes [c1] : E1 → E2 and [c2] : E′2 → E3. Let Σ

′ be the marked surface obtained

from Σ by gluing the boundary intervals E2 and E ′2. Then the concatenation of the arc

classes [c1] and [c2] give an arc class [c] : E1 → E3 on Σ′. See Figures 7 and 8. Recall the

gluing morphism qE2,E′2: PG,Σ → PG,Σ′.

Proposition 3.27. We have q∗E2,E′2g[c] = g[c1] · g[c2].

Proof. Recall the presentation of the gluing morphism given in Section 3.2. We may

assume that Σ′ is connected without loss of generality, and divide the argument into the

two cases.

(1) Disconnected case: In this case, we have

q∗E2,E′2g[c] = g−1

E1(gE2w0g

−1E′

2· gE3)w0

= g−1E1gE2w0 · g

−1E′

2gE3w0

= g[c1] · g[c2].

(2) Connected case: In this case, consider the based loop αx := ǫE2 ∗ ǫ−1E′

2∈

π1(Σ′, x). See Figure 8. Then we have g[c] = g−1

E1ρ(αx)gE3w0 and q∗E2,E′

2ρ(αx) =

gE2w0g−1E′

2. Hence

q∗E2,E′2g[c] = g−1

E1(gE2w0g

−1E′

2)gE3w0

= g−1E1gE2w0 · g

−1E′

2gE3w0

= g[c1] · g[c2].

�

As a variant of the above argument, we can describe the monodromy homomorphism

in terms of the Wilson lines. Given a G-local system L on Σ , a base point x ∈ Σ and a

local trivialization s at x, we get the monodromy homomorphism ρs•(L) : π1(Σ, x) → G,

[γ] 7→ ρs[γ](L). The set of conjugacy classes in π1(Σ, x) is identified with the set π(Σ) :=

[S1,Σ] of free loops on Σ. The free homotopy class of a loop γ is denoted by |γ|, so that

we have a canonical projection π1(Σ, x) → π(Σ), [γ] 7→ |γ|. It is a classical fact that the

conjugacy class

ρ•(L) := [ρs•(L)] ∈ Hom(π1(Σ, x), G)/AdG ⊂ Map(π(Σ), G/AdG)

of the monodromy homomorphism depends only on the gauge-equivalence class of L, andconversely determines the latter. We call the function ρ|γ| : PG,Σ → G/AdG the Wilson

loop along a free loop |γ| ∈ π(Σ).


E1 E2

[c1]

E ′2

E3

[c2] Glue E2 and E ′2

E1

[c]

E3

Figure 7. Multiplicativity of Wilson lines, disconnected case.

E1

E2

[c]Glue E1 and E2

|γ|

Figure 8. Multiplicativity of Wilson lines, connected case.

The Wilson loop can be defined as a morphism ρ|γ| : PG,Σ → LocG,Σ → [G/AdG],

where LocG,Σ := [Hom(π1(Σ, x), G)/G] denotes the moduli stack of G-local systems and

the first morphism is induced by the projection PG,Σ → Hom(π1(Σ, x), G), (ρ, λ, φ) 7→ ρ.

The second morphism is induced by the G-equivariant morphism Hom(π1(Σ, x), G) → G

given by the evaluation at the based loop [γx] ∈ π1(Σ, x) presenting the free loop |γ|. Letρ|γ| : PG,Σ → G denote the composite of these morphisms on atlases.

Proposition 3.28. Let Σ be a marked surface, [c] : E1 → E2 an arc class. Let Σ′ be

the marked surface obtained from Σ by gluing the boundary intervals E1 and E2, and

|γ| ∈ π(Σ′) be the free loop arising from [c]. Then we have the following commutative


diagram of morphisms of stacks:

PG,Σ G

PG,Σ′ [G/AdG],

g[c]

qE1,E2

ρ|γ|

(3.7)

where the right vertical morphism is the canonical projection.

Proof. From the presentation of the gluing morphism given in Section 3.2, we have

q∗E1,E2ρ|γ| = gE1w0g

−1E2

= AdgE2(g[c]).

In other words, we have the commutative diagram

PG,Σ G [G/AdG]

PG,Σ′ G [G/AdG]

g[c]

qE1,E2AdgE2

ρ|γ|

and thus we get the desired assertion. �

Remark 3.29 (Twisted Wilson lines). Let ΠB1 (Σ) be the groupoid whose objects are

boundary intervals of Σ and morphisms are arc classes with the composition rule given

by concatenations. Then each point [L, β; p] ∈ PG,Σ defines a functor

gtw• ([L, β; p]) : ΠB1 (Σ) → G, [c] 7→ gtw[c] ([L, β; p]),

where gtw[c] ([L, β; p]) := g[c]([L, β; p])w0 denotes the twisted Wilson line and the group G

is naturally regarded as a groupoid with one object. Note that an automorphism [c] of

a boundary interval E in ΠB1 (Σ) can be represented by a loop γ based at x ∈ E, and

the conjugacy class of the twisted Wilson line gtw[c] coincides with the Wilson loop ρ|γ|.

Although the Wilson lines themselves do not induce such a functor, we will see that they

possess a nice positivity property as well as the mulitplicativity for gluings explained

above.

3.5. Generation of O(PG,Σ) by matrix coefficients of (twisted) Wilson lines.

We are going to obtain an explicit presentation of the Betti algebra O(PG,Σ) by using

the (twisted) Wilson lines when Σ has no punctures. In the contrary case ∂Σ = ∅, thedescription of the Betti algebra O(PG,Σ) = O(XG,Σ) as an O(LocG,Σ)-module has been

already obtained in [FG06, Section 12.5].

Choose a generating set S = {(αi, βi)gi=1, (γa)a∈P, (δk)

bk=1} of π1(Σ, x), a collection {mk}

of distinguished marked points, and paths ǫ(k)j = ǫ

E(k)j

as in Section 3.1.1. Then we get a

Betti atlas P({mk})G,Σ , which consists of triples (ρ, λ, φ) satisfying certain conditions described

in Lemma 3.5.

Assume that Σ has no punctures, and choose one boundary interval, say, E0 := E(1)1 .

Write φE = gE.pstd, gE ∈ G for E ∈ B and set g0 := gE0. Then we have a G-invariant


morphism

Φ′E0

: PG,Σ → G2g+b ×GB\{E0}

which sends (ρ, λ, φ) to the tuple((ρE0(αi), ρE0(βi), ρE0(δk))i=1,...,g

k=1,...,b, (gE0,E)E 6=E0

).

Here ρE0(γ) := g−10 ρ(γ)g0 is the monodromy along γ for the local trivialization given by

the pinning φE0 for γ ∈ S, and gE0,E := g−10 gEw0 is the Wilson line along the arc class

[ǫE0,E] = [ǫ−1E0

∗ ǫE ] : E0 → E. Then it descends to an embedding

Φ′E0

: PG,Σ = P({mk})G,Σ /G→ G2g+b ×GB\{E0} (3.8)

of varieties. Note from Remark 3.29 that ρE0(γ) for γ ∈ S can be regarded as the twisted

Wilson line along the based loop γx at x ∈ E0. We can take their matrix coefficients, not

only their traces.

For each k = 1, . . . , b, consider the paths ǫ(k)j,j−1 which are based-homotopic to boundary

arcs which contain exactly one marked point m(k)j , for j = 1, . . . , Nk. Here the indices are

read modulo Nk.

Lemma 3.30. The Wilson line g(k)j,j−1 along the arc class [ǫ

(k)j,j−1] : E

(k)j → E

(k)j−1 takes

values in B+.

Proof. Let E1 (resp. E2) denote the boundary interval having m(k)j as its initial (resp.

terminal) point. Let φi = gi.pstd be the pinning assigned to Ei for i = 1, 2. It can happen

that E1 = E2: in that case, we have g2 = ρ(δk)−1g1. From the condition π+(φ1) = λ

m(k)j

=

π−(φ2), we get g1.B+ = g2.B

− = g2w0.B+. Hence g

(k)j,j−1 = g−1

1 g2w0 ∈ B+. �

Since ǫE0,E

(k)j

= ǫE0,E

(k)Nk

∗ ǫ(k)Nk,Nk−1 ∗ · · · ∗ ǫ

(k)j+1,j, we have

gE0,E

(k)j

= (g(k)w0)(g(k)Nk,Nk−1w0) · · · (g

(k)j+2,j+1w0)g

(k)j+1,j (3.9)

for k = 1, . . . , b and j = 2, . . . , Nk − 1. Here g(k) := gE0,E

(k)Nk

denotes the Wilson line along

the arc class [ǫE0,E

(k)Nk

] : E0 → E(k)Nk

for k = 2, . . . , b, and g(1) := 1. See Figure 9. Therefore

the embedding (3.8) gives rise to another embedding

ΦE0 : PG,Σ → G2g+b ×Gb−1 × (B+)∑b

k=1Nk ,

which sends a G-orbit of (ρ, λ, φ) to the tuple((ρE0(αi), ρE0(βi), ρE0(δk))i=1,...,g

k=1,...,b, (g(k))k=2,...,b, (g

(k)j,j−1) k=1,...,b

j=1,...,Nk

). (3.10)

Theorem 3.31. The image of the embedding ΦE0 is the closed subvariety which consists

of the tuples (3.10) satisfying the following conditions:

• Monodromy relation:∏g

i=1[ρE0(αi), ρE0(βi)] ·∏b

k=1 ρE0(δk) = 1;

• Boundary relation: (g(k)Nk,Nk−1w0) · · · (g

(k)2,1w0)(g

(k)1,Nk

w0) = ρE0(δk)−1 for k = 1, . . . , b.


E(1)1

=

E0

E(1)2

E(1)3 g

(1)2,1

g(1)3,2

g(1)4,3

g(k) E(k)1

E(k)2

E(k)3

E(k)4

g(1)2,1

g(1)3,2g

(1)4,3

g(1)1,4

Figure 9. Some Wilson lines.

Proof. It is clear from the previous discussion and the multiplicative property of the

twisted Wilson lines gtw[c] = g[c]w0 that the image of ΦE0 satisfies the conditions. Conversely,

given a tuple (3.10) which satisfies the conditions, we can reconstruct the G-orbit of a

triple (ρ, λ, φ) ∈ P({mk})G,Σ , as follows. We first get the monodromy homomorphism ρE0

normalized at the boundary interval E0, and the pinning φE0 = pstd. The other pinnings

are given by φE := (gE0,E.pstd)∗ for E ∈ B \ {E0}, where gE0,E ∈ G is determined by the

formula (3.9). The collection λ of the underlying flags is given by

λm

(k)j

:= π+(φE(k)j

)

= (g(k)w0)(g(k)Nk,Nk−1w0) · · · (g

(k)j+2,j+1w0)g

(k)j+1,j.B

−

= (g(k)w0)(g(k)Nk,Nk−1w0) · · · (g

(k)j+2,j+1w0)(g

(k)j+1,jw0)g

(k)j,j−1.B

+

= π−(φE(k)j−1

).

Each consecutive pair of flags is generic, since

[λm

(k)j

, λm

(k)j−1

] = [B−, w0g(k)j,j−1.B

−] = [B+, B−]

by g(k)j,j−1 ∈ B+. Thus we get (ρ, λ, φ) ∈ P

({mk})G,Σ normalized as φE0 = pstd. �

Corollary 3.32. When Σ has no punctures, we have

O(PG,Σ) ∼= (O(G)⊗(2g+b) ⊗O(G)⊗(b−1) ⊗O(B+)⊗∑b

k=1Nk)/I ,

where I is the ideal which gives the two relations described in Theorem 3.31. In particular,

the Betti algebra O(PG,Σ) is generated by the matrix coefficients of (twisted) Wilson lines.

Remark 3.33. When Σ has punctures and non-empty boundary, we have

O(PG,Σ) ∼= (O(G)⊗(2g+b) ⊗O(G)⊗p ⊗O(G)⊗(b−1) ⊗O(B+)⊗∑b

k=1Nk)/I ′,

where p is the number of punctures, G := {(g, B) ∈ G × BG | g ∈ B} denotes the

Grothendieck–Springer resolution, to which the pair (ρE0(γa), λa) for a ∈ P belongs. The


E1

E2

E3

E4g1,3

g1,3 ∈ Gw0,w0

E1

E2

E3

E4g1,3

g1,3 ∈ B+w0B+

E1

E2

E3

E4g1,3

g1,3 ∈ G

Figure 10. Some Wilson lines on the moduli space PG,Q;Ξ. The boundary

intervals not belonging to Ξ are shown by dashed lines.

ideal I ′ gives the monodromy relation

g∏

i=1

[ρE0(αi), ρE0(βi)] ·

p∏

j=1

ρE0(γaj ) ·b∏

k=1

ρE0(δk) = 1

and the same boundary relation, where we fixed an appropriate enumeration P = {a1, . . . , ap}.

In particular, O(G) contains some functions not coming from the matrix coefficients of

the twisted Wilson line ρE0(γa): see [FG06, Section 12.5] and [She20, Section 4.2] for a

detail.

Example 3.34. When Σ = T is a triangle, we have

O(PG,T ) ∼= O(B+)⊗3/〈 g3,2w0g2,1w0g1,3w0 = 1 〉.

As we have seen in Corollary 2.13, the images of the Wilson lines gj,j−1 are in fact restricted

to the double Bruhat cell B+∗ . This can be seen as g3,2 = (w0g

−11,3w0)w0(w0g

−12,1w0) ∈

B−w0B− and the cyclic symmetry.

Example 3.35. When Σ = Q is a quadrilateral, we have

O(PG,Q) ∼= O(B+)⊗4/〈 g4,3w0g3,2w0g2,1w0g1,4w0 = 1 〉.

Let us consider the Wilson line g1,3 on PG,Q shown in the left of Figure 10. In the

notation of Corollary 2.17, it corresponds to bS. Letting g3,1 := w0g−11,3w0 = (g∗1,3)

T, we get

the relations

g1,3w0g3,2w0g2,1w0 = 1,

g3,1w0g1,4w0g4,3w0 = 1.

Then similarly to the previous example, we get g1,3 ∈ B−w0B− and g3,1 ∈ B−w0B

−,

the latter being equivalent to g1,3 ∈ B+w0B+. Thus we get g1,3 ∈ Gw0,w0 = B+w0B

+ ∩B−w0B

−.

Example 3.36 (partially generic cases). The restriction g1,3 ∈ Gw0,w0 in the previous

example can be viewed as a consequence of the genericity condition for the consecutive


flags. Let us consider the moduli space PG,Q:Ξ with Ξ = {E1, E2, E3} and Ξ = {E1, E3},which are schematically shown in the middle and in the right in Figure 10, respectively.

In these cases we have less Wilson lines and less restrictions for the values of g1,3: it can

take an arbitrary value in B+w0B+ and in G, respectively.

In particular, we have PG,Q;{E1,E3}∼= G. The configuration of flags is parametrized as

[B+, B−, g1,3.B+, g1,3.B

−]. Our discussion shows that the image of the dominant mor-

phism PG,Q → PG,Q;{E1,E3}∼= G is exactly the double Bruhat cell Gw0,w0.

3.6. Decomposition formulae for Wilson lines and Wilson loops. Our goal in this

subsection is to obtain a decomposition formula for the Wilson lines and Wilson loops

similar to Proposition 3.23. Assume ∂Σ 6= ∅ and choose an arc class [c] : Ein → Eout. Let

: Σ → Σ be the universal cover, and take a representative c and its lift c to Σ. Fix an

ideal triangulation ∆ of Σ, which is also lifted to a tesselation ∆ of the universal cover.

Applying an isotopy if necessary, we may assume that the intersections of c with ∆ and c

with ∆ are minimal. Let Πc;∆ ⊂ Σ be the smallest polygon which is a union of triangles

in ∆ and contains c. The two endpoints of c lies on lifts of the edges Ein, Eout, which are

denoted by Ein and Eout, respectively. By definition the polygon Πc;∆ is equipped with an

ideal triangulation induced from ∆, which we denote by ∆c. Let us write the associated

turning pattern as τ∆([c]) := τ∆c(Ein, Eout) = (τ1, . . . , τM) ∈ {L,R}M , which we call the

turning pattern of the arc class [c] with respect to ∆. Let T1, . . . , TM be the sequence of

triangles of ∆c which are traversed by c in this order.

Let πc := |Πc;∆: Πc;∆ → Σ, which is a covering map over its image. It induces a

map π∗c : PG,Σ → PG,Πc;∆;{Ein,Eout}

via pull-back. Here recall Definition 3.3. From the

definitions and Remark 3.21, we have:

Lemma 3.37. The following diagram commutes:

PG,Σ PG,Πc;∆;{Ein,Eout}

G.

π∗c

g[c]gEin,Eout

Combined with Proposition 3.23, we would obtain a decomposition formula for Wilson

lines. In order to write it down explicitly, let us prepare some notations.

Fix a dotted triangulation ∆∗ of Σ. Let mT denote the dot assigned to a triangle

T ∈ t(∆). Note that ∆∗ determines a dotted triangulation ∆lift∗ of the polygon Πc;∆ over

∆c by lifting the dots, which may not agree with the “canonical” dotted triangulation

∆can∗ := ∆∗(Ein, Eout) associated with the turning pattern (Ein, Eout). The only difference

is the position of dots: denote the dot on the triangle Tν for the triangulation ∆can∗ (resp.

∆lift∗ ) by mν (resp. nν) for ν = 1, . . . ,M . Then the disagreement of the two dots mν and

nν results in a relation fmν= Stν3 ◦ fnν

for some tν ∈ {0,±1}.

For ν = 1, . . . ,M , let Tν := πc(Tν) ∈ t(∆) denote the projected image of the ν-th

triangle, which do not need to be distinct. Finally, set fnν:= fmTν

◦prTν :∏

T∈t(∆) PG,T →Conf3PG.


Theorem 3.38 (Decomposition formula for Wilson lines). For an arc class [c] : Ein →Eout and a dotted triangulation ∆∗, let the notations as above. Then we have

q∗∆g[c] = µM ◦M∏

ν=1


).

See Example 3.39 and Figure 11 for an example.

Proof. Since the pull-back via the covering map πc commutes with the gluing morphisms,

we have the commutative diagram

∏T∈t(∆) PG,T

∏Mν=1PG,Tν

P∆G,Σ P∆c

G,Πc;∆;{Ein,Eout},

π∗c

q∆ q∆c

π∗c

(3.11)

where π∗c :=

∏Mν=1(πc|Tν )

∗, and the right vertical map is the composite of the gluing

morphism and the projection forgetting the pinnings except for those assigned to Ein and

Eout. From the definition of tν and fnν, the following diagram commutes:

∏Mν=1Conf3PG

∏Mν=1Conf3PG

∏T∈t(∆) PG,T

∏Mν=1PG,Tν .

∏ν Stν

3

(fnν)ν

π∗c

∏ν fmν∏

ν fnν

(3.12)

Combining together, we get

q∗∆g[c] = gEin,Eout◦ π∗

c ◦ q∆ (by Lemma 3.37) (3.13)

= gEin,Eout◦ q∆c

◦ π∗c (by (3.11)) (3.14)

= µM ◦

(M∏

ν=1

gν

)◦ π∗

c (by Proposition 3.23) (3.15)

= µM ◦

(M∏

ν=1

bτν ◦ fmν

)◦ π∗

c (3.16)

= µM ◦M∏

ν=1


) (by (3.12)), (3.17)

as desired.

�

As a slight generalization, we can obtain a similar decomposition formula for the pull-

back q∗∆;Eg[c] for an interior edge E ∈ eint(∆). For simplicity, let us consider the case where

π−1c (E) is the diagonal of the quadrilateral Q12 := T1 ∪ T2. Then we get have two dotted


cell decompositions with a common ideal cell decomposition of Πc;∆, equipped with the

dots (m12, m3, . . . , mM) and (n12, n3, . . . , nM). Similarly as above we get

q∗∆;Eg[c] = µM−1 ◦

((bτ12 ◦ fm12)×

M∏

ν=3

(bτν ◦ fmν)

)◦ π∗

c .

As before one can find t12 ∈ {0, 1, 2, 3} so that

fm12 = St124 ◦ fn12 .

Then we get

q∗∆;Eg[c] = µM−1 ◦

((bτ12 ◦ S

t124 ◦ fn12

)×M∏

ν=3


)

).

Note that each matrix coefficient cVf,v(g[c]) in any finite-dimensional representation V of G

gives a regular function on the moduli stack PG,Σ, namely an element of O(PG,Σ). Thenthese decomposition formulae for ideal cell decompositions give an explicit presentation

of its image in OPt(PG,Σ) via the isomorphism (3.6).

Note also that by Proposition 3.28, the decomposition formula given in Theorem 3.38

also gives a formula for the presentation morphisms ρ|γ| of Wilson lines mod AdG.

Example 3.39 (A Wilson line on a marked annulus). Let Σ be a marked annulus with

one marked point on each boundary component, equipped with the dotted triangulation

∆ shown in Figure 11. Consider the arc class [c] : E0 → E4 shown there. The turning

pattern is τ∆([c]) = (L, L,R,R). Under the projection πc : Πc;∆ → Σ, we have T =

πc(T1) = πc(T3), T′ = πc(T2) = πc(T4). By comparing the two pictures in the right, we

have t1 = 0, t2 = 0, t3 = −1, t4 = 1. Thus we have

q∗∆g[c] = µ4 ◦ ((bL ◦ fmT)× (bL ◦ fmT ′ )× (bR ◦ S−1

3 ◦ fmT)× (bR ◦ S3 ◦ fmT ′ )). (3.18)

Of course, we could have chosen another triangulation ∆′ obtained from ∆ by the flip

along the edge E2. In this case we have τ∆′([c]) = (L,R) and we need no cyclic shifts to

express g[c].

Example 3.40 (A Wilson loop on a once-punctured torus). A once-punctured torus

ΣE0,E4 is obtained by gluing the boundary intervals E0 and E4 of the marked annulus Σ

considered above. The arc class [c] : E0 → E4 descends to a free loop |γ| ∈ π(ΣE0,E4).

Then by the proof of Proposition 3.28, a suitable conjugate of the Wilson line g[c] : PG,Σ →G gives a presentation of the Wilson loop ρ|γ| : PG,ΣE0,E4

→ [G/AdG].

4. Factorization coordinates and their relations

As a preparation for the subsequent sections, we recall several parametrizations and

coordinates of factorizing nature: Lusztig parametrizations on unipotent cells, coweight

parametrizations of double Bruhat cells, and Goncharov–Shen coordinates on the config-

uration space Conf3PG. A necessary background on the cluster algebra is reviewed in

Appendix C.


c

∗∗

T

T ′

(Σ,∆∗)

E0

E1

E2

E4

Ein

Eout

cT1

T2

T3

T4

∗∗

∗∗

(Πc;∆,∆lift∗ )

Ein

Eout

cT1

T2

T3

T4

∗∗

∗∗

(Πc;∆,∆can∗ )

Figure 11. A marked annulus Σ with a dotted triangulation ∆∗ (left), a

polygon Πc;∆ with the dotted triangulations ∆lift∗ (right top) and ∆can

∗ (right

bottom). Glued edges are marked by double-head arrows.

Notation 4.1. For a torus T = (C∗)N equipped with a coordinate system X = (Xk)Nk=1

and a map f : T → V to a variety V , we occasionally write f = f(X) as in the usual

calculus.

4.1. Lusztig parametrizations on the unipotent cells and the Goncharov–Shen

potentials. For w ∈ W (g), let U±w := U± ∩ B∓wB∓ denote the unipotent cell.

Proposition 4.2. Given a reduced word s = (s1, . . . , sl) of w, the maps

xs : (C∗)l → U+w , (t1, . . . , tl) 7→ xs1(t1) . . . xsl(tl),

ys : (C∗)l → U−w , (t1, . . . , tl) 7→ ys1(t1) . . . ysl(tl)

are open embeddings.

We call these parametrizations Lusztig parametrizations. These maps induce injective

C-algebra homomorphisms

(xs)∗ : O(U+w ) → O((C∗)l) = C[t±1

1 , . . . , t±1l ], (4.1)

(ys)∗ : O(U−w ) → O((C∗)l) = C[t±1

1 , . . . , t±1l ]. (4.2)

When w = w0 is the longest element, we have U±w0

= U±∗ .

Goncharov–Shen potentials. Let us consider the configuration space

Conf∗(AG,BG,BG) := G\{(B1, B2, B3) | (B1, B2) and (B1, B3) are generic}.


It has a parametrization

β3 : U+ ∼−→ Conf(AG,BG,BG), u+ 7→

[[U+], B−, u+.B

−].

The map β−13 pulls-back the additive characters χs : U

+ → C, χs(u+) := ∆s,rss(u+) for

s ∈ S to give a function

Ws := χs ◦ β−13 : Conf∗(AG,BG,BG) → C,

which we call the Goncharov–Shen potential. Note that u+ ∈ U+w if and only if w(B2, B3) =

w∗, when we write β3(u+) = (A1, B2, B3). Here W → W,w 7→ w∗ is an involution given

by

w∗ = w0ww0.

The following relation will be used later:

Lemma 4.3. For a reduced word s = (s1, . . . , sl) of w, let u+ = xs1(t1) . . . xsl(tl) ∈ U+w

be the corresponding Lusztig parametrization. Then we have

χs(u+) =∑

k:sk=s

tk.

4.2. Coweight parametrizations on double Bruhat cells. The coweight parametriza-

tions on double Bruhat cells are introduced in [FG06] and further investigated in [Wil13].

Let G be an adjoint group. For each u, v ∈ W (g), the double Bruhat cell is defined to

be Gu,v := B+uB+ ∩ B−vB−. It is a subvariety of G. In this paper, we only treat with

the special cases u = e or v = e. See [FG06, Wil13] for the general construction3.

Let us write B+v := Ge,v and B−

u := Gu,e. First consider B+v . Let s = (s1, . . . , sl) be a

reduced word for v. Then the evaluation map ev+s: (C∗)n+l → B+

v is defined by

ev+s(x) :=

(n∏

s=1

Hs(xs)

)·

−→∏

k=1,...,l

(EskHsk(xn+k)),

where x = (xk)n+lk=1 and the symbol

−→∏k=1,...,l means that we multiply the elements suc-

cessively from the left to the right, namely−→∏

k=1,...,lgk := g1 . . . gl. Similarly in the case

v = e, we take a reduced word s for u and define ev−s: (C∗)n+l → B−

u by replacing each

E with F. We call the variables x = (xk)k the coweight parameters.

The following indexing for the coweight parameters x will turn out to be useful: for a

reduced word s = (s1, . . . , sl) of an element of W (g), let k(s, i) denote the i-th number k

such that sk = s. Let ns(s) be the number of s which appear in the word s. If we relabel

the variables as

xsi := xk(s,i) (4.3)

for s ∈ S, i = 0, . . . , ns(s), then they always appear in the form Hs(xsi ) in the expression

of ev±s(x). Let

I(s) := {(s, i) | s ∈ S, i = 0, . . . , ns(s)}. (4.4)

3Indeed, the general case is obtained by a suitable amalgamation from the cases u = e and v = e.


Then for a reduced word s of u ∈ W (g), the evaluation maps give open embeddings

ev±s

: (C∗)I(s) → B±u where the variable assigned to the component (s, i) ∈ I(s) is

substituted to the k(s, i)-th position.

Example 4.4 (Type A3). Let g = A3. The evaluation map associated with the reduced

word s = (1, 2, 3, 1, 2, 1) is given by

ev+s(x)

= H1(x10)H2(x20)H

3(x30)E1H1(x11)E

2H2(x21)E3H3(x31)E

1H1(x12)E2H2(x22)E

1H1(x13).

The evaluation maps are compatible with group multiplication. For example, let us

consider s := (1, 2, 3) and s′ := (1). Then we have

ev+s(x10, x

20, x

30, x

11, x

21, x

31) · ev

+s′(y

10, y

20, y

30, y

11)

= H1(x10)H2(x20)H

3(x30)E1H1(x11)E

2H2(x21)E3H3(x31) ·H

1(y10)H2(y20)H

3(y30)E1H1(y11)

= H1(x10)H2(x20)H

3(x30)E1H1(x11y

10)E

2H2(x21y20)E

3H3(x31y30)E

1H1(y11)

= ev+s′′(x

10, x

20, x

30, x

11y

10, x

21y

20, x

31y

30, y

11).

with s′′ := (1, 2, 3, 1). Here in the third line, we used the fact thatHs(x) and H t(y) always

commutes with each other, and that Es and H t(x) commutes with each other when s 6= t.

If we denote the variable assigned to the component (i, s) ∈ I(s′′) by zsi , then

z10 = x10, z20 = x20, z30 = x30,

z11 = x11y10, z21 = x21y

20, z31 = x31y

30, z12 = y11.

For a reduced word s of w ∈ W (g) and ǫ ∈ {+,−}, each variable xsi of the coweight

parametrization evǫsis assigned to the vertex vsi of the weighted quiver J ǫ(s). See Appen-

dix C. The group multiplication corresponds to an appropriate amalgamation of quivers.

For example, the multiplication considered in Example 4.4 corresponds to the quiver amal-

gamation shown in Figure 12. The pair Sǫ(s) := (J ǫ(s), (xsi )(s,i)∈I(s)) forms an X-seed in

the ambient field F = K(Bǫw).

Theorem 4.5 (Fock-Goncharov [FG06], Williams [Wil13]). For an element w ∈ W (g)

and ǫ ∈ {+,−}, the seeds Sǫ(s) associated with reduced words s of w are mutation-

equivalent to each other. Hence the collection(Sǫ(s))s is a cluster Poisson atlas (Defini-

tion C.2) on the double Bruhat cell Bǫw.

The following lemma directly follows from the definition of the Dynkin involution and

Lemma 2.4, which will be useful in the sequel.

Lemma 4.6. We have the following relations:

w0−1Hs(x)−1w0 = Hs∗(x),

w0−1(Es)−1w0 = Fs

∗

,

w0−1(Fs)−1w0 = Es

∗

.

Since the map g 7→ w0−1g−1w0 is an anti-homomorphism, we get the following:


x10 x11

x20 x21

x30 x31

y10 y11

y20

y30

x10 x11y10 x12

x20 x21y20

x30 x31y30

Figure 12. Amalgamation of the quivers J+(1, 2, 3) and J+(1) for type

A3 produces the quiver J+(1, 2, 3, 1).

Corollary 4.7. For a reduced word s = (s1, . . . , sN) of w0 ∈ W (g), let s∗op := (s∗N , . . . , s∗1).

Then we have

w0−1ev+

s(x)−1w0 = ev−

s∗op◦ ι∗(x),

where ι∗ : (C∗)I(s) → (C∗)I((s)∗op) is an isomorphism induced by the bijection

ι : I(s∗op) → I(s), (s∗, i) 7→ (s, ns(s)− i). (4.5)

4.3. Goncharov–Shen coordinates on Conf3PG. We recall the Goncharov–Shen’s clus-

ter Poisson coordinate system on Conf3PG associated with a reduced word s = (s1, . . . , sN)

of w0 ∈ W (g). See [GS19] for a detail. Let [B1, B2, B3; p12, p23, p31] ∈ Conf3PG. Using

Corollary 2.8, we take the decomposition of the generic pair (B2, B3) with respect to s:

B2 = B02

s∗1−→ B12

s∗2−→ . . .s∗N−→ BN

2 = B3,

where w(Bk−12 , Bk

2 ) = s∗k for k = 1, . . . , N . Suppose that the triple (B1, B2, B3) is “suffi-

ciently generic” so that each pair (B1, Bk2 ) is generic for k = 0, . . . , N . Let B1, B

′1 be two

lifts of B1 determined by the pinnings p12, p∗31, respectively. Now we define:

X(si):=

Ws

(B1, B2, B

k(s,1)2

)(i = 0),

Ws

(B1, B

k(s,i)2 , B

k(s,i+1)2

)/Ws

(B1, B

k(s,i−1)2 , B

k(s,i)2

)(i = 1, . . . , ns(s)− 1),

Ws

(B′

1, Bk(s,ns(s)−1)2 , B

k(s,ns(s))2

)−1

(i = ns(s)).

Here as before, k(s, i) denotes the i-th number k such that sk = s in s.

Let G be the simply-connected group which covers G and take a lift

B2 = B02

s∗1−→ B12

s∗2−→ . . .s∗N−→ BN

2 = B3 (4.6)

of the above chain to AG := G/U+ so that the pair (B2, B3) is determined by the pinning

p23 and the conditions h(Bj2, B

j−12 ) = 1 for j = 1, . . . , N hold. Here the h-invariant and

the w-distance of the elements of A×2

Gare defined in the same way as those for A×2

G .

Such a lift exists thanks to [GS19, Lemma-Definition 5.3]. Then we have the primary

coordinates

Ps,k :=Λsk(B1, B

k2 )

Λsk(B1, Bk−12 )

,


where B1 is an arbitrary lift of B1 ∈ AG to AG and Λs : AG × AG → C is the unique

G-invariant rational function given by

Λs(h.[U+], w0.[U

+]) := hs .

Note that Ps,k does not depend on the choice of the lifts B1, B2 and it gives a well-defined

regular function on Conf3PG. See also Lemma 4.17.

Let βs

k := rsN . . . rsk+1(α∨

sk) be a sequence of coroots associated with s. For each s ∈ S,

there exists a unique k = k(s) such that βs

k = α∨s . Then we set

X( s−∞)

:= Ps,k(s).

Definition 4.8. The rational functions X(si)(s ∈ S, i = −∞, 0, 1, . . . , ns(s)) are called

the Goncharov–Shen coordinates (GS coordinates for short) on Conf3PG, associated with

the reduced word s. When we want to emphasize the dependence on the reduced word

s, we write X(si)=: Xs

(si).

Conversely, we can construct an embedding

ψs : (C∗)I∞(s) → Conf3PG (4.7)

from given set of GS coordinates, where I∞(s) := {(s, i) | s ∈ S, i = −∞, 0, 1, . . . , ns(s)}.If G = PGLn+1 and the reduced word s = sstd(n) is the one defined inductively by

sstd(n) = (1, 2, . . . , n)sstd(n− 1), sstd(1) = (1), (4.8)

then the GS coordinates are nothing but the Fock–Goncharov coordinates introduced in

[FG06, Section 9].

Lemma 4.9 ([GS19, Lemma 9.2]). Let (k1, k2, k3) ∈ H3 and denote the action of (k1, k2, k3)

on Conf3PG described in Lemma 3.10 by αk1,k2,k3 : Conf3PG → Conf3PG. Then for s ∈ S,

we have

α∗k1,k2,k3

X(si)=

k−αs

1 X(s0)if i = 0,

k−αs∗

3 X( sns(s))

if i = ns(s),

k−αs

2 X( s−∞)

if i = −∞,

X(si)otherwise.

Proof. The first three equalities are given in [GS19, Lemma 9.2]. The last one straight-

forwardly follows from the definition of the H3-action and Ws. �

For s ∈ S, the GS coordinate X(si)for i = 0, 1, . . . , ns(s) (resp. i = −∞) is assigned

to the vertex vsi (resp. ys) of the weighted quiver J+(s). See Appendix C. Then the pair

S(s) := (J+(s), (Xsi )(s,i)∈I∞(s)) forms a seed in the ambient field F = K(Conf3PG).

Theorem 4.10 (Goncharov–Shen [GS19, Theorem 7.2]). The seeds S(s) associated with

the reduced words s of the longest element w0 are mutation-equivalent to each other. Hence

the collection (S(s))s is a cluster Poisson atlas (Definition C.2) on Conf3PG.


Note that the frozen coordinates X(s0), X( s

ns(s))and X( s

−∞)depend only on one of the

three pinnings. Later we use the following labellings:

s12 := (s, 0), s23 := (s,−∞), s31 := (s∗ ns(s∗)) ∈ I∞(s). (4.9)

We call the coordinates Xsj,j+1for j = 1, 2, 3 the GS coordinates of Dynkin index s.

Remark 4.11. Oh the other hand, the unfrozen coordinates X(si)for i = 1, . . . , ns(s)−1

only depend on the underlying flags (B1, B2, B3). Hence we have the following birational

charts for the configuration spaces with some of the pinnings dropped:

(C∗)I∞(s)\{(s,−∞)|s∈S} → [G\{(B1, B2, B3; p12, p31)}],

(C∗)I∞(s)\{(s,0),(s,−∞)|s∈S} → [G\{(B1, B2, B3; p31)}],

and so on. Here a pair of flags over which no pinning is assigned is not required to be

generic. For example, in the second configuration space only the pair (B3, B1) is required

to be generic. These configuration spaces are building blocks for the moduli space PG,Σ;Ξ

with partial genericity (recall Definition 3.3).

Remark 4.12. The above definition of coordinates is the same as the original one given

in [GS19]. It can be verified as follows. For each s ∈ S, take the unique flag Bk2,s such that

w(Bk2 , B

k2,s) = w0rs and w(B

k2,s, B1) = rs. Then actually we have Bk−1

2,s = Bk2,s whenever

sk 6= s [GS19, Lemma 7.9]. See also Remark 4.16. We collect all the distinct flags among

Bk2,s and relabel them as B(si)

(s ∈ S, i = 0, . . . , ns(s)). Then the triple(B1, B(si)

, B( si+1)

)

determines a configuration of SL2-flags, namely an element of Conf∗(ASL2,BSL2,BSL2).

See [GS19, (291)]. Then by [GS19, Proposition 7.10], we have

Ws

(B1, B

k(s,i)2 , B

k(s,i+1)2

)=W

(B1, B(si)

, B( si+1)

),

where the right-hand side is the potential of a configuration of SL2-flags.

4.4. Relation between Goncharov–Shen coordinates and coweight parametriza-

tions. In this section, we give an expression of the basic Wilson lines bL, bR (Defini-

tion 2.12) in terms of the GS coordinates, relating the coweight parametrizations on the

double Bruhat cells B+∗ := Ge,w0, B−

∗ := Gw0,e with the GS coordinates. The index set

I(s) introduced in (4.4) is naturally regarded as a subset of I∞(s).

Theorem 4.13 (cf. [GS19, Lemma 7.29]). For each reduced word s of w0 ∈ W (g) we

have

ψ∗sbL = ev+

s, ψ∗

sbR = ev−

s∗op◦ ι∗,

where ι∗ : (C∗)I(s) → (C∗)I(s∗op) is the isomorphism induced by (4.5).

Below we give a proof of this theorem based on the standard configuration (Lemma 2.11).

Let us write

C−13 ◦ ψs = (h1(X), h2(X), u+(X)) : (C∗)I∞(s) → H ×H × U+

∗ . (4.10)


Then from Corollary 2.13, we have

ψ∗sbL = u+(X)h2(X), ψ∗

sbR = w0

−1(u+(X)h2(X))−1w0.

We are going to compute the functions u+(X) and h2(X).

In the following, we use the short-hand notations xs[i j](t) := xsi(ti) . . . xsj (tj) and

ys[i j](t) := ysi(ti) . . . ysj(tj) for a reduced word s = (s1, . . . , sN) of w0 ∈ W (g) and

1 ≤ i < j ≤ N .

Lemma 4.14. For a configuration C = C3(h1, h2, u+) ∈ Conf3PG and its representative

as in Lemma 2.11, write u+ = xs1(t1) . . . xsN (tN) = xs[1 N ](t) using the Lusztig coordinates

associated with s. Let (X(si)) ∈ (C∗)I∞(s) be the GS coordinates of C associated with s.

Then we have the followings:

(1) Bk2 = xs[1 k](t)B

−.

(2) For each s ∈ S and i = 1, . . . , ns(s), we have

tk(s,i) = X(s0). . . X( s

i−1). (4.11)

Here k(s, i) is the i-th number k with sk = s in s from the left.

Substituting (4.11) into xs[1,N ](t), we get an expression of the function u+(X).

Proof. To check that the right-hand side of the first statement indeed gives Bk2 , let us

compute

w(xs[1 k−1](t)B−, xs[1 k](t)B

−) = w(B−, xsk(tk)B−) = rs∗

k,

where we used the relation xs(t) = ys(t−1)α∨

s (t)r−1s ys(t

−1). Then the uniqueness statement

of Corollary 2.8 and an induction on k implies Bk2 = xs[1 k](t)B

−.

To prove the second statement, we compute

[B1, Bk(s,i)2 , B

k(s,i+1)2 ] =

[[U+], xs[1 k(s,i)](t)B

−, xs[1 k(s,i+1)](t)B−]

=[[U+], B−, xs[k(s,i)+1 k(s,i+1)](t)B

−].

Thus we have Ws(B1, Bk(s,i)2 , B

k(s,i+1)2 ) = χs(x

s

[k(s,i)+1 k(s,i+1)](t)) = tk(s,i+1) by Lemma 4.3.

A similar computation shows that X(s0)= Ws(B1, B2, B

k(s,1)2 ) = tk(s,1) and we get tk(s,i) =

X(s0). . .X( s

i−1)by induction on i. �

Lemma 4.15. We have h2(X) =∏

s∈S Hs(Xs), where Xs :=

∏ns(s)i=0 X(si)

.

Proof. Again fix a configuration C = C3(h1, h2, u+). Recall that the pinning p12 = pstdcorresponds to the lift B1 = [U+] of B1. On the other hand, the pinning p∗31 = u+h2.pstdcorresponds to the lift B′

1 = h2.[U+]. Then we can compute:

[B′

1, Bk(s,ns(s)−1)2 , B

k(s,ns(s))2

]=[h2.[U

+], B−, xs[k(s,ns(s)−1)+1 k(s,ns(s))](t).B−]

=[[U+], B−,Ad−1

h2(xs[k(s,ns(s)−1)+1 k(s,ns(s))](t)).B

−].


With a notice that Ad−1h2(xs(tk(s,ns(s)))) = xs(h

−αs

2 tk(s,ns(s))), we get

X( sns(s))

= Ws

(B′

1, Bk(s,ns(s)−1)2 , B

k(s,ns(s))2

)−1

= hαs

2 t−1k(s,ns(s)).

Hence from Lemma 4.14 we get hαs

2 = tk(s,ns(s))X( sns(s))

= Xs, which implies h2 =∏

s∈SHs(Xs). �

Proof of Theorem 4.13. From the definition of the one-parameter subgroup xu, we have

the relation Hs(a)xu(b)Hs(a)−1 = xu(a

δsub). In particular xs(t) = Hs(t)EsHs(t)−1. Com-

bining with Lemmas 4.14 and 4.15, we get

ψ∗sbL = u+(X)h2(X) =

−→∏

k=1,...,N

(Hsk(tk)EskHsk(tk)−1) ·

∏

s∈S

Hs(Xs),

where each tk = tk(X) is a monomial given by (4.11). Since Hs commutes with Eu for

s 6= u, we can obtain the following expression by the relabeling as in (4.2):

ψ∗sbL =

∏

s∈S

Hs(tk(s,1))−→∏

s∈Si=1,...,ns(s)

(EsHs(tk(s,i))−1Hs(tk(s,i+1)))

∏

s∈S

Hs(Xs)

where tk(s,ns(s)+1) := 1 for s ∈ S. Note that it already has the form of the coweight

parametrization ev+s(x), where the parameter x = (xsi )s∈S, i=0,...,ns(s) is computed as fol-

lows:

xsi =

tk(s,1) = X(s0)(i = 0),

t−1k(s,i)tk(s,i+1) = X(si)

(i = 1, . . . , ns(s)− 1),

t−1k(s,ns(s))Xs = X( s

ns(s))(i = ns(s)),

where we used Lemma 4.14 for the second steps. Thus we have ψ∗sbL(X) = ev+

s(X), as

desired. The second statement follows from Corollary 4.7.

�

Remark 4.16. Similarly to the proof of Lemma 4.14 we can compute the flags defined

in Remark 4.12, as follows:

Bk2,s = xs[1 k](t)w0rs∗ .B

−,

B(sk)= xs[1 k](t)w0rs∗

k.B−.

4.5. Primary coordinates in the standard configuration. Let s = (s1, . . . , sN) be a

reduced word of w0 ∈ W (g). The following computation of the primary coordinates Ps,k

in terms of the standard configuration will be used in Section 6.

Lemma 4.17. For a configuration C = C3(h1, h2, u+) ∈ Conf3PG and its representative

as in Lemma 2.11, write u+ = xs1(t1) . . . xsN (tN) = xs[1 N ](t) using the Lusztig coordinates

associated with s. Then we have the following:


(1) B2 = h1w0.[U+] and Bk

2 = xs[1 k](t)h(k)1 w0.[U

+] for k = 0, 1, . . . , N . Here h ∈ G is

a lift of h1, and

h(k)1 := rsk · · · rs1(h1)

k∏

j=1

rsk · · · rsj+1(α∨

sj(tj)).

(2) Ps,k(C3(h1, h2, u+)) = tk((h(k)1 )−αsk ).

Observe that the decorated flags given in (1) are indeed projected to those given in

Lemma 4.14 (1). Also note that the right-hand side of (2) does not depend on the choice

of the lift h1.

Proof. Since the second component in the representative of C is φ′(u+)h1w0.pstd, the dec-

orated flag B2 must be a lift of

φ′(u+)h1w0.[U+] = h1w0Ad(h1w0)−1(φ′(u+)).[U

+] = h1w0.[U+].

Such an element is written as h1w0.[U+] for some lift of h1 to G, which proves the first

statement of (1). Set B(k) := xs[1 k](t)h(k)1 w0.[U

+]. In order to show the second statement

of (1), it suffices to see that

(i) B(0) = B2,

(ii) w(B(k), B(k−1)) = rs∗kand h(B(k), B(k−1)) = 1 for k = 1, . . . , N .

The statement (i) is immediate from the definition. In Conf2AG, we have[B(k), B(k−1)

]

=[xsk(tk)h

(k)1 w0.[U

+], h(k−1)1 w0.[U

+]]

=[ysk(t

−1k )α∨

sk(tk)r

−1skysk(t

−1k )h

(k)1 w0.[U

+], h(k−1)1 w0.[U

+]]

=[α∨sk(tk)r

−1skh(k)1 w0Ad(h

(k)1 w0)−1(ysk(t

−1k )).[U+], h

(k−1)1 w0Ad(h

(k−1)1 w0)−1(ysk(−t

−1k )).[U+]

]

=[α∨sk(tk)rsk(h

(k)1 )r−1

skw0.[U

+], h(k−1)1 w0.[U

+]].

Moreover we have

α∨sk(tk)rsk(h

(k)1 ) = α∨

sk(tk)rsk−1

· · · rs1(h1)α∨sk(t−1k )

k−1∏

j=1

rsk−1· · · rsj+1

(α∨sj(tj))

= rsk−1· · · rs1(h1)

k−1∏

j=1

rsk−1· · · rsj+1

(α∨sj(tj)) = h

(k−1)1 . (4.12)

Thus we get[B(k), B(k−1)

]=[r−1skw0.[U

+], w0.[U+]]=[[U+], rs∗

k.[U+]

],

which shows (ii).


For the computation of Ps,k(C3(h1, h2, u+)), we may take a lift of the first flag B1 as

[U+]. Then,

Ps,k(C3(h1, h2, u+)) =Λsk([U

+], xs[1 k](t)h(k)1 w0.[U

+])

Λsk([U+], xs[1 k−1](t)h

(k−1)1 w0.[U+])

=Λsk((h

(k)1 )−1.[U+], w0.[U

+])

Λsk((h(k−1)1 )−1.[U+], w0.[U+])

=Λsk((h

(k)1 )−1.[U+], w0.[U

+])

Λsk((α∨sk(tk)rsk(h

(k)1 ))−1.[U+], w0.[U+])

(by (4.12))

= tk(h

(k)1 )−sk

rsk(h(k)1 )−sk

= tk(h(k)1 )−sk

+rsk (sk) = tk(h

(k)1 )−αsk

as desired. �

Corollary 4.18. For s ∈ S, we have

(C∗3X( s

−∞))(h1, h2, u+) = tk(s)h

αs∗

1

k(s)∏

j=1

t〈rs1 ···rsj (α

∨sj

),αs∗〉

j

for (h1, h2, u+) ∈ H ×H ×U+∗ with u+ = xs[1 N ](t). Here recall that k(s) is determined by

rsN . . . rsk(s)+1(α∨

sk(s)) = α∨

s .

Proof. By the definition of X( s−∞)

and Lemma 4.17 (2), the desired statement follows from

the following calculation:

rsk(s) · · · rs1(h1)−αsk(s) = h

rs1 ···rsk(s)−1(αsk(s)

)

1 = h−w0rsN ...rsk(s)+1

(αsk(s))

1 = h−w0αs

1 = hαs∗

1 .

�

The results in Lemmas 4.14 and 4.15 and Corollary 4.18 gives explicit forms of h1, h2 :

GI∞(s)m → H and u+ : GI∞(s)

m → U+∗ as follows:

Lemma 4.19. Write tk(s,i) := X(s0). . .X( s

i−1)for (i, s) ∈ I(s) (recall the notation in

(4.11)). Then we have the following:

(1) h1(X) =∏

s∈SHs(Xs), where

Xs = X( s∗

−∞)t−1k(s∗)

k(s∗)∏

k=1

t〈rs1 ···rsk−1

(α∨sk

),αs〉

k

where k(s∗) is determined by rs1 . . . rsk(s∗)−1(α∨

sk(s∗)) = α∨

s .

(2) h2(X) =∏

s∈SHs(Xs), where Xs :=

∏ns(s)i=0 X(si)

.

(3) u+(X) = xs1(t1) . . . xsN (tN ).


4.6. Goncharov–Shen coordinates on PG,Σ via amalgamation. Here we recall the

GS coordinates on PG,Σ, which are constructed via amalgamation of those on Conf3PG.Let us begin with a triangle Σ = T . Recall that we have the isomorphisms (3.2)

determined by choosing a dot mj , j ∈ {1, 2, 3}. Via this isomorphism, we define the GS

coordinates on PG,T to be X(T,mj ,s)

(si):= f ∗

mjXs

(si), where Xs

(si)denote the GS coordinates

on Conf3PG associated with the reduced word s of w0. The accompanying quiver QT,mj ,s

is defined to be the quiver J+(sT ) placed on T so that for s ∈ S,

• the vertices vs0 lie on the edge connecting mj and mj+1,

• the vertices vsns(sT ) lie on the edge connecting mj and mj+2, and

• the vertices ys lie on the edge connecting mj+1 and mj+2.

See Figure 13 for an example. Here the isotopy class of the embedding of the quiver

J+(sT ) into the triangle T relative to the boundary intervals is included in the defining

data of QT,mj ,s. Then the pair S(T,mj ,s) :=

(QT,mj ,s, (X

(T,mj ,s)

(si))(s,i)∈I∞(s)

)is a seed in

K(PG,T ). Recall the cyclic shift given in Lemma 3.9.

Theorem 4.20 (Goncharov–Shen [GS19, Theorem 5.11]). The cyclic shift S3 is realized

as a sequence of cluster transformations. In particular, the seeds S(T,mj ,s) for any choice

of a dot mj and a reduced word s of w0 are mutation-equivalent to each other. Hence the

collection (S(T,mj ,s))j∈{1,2,3},s is a cluster Poisson atlas (Definition C.2) on PG,T .

m2 m3

m1

∗

2

v302

v312

v322

v33

v20

v10

v21

v11

v22

v12

v23

v13

y1 y22y3

Figure 13. Placement of the weighted quiver J+((123)3) with g = C3 on

a dotted triangle T .

Now let us define the GS coordinates on PG,Σ for a general marked surface Σ. A

decorated triangulation of Σ consists of the following data ∆ = (∆∗, s∆):

• An oriented dotted triangulation ∆∗, which is a dotted triangulation ∆∗ of Σ

equipped with an orientation for each edge 4.

4We need the orientation only to fix a bijection between the GS coordinates on an edge and the index

set S.


• A choice s∆ = (sT )T of reduced words sT of the longest element w0 ∈ W (g), one

for each triangle T of ∆∗.

We simply call ∆∗ the underlying dotted triangulation of ∆. Then the GS coordinate

system on P∆G,Σ ⊂ PG,Σ associated with ∆ is defined to be

X∆ := {X(T,mT ,sT )

(si)}T∈t(∆), s∈S, i=1,...,ns(s)−1 ∪ {X(E,s∆)

s }E∈e(∆), s∈S,

where mT denotes the marked point corresponding to the dot on T , and the coordi-

nates X(T,mT ,sT )

(si)on PG,T naturally descend to functions on P∆

G,Σ by the H∆-invariance

(Lemma 4.9). The coordintes X(E,s∆)s assigned to an oriented edge E ∈ e(∆) are defined

via amalgamation, as follows.

When E is an interior edge shared by two triangles T1, T2 ∈ t(∆), where T1 is on the

left side with respect to the orientation of E. For ℓ = 1, 2, label the marked points of Tℓas m

(ℓ)1 , m

(ℓ)2 , m

(ℓ)3 in the counter-clockwise order so that m

(ℓ)1 corresponds to the dot. See

Figures 14 and 15.

m(1)3 = m

(2)3

m(1)2

m(1)1 = m

(2)2

m(2)1

∗

∗

Figure 14. The quiver on T1 ∪E T2 with g = A3 and sT1 = sT2 =

(1, 2, 3, 1, 2, 1). We glue the vertices as (v13)(1) = y

(2)1 , (v22)

(1) = y(2)2 and

(v31)(1) = y

(2)3 .

Then there exists a, b ∈ {1, 2, 3} such that the edge E corresponds to the intervals

[m(1)a , m

(1)a+1] in T1 and [m

(2)b , m

(2)b+1] in T2 (indices should be read modulo 3). Amalgamate

the two seeds S(Tℓ,m

(ℓ)1 ,sTℓ)

for ℓ = 1, 2 with the gluing data

F := {s(1)a,a+1 | s ∈ S}, F ′ := {s

(2)b,b+1 | s ∈ S},

φ : F → F ′, s(1)a,a+1 7→ (s∗b,b+1)

(2) for s ∈ S. (4.13)


m(1)3 = m

(2)3

m(1)2

m(1)1 = m

(2)2

m(2)1

∗

∗

22

22

2

2

2

2

2

2

Figure 15. The quiver on T1∪E T2 with g = C3 and sT1 = sT2 = (1, 2, 3)3.

We glue the vertices as (vs3)(1) = y

(2)s .

Here we denote the indices in I∞(sTℓ) related to the triangle Tℓ with the superscript (ℓ),

and employed the notation (4.9). See Figures 14 and 15 for examples. In particular, the

edge coordinates on E are defined by

X(E,s∆)s := X(1)

sa,a+1·X

(2)s∗b,b+1

, (4.14)

where X(ℓ)

(si):= X

(Tℓ,m(ℓ)1 ,sTℓ )

(si), and the labeling (4.9) is used. Then Lemma 4.9 ([GS19,

Lemma 9.3]) tells us that these edge functions are indeed H∆-invariant. If Eop is the

same edge with the reversed orientation, then the resulting quiver is the same (under an

isotopy) while we have X(E,s∆)s = X

(Eop,s∆)s∗ .

When E is a boundary interval oriented along the boundary, it belongs to a triangle

T1 ∈ t(∆) on its left. Then under the notation above, E connects the marked points ma

and ma+1 for some a ∈ {1, 2, 3}. Define

X(E,s∆)s := X(T,mT ,sT )

sa,a+1.

When E is oriented against the boundary, define X(E,s∆)s := X

(Eop,s∆)s∗ .

Applying this procedure for each edge, we get a weighted quiver Q∆ drawn on the

surface Σ as well as a desired collection X∆ of coordinates. In a light of Theorem 3.13,

the collection X∆ of functions provide an open embedding ψ∆ : (C∗)I∆ → PG,Σ, whose


image is contained in P∆G,Σ and the index set is given by

I(∆) := {(s, i;T ) | T ∈ t(∆), s ∈ S, i = 1, . . . , ns(s)} ⊔ {(s;E) | E ∈ e(∆), s ∈ S}.

Thus the pair S(∆) := (X∆, Q∆) forms a seed in the ambient field K(PG,Σ).

Theorem 4.21 ([FG06, Le16, GS19]). The seeds S(∆) associated with the decorated

triangulations ∆ of Σ are mutation-equivalent to each other. Hence the collection (S(∆))∆is a cluster Poisson atlas on PG,Σ.

Comparison with the cluster Poisson algebra. Let SG,Σ denote the cluster Poisson

structure on PG,Σ which includes the cluster Poisson atlas (S(∆))∆. Then Theorem 4.21

tells us that our moduli space PG,Σ is birationally isomorphic to the cluster Poisson variety

XSG,Σ, and hence their fields of rational functions are isomorphic. Slightly abusing the

notation, let us denote the cluster Poisson algebra by Ocl(PG,Σ) := O(XSG,Σ). Shen proved

the following stronger result:

Theorem 4.22 (Shen [She20]). We have an isomorphism Ocl(PG,Σ) ∼= O(PG,Σ) of C-algebras.

In particular, we have:

Corollary 4.23. The matrix coefficients of Wilson lines and the traces of Wilson loops

are universally Laurent polynomials:

cVf,v(g[c]), trV (ρ|γ|) ∈ Ocl(PG,Σ)

for any representation V , f ∈ V ∗, v ∈ V , arc class [c] and a free loop |γ|.

Our aim in the sequel is to obtain an explicit formula for these Laurent polynomials,

and prove the positivity of coefficients when the coordinate system is associated with a

decorated triangulation.

Partially generic case. For a subset Ξ ⊂ B, consider the moduli stack PG,Σ;Ξ of Ξ-

generic framed G-local systems with Ξ-pinnings (recall Definition 3.3 and (3.1)). For a

decorated triangulation ∆, set

IΞ(∆) := I(∆) \ {(s;E) | E ∈ B \ Ξ, s ∈ S}. (4.15)

Then by Remark 4.11, we have an open embedding (C∗)IΞ(∆) → PG,Σ;Ξ which fits into

the commutative diagram

(C∗)I(∆) PG,Σ

(C∗)IΞ(∆) PG,Σ;Ξ.

ψ∆

Here the left vertical map is induced from the inclusion IΞ(∆) ⊂ I(∆), and the right

vertical map is the projection forgetting the pinnings except for those assigned to the

boundary intervals in Ξ. Since these embeddings differ only in frozen variables, the

moduli space PG,Σ;Ξ also has a canonical cluster Poisson atlas so that the projections

PG,Σ → PG,Σ;Ξ are cluster projections.


5. Coordinate expressions of Wilson lines and loops

Our goal in this section is to obtain formulae for Wilson lines and Wilson loops in terms

of the GS coordinates on the moduli space PG,Σ.

5.1. Coordinate expressions of side pairings. First let us compute the side pairings

in terms of GS coordinates. Recall the formula given in Proposition 3.23. Since each

gν admits a coweight parametrization (Section 4.2), we are lead to introduce a slightly

generalized coweight parametrization for their product. Let τ1, . . . , τM ∈ {L,R} be a

sequence of alphabets L and R, w1, . . . , wM ∈ W (g) and take a reduced word sν of wν for

ν = 1, . . . ,M .

Recall the index set (4.4) associated with a reduced word. Here we consider Iν :=

I(sτνν ) = {(s, i; ν) | s ∈ S, i = 0, . . . , ns(sτνν )}, where the modified reduced word

sτνν :=

{sν if τν = L,

(sν)∗op if τν = R

of wν will be useful in the sequel. For ν = 1, . . . ,M , define evτνsν

: (C∗)Iν → G by

evτνsν

:=

{ev+

sνif τν = L,

ev−(sν )∗op if τν = R.

Let I1 ∗ · · · ∗ Iν be the index set obtained from the product I1 × · · · × Iν by identifying

(ns(sτσσ ), s; σ) and (0, s; σ + 1) for s ∈ S and σ = 1, . . . , ν − 1.

Definition 5.1. We inductively define evτ1,...,τνs1,...,sν : (C∗)I1∗···∗Iν → G so that the following

diagram commutes for ν = 2, . . . ,M :

(C∗)I1∗···∗Iν−1 × (C∗)Iν(ev

τ1,...,τν−1s1,...,sν−1

, evτνsν )

−−−−−−−−−−−→ G×G

αν

yyµ2

(C∗)I1∗···∗Iν −−−−−→ev

τ1,...,τνs1,...,sν

G,

where the map µ2 : G×G→ G is the group multiplication. The map αν : (C∗)I1∗···∗Iν−1 ×(C∗)Iν → (C∗)I1∗···∗Iν is the amalgamation map with the gluing data

F := {(s, ns(sτν−1

ν−1 ); ν − 1) | s ∈ S}, F ′ := {(s, 0; ν) | s ∈ S},

φ : F → F ′, (s, ns(sτν−1

ν−1 ); ν − 1) 7→ (s, 0; ν) for s ∈ S. (5.1)

We call the maps evτ1,...,τνs1,...,sν

the generalized evaluation map. See Example 4.4 for a small

example.

Let Π be a polygon with an ideal triangulation ∆. Let Ein, Eout be a pair of side edges

of Π such that the associated path c traverses every triangles of ∆. Let τ∆(Ein, Eout) =

(τν)Mν=1 be the associated turning pattern. Recall from Definition 3.22 the dotted trian-

gulation ∆∗(Ein, Eout) associated with (Ein, Eout). Let Ein = E1, . . . , EM+1 = Eout be the

sequence of edges which c traverses, and endow them the orientations so that the algebraic


intersection number i(Eν , c) = +1 for all ν = 1, . . . ,M + 1. Choose a reduced word sν of

w0 ∈ W (g) for ν = 1, . . . ,M and let ∆ := (∆∗(Ein, Eout), (sν)Mν=1).

We order the GS coordinates on PG,Π in such a way that they are substituted into the

generalized evaluation map in a correct order. The chain of GS coordinates associated

with (Ein, Eout) is the tuple XEin,Eout = (Xsi [ν])(s,i;ν)∈I1∗···∗IM of coordinate functions on

the torus (C∗)I(∆) defined as follows.

Edges: For ν = 1, . . . ,M and s ∈ S, define

Xs0 [ν] := (ψ∆)∗X(Eν ,s∆)

s , Xsns(sτνν )[ν] := (ψ∆)∗X(Eν+1,s∆)

s .

Faces: For (s, i; ν) ∈ I1 ∗ · · · ∗ IM with 0 < i < ns(sτνν ), define

Xsi [ν] :=

(ψ∆)∗X

(Tν ,mν ,sν)

(si)if τν = L,

(ψ∆)∗X(Tν ,mν ,sν)

( s∗

ns∗ (sν)−i)if τν = R.

(5.2)

Here notice that the right-hand side refers to the original word sν , and we have

(s∗, ns∗(sν)− i) ∈ I(sν) when τν = R.

In other words, some of the GS coordinates are partially ordered in XEin,Eout as “scanned”

by the path c. This reordering rule fixes an inclusion ιEin,Eout : I1∗· · ·∗IM → I(∆). Recall

Notation 4.1.

Theorem 5.2 (Evaluation formula for side pairings). Let the notations as above. Then

for any decorated triangulation ∆ = (∆∗(Ein, Eout), s∆ = (sν)Mν=1) with the underlying

dotted triangulation ∆∗(Ein, Eout), we have

(ψ∆)∗gEin,Eout = evτ1,...,τMs1,...,sM

◦ ι∗Ein,Eout,

where ι∗Ein,Eout: (C∗)I(∆) → (C∗)I1∗···∗IM is the projection induced by the inclusion ιEin,Eout.

In other words, the side pairing gEin,Eout : P∆G,Π → G is expressed as

(ψ∆)∗gEin,Eout = evτ1,...,τMs1,...,sM

(XEin,Eout).

Here XEin,Eout denotes the chain of GS coordinates associated with (Ein, Eout).

Proof. Let α∆ :∏M

ν=1(C∗)I∞(sν) → (C∗)I(∆) be the amalgamation map which fits into the

following commutative diagram:

∏Mν=1(C

∗)I∞(sν)∏M

ν=1PG,Tν

(C∗)I(∆) PG,Π.

∏ν(f

−1mν ◦ψsν )

α∆ q∆

ψ∆


Let us denote the indices in the index set I∞(sν) related to the triangle Tν with the

superscript (ν). Then we have

gEin,Eout ◦ ψ∆ ◦ α∆ = gEin,Eout ◦ q∆ ◦M∏

ν=1

(f−1mν

◦ ψsν)

= µM ◦M∏

ν=1

(bτν ◦ ψsν) (Proposition 3.23)

= µM ◦M∏

ν=1

(evτνsν

◦ ι∗ν) (Theorem 4.13)

= evτ1,...,τMs1,...,sM

◦ α′∆◦

M∏

ν=1

ι∗ν (Definition 5.1),

where ι∗ν : (C∗)I∞(sν ) → (C∗)Iν is the projection induced by the inclusion

ιν : Iν → I∞(sν), (s, i; ν) 7→

{(s, i)(ν) if τν = L,

(s∗, ns∗(sν)− i)(ν) if τν = R,

and α′∆

:

M−→∏ν=1

(C∗)Iν → (C∗)I1∗···∗IM is the amalgamation map induced by the projection

I1 × · · · × IM → I1 ∗ · · · ∗ IM . Then it suffices to prove α′∆◦∏

ν ι∗ν = ι∗Ein,Eout

◦ α∆. In

other words, it suffices to check that the amalgamation rule (5.1) agrees with (4.13) via

the permutations ιν . Fix ν ∈ {1, . . . ,M − 1}.

• When (τν , τν+1) = (L, L), we glue the pair

((s∗31)(ν), s

(ν+1)12 ) = ((s, ns(sν))

(ν), (s, 0)(ν+1)) = (ιν(s, ns(sν)), ιν+1(s, 0)),

which is exactly one of the pairs identified in I1 ∗ · · · ∗ IM .

The argument for the remaining three cases are similar:

• When (τν , τν+1) = (L,R), we glue

((s∗31)(ν), s

(ν+1)31 ) = ((s, ns(sν))

(ν), (s∗, ns∗

(sν+1))(ν+1)) = (ιν(s, n

s(sν)), ιν+1(s, 0)).

• When (τν , τν+1) = (R,R), we glue

((s∗12)(ν), s

(ν+1)31 ) = ((s∗, 0)(ν), (s∗, ns

∗

(sν+1))(ν+1)) = (ιν(s, n

s∗(sν)), ιν+1(s, 0)).

Notice that ns∗(sν) = ns((sν)

∗op) = ns(sRν ).

• When (τν , τν+1) = (R,L), we glue

((s∗12)(ν), s

(ν+1)12 ) = ((s∗, 0)(ν), (s, 0)(ν+1)) = (ιν(s, n

s∗(sν)), ιν+1(s, 0)).

Then for any case, we have α′∆

◦∏

ν ιν = ι∗Ein,Eout◦ α∆. Thus we get (ψ∆)∗gEin,Eout =

evτ1,...,τMs1,...,sM

◦ ι∗Ein,Eoutas desired. �


Remark 5.3. Notice that the image of the inclusion I1 ∗ · · · ∗ IM → I(∆) is contained

in the subset Iin,out(∆) := I{Ein,Eout}(∆). Here recall (4.15). In other words, the chain of

GS coordinates do not contain frozen variables except for those assigned to the edges Ein

and Eout. Therefore the evaluation formula also gives the coordinate expression of the

side pairing as a morphism gEin,Eout : PG,Σ;{Ein,Eout} → G.

5.2. Coordinate expressions of Wilson lines and Wilson loops. From the formulae

obtained in the previous subsection, we can deduce a formula for the Wilson line g[c]associated with an arc class [c] : Ein → Eout in terms of GS coordinates on P∆

G,Σ. Take

a decorated triangulation ∆ = (∆∗, s∆) of Σ and let the notations as in the proof of

Theorem 3.38. In particular, we have a cyclic shift Stν3 adjusting the disagreement of

the dots mν and nν . By setting sν := sπc(Tν)for ν = 1, . . . ,M , we get two decorated

triangulations ∆lift := (∆lift∗ , (sν)

Mν=1) and ∆can := (∆can

∗ , (sν)Mν=1) of Πc;∆.

Let us denote byXEin,Eout = (Xsi [ν])(i,s;ν)∈I1∗···∗IM the chain of GS coordinates associated

with (Ein, Eout) and the decorated triangulation ∆can. Define the twisted chain X[c] =

(Xsi [ν])(i,s;ν)∈I1∗···∗IM of GS coordinates along [c] by

Xsi [ν] := (πxc )

∗Xsi [ν],

here πxc : (C∗)I(∆) → (C∗)Iin,out(∆can) denotes the coordinate expression of π∗

c : PG,Σ →PG,Πc;∆;{Ein,Eout}

with Iin,out(∆can) := I{Ein,Eout}

(∆can).

Theorem 5.4 (Evaluation formula for Wilson lines). Let ∆ be a decorated triangulation,

[c] : Ein → Eout a morphism, and the notations as above. Then we have

ψ∗∆g[c] = evτ1,...,τM

s1,...,sM◦ ι∗

Ein,Eout◦ πxc .

In other words, the Wilson line g[c] : PG,Σ → G is expressed as

(ψ∆)∗g[c] = evτ1,...,τMs1,...,sM

(X[c]), (5.3)

where X[c] = (Xsi [ν])(i,s;ν)∈I1∗···∗IM is the twisted chain of GS coordinates on PG,Σ along [c]

associated with the decorated triangulation ∆.

Proof. By Theorem 5.2, we have

(ψ∆can)∗gEin,Eout= evτ1,...,τM

s1,...,sM◦ ι∗

Ein,Eout.

Combining it with Lemma 3.37, we get

(ψ∆)∗g[c] = gEin,Eout◦ π∗

c ◦ ψ∆

= gEin,Eout◦ ψ∆can ◦ πxc

= evτ1,...,τMs1,...,sM

◦ ι∗Ein,Eout

◦ πxc

as desired. �

We are going to see that the functions Xsi [ν] are positive rational functions of the GS

coordinate functions X∆ (or more precisely, their pull-backs via ψ∆ to the coordinate

torus (C∗)I(∆)) on PG,Σ associated with the decorated triangulation ∆. Indeed, these

rational functions are obtained through cluster transformations. By abuse of notation,


the GS coordinate X on PG,Πc;∆satisfying (ψ∆can)∗(X) = Xs

i [ν] is again denoted by

Xsi [ν]. Then by their definitions and ψ∆can ◦ πxc = π∗

c ◦ ψ∆, we need to express the

functions (π∗c )

∗Xsi [ν] in terms of the GS coordinate system X∆ on PG,Σ associated with

∆.

From the commutative diagram (3.11), we have

q∗∆(π∗c )

∗Xsi [ν] = (π∗

c )∗q∗∆c

Xsi [ν].

By the definition of GS coordinate system, q∗∆cXsi [ν] is equal to either a GS coordinate

on∏M

ν=1PG,Tν (when Xsi [ν] is located in a face), or the product of two GS coordinates on∏M

ν=1PG,Tν (when Xsi [ν] is located on an edge). Therefore we now calculate the pull-back

(π∗c )

∗X(Tν ,mν ,sν)

(si)of the GS coordinates on

∏Mν=1PG,Tν associated with ∆can.

From the commutative diagrams (3.12), we have

(π∗c )

∗X(Tν ,mν ,sν)

(si)= (fnν

)∗(S∗3 )tνXsν

(si)

= (Stν3 ◦ fmTν◦ prTν )

∗Xsν

(si)

= (fmTν◦ f−1

mTν◦ Stν3 ◦ fmTν

◦ prTν )∗Xsν

(si)

= (prTν )∗(f−1

mTν◦ Stν3 ◦ fmTν

)∗X(Tν ,mTν ,sν)

(si)

for (s, i) ∈ I∞(sν), ν = 1, . . . ,M . Then by Theorem 4.20, the rational expression of the

function (f−1mTν

◦Stν3 ◦fmTν)∗X

(Tν ,mTν ,sν)

(si)on PG,Tν in {X

(Tν ,mTν ,sν)

(si)}(s,i)∈I∞(sν) is obtained by

a composition of cluster transformations. Hence (π∗c )

∗X(Tν ,mν ,sν)

(si)has a positive rational

expression in the rational functions X(Tν ,mTν ,sν)

(si)for (s, i) ∈ I∞(sν) on

∏T∈t(∆) PG,T , and

it is obtained via cluster transformations.

As we mentioned above, (π∗c )

∗q∗∆cXsi [ν] = q∗∆(π

∗c )

∗Xsi [ν] is equal to such a rational

function or their product. Moreover, the amalgamation q∗∆ commutes with the mutation

sequence appearing in the above cluster transformation at the ν-th triangle Tν , since

the latter consists of mutations at vertices only on the face of Tν . Hence this cluster

transformation descends to a cluster transformation µrot[ν] on PG,Σ. Thus (π∗c )

∗Xsi [ν] is

also a cluster coordinate on PG,Σ, and related with the GS coordinate system X∆ on PG,Σvia the composite of the cluster transformations µrot[ν

′] for ν ′ = 1, . . . ,M .

Example 5.5 (A Wilson line on a marked annulus). Recall Example 3.39. Consider the

type C2 case, and choose the reduced words of w0 to be sT = sT ′ = s := (1, 2, 1, 2). The

evaluation maps for s are given by

ev+s(x10, x

20, x

11, x

21, x

12, x

22) = H1(x10)H

2(x20)E1H1(x11)E

2H2(x21)E1H1(x12)E

2H2(x22),

ev−s∗op(y10, y

20, y

11, y

21, y

12, y

22) = H1(y10)H

2(y20)F2H2(y21)F

1H1(y11)F2H2(y22)F

1H1(y12).

The weighted quivers associated with the decorated triangulations ∆lift := (∆lift∗ , (sν)

4ν=1)

and ∆can := (∆can∗ , (sν)

4ν=1) of the polygon Πc;∆ are shown in Figure 16. It is a general

feature of the quiver Q∆can that the graph obtained by concatenating its red arrows is


∗∗

Q∆

2

2

2

2

X1

X2

X3

X6

X7

X8

X11

X12

2 2 2

2

X4X5

X9

X10

X13 X14

∗∗

∗∗

Q∆lift

2

2

2

2

2 2 2

2

2

2

22

4

32 2 2

2

1

2

∗∗

∗∗

Q∆can

2

2

2

2

X20 [1]X1

0 [1]

X11 [1]

X10 [2]

2 2 2

2

222

2

X21 [3]

2

2

2 2

X21 [4]

X20 [4]

X22 [4]X1

2 [4]

µrot[3]−1µrot[4]

Figure 16. The weighted quivers associated with ∆, ∆lift and ∆can for

type C2 with the reduced word s = (1, 2, 1, 2).

homotopic to s parallel copies of the path c. Also observe that by forgetting the blue

vertices and arrows, we get the subquiver J+(s) ∗ J+(s) ∗ J−(s∗op) ∗ J−(s∗op).

Enumerate the GS coordinates on PG,Σ associated with ∆ as shown in the left of the

figure. In the right bottom, some of the twisted GS coordinates are shown at the position

where they are evaluated under the map evL,L,R,Rs,s,s,s , though they actually are functions on

PG,Σ rather than PG,Πc;∆. Here recall the convention Xs

2 [ν] = Xs0 [ν + 1] for s = 1, 2 and

ν = 1, 2, 3.

The cyclic shift µrot[3]−1µrot[4] transforms the quiver Q∆lift toQ∆can , which is realized by

the composite of mutations at the four vertices shown in green, in the order shown there.

Then from the formula (C.1) for the cluster Poisson transformation we can compute:

X20 [1] = X1, X2

1 [1] = X2, X22 [1] = X2

0 [2] = X3, X21 [2] = X4,

X10 [1] = X6, X1

1 [1] = X7, X12 [1] = X1

0 [2] = X8, X11 [2] = X9,


and

X22 [2] = X2

0 [3] = (X2X7 +X2 + 1)X11, X21 [3] =

(X2X7 +X2 + 1)2

X7,

X22 [3] = X2

0 [4] =X2(X4 + 1)(X7 + 1)

X2X7 +X2 + 1X3, X2

1 [4] =X9

X24X9 +X2

4 + 2X4 + 1X4,

X22 [4] =

(X24X9 +X2

4 + 2X4 + 1)

X4 + 1X14, X1

2 [2] = X10 [3] =

X22 (X7 + 1)X7

(X2X7 +X2 + 1)2X12,

X11 [3] =

(X2X7 +X2 + 1)2

X7

, X12 [3] = X1

0 [4] =(X2

4X9 +X24 + 2X4 + 1)X7

(X4 + 1)2(X7 + 1)X8,

X11 [4] =

(X4 + 1)2

X24X9

, X12 [4] =

X24X9

X24X9 +X2

4 + 2X4 + 1X13.

Substituting these rational expressions to evL,L,R,Rs,s,s,s = ev+

s· ev+

s· ev−

s∗op

· ev−s∗op

so that

Xsi [k] 7→ xsi for k = 1, 2 and Xs

i [k] 7→ ysi for k = 3, 4, we get the coordinate expression of

the Wilson line g[c].

It is easy to get an expression for a Wilson loop by using Proposition 3.28 and by

amalgamating some of variables in the corresponding expression (5.3).

6. Laurent positivity of Wilson lines and Wilson loops

6.1. The statements. In this section, we show that Wilson lines and Wilson loops have

a remarkable positivity nature. Let us first clarify the positivity properties which we will

deal with, and state the main theorems of this section.

Let Σ be a marked surface (See Section 3.1 for our assumption on the marked surface).

Let FG,Σ := K(PG,Σ) be the field of rational functions on PG,Σ. We say that a rational

function f ∈ FG,Σ is a GS-universally positive Laurent polynomial on PG,Σ if it is expressed

as a Laurent polynomial with non-negative integral coefficients in the GS coordinate

system associated with any decorated triangulation ∆. A rational G-valued morphism

F : PG,Σ → G is called a GS-universally positive G-valued Laurent polynomial on PG,Σ if

for any finite-dimensional representation V of G, there exists a basis B of V such that

cVf,v ◦ F : PG,Σ → C

is a GS-universally positive Laurent polynomial on PG,Σ for all v ∈ B and f ∈ F, whereF is the basis of V ∗ dual to B (see (2.4)).

Remark 6.1. In [FG06], Fock and Goncharov introduced the notion of special good

positive Laurent polynomials on XPGLn+1,Σ. Our notion of GS-universally positive Laurent

polynomial on PG,Σ is a straightforward generalization of their notion. Indeed, if we set

G = PGLn+1 and sstd(n) = (1, 2, . . . , n, . . . , 1, 2, 3, 1, 2, 1) as in (4.8) for all triangles T

of the decorated triangulation ∆ = (∆∗, (sstd(n))T ) of Σ, the GS coordinate system on

PPGLn+1,Σ associated with ∆ is the special atlas on XPGLn+1,Σ in [FG06, Definiton 9.1]

(modulo the the difference between PPGLn+1,Σ and XPGLn+1,Σ).


We should remark that the definition of GS-universally positive G-valued Laurent poly-

nomial does not change if we replace “any finite-dimensional representation V ” in its def-

inition with “any simple finite-dimensional representation V (λ), λ ∈ X∗(H)+” because

of the complete reducibility of finite-dimensional representations.

The following theorems is the main result in this section.

Theorem 6.2. Let G be a semisimple algebraic group of adjoint type, and assume that

our marked surface Σ has non-empty boundary. Then for any arc class [c] : Ein → Eout,

the Wilson line g[c] : PG,Σ → G is a regular GS-universally positive G-valued Laurent

polynomial.

Combining with Proposition 3.28, we immediately get the following:

Corollary 6.3. Let G be a semisimple algebraic group of adjoint type, and |γ| ∈ π(Σ)

a free loop. Then, for any finite dimensional representation V of G, the trace of the

Wilson loop trV (ρ|γ|) := trV ◦ρ|γ| : PG,Σ → C is a regular GS-universally positive Laurent

polynomial.

Corollary 6.3 is a generalization of [FG06, Theorem 9.3, Corollary 9.2]. The rest of this

section is devoted to the proof of Theorem 6.2.

6.2. A basis of O(PG,T ) with positivity. Our computation is performed locally on

each triangle T of an arbitrarily fixed decorated triangulation ∆ of Σ. An important fact

is the existence of a basis of O(PG,T ) with an appropriate positivity. In this subsection, we

explain a construction of such a nice basis. Fix a triangle T and label the marked points

of T as m1, m2, m3 in the counter-clockwise order. Recall the standard configuration

C3 : H ×H × U+∗

∼−→ Conf3PG in Lemma 2.11, and the map fmi

: PG,T∼−→ Conf3PG given

in (3.2). Then we have an isomorphism

C3,mi:= f−1

mi◦ C3 : H ×H × U+

∗∼−→ PG,T

for i ∈ {1, 2, 3}, which induces an isomorphism of the coordinate rings

C∗3,mi

: O(PG,T )∼−→ O(H ×H × U+

∗ ) = O(H)⊗O(H)⊗O(U+∗ ).

The coordinate ring O(H) is identified with the group algebra C[X∗(H)] =∑

µ∈X∗(H) Ceµ.

To distinguish the first component of O(H) ⊗ O(H) ⊗ O(U+∗ ) from its second compo-

nent, we write the element eµ, µ ∈ X∗(H) in the first (resp. second) component as eµ1(resp. eµ2 ). Recall that the coordinate algebra O(U+

∗ ) has a X∗(H)-grading O(U+∗ ) =⊕

β∈X∗(H) O(U+∗ )β such that

F ◦ Adh|U+∗= hβF

for h ∈ H and F ∈ O(U+∗ )β. For ξ ∈ X∗(H), set

∆+w0,ξ

:= (∆λ1,w0λ1 |U+∗)−1∆λ2,w0λ2 |U+

∗∈ O(U+

∗ ) (6.1)

with λ1, λ2 ∈ X∗(H)+ such that −λ1 + λ2 = ξ. Note that ∆+w0,ξ

is a well-defined element.


The description of the cyclic shift S3 on Conf3PG (Lemma 3.9) in terms of the standard

configuration is important in the sequel. In the description, we use the Berenstein–Fomin–

Zelevinsky twist automorphism ηw0 [BFZ96, BZ97] (we call it the twist automorphism for

short) defined by

ηw0 : U+∗ → U+

∗ , u+ 7→ [w0uT

+]+.

This is a regular automorphism of U+∗ . The properties of ηw0 are collected in Appendix A.

Lemma 6.4. We have S3(C3(h1, h2, u+)) = C3(h′1, h

′2, u

′+), where

h′1 = h∗1h2w0([u+w0]0),

h′2 = h∗1,

u′+ = Adh∗1(ηw0(u+)∗).

Hence the isomorphism C−13 ◦ S3 ◦ C3 : H ×H × U+

∗∼−→ H ×H × U+

∗ is expressed as

(C−13 ◦ S3 ◦ C3)

∗(eµ1 ⊗ eν2 ⊗ F ) = eµ∗+ν∗+β∗

1 ⊗ eµ2 ⊗∆+w0,−µ∗−β

· (η∗w0)4(F ) (6.2)

for µ, ν ∈ X∗(H) and F ∈ O(U+∗ )β.

Proof. We have

S3(C3(h1, h2, u+)) = [φ′(u+)h1w0.pstd, u+h2w0.pstd, pstd]

= [pstd, (φ′(u+)h1w0)

−1u+h2w0.pstd, (φ′(u+)h1w0)

−1.pstd].

The third component is rewritten as

w0h−11 φ′(u+)

−1.pstd = Adw0h−11(φ′(u+)

−1)w0(h−11 )w0.pstd

= Adh∗1((φ′(u+)

T)∗)h∗1w0.pstd

= Adh∗1(ηw0(u+)∗)h∗1w0.pstd. (by Lemma A.4)

Hence we have h′2 = h∗1 and u′+ = Adh∗1(ηw0(u+)∗). The second component is rewritten as

w0h−11 φ′(u+)

−1u+h2w0.pstd = w0h−11 [u+w0]0w0

−1φ(u+)h2w0.pstd (by Proposition A.3)

= h∗1w0([u+w0]0)φ(u+)h2w0.pstd

= Adh∗1w0([u+w0]0)(φ(u+))h∗1w0([u+w0]0)h2w0.pstd.

Hence, by reading the Cartan part off, we see that h′1 = h∗1h2w0([u+w0]0).

From the computation of h′1.h′2 and u′+ above, it follows that

((C−1

3 ◦ S3 ◦ C3)∗(eµ1 ⊗ eν2 ⊗ F )

)(h1, h2, u+) = (eµ1 ⊗ eν2 ⊗ F )(h′1, h

′2, u

′+)

= hµ∗+ν∗+β∗

1 hµ2 (w0([u+w0]0))µ(F ◦ ∗)(ηw0(u+)).


for µ, ν ∈ X∗(H) and F ∈ O(U+∗ )β. Moreover, for µ1, µ2 ∈ X∗(H)+ with −µ1+µ2 = −µ∗,

we have

(w0([u+w0]0))µ = [u+w0]

−µ∗

0

= (∆µ1,µ1([u+w0]0))−1∆µ2,µ2([u+w0]0)

= (∆µ1,µ1(u+w0))−1∆µ2,µ2(u+w0)

= (∆µ1,w0µ1(u+))−1∆µ2,w0µ2(u+)

= ∆+w0,−µ∗(u+).

By Lemma A.5, we have

(F ◦ ∗)(ηw0(u+)) = (∆+w0,β

· (η∗w0)3(F ))(ηw0(u+))

= (η∗w0(∆+

w0,β))(u+)((η

∗w0)4(F ))(u+)

= (∆+w0,−β

· (η∗w0)4(F ))(u+).

Therefore, we obtain

(C−13 ◦ S3 ◦ C3)

∗(eµ1 ⊗ eν2 ⊗ F ) = eµ∗+ν∗+β∗

1 ⊗ eµ2 ⊗∆+w0,−µ∗−β

· (η∗w0)4(F )

as desired. �

We now construct a “canonical” basis of O(PG,T ) from that of O(U+∗ ) with some nice

properties (G), (T), and (M).

Lemma 6.5. Let F be a basis of O(U+∗ ) such that

(G) the elements of F are homogeneous with respect to the X∗(H)-grading O(U+∗ ) =⊕

β∈X∗(H) O(U+∗ )β,

(T) F is preserved by the twist automorphism η∗w0: O(U+

∗ ) → O(U+∗ ) as a set, and

(M) ∆+w0,ξ

· F ∈ F for any ξ ∈ X∗(H) and F ∈ F.

Then the basis FT of O(PG,T ) given by

FT := {(C∗3,mi

)−1(eµ1 ⊗ eν2 ⊗ F ) | µ, ν ∈ X∗(H), F ∈ F} (6.3)

does not depend on the choice of i ∈ {1, 2, 3}.

Proof. Using the isomorphisms

C∗3 : O(Conf3PG)

∼−→ O(H)⊗O(H)⊗O(U+

∗ ), f∗mi

: O(Conf3PG)∼−→ O(PG,T )

for i ∈ {1, 2, 3}, we set

F3 := {(C∗3)

−1(eµ1 ⊗ eν2 ⊗ F ) | µ, ν ∈ X∗(H), F ∈ F},

Fmi:= f ∗

mi(F3).


By the rotational symmetry, it suffices to show that Fm1 = Fm2 . By Lemma 3.9, we have

Fm2 = f ∗m2

(F3)

= (f ∗m1

◦ (fm2 ◦ f−1m1

)∗)(F3)

= (f ∗m1

◦ S∗3 )(F3).

Therefore, it remains to show that S∗3 (F3) = F3. For µ, ν ∈ X∗(H) and F ∈ F with

F ∈ O(U+∗ )β, we have

S∗3 ((C

∗3)

−1(eµ1 ⊗ eν2 ⊗ F ))

= (C∗3)

−1((C−13 ◦ S3 ◦ C3)

∗(eµ1 ⊗ eν2 ⊗ F ))

= (C∗3)

−1(eµ∗+ν∗+β∗

1 ⊗ eµ2 ⊗∆+w0,−µ∗−β

· (η∗w0)4(F ))

by Lemma 6.4. The assumptions (T) and (M) imply ∆+w0,−µ∗−β

·(η∗w0)4(F ) ∈ F. Therefore,

S∗3 ((C

∗3)

−1(eµ1 ⊗ eν2 ⊗ F )) ∈ F3, which proves S∗3 (F3) = F3. �

Remark 6.6. There are several examples of bases of O(U+∗ ) which satisfy the properties

(G), (T), and (M):

• the dual semicanonical basis, in the case when g is of symmetric type [GLS11,

Theorem 15.10], [GLS12, Theorem 6].

• the dual canonical basis (specialized at q = 1) [KO, Definition 4.6, Theorem 6.1].

• the simple object basis arising from the monoidal categorification via quiver Hecke

algebras [KKOP19, Corollary 5.4]. See Section B.3 below.

In this paper, we mainly use the last one because it has a convenient positivity.

We use the following strong fact in order to construct a basis of O(PG,T ) with an

appropriate positivity. See Appendix B for a proof.

Theorem 6.7. There exist

• a basis Fpos of O(U+∗ ), and

• two bases B(λ) := {Gλ(b) | b ∈ B(λ)} and Bup(λ) := {Gupλ (b) | b ∈ B(λ)} of

V (λ) for each λ ∈ X∗(H)+ (here B(λ) is just an index set)

satisfying the following properties:

(Grep) B(λ) and Bup(λ) consist of weight vectors of V (λ), and we have

(Gλ(b), Gupλ (b′))λ = δb,b′

for b, b′ ∈ B(λ).

(G) the elements of Fpos are homogeneous with respect to the X∗(H)-grading O(U+∗ ) =⊕

β∈X∗(H) O(U+∗ )β.

(T) Fpos is preserved by the twist automorphism η∗w0: O(U+

∗ ) → O(U+∗ ) as a set.

(M) ∆+w0,ξ

· F ∈ Fpos for any ξ ∈ X∗(H) and F ∈ Fpos.

(P1) Recall the notation (2.3). For b, b′ ∈ B(λ), we have

cλGλ(b)∨,Gupλ

(b′)|U+∗∈∑

F∈Fpos

Z≥0F, (cλGλ(b)∨,Gupλ

(b′) ◦ T)|U+∗∈∑

F∈Fpos

Z≥0F.


(P2) Recall the notation (4.1). For any reduced word s = (s1, . . . , sN) of w0, we have

(xs)∗(F ) ∈ Z≥0[t±11 , . . . , t±1

N ]

for all F ∈ Fpos.

In the following, we write the weight of Gλ(b) (and Gupλ (b′)) as wt b.

Remark 6.8. In (P1), either cλGλ(b)∨,Gupλ

(b′) or (cλGλ(b)∨,Gupλ

(b′) ◦ T)|U+∗is equal to 0 if the

weights of Gλ(b) and Gupλ (b′) are distinct. Since cλGλ(b)∨,G

upλ

(b′) ◦T = cλGupλ

(b′)∨,Gλ(b), we can

interchange the roles of B(λ) and Bup(λ).

Remark 6.9. Theorem 6.7 is highly non-trivial. An example of such bases can be ob-

tained from the theory of categorification of O(U+∗ ) via quiver Hecke algebras, developed

in [KL09, Rou08, KL11, Rou12, KK12, KKKO18, KKOP18, KKOP19]. We give a proof

of Theorem 6.7 in Appendix B based on their results.

In the following, the notations Fpos, B(λ), and Bup(λ) always stand for bases satisfying

the properties (Grep), (G), (T), (M), (P1), and (P2) (which are not necessarily the ones

given in Appendix B). Moreover, let Fpos,T be the basis of O(PG,T ) defined from Fpos as

in (6.3).

Theorem 6.10. The basis Fpos,T consists of regular GS-universally positive Laurent poly-

nomials on PG,T .

Proof. Recall that a decorated triangulation ∆ of T is determined by the choice of a dot

mi on T and a reduced word s of w0. The associated GS coordinates on PG,T are defined

as X(T,mi,s)

(si):= f ∗

miXs

(si), where the right-hand side is the pull-back of the GS coordinate

on Conf3PG associated with s. By Lemma 6.5, given any decorated triangulation ∆ of

T , we may regard Fpos,T as f ∗mi(Fpos,3), where mi is the dot of ∆ and

Fpos,3 := {(C∗3)

−1(eµ1 ⊗ eν2 ⊗ F ) | µ, ν ∈ X∗(H), F ∈ Fpos}.

Therefore, it suffices to show that Fpos,3 is expressed as a Laurent polynomial with non-

negative integral coefficients in terms of the GS coordinates on Conf3PG associated with

s. Recall the map ψs in (4.7) and the maps h1, h2, u+ in (4.10), whose explicit descriptions

are given in Lemma 4.19. For µ, ν ∈ X∗(H) and F ∈ Fpos, we have

ψ∗s

((C∗

3)−1(eµ1 ⊗ eν2 ⊗ F )

)= (eµ1 ⊗ eν2 ⊗ F )(h1(X), h2(X), u+(X)).

By Lemma 4.19 and the property (P2) of Fpos, the right-hand side is a Laurent polynomial

with non-negative integral coefficients in {Xs

(si)}(s,i)∈I∞(s). This completes the proof. �

Remark 6.11. In the proof of Theorem 6.10, we do not use the property (P1) of Fpos.

Lemma 6.12. We have F ◦ ∗ ∈ Fpos for all F ∈ Fpos.

Proof. From Lemma A.5, we have F ◦ ∗ = ∆+w0,β

· (η∗w0)3(F ) for F ∈ Fpos ∩ O(U+

∗ )β. We

have (η∗w0)3(F ) ∈ Fpos by the property (T), and then ∆+

w0,β· (η∗w0

)3(F ) ∈ Fpos by the

property (M). �


Recall the basic Wilson lines bL : Conf3PG → B+∗ and bR : Conf3PG → B−

∗ from

Definition 2.12. For i ∈ {1, 2, 3}, we set

gmi,L : PG,Tfmi−−→ Conf3PG

bL−→ B+∗

ι→ G,

gmi,R : PG,Tfmi−−→ Conf3PG

bR−→ B−∗

ι→ G,

where the last maps ι are the inclusion maps.

Theorem 6.13. For λ ∈ X∗(H)+, b, b′ ∈ B(λ), i ∈ {1, 2, 3} and τ ∈ {L,R}, the map

cλGλ(b)∨,G

upλ

(b′)◦ bmi,τ : PG,T → C is written as a Z≥0-linear combination of elements of

Fpos,T . In particular, it is a regular GS-universally positive Laurent polynomial, and bmi,τ

is a regular GS-universally positive G-valued Laurent polynomial.

Proof. We have

cλGλ(b)∨,Gupλ

(b′) ◦ gmi,L = cλGλ(b)∨,Gupλ

(b′) ◦ ι ◦ bL ◦ fmi

= cλGλ(b)∨,Gupλ

(b′) ◦ ι ◦ bL ◦ C3 ◦ C−13 ◦ fmi

= (1⊗ ewt b′

2 ⊗ cλGλ(b)∨,Gupλ

(b′)|U+∗) ◦ C−1

3 ◦ fmi(by Corollary 2.13)

∈∑

F∈Fpos

Z≥0(C∗3,mi

)−1(1⊗ ewt b′

2 ⊗ F ) (by the property (P1))

⊂∑

F∈Fpos,T

Z≥0F .

cλGλ(b)∨,Gupλ

(b′) ◦ gmi,R = cλGλ(b)∨,Gupλ

(b′) ◦ ι ◦ bR ◦ fmi

= cλGλ(b)∨,Gupλ

(b′) ◦ ι ◦ bR ◦ C3 ◦ C−13 ◦ fmi

= (1⊗ e(wt b)∗

2 ⊗ cλGλ(b)∨,Gupλ

(b′) ◦ T ◦ ∗|U+∗) ◦ C−1

3 ◦ fmi(by Corollary 2.13)

∈∑

F∈Fpos

Z≥0(C∗3,mi

)−1(1⊗ e(wt b)∗

2 ⊗ (F ◦ ∗)) (by the property (P1))

=∑

F∈Fpos

Z≥0(C∗3,mi

)−1(1⊗ e(wt b)∗

2 ⊗ F ) (by Lemma 6.12)

⊂∑

F∈Fpos,T

Z≥0F .

The remaining statements immediately follow from Remark 6.1 and Theorem 6.10. �

6.3. A proof of Theorem 6.2. Let Σ be a marked surface with non-empty boundary,

and fix an arbitrary decorated triangulation ∆ = (∆∗, s∆) of Σ (recall our assumption on

the marked surface in Section 3.1). Recall P∆G,Σ defined after Theorem 3.13, where ∆ is

the underlying triangulation of ∆∗. Fix an arc class [c] : Ein → Eout. For our purpose, it

suffices to show that

cλGλ(b)∨,Gupλ

(b′) ◦ g[c] : PG,Σ → C


is a GS-universally positive Laurent polynomial for any λ ∈ X∗(H)+ and b, b′ ∈ B(λ)

(see Remark 6.1).

Let q∆ :∏

T∈t(∆) PG,T → P∆G,Σ be the gluing map in Theorem 3.13, and prT :

∏T ′∈t(∆) PG,T ′ →

PG,T ′ the projection for T ∈ t(∆). Recall that

q∗∆(X(T,mT ,sT )

(si)) = pr∗T (f

∗mT

(XsT

(si))),

q∗∆(X(E,s∆)s ) =

{pr∗

T (1)(f∗m

T (1)(X

sT (1)sa,a+1)) · pr

∗T (2)(f

∗m

T (2)(X

sT (2)

s∗b,b+1

)) if E is an interior edge,

pr∗TE(f∗mTE

(X(1)sa,a+1)) if E is a boundary interval,

for T ∈ t(∆), s ∈ S, i = 1, . . . , ns(s) − 1 and E ∈ e(∆). Here T (1) and T (2) are two

triangles containing E where T (1) is on the left side with respect to the orientation of

E, and TE is a triangle containing E. Recall Section 4.6 for the definition of sa,a+1 and

sb,b+1 (a and b obviously depend on E, but it is omitted from the notation). By the

correspondence above, it suffices to show that q∗∆(cλGλ(b)∨,G

upλ

(b′)◦ g[c]) is expressed as a

Laurent polynomial with non-negative integral coefficients in any GS coordinate system

on∏

T∈t(∆) PG,T .Henceforth, we follow the notation in the beginning of Section 3.6. For ν = 1, . . . ,M ,

denote by mν the dot on Tν which is associated with the turning pattern (τ1, . . . , τM) of

c. Moreover, we have the commutative diagram

∏T∈t(∆) PG,T

∏Mν=1PG,Tν

P∆G,Σ P∆c

G,Πc;∆,

π∗c

q∆ q∆c

π∗c

where π∗c :=

∏Mν=1(πc|Tν )

∗ (see the proof of Theorem 3.38). Then, by (3.16), we have

q∗∆(cλGλ(b)∨,G

upλ

(b′) ◦ g[c]) = cλGλ(b)∨,Gupλ

(b′) ◦ µM ◦

(M∏

ν=1

gmν ,τν

)◦ π∗

c

= (π∗c )

∗

(cλGλ(b)∨,G

upλ

(b′) ◦ µM ◦M∏

ν=1

gmν ,τν

)

= (π∗c )

∗

∑

b1,...,bM−1∈B(λ)

M∏

ν=1

(cλGλ(bν−1)∨,G

upλ

(bν)◦ gmν ,τν ◦ prTν

) ,

where b0 := b and bM := b′. By Theorem 6.13, each cλGλ(bν−1)∨,G

upλ

(bν)◦ gmν ,τν is a GS-

universally positive Laurent polynomial on PG,Tν . Moreover,

(π∗c )

∗(pr∗Tν(X

(Tν ,m,s)

(si))) = pr∗

πc(Tν)(X

(πc(Tν ),πc(m),s)

(si))

for any dotm on Tν , any (s, i) ∈ I∞(s), and any reduced word s of w0. Thus q∗∆(c

λGλ(b)∨,G

upλ

(b′)◦

g[c]) is expressed as a Laurent polynomial with non-negative integral coefficients in the

GS coordinate system on∏

T∈t(∆) PG,T , which completes the proof of Theorem 6.2.


Appendix A. Some maps related to the twist automorphism

In this Appendix, we collect some useful properties of the Berenstein–Fomin–Zelevinsky

twist automorphism [BFZ96, BZ97]

ηw0 : U+∗ → U+

∗ , u+ 7→ [w0uT

+]+,

and its related maps.

Let B± := {u±.B∓ | u± ∈ U±} ⊂ BG be the open Schubert cells. Consider the

intersection B∗ := B+ ∩ B−.

Lemma A.1 ([FG06, Lemma 5.2]). There are bijections

α±∗ : U±

∗ → B∗, u± 7→ u±.B∓.

Then we have a bijection φ′ := (α−∗ )

−1 ◦ α+∗ : U+

∗ → U−∗ , which satisfies φ′(u+).B

+ =

u+B− for all u+ ∈ U+

∗ . Let us consider another map φ : U+∗ → U−

∗ defined by φ(u+) :=

(φ′−1(uT+))T for u+ ∈ U+, which satisfies the following property:

Lemma A.2. We have φ(u+)−1.B+ = u−1

+ .B− for all u+ ∈ U+∗ .

Proof. Let v+ := φ′−1(uT+) ∈ U+∗ . Then

φ(u+)−1B+φ(u+) = (vT+)

−1B+vT+ = (v+B−v−1

+ )T

= (φ′(v+)B+φ′(v+)

−1)T = u−1+ B−u+.

�

Using these maps, we get the following decomposition of an element of the unipotent

cell U+∗ . Recall the triangular decomposition G0 = U−HU+, g = [g]−[g]0[g]+.

Proposition A.3. Let u+ ∈ U+∗ , which can be written as u+ = u−hw0

−1u′− with u−, u′− ∈

U− and h ∈ H. Then

u− = φ′(u+),

u′− = φ(u+),

h = [u+w0]0.

In other words, we have u+ = φ′(u+)[u+w0]0w0−1φ(u+) for all u+ ∈ U+

∗ .

Proof. The first equality follows from

u+B−u−1

+ = u−hw0−1B−w0h

−1u−1− = u−B

+u−1− .

By the same argument, if we write v− ∈ U−∗ as v− = v+w0b

′+ with v+ ∈ U+ and b′+ ∈ B+,

then v+ = φ′−1(v−). Using this for uT+ = (u′−)Tw0hu

T−, we get (u′−)

T = φ′−1(uT+). Hence

u′− = (φ′−1(uT+))T = φ(u+).

For the third equality, note that w0−1uT+ = w0

−1(u′−)Tw0hu

T− ∈ U−HU+. Thus we get

h = [w0−1uT+]0 = [u+w0]0. �

The maps φ and φ′ are related to ηw0 as follows:


Lemma A.4. Let ηw0 : U+∗ → U+

∗ , u+ 7→ [w0uT+]+ be the twist automorphism. Then we

have φ(u+) = (η−1w0(u+))

T and φ′(u+) = (ηw0(u+))T.

Proof. Let us write u+ = b−w0u′− with b− ∈ B− and u′− ∈ U−. Then

ηw0(φ(u+)T) = ηw0((u

′−)

T) = [w0u′−]+ = [b−1

− u+]+ = u+.

The second equality immediately follows from the first one. �

The ∗-involution ∗ : G→ G (Lemma 2.4) restricts to an involution ∗ : U+∗ → U+

∗ .

Lemma A.5. Let β ∈ X∗(H). For F ∈ O(U+∗ )β, we have

(η∗w0)3(F ) = ∆+

w0,−β· (F ◦ ∗).

Proof. See the proof of [KO, Theorem 8.1]. �

Appendix B. Proof of Theorem 6.7

In this appendix, we prove Theorem 6.7 using the categorification of O(U+∗ ) via quiver

Hecke algebras. To keep the length of the paper reasonable, we do not repeat the detailed

definitions of notions concerning quiver Hecke algebras. We always follow the notation in

[KKOP19] unless otherwise specified. See Remark 6.9 for further references.

B.1. Bases with positivity. In this subsection, we give preliminary results for proving

that the bases in the subsequent sections have the property (P1) in Theorem 6.7.

Let n+ be the Lie algebra of U+, and U(n+) its universal enveloping algebra (over C).Then we can define a map ⊙ : n+×n+ → n+ by Baker–Campbell–Hausdorff formula, and

it induces a group structure on n+. Moreover, there is an isomorphism exp: (n+,⊙)∼−→ U+

of algebraic groups such that exp(tes) = xs(t) for t ∈ C.Set X∗(H)≥0 :=

∑s∈S Z≥0αs. Since we have n+ =

∑β∈Φ+

gβ , the algebra U(n+) is

X∗(H)≥0-graded: U(n+) =⊕

β∈X∗(H)≥0U(n+)β. Moreover U(n+) is a cocommutative

Hopf algebra with the comultiplication

∆: U(n+) → U(n+)⊗C U(n+), X 7→ X ⊗ 1 + 1⊗X

for X ∈ n+. Define U(n+)∗gr as the graded dual

U(n+)∗gr :=⊕

β∈X∗(H)≥0

HomC(U(n+)β,C) ⊂ U(n+)∗.

Then ∆ induces a commutative algebra structure on U(n+)∗gr. Then we have an isomor-

phism of algebras

U(n+)∗gr∼−→ O(U+), F 7→ Fgrp := (u+ 7→

∑

k≥0

1

k!F (exp−1(u+)

k)). (B.1)

Here we consider exp−1(u+) as an element of n+ ⊂ U(n+), and its power exp−1(u+)k in

U(n+). See [GLS11, Section 5] and references therein for more details. We sometimes

omit the subscript grp when it does not cause any confusion.


Remark B.1. Let V be a G-module. Then V carries a g-module structure. Therefore,

for v ∈ V and f ∈ V ∗, we can define cV,nilf,v ∈ U(n+)∗gr by

U(n+) ∋ X 7→ 〈f, X.v〉 ∈ C.

Then we have (cV,nilf,v )grp = cVf,v|U+ ∈ O(U+).

For λ ∈ X∗(H)+, let us denote by V (λ) the irreducible highest weight g-module of

highest weight λ. We fix a highest weight vector vλ. Then it is well-known that there

uniquely exists a symmetric non-degenerate C-bilinear form ( , )λ : V (λ) × V (λ) → Csuch that

(vλ, vλ)λ = 1 (X.v, w)λ = (v,XT.w)λ

for all v, w ∈ V (λ) and X ∈ g, where X 7→ XT is an anti-involution of g given by eTs = fs,

fTs = es, and (α∨

s )T = α∨

s for s ∈ S. For v ∈ V (λ), set

v∨ := (v′ 7→ (v, v′)λ) ∈ V (λ)∗.

Remark B.2. For λ ∈ X∗(H)+, the notations V (λ), ( , )λ, and v∨ are compatible with

the ones defined in Section 2.

Assumption B.3. We assume that there exist

• a basis B := {B(b) | b ∈ B(∞)} of U(n+) (B(∞) is just an index set), and

• a basis B(λ) := {Bλ(b) | b ∈ B(λ)} of a g-module V (λ) for each λ ∈ X∗(H)+,

satisfying

B(b).Bλ(b) ∈∑

b′∈B(λ)

Z≥0Bλ(b′), B(b).Bup

λ (b) ∈∑

b′∈B(λ)

Z≥0Bupλ (b′) (B.2)

for all b ∈ B(∞) and b ∈ B(λ). Here Bupλ (b) ∈ V (λ) is defined by the condition

(Bupλ (b), Bλ(b

′))λ = δb,b′ for b′ ∈ B(λ).

Let D := {D(b) | b ∈ B(∞)} be a basis of U(n+)∗gr such that

〈D(b), B(b′)〉 = δb,b′ (B.3)

for all b, b′ ∈ B(∞).

Proposition B.4. Let λ ∈ X∗(H)+ and b, b′ ∈ B(λ). Then

cλBλ(b)∨,Bupλ

(b′)|U(n+) ∈∑

b∈B(∞)

Z≥0D(b), (cλBλ(b)∨,Bupλ

(b′) ◦ T)|U(n+) ∈∑

b∈B(∞)

Z≥0D(b).

Proof. By definition of D, it suffices to show that

〈cλBλ(b)∨,Bupλ

(b′), B(b)〉 ∈ Z≥0 and 〈cλBλ(b)∨,Bupλ

(b′), B(b)T〉 ∈ Z≥0

for all b ∈ B(∞). By the property (B.2), we have


(b′), B(b)〉 = (Bλ(b), B(b).Bupλ (b′))λ ∈ Z≥0,


(b′), B(b)T〉 = (Bλ(b), B(b)T.Bupλ (b′))λ = (B(b).Bλ(b), B

upλ (b′))λ ∈ Z≥0.

�


It is well-known that

∆λ,w0λ∆λ′,w0λ′ = ∆λ+λ′,w0(λ+λ)

holds for λ, λ′ ∈ X∗(H)+, and

U+∗ = {u+ ∈ U+ | ∆λ,w0λ(u+) 6= 0 for all λ ∈ X∗(H)+}

= {u+ ∈ U+ | ∆λ0,w0λ0(u+) 6= 0}

if λ0 ∈ X∗(H) satisfies 〈λ0, α∨s 〉 > 0 for all s ∈ S. Therefore D := {∆λ,w0λ|U+ | λ ∈

X∗(H)+} is a multiplicatively closed subset of O(U+), and we have a canonical injective

homomorphism

O(U+) → O(U+)D−1 = O(U+∗ ).

If the set

D := {D(b)∆+w0,−λ

| b ∈ B(∞), λ ∈ X∗(H)+}(⊃ D)

forms a basis of O(U+∗ ) (recall (6.1)), then D and B(λ), λ ∈ X∗(H) are bases ofO(U+

∗ ) and

V (λ), respectively, satisfying (P1) in Theorem 6.7 by Remark B.1 and Proposition B.4.

B.2. Bases arising from categorification via quiver Hecke algebras. In this sub-

section, we show that bases arising from categorification via quiver Hecke algebras satisfy

(Grep), (G), (P1), (P2). More precisely, we give bases L(λ) and P(λ) of V (λ) and a set

L which generates O(U+∗ ) as a vector space, which satisfy (Grep), (G), (P1), (P2). The

linear independence of L will be explained in subsection B.3. Although these follow im-

mediately from known results, we collect them and give supplementary explanations for

the reader’s convenience. We refer to [KKOP19] for all missing definitions.

Bases of universal enveloping algebras. Let Uq(g) be the quantum group associated

with (C(g), X∗(H), {αs}s∈S, {α∨s }s∈S, ( , )) over Q(q), and U−

q (g) (resp. U+q (g)) its negative

(resp. positive) half. We will denote by U−Z[q±1](g) (resp. U

+Z[q±1](g)) the Z[q±1]-subalgebra

of U−q (g) generated by the divided powers {f

(k)s | s ∈ S, k ∈ Z≥0} (resp. {e

(k)s | k ∈ s ∈

S,Z≥0}) of generators of U−q (g). Here, by abuse of notation, we again write Chevalley-

type generators of Uq(g) as es, fs (s ∈ S) since it will cause no confusion. Let Aq(n+) be

the unipotent quantum coordinate ring, and Aq(n+)Z[q±1] its Z[q±1]-form.

Remark B.5. If we regard C as a Z[q±1]-module via q 7→ 1, then, as C-algebras, we haveisomorphisms

C⊗Z[q±1] U+Z[q±1](g) ≃ U(n+), 1⊗ e(k)s 7→ eks/k!,

C⊗Z[q±1] U−Z[q±1](g) ≃ U(n−) ≃ U(n+), 1⊗ f (k)

s 7→ fks /k! 7→ eks/k!,

C⊗Z[q±1] Aq(n+)Z[q±1] ≃ U(n+)∗gr ≃ O(U+) (recall Subsection B.1),

where n− is the Lie algebra of U−.

For β ∈ X∗(H)≥0, let R(β) denote the quiver Hecke algebra over a field k associated

with C(g) and some set of polynomials (Qi,j(u, v))i,j∈I given in [KKOP19, Subsection

1.2]. Let R(β)-Mod be the abelian category of Z-graded R(β)-modules. Note that the

homomorphisms in R(β)-Mod preserve the degree. Let us denote by R(β)-gmod the full

subcategory of R(β)-Mod consisting of modules which are finite-dimensional over k, and


by R(β)-proj the full subcategory of R(β)-gMod consisting of finitely generated projective

modules. Set

R-Mod :=⊕

β∈X∗(H)≥0

R(β)-Mod, R-gmod :=⊕

β∈X∗(H)≥0

R(β)-gmod, R-proj :=⊕

β∈X∗(H)≥0

R(β)-proj.

There exists an exact bifunctor, called the convolution product,

− ◦ − : R(β)-Mod× R(γ)-Mod → R(β + γ)-Mod

which makes R-Mod into a monoidal category whose unit object is 1 := k ∈ R(0)-gmod.

Moreover, it induces bifunctors

− ◦− : R(β)-gmod× R(γ)-gmod → R(β + γ)-gmod,

− ◦− : R(β)-proj× R(γ)-proj → R(β + γ)-proj.

Therefore the Grothendieck group K(R-gmod) of R-gmod and the split Grothendieck

groupK(R-proj) of R-proj have Z[q±1]-algebra structures, where the action of q is induced

from the grading shift functor given by (qM)i = Mi−1 for M =⊕

i∈ZMi ∈ R-Mod. The

following states that R(β)-gmod and R(β)-proj categorify Aq(n+)Z[q±1] and U−

Z[q±1](g),

respectively.

Theorem B.6 ([KL09, KL11, Rou08]). There exist isomorphims of Z[q±1]-algebras

K(R-gmod) ≃ Aq(n+)Z[q±1], and K(R-proj) ≃ U−

Z[q±1](g).

Through these isomorphisms, the decompositionsK(R-gmod) =⊕

β∈X∗(H)≥0K(R(β)-gmod)

and K(R-proj) =⊕

β∈X∗(H)≥0K(R(β)-proj) coincide with the natural X∗(H)≥0-gradings

of Aq(n+)Z[q±1] =

⊕β∈X∗(H)≥0

Aq(n+)Z[q±1],β and U−

Z[q±1](g) =⊕

β∈X∗(H)≥0U−Z[q±1](g)−β.

Remark B.7. In [KKOP19], the weight of a module M ∈ R(β)-Mod is defined as

−β. However, in the present paper, we consider K(R-gmod) and K(R-proj) simply

as X∗(H)≥0-graded algebras. By the specialization q → 1 in Remark B.5, the spaces

Aq(n+)Z[q±1],β and U−

Z[q±1](g)−β specialize to HomC(U(n+)β,C)(⊂ U(n+)∗gr) and U(n−)−β(≃

U(n+)β), respectively.

For M ∈ R(β)-gmod, its dual space M∗ := Homk(M, k) admits a graded R(β)-module

structure given by

(r.f)(m) := f(ψ(r).m) for r ∈ R(β), f ∈M∗ and m ∈M,

where ψ denotes the k-algebra anti-involution on R(β) which fixes the usual generators.

A simple graded R(β)-module M is said to be self-dual if M ≃M∗. Indeed, every simple

graded R(β)-module is isomorphic to a grading shift of a self-dual simple module [KL09,

Section 3.2]. It is known that

(M ◦N)∗ ≃ q(β,γ)N∗ ◦M∗ (B.4)

for M ∈ R(β)-gmod and N ∈ R(γ)-gmod (see, for example, [Bru13]).


Let Lq be the set of the classes of self-dual simple objects in R-gmod, and Pq the set ofthe classes of projective covers of self-dual simple objects in R-Mod. Then Lq (resp. Pq)forms a Z[q±1]-basis of Aq(n

+)Z[q±1] (resp. U−Z[q±1](g)). We remark that

f (k)s ∈ Pq for s ∈ S, k ∈ Z≥0, (B.5)

[P1] · [P2] ∈∑

[P ]∈Pq

Z≥0[q±1][P ], and [L1] · [L2] ∈

∑

[L]∈Lq

Z≥0[q±1][L]. (B.6)

By Remark B.5 and Theorem B.6, Lq (resp. Pq) induces a basis L (resp. bases P and P+)

of O(U+) (resp. U(n−) and U(n+)). Moreover, L satisfies (G) in Theorem 6.7, and it is

dual to P+ as in (B.3).

Remark B.8. When R is symmetric (in particular, C(g) is symmetric), P (resp. L)corresponds to the canonical bases/global bases (resp. dual canonical bases/upper global

bases) in the sense of Lusztig and Kashiwara through the isomorphism in Theorem B.6.

Bases of representations. For λ ∈ X∗(H)+, let us denote by Vq(λ) the highest weight

Uq(g)-module of highest weight λ. Fix a highest weight vector vλ,q. Then there uniquely

exists a symmetric non-degenerate Q(q)-bilinear form ( , )λ : Vq(λ)× Vq(λ) → Q(q) such

that

(vλ,q, vλ,q)λ = 1 (X.v, w)λ = (v, XT.w)λ

for all v, w ∈ Vq(λ) and X ∈ Uq(g), where X 7→ XT is an anti-involution of Uq(g) givenby eTs = fs, f

Ts = es, and (qα

∨s )T = qα

∨s for s ∈ S. Set

VZ[q±1](λ) := U−Z[q±1](g).vλ,q,

VZ[q±1](λ)∨ := {v ∈ Vq(λ) | (v, w)λ ∈ Z[q±1] for all w ∈ VZ[q±1](λ)}.

Remark B.9. For λ ∈ X∗(H)+, VZ[q±1](λ) and VZ[q±1](λ)∨ are free Z[q±1]-modules, and

we have

C⊗Z[q±1] VZ[q±1](λ) ≃ C⊗Z[q±1] VZ[q±1](λ)∨ ≃ V (λ)

as U(g)-modules, where the actions of U(n±) on the first two modules are compatible with

the specialization in Remark B.5.

For λ ∈ X∗(H)+ and µ ∈ λ−X∗(H)≥0, let Rλ(µ) denote the corresponding cyclotomic

quiver Hecke algebra over k. Note that Rλ(µ) is defined as a quotient of R(λ−µ) and finite

dimensional over k [KK12, Corollary 4.4]. Let Rλ(µ)-Mod be the abelian category of Z-graded Rλ(µ)-modules. Let us denote by Rλ(µ)-gmod the full subcategory of Rλ(µ)-Mod

consisting of modules which are finite-dimensional over k, and by Rλ(µ)-proj the full

subcategory of Rλ(µ)-Mod consisting of finitely generated projective modules. Set

Rλ-gmod :=⊕

β∈X∗(H)≥0

Rλ(λ− β)-gmod, Rλ-proj :=⊕

β∈X∗(H)≥0

Rλ(λ− β)-proj.


For s ∈ S and β ∈ X∗(H)≥0, we can define exact functors,

Eλs : R

λ(λ− β)-Mod → Rλ(λ− β + αs)-Mod,

F λs : R

λ(λ− β)-Mod → Rλ(λ− β − αs)-Mod

by

Eλs (M) = e(αs, β − αs)M, F λ

s (M) = Rλ(λ− β − αs)e(αs, β)⊗Rλ(λ−β) M.

Here e(αs, β − αs) and e(αs, β) denote certain idempotents. Then, both of them induce

functors Rλ-gmod → Rλ-gmod and Rλ-proj → Rλ-proj. They categorify the Z[q±1]-forms

VZ[q±1](λ) and VZ[q±1](λ)∨.

Theorem B.10 ([KK12]). For λ ∈ X∗(H)+, there exist isomorphims of U±Z[q±1](g)-

modules

K(Rλ-gmod) ≃ VZ[q±1](λ)∨, and K(Rλ-proj) ≃ VZ[q±1](λ),

where the action of es and fs (s ∈ S) on the Grothendieck groups are given by Eλs

and F λs , respectively. Through these isomorphisms, the decompositions K(Rλ-gmod) =⊕

β∈X∗(H)≥0K(Rλ(λ − β)-gmod) and K(Rλ-proj) =

⊕β∈X∗(H)≥0

K(Rλ(λ − β)-proj) co-

incide with the weight space decompositions.

Since Rλ(λ−β) is a quotient algebra of R(β) for β ∈ X∗(H)≥0, We have a fully faithful

exact functorRλ-gmod → R-gmod, which induces a Z[q±1]-linear injective homomorphism

ι : K(Rλ-gmod) → K(R-gmod). Then there exists a subset L(λ)q of Lq which forms a

basis of ι(K(Rλ-gmod)). Hence we obtain a Z[q±1]-basis Lq(λ) of K(Rλ-gmod) satisfying

ι(Lq(λ)) = L(λ)q . Clearly, Lq(λ) consists of the classes of simple modules in Rλ-gmod.

Let Pq(λ) be the set of classes of projective covers of simple modules occurring in

Lq(λ). Then Pq(λ) forms a Z[q±1]-basis of K(Rλ-proj). Moreover Pq(λ) is dual to Lq(λ)with respect to ( , )λ (see [KK12, Section 6]). Then by Remark B.9, Lq(λ) and Pq(λ)induce bases L(λ) and P(λ) of V (λ) respectively, which are dual to each other with

respect to ( , )λ. The bases L(λ) and P(λ) satisfy the condition (Grep) in Theorem 6.7 by

Theorem B.10.

Next we show that the bases P+, L(λ) and P(λ) satisfy Assumption B.3. The authors

learned the following argument from Ryo Fujita. Let γ ∈ X∗(H)≥0 and P ∈ R(γ)-proj.

Then for β ∈ X∗(H)≥0, we can define a functor

F λP : R

λ(λ− β)-Mod → Rλ(λ− β − γ)-Mod

given by F λP (M) = Rλ(λ−β− γ)e(γ, β)⊗R(γ)⊗kRλ(λ−β) (P ⊗kM). Note that F λ

R(αs)= F λ

s .

Lemma B.11. The followings hold.

(1) The functor F λP is exact and sends finitely generated projective modules to finitely

generated projective modules for any P ∈ R(γ)-proj.

(2) For P ∈ R(γ)-proj and P ′ ∈ R(γ′)-proj, we have F λP ◦ F λ

P ′ ≃ F λP◦P ′.

Proof. Let us prove (1). Since P is a finitely generated projective module, P is a direct

summand of a free R(γ)-module of finite rank. Therefore, it suffices to prove that F λR(γ)


is exact and sends finitely generated projective modules to finitely generated projective

modules. By definition, R(γ) has idempotents {e(ν) | ν = (s1, · · · , sl),∑l

k=1 αsk = γ}such that

e(ν)e(ν ′) = δν,ν′e(ν) and∑

ν=(s1,··· ,sl),∑l

k=1 αsk=γe(ν) = 1.

Therefore we have only to show that F λR(γ)e(ν) is exact and sends finitely generated

projective modules to finitely generated projective modules for ν = (s1, · · · , sl) with∑lk=1 αsk = γ. Since F λ

R(γ)e(ν) ≃ F λs1 ◦ · · · ◦F

λslby definition, the desired statement follows

from [KK12, Corollary 4.6].

Let M ∈ Rλ(λ− β)-Mod. The statement (2) follows form the following isomorphisms.

(F λP ◦ F λ

P ′)(M)

=Rλ(λ− β − γ − γ′)e(γ, β + γ′)⊗R(γ)⊗kRλ(λ−β−γ′)[P ⊗k

(Rλ(λ− β − γ′)e(γ′, β)⊗R(γ′)⊗kRλ(λ−β) (P

′ ⊗k M))]

≃Rλ(λ− β − γ − γ′)e(γ, γ′, β)⊗R(γ)⊗kR(γ′)⊗kRλ(λ−β) (P ⊗k P′ ⊗k M)

≃Rλ(λ− β − γ − γ′)e(γ + γ′, β)⊗R(γ+γ′)⊗kRλ(λ−β)[(R(γ + γ′)e(γ, γ′)⊗R(γ)⊗kR(γ′) (P ⊗k P

′))⊗k M

]

=Rλ(λ− β − γ − γ′)e(γ + γ′, β)⊗R(γ+γ′)⊗kRλ(λ−β) [(P ◦ P ′)⊗k M ]

=F λP◦P ′(M).

�

By Theorem B.10, Lemma B.11 and and the equality F λR(αs)

= F λs , the actions of the

class [P ] ∈ K(R-proj) ≃ U−Z[q±1](g) on K(Rλ-gmod) ≃ VZ[q±1](λ)

∨ and K(Rλ-proj) ≃

VZ[q±1](λ) are induced from the functor F λP . Therefore, for [P ] ∈ Pq, [Lλ0 ] ∈ Lq(λ) and

[P λ0 ] ∈ Pq(λ), we have

[P ].[Lλ0 ] = [F λP (L

λ0)] ∈

∑[Lλ]∈Sq(λ)

Z≥0[q±1][Lλ],

[P ].[P λ0 ] = [F λ

P (Pλ0 )] ∈

∑[Pλ]∈Pq(λ)

Z≥0[q±1][P λ].

By the duality with respect to ( , )λ, we also have

[P ]T.[Lλ0 ] ∈∑

[Lλ]∈Lq(λ)Z≥0[q

±1][Lλ] and [P ]T.[P λ0 ] ∈

∑[Pλ]∈Pq(λ)

Z≥0[q±1][P λ].

Hence, by the specialization in Remarks B.5 and B.9, the bases P+, L(λ) and P(λ) satisfyAssumption B.3. Therefore, the argument in subsection B.1 implies that

L := {[L]q=1∆+w0,−λ

| [L]q=1 ∈ L, λ ∈ X∗(H)+} ⊂ O(U+∗ )

satisfies (P1) if L is a basis of O(U+∗ ).


The generating set of O(U+∗ ) satisfying (G), (P1), (P2). The argument in subsec-

tion B.1 and Theorem B.6 imply that L generates O(U+∗ ) as a vector space and consists

of homogeneous elements, that is, satisfies (G) in Theorem 6.7. Next, we show that

[L]q=1D−1λ,w0λ

∈ L satisfies

(xs)∗([L]q=1∆+w0,−λ

) ∈ Z≥0[t±11 , . . . , t±1

N ] (B.7)

for any reduced word s = (s1, . . . , sN) of w0 (recall the notation (4.1)). This statement

corresponds to (P2) in Theorem 6.7 (the linear independence of the set L will be explained

in the next subsection). By the identification (B.1), (B.5) and (B.6), we have

(xs)∗([L]q=1)

=∑

(k1,...,kN )∈ZN≥0

⟨[L]q=1,

ek1s1k1!

· · ·ekNsNkN !

⟩tk11 · · · tkNN

∈∑

(k1,...,kN )∈ZN≥0

(∑[P ]Tq=1∈P

+Z≥0

⟨[L]q=1, [P ]

T

q=1

⟩)tk11 · · · tkNN = Z≥0[t1, . . . , tN ].

Moreover, by [BZ97, Lemma 6.4],

(xs)∗(∆+w0,λ

) = tb11 · · · tbNN ,

where bi = 〈λ, rs1 · · · rsi−1α∨si〉 for i = 1, . . . , N . Thus (B.7) follows.

B.3. Categorification of quantum twist automorphisms. In this subsection, we

complete our proof of Theorem 6.7 by proving that the bases L, L(λ) and P(λ) provide anappropriate example of F, B(λ) and Bup(λ), respectively, in Theorem 6.7. We have already

checked the conditions (Grep), (G), (P1), and (P2) except for the linear independence of

L. In this subsection, we check the following statements.

(LI) L is a C-basis of O(U+∗ ).

(LII) ∆+w0,ξ

· F ∈ L for any ξ ∈ X∗(H) and F ∈ L.

(LIII) As a set, L is preserved by the twist automorphism η∗w0: O(U+

∗ ) → O(U+∗ ).

These statements are proved in [KKOP19] by categorifying L and η∗w0. In [KKOP19], they

constructed the localization of R-gmod, and proved that the localized category is rigid

as a monoidal category. Indeed, a dualization functor of this category categorifies the

twist automorphism. While this statement was mentioned in [KKOP19, Introduction],

its proof seems to be implicit in [KKOP19]. In this subsection, we give an explicit proof

of this fact. Since this fact is important itself, we work on a unipotent cell associated to

an arbitrary w ∈ W (g).

Remark B.12. We do not need to assume that C(g) is of finite type. All of the following

arguments are valid when g is an arbitrary symmetrizable Kac–Moody Lie algebra.

For w ∈ W (g), set

U+w := U+ ∩ B−w−1B−, U+(w) := U+ ∩ wU−w−1.

Then U+w (resp. U+(w)) is called the unipotent cell (resp. the unipotent subgroup) asso-

ciated to w. Note that U+w0

= U+∗ and U+(w0) = U+. Let Dw := {∆λ,wλ|U+(w) | λ ∈


X∗(H)+}, which is a multiplicatively closed subset of O(U+(w)) and we have a C-algebraisomorphism

O(U+(w))D−1w ≃ O(U+

w ).

See [GLS11, Proposition 8.5], [KO, Corollary 2.22] for explicit descriptions of the isomor-

phism.

Remark B.13. When w = w0, this isomorphism coincides with the equalityO(U+)D−1 =

O(U+∗ ) explained in Appendix B.1. However note that U+

w is not a subset of U+(w) in

general.

We recall the twist automorphism ηw on U+w .

Theorem B.14 ([BFZ96, Lemma 1.3] and [BZ97, Theorem 1.2]). There exists a regular

automorphism ηw : U+w

∼−→ U+

w given by

u 7→ [w−1uT]+.

We have quantum analogues of the notions above. Let Aq[U+w ] (resp. Aq(n

+(w))) be

the quantum unipotent cell (resp. the quantum unipotent subgroup) over Q(q) associated

to w. Write Aq[U+w ]Z[q±1] and Aq(n

+(w))Z[q±1] for appropriate Z[q±1]-forms of Aq[U+w ] and

Aq(n+(w)), respectively. See, for example, [KO, Definition 3.26, Theorem 3.29, Definition

4.3 and Corollary 4.11] for their precise definitions. The followings are the important

properties of these algebras:

• Aq[U+w ]Z[q±1] and Aq(n

+(w))Z[q±1] are free Z[q±1]-modules, and we have

C⊗Z[q±1] Aq[U+w ]Z[q±1] ≃ O(U+

w ), C⊗Z[q±1] Aq(n+(w))Z[q±1] ≃ O(U+(w)).

The natural algebra homomorphisms Aq[U+w ]Z[q±1] → C ⊗Z[q±1] Aq[U

+w ]Z[q±1] ≃

O(U+w ) and Aq(n

+(w))Z[q±1] → C⊗Z[q±1]Aq(n+(w))Z[q±1] ≃ O(U+(w)) are described

as F 7→ F |q=1.

• There is a quantum analogue Dqλ,wλ ∈ Aq(n

+(w))Z[q±1] of ∆λ,wλ|U+(w) for each

λ ∈ X∗(H)+ (in particular, Dqλ,wλ|q=1 = ∆λ,wλ|U+(w)). Then Dq

w := {qmDqλ,wλ |

m ∈ Z, λ ∈ X∗(H)+} is an Ore set of Aq(n+(w))Z[q±1], and we have a Q(q)-algebra

isomorphism

ιw : Aq(n+(w))Z[q±1](D

qw)

−1 ∼−→ Aq[U

+w ]Z[q±1],

which specializes to the isomorphism O(U+(w))D−1w ≃ O(U+

w ) above [KO, Theo-

rem 4.13].

• Aq(n+(w))Z[q±1] is a subalgebra of Aq(n

+)Z[q±1].

Remark B.15. The algebra Aq(n+)Z[q±1], which is the same as the one in [KKKO18,

KKOP18], is isomorphic to AZ[q±1][N−] in [KO] (where Q[q±1] can be replaced with Z[q±1]

in an obvious way) as a Z[q±1]-algebra by the correspondence satisfying

DKKKO(µ, ν) 7→ DKiOµ,ν

for λ ∈ X∗(H)+ and µ, ν ∈ Wλ. Here DKKKO(µ, ν) is the element D(µ, ν) in [KKKO18,

KKOP18] and DKiOµ,ν is the element Dµ,ν in [KO]. Indeed, Dq

λ,wλ in this paper is equal to


DKKKO(wλ, λ). Moreover, this isomorphism induces an identification of Aq(n+(w))Z[q±1]

in this paper (which is the same as the one in [KKKO18]) with AZ[q±1][N−(w)], and an

identification of Aq[U+w ]Z[q±1] in this paper with Aq[N

w− ] in [KO].

For µ, ν ∈ X∗(H), we write µ ≺ ν if there exists a sequence of positive roots β1, . . . , βt ∈Φ+ such that µ = rβt · · · rβ1ν and 〈β∨

k , rβk−1· · · rβ1ν〉 > 0 for all k = 1, . . . , t. Here, when

β = wαs for some w ∈ W and s ∈ S, we set rβ := wrsw−1 and β∨ := wα∨

s . For

λ ∈ X∗(H)+ and µ, ν ∈ Wλ, we write

Dµ,ν := ∆µ,ν |U+.

If ν 6� µ, then Dµ,ν = 0. When ν � µ, by [KKOP18, Proposition 4.1], we have a self-dual

simple object M(µ, ν) in R-gmod, which is unique up to isomorphism, such that

[M(ν, µ)]|q=1 = Dµ,ν .

Let Cw be a subcategory which is the smallest subcategory of R-gmod satisfying the

following conditions:

• Cw is stable under taking extensions, kernels, cokernels, convolutions ◦ and grading

shifts,

• Cw contains M(w≤ksk , w≤k−1sk) for all k = 1, . . . , l, where (s1, . . . , sl) is a

reduced word of w and w≤k = rs1 · · · rsk .

We remark that Cw0 = R-gmod.

Theorem B.16 ([KKKO18], [KKOP18, Theorem 2.20]). The isomorphism in Theo-

rem B.6 restricts to an isomorphism

Ψw : K(Cw)∼−→ Aq(n

+(w))Z[q±1].

In [KKOP19], Kashiwara, Kim, Oh, and Park constructed a localizations Cw of Cw

by the non-degenerate graded braiders {M(ws, s) | s ∈ S}. It has the following nice

properties [KKOP19, Section 5]:

• Cw is a graded monoidal abelian category, and each object has finite length.

• There exists a canonical exact monoidal functor Φ: Cw → Cw such that

(i) The objects Φ(M(ws, s)) (s ∈ S) are invertible with respect to the monoidal

structure in Cw,

(ii) (◦s∈SΦ(M(ws, s))◦ms) ◦Φ(L) is a simple object in Cw for all simple object

L of Cw and ms ∈ Z (s ∈ S). We remark that ◦s∈SΦ(M(ws, s))◦ms is

well-defined up to grading shifts.

Theorem B.17 ([KKOP19, Corollary 5.4]). The isomorphism Ψw induces a C-algebraisomorphism

Ψw : K(Cw)∼−→ Aq(n

+(w))Z[q±1](Dqw)

−1.

Since we have ιw : Aq(n+(w))Z[q±1](D

qw)

−1 ∼−→ Aq[U

+w ]Z[q±1], we can conclude that K(Cw)

categorifies Aq[U+w ]Z[q±1].

The contravariant functor M 7→M∗ on Cw can be extended to a contravariant functor

on Cw, and we can again define the notion of self-dual objects in Cw. By [KKOP19,


Corollary 5.4], each simple object in Cw is isomorphic to a self-dual simple object in Cw.

Therefore, the set of the classes of self-dual simple objects in Cw forms a Z[q±1]-basis of

K(Cw) ≃ Aq[U+w ]Z[q±1]. Therefore, it induces a basis of O(U+

w ). When w = w0, this basis

is equal to the set L and it satisfies (M) in Theorem 6.7 by the property (ii) of Cw. It

proves (LI) and (LII).

Next we move on to a categorification of the quantum twist automorphism ηw,q.

Theorem B.18 ([KO, Theorem 6.1, Corollary 6.2]). There exists a Z[q±1]-algebra auto-

morphism ηw,q : Aq[U+w ]Z[q±1] → Aq[U

+w ]Z[q±1] such that its specialization

ηw,q|q=1 : O(U+w ) → O(U+

w )

coincides with the algbra automorphism η∗w induced from the twist automorphism ηw.

By the formula in [KO, Theorem 6.1], we have

ηw,q([Φ(M(w′λ, λ))]) = q−(λ,w′λ−λ)[Φ(M(wλ, λ))]−1[Φ(M(wλ,w′λ))] (B.8)

for λ ∈ X∗(H)+ and w,w′ ∈ W with wλ � w′λ � λ.

Remark B.19. The Z[q±1]-algebra K(Cw) (therefore, Aq[U+w ]Z[q±1]) is isomorphic to

AZ[q±1][Nw− ] in [KO] (where Q[q±1] can be replaced with Z[q±1] in an obvious way) as

a Z[q±1]-algebra by the correspondence

[Φ(M(µ, ν))] 7→ [DKiOµ,ν ]

for λ ∈ X∗(H)+ and µ, ν ∈ Wλ (cf. Remark B.15).

A categorification of ηw,q is given by the left dualizing functor D in Cw:

Theorem B.20 ([KKOP19, Corollary 5.11]). The monoidal category Cw is left rigid, that

is, every object M has a left dual D(M) in Cw.

Remark B.21. When w = w0, Cw is left and right rigid [KKOP19, Theorem 5.13].

Since the tensor product ◦ in Cw is exact [KKOP19, Proposition 2.13], the functor D

is exact. Moreover we have

• M1 ◦M2 ≃ D(M2) ◦D(M1) for M1,M2 ∈ Cw,

• D(qM) = q−1D(M) for M ∈ Cw.

Hence D induces an algebra anti-homomomorphism

[D] : K(Cw) → K(Cw), [M ] 7→ [D(M)].

By (B.4),

σ : K(R-gmod) → K(R-gmod), [M ] 7→ q−(β,β)/2[M∗] for M ∈ R(β)-gmod (B.9)

is an algebra anti-homomomorphism such that σ(qx) = q−1σ(x) for x ∈ K(R-gmod). Note

that, if we consider the specialization at q = 1, the map σ|q=1 : O(U+) ≃ K(R-gmod)|q=1 →K(R-gmod)|q=1 ≃ O(U+) become equal to the identity map. The anti-homomorphism σ

in (B.9) induces an anti-homomorphism K(Cw)∼−→ K(Cw), which is again denoted by σ.

The following is the main statement of this subsection:


Theorem B.22. The Z[q±1]-algebra homomorphism [D] ◦σ : K(Cw) → K(Cw) coincides

with ηw,q : Aq[U+w ]Z[q±1] → Aq[U

+w ]Z[q±1] through the isomorphism ιw ◦ Ψw.

Proof. We always identify K(Cw) with Aq[U+w ]Z[q±1] via ιw ◦ Ψw. By extending the base

ring, we may work on Aq[U+w ]. It is well-known that Aq[U

+w ] can be realized as a subalgebra

of a quantum Laurent polynomial ring over Q(q) (see, for example, [Ber96]). Therefore, we

can consider the skew-field Fq of fractions of Aq[U+w ], and Aq[U

+w ] is canonically embedded

to Fq. Let s = (s1, . . . , sl) be a reduced word of w, and w≤k := rs1 · · · rsk for k = 1, . . . , l.

Then it is well-known that the set

Ss := {qm[Φ(M(w≤1s1, s1))]m1 · · · [Φ(M(w≤lsl, sl))]

ml | m ∈ Z, m1, . . . , ml ∈ Z≥0}

is an Ore set of Aq[U+w ], and Aq[U

+w ]S

−1s

is canonically isomorphic to the quantum Laurent

polynomial ring Ts overQ(q) in the variables [Φ(M(w≤1s1, s1))], . . . , [Φ(M(w≤lsl, sl))].

See, for example, [GLS13, GY17] or [Oya19, Proposition A.4] and the references therein.

Then ηw,q and [D]◦σ are considered as Q(q)-algebra homomorphisms from Aq[U+w ] to Fq,

and if they satisfy

([D] ◦ σ)([Φ(M(w≤ksk , sk))]) = ηw,q([Φ(M(w≤ksk , sk))]) (B.10)

for k = 1, . . . , l, then we can conclude that [D] ◦ σ = ηw,q. Indeed, we can extend

[D] ◦ σ and ηw,q to Q(q)-algebra homomorphisms from Ts to Fq, and (B.10) tells us their

coincidence on the generators of Ts. Therefore we are going to show (B.10).

By the formulas (B.8), we have

ηw,q([Φ(M(w≤ksk , sk))])

= q−(sk,w≤ksk

−sk)[Φ(M(wsk , sk))]

−1[Φ(M(wsk , w≤ksk))]

for k = 1, . . . , l. On the other hand, by [KKOP18, Proposition 4.6], we have an epimor-

phism

M(wsk , w≤ksk) ◦M(w≤ksk , sk) ։ M(wsk , sk)

in Cw, with a notice that M(wsk , w≤ksk), M(w≤ksk , sk), M(wsk , sk) are objects

in Cw again by [KKOP18, Proposition 4.6]. Then by [KKOP19, Proposition 4.11,Lemma

5.6], we have

D(M(w≤ksk , sk)) =M(wsk , sk)◦−1 ◦M(wsk , w≤ksk).

Since M(w≤ksk , sk) is a self-dual R(sk − w≤ksk)-module, we have

([D] ◦ σ)([Φ(M(w≤ksk , sk))])

= [D](q−(sk−w≤ksk

,sk−w≤ksk

)/2[Φ(M(w≤ksk , sk))])

= q(sk−w≤ksk

,sk−w≤ksk

)/2[M(wsk , sk)]−1[M(wsk , w≤ksk)]

= q−(sk,w≤ksk

−sk)[M(wsk , sk)]

−1[M(wsk , w≤ksk)].

Thus we obtain (B.10). �

By Theorem B.18 and Theorem B.22, we obtain the following:


Corollary B.23. The C-algebra homomorphism [D]|q=1 : C⊗Z[q±1] (K(Cw)) → C⊗Z[q±1]

(K(Cw)) coincides with η∗w : O(U+

w ) → O(U+w ) through the isomorphism ιw ◦ Ψw|q=1.

SinceD sends a simple object to a simple object in Cw (cf. [KKOP19, Theorem 5.7]), the

set of the isomorphism classes of simple objects in Cw is preserved by [D]. In particular,

when w = w0, the specialization [D]|q=1 = η∗w0preserves L, which proves (LIII).

Appendix C. Cluster varieties, weighted quivers and their amalgamation

Here we recall weighted quivers and their mutations, and the amalgamation procedure

which produces a new weighted quiver from a given one by “gluing” some of its vertices.

This procedure naturally fits into both the gluing map (Theorem 3.13) via Goncharov–

Shen coordinates, and the group multiplication of G via coweight parametrizations (Def-

inition 5.1). We also recall the construction of weighted quivers from reduced words,

following [FG06] and [GS19].

C.1. Weighted quivers and the cluster Poisson varieties. We use the conventions

for weighted quivers in [IIO19]. Recall that a weighted quiver Q = (I, I0, σ, d) is defined

by the following data:

• I0 ⊂ I are finite sets.

• σ = (σij)i,j∈I is a skew-symmetric Z/2-valued matrix such that σij ∈ Z unless

(i, j) ∈ I0 × I0.

• d = (di)i∈I ∈ ZI>0 is a tuple of positive integers.

Diagrammatically, I is the set of vertices of the quiver, d is the tuple of weights assigned

to vertices, and the data of arrows are encoded in the matrix σ as

σij := #{arrows from i to j} −#{arrows from j to i}.

Here we have “half ” arrows when σij ∈ Z/2 (shown by dashed arrows in figures). The

quiver has no loops nor 2-cycles by definition. The subset I0 is called the frozen set, and

mutations will be allowed only at the vertices in the complement Iuf := I \ I0. The ciral

dual of Q is defined by Qop := (I, I0,−σ, d).We define the exchange matrix ε = (εij)i,j∈I of Q to be εij := diσij gcd(di, dj)

−1. Since

we can reconstruct the skew-symmetric matrix σ from the pair (ε, d), we sometimes write

Q = (I, I0, ε, d). The following is a reformulation of the matrix mutation (see e.g., [FG09,

(12)]) in terms of the weighted quiver:

Definition C.1. For k ∈ Iuf , let Q′ = (I, I0, σ

′, d) be the weighted quiver given by

σ′ij =

−σij i = k or j = k,

σij +|σik|σkj + σik|σkj|

2αkij otherwise,

where αkij = dk gcd(di, dj) gcd(dk, di)−1 gcd(dk, dj)

−1. The operation µk : Q 7→ Q′ is called

the mutation at the vertex k. Then the exchange matrix ε′ of Q′ is given by the matrix

mutation.


Let F be a field isomorphic to the field of rational functions on |I| independent variableswith coefficients in C. An (X -)seed is a pair (Q,X), where X = (Xi)i∈I is a tuple of

algebraically independent elements of F and Q is a weighted quiver. For k ∈ I \ I0,let (Q′,X ′) be another seed where Q′ = µk(Q) is obtained from Q by the mutation

at k, and X ′ = (X ′i)i∈I is given by the cluster Poisson transformation (or the cluster

X -transformation):

X ′i =

{X−1k i = k,

Xi(1 +X−sgn(εik)k )−εik i 6= k.

(C.1)

The operation µk : (Q,X) 7→ (Q′,X ′) is called the seed mutation at k. It is not hard to

see that seed mutations are involutive: µkµk = id. We say that two seeds are mutation-

equivalent if they are connected by a sequence of seed mutations and seed permutations

(bijections of I preserving I0 setwise).

Let TIuf be the regular |Iuf |-valent tree, each of whose edge is labeled by an index in Iufso that two edges sharing a vertex have different labels. An assignment S = (S(t))t∈TIuf

of a seed S(t) = (Q(t),X(t)) to each vertex t of TIuf is called a seed pattern if for two

vertices t, t′ sharing an edge labeled by k ∈ Iuf , the corresponding seeds are related as

S(t′) = µkS(t).

The cluster Poisson variety XS =⋃t∈TIuf

X(t) is defined by patching the coordinate tori

X(t) = (C∗)I corresponding to seeds S(t) by the rational transformations

µ∗k : O(X(t′)) = C[X(t′)

i | i ∈ I] → O(X(t)) = C[X(t)i | i ∈ I]

given by the formula (C.1) whenever t and t′ shares an edge labeled by k ∈ Iuf . The cluster

Poisson variety has a natural Poisson structure given by {X(t)i , X

(t)j } := εijX

(t)i X

(t)j .

The ring O(XS) =⋂t∈TIuf

O(X(t)) of regular functions is called the cluster Poisson

algebra, whose elements are called universally Laurent polynomials. An element of the

sub-semifield L+(XS) :=⋂t∈TIuf

Z≥0[X(t)i | i ∈ I] ⊂ O(XS) is called a universally positive

Laurent polynomial.

Definition C.2. A cluster Poisson atlas on a variety (or scheme, stack) V over C is a

collection (Sα)α∈A of seeds (here A is an index set) in the field K(V ) of rational functions

on V such that

• each seed Sα = (Qα,Xα) gives rise to a birational isomorphism Xα : V 99K CI

which admits an open embedding ψα : CI → V as a birational inverse;

• the seeds Sα for α ∈ A are mutation-equivalent to each other.

From the second condition, the collection (Sα)α∈A can be extended to a unique seed

pattern S = (S(t))t∈TIuf. In particular we get a birational isomorphism V ∼= XS. We call

the seed pattern S a cluster Poisson structure on V , as it is a maximal cluster Poisson

atlas. Note that the conditions do not imply an existence of an open embedding XS → V

when (Sα)α∈A ( S.

A rational function on V can be regarded as a rational function on XS, and we can ask

whether it is a universally (positive) Laurent polynomial.


C.2. Amalgamations. We recall the amalgamation procedure of weighted quivers, fol-

lowing [FG06].

Definition C.3. Let Q = (I, I0, σ, d), Q′ = (I ′, I ′0, σ

′, d′) be two weighted quivers. Fix

two subsets F ⊂ I0, F′ ⊂ I ′0 and a bijection φ : F → F ′ such that d′(φ(i)) = d(i) for all

i ∈ I, which we call the gluing data. Then the amalgamation of Q and Q′ with respect to

the gluing data (F, F ′, φ) produces the weighted quiver Q ∗φ Q′ = (J, J0, τ, c) defined as

follows

• J := I ∪φ I′, J0 ⊂ I0 ∪φ I

′0.

• c(i) :=

{d(i) if i ∈ I,

d′(i) if i ∈ I ′ \ F ′.

• The entry τij is given by:

j ∈ I \ F j ∈ I ′ \ F ′ j ∈ F

i ∈ I \ F σij 0 σiji ∈ I ′ \ F ′ 0 σ′

ij σ′ij

i ∈ F σij σ′ij σij + σ′

ij

Here we can choose any subset J0 of I0∪φ I′0 such that σij is integral unless (i, j) ∈ J0×J0.

In this paper, we consider the minimal J0 given by

J0 = I0 ∪φ I′0 \ J1, J1 := {i ∈ I0 ∪φ I

′0 | σij ∈ Z for all j ∈ J}.

The amalgamation procedure can be upgraded to that for two seeds. Let (Q,X) and

(Q′,X ′) be two seeds, (F, F ′, φ) a gluing data as above. Then we define a new seed

(Q ∗φ Q′,Y ), where the weighted quiver Q ∪φ Q

′ is given as above and the variables

Y = (Yi)i∈J is defined by

Yi :=

Xi if i ∈ I \ F

X ′i if i ∈ I ′ \ F ′

Xi ·X′φ(i) if i ∈ F.

Then it is not hard to check that the amalgamation of seeds commutes with the mutation

at any vertex k ∈ (I \ I0) ⊔ (I ′ \ I ′0). Thus for two seed patterns S and S′ and a gluing

data as above, we have a dominant morphism

αφ : XS × XS′ → XS∗φS′ .

Here the seed pattern S ∗φ S′ is obtained by the amalgamation of the seeds S(t) and S′(t)

for t ∈ TIuf .

C.3. Weighted quivers from reduced words. Let us fix a finite dimensional complex

semisimple Lie algebra g. Let C(g) = (Cst)s,t∈S be the associated Cartan matrix. For

s ∈ S, we define a weighted quiver J+(s) = (J(s), J0(s), σ(s), d(s)) as follows.

• J(s) = J0(s) := (S \ {s}) ∪ {sL, sR}, where sL, sR are new elements.


• The skew-symmetric matrix σ(s) = (σtu)t,u∈J(s) is given by

σsR,sL = 1,

σsL,u = σu,sR =

{1/2 if u 6= s and Csu 6= 0,

0 if u 6= s and Csu = 0.

Note that other entries are determined by the skew-symmetricity.

• d(s) is given by d(s)sL,R := ds, d(s)t := dt for t 6= s.

Let J−(s) := J+(s)op. We call J±(s) the elementary quivers associated with g.

For each elementary quiver J ǫ(s) with s ∈ S and ǫ ∈ {+,−}, we define a function

δ(s) : J(s) → S on the set of vertices by δ(s)sL,R := s and δ(s)t := t for t ∈ S \ {sL, sR}.We call δ the Dynkin labeling of vertices of J ǫ(s).

Example C.4. Here are some examples of the elementary quivers.

(1) Type A3: S = {1, 2, 3} and the Cartan matrix is given by

C(A3) =

2 −1 0

−1 2 −1

0 −1 2

.

The elementary quivers J+(1), J+(2) and J+(3) are given as follows:

1R1L

2

3

J+(1)

2R2L

1

3

J+(2)

3R3L

2

1

J+(3)

(2) Type C3: S = {1, 2, 3} and the Cartan matrix is given by5

C(C3) =

2 −1 0

−1 2 −2

0 −1 2

.

The elementary quivers J+(1), J+(2), J+(3) are given as follows.

1R1L

2

23

J+(1)

2R2L

1

23

J+(2)

23R

23L

2

1

J+(3)

5Here we changed the convention from [IIO19]: for type Cn, the long root is chosen to be αn. Similarly

for type Bn, the short root is chosen to be αn.


Note that the vertices with the same Dynkin label are drawn on the same level in the

pictures.

For a reduced word s = (s1 . . . sl) of u ∈ W (g) and ǫ ∈ {+,−}, we construct a weighted

quiver J ǫ(s) = J ǫ(s1, . . . , sl) by amalgamating the elementary quivers J ǫ(s1), · · · ,Jǫ(sl)

in the following way: for k = 1, . . . , l− 1, amalgamate J ǫ(sk) and J ǫ(sk+1) by setting the

gluing data in Definition C.3 as

F := J(sk) \ {sLk }, F ′ := J(sk+1) \ {s

Rk+1},

φ : F → F ′, sRk 7→ sk, sk+1 7→ sLk+1, t 7→ t for t 6= sk, sk+1.

Note that the Dynkin labelings δ(sk) are preserved under this amalgamation. Hence, these

functions combine to give an S-valued function on the set of vertices of J ǫ(s), which we

call the Dynkin labeling again. In the weighted quiver J+(s) = J+(s1)∗· · ·∗J+(sl), let v

si

be the (i+ 1)-st vertex with Dynkin label s from the left, for s ∈ S and i = 0, . . . , ns(s).

Here ns(s) is the number of s which appear in the word s. We also use the labelling

vsi =: vk(s,i) for s ∈ S and i = 1, . . . , ns(s), where k(s, i) ∈ {1, . . . , l} denotes the i-th

number k such that sk = s in the word s. Similarly, the vertices of J−(s∗op) are labeled

as vsi = vk(s,i) where the index i runs from the left to the right.

Example C.5. Here are some examples of the weighted quiver J+(s) (left) and the

corresponding quiver J−(s∗op) (right).

(1) Type A3, s = (1, 2, 3, 1, 2, 1) and s∗op = (3, 2, 3, 1, 2, 3).

v10 v11 v12 v13

v20 v21 v22

v30 v31 v30 v31 v32 v33

v20 v21 v22

v10 v11

Here the vertices v11, v12, v

21, v

31 , v

21 and v32 are mutable.

(2) Type C3, s = (1, 2, 3, 1, 2, 3, 1, 2, 3) and s∗op = (3, 2, 1, 3, 2, 1, 3, 2, 1).

2

v302

v312

v322

v33

v10

v20

v11

v21

v12

v22

v13

v23

2

v302

v312

v322

v33

v10

v20

v11

v21

v12

v22

v13

v23

Here the vertices vsi and vsi for s = 1, 2, 3, i = 1, 2 are mutable.

Note that the weighted quivers J+(s) and J−(s∗op) are isomorphic.


Next we recall the quiver J+(s), which is called the decorated quiver in [IIO19] in special

cases. We follow a more general construction given in [GS19].

Let s = (s1, . . . , sl) be a reduced word of u ∈ W (g). For k = 1, . . . , l, let αs

k (resp. βs

k)

be the root (resp. coroot) defined by

αs

k := rsl . . . rsk+1αsk , βs

k := rsl . . . rsk+1α∨sk.

Then for each s ∈ S, there exists a unique k = k(s) such that βs

k = α∨s . Note that from

[Kac, (3.10.3)], we have αs

k(s) = αs at the same time. Let H(s) = (H(s), H0(s), ε(s), d(s))

be the weighted quiver defined as follows.

• H(s) = H0(s) := S.

• The exchange matrix ε(s) = (εst)s,t∈S is given by

εst :=sgn(k(t)− k(s))

2Cst.

• d(s) = (d(s)s)s∈S is given by d(s)s := ds.

The vertex of H(s) corresponding to s ∈ S is denoted by ys. Then we define J+(s) to

be the weighted quiver obtained from the disjoint union of the quivers J+(s) and H(s)

by adding the arrows vk(s)−1 → ys and ys → vk(s) for each s ∈ S. As a convention, these

additional arrows and the quiver H(s) are shown in blue in figures.

Example C.6. Here are some examples of the quiver J+(s).

(1) Type A3, s = (1, 2, 3, 1, 2, 1). The sequence (αs

k)k=1,...,6 is computed as

αs

6 = α1, αs

5 = α1 + α2, αs

4 = α2,

αs

3 = α1 + α2 + α3, αs

2 = α2 + α3, αs

1 = α3.

Thus we get k(1) = 6, k(2) = 4, k(3) = 1, and the quiver J+(1, 2, 3, 1, 2, 1) is

given by

v10 v11 v12 v13

v20 v21 v22

v30 v31

y1y2y3

(2) Type C3, s = (1, 2, 3, 1, 2, 3, 1, 2, 3). The sequence (αs

k)k=1,...,9 is computed as

αs

9 = α3, αs

8 = α2 + α3, αs

7 = α1 + α2 + α3,

αs

6 = 2α2 + α3, αs

5 = α1 + 2α2 + α3, αs

4 = α2,

αs

3 = 2(α1 + α2) + α3, αs

2 = α1 + α2, αs

1 = α1.


Thus we get k(1) = 1, k(2) = 4,= k(3) = 9, and the quiver J+((1, 2, 3)3) is given

by

2v30

2v31

2v32

2v33

v10

v20

v11

v21

v12

v22

v13

v23

y1 y2

2y3

Appendix D. A short review on quotient stacks

We shortly recall some basic facts on the quotient stacks, to the minimal extend we

need to recognize the moduli spaces PG,Σ correctly. We refer the reader to [Go, Sta] for

a self-contained presentation of the general theory of (algebraic) stacks. The lecture note

[Hei] will be also useful to get an intuition for stacks for the readers more familier with

the differential geometry or the algebraic topology than the algebraic geometry.

Let X be an algebraic variety (or more generally, a scheme), and G an affine algebraic

group acting on X algebraically. In order to study the quotient of X by G from the

viewpoint of algebraic geometry, a good way is to define it as a quotient stack [X/G].

Morally, the geometry of [X/G] is the G-equivariant geometry of X . Several lemmas

below will justify this slogan. When the action of G is free, one can think of [X/G] as

an algebraic variety (Lemma D.4); in general, the quotient stack [X/G] also contains the

information on the stabilizers.

Let X be a scheme over C and G an affine algebraic group acting on X . Then the

quotient stack X = [X/G] is a category fibered on groupoids ([Go, Definition 2.12])

where the objects over a scheme B are pairs (E, f) of a principal G-bundle E → B and

a G-equivariant morphism f : E → X ; morphisms over B are Cartesian diagrams of

G-bundles which respect the equivariant morphisms to X .

Note that an object (E, f) over B = SpecC can be viewed as a G-orbit in X . Thus the

set X/G of orbits is recovered as the set of images of f : E → X , that is, the isomorphism

classes of the objects of X (SpecC). Yoneda’s lemma for stacks implies that a morphism

u : B → X from a scheme B corresponds to an object of X (B).

It is known that X is an Artin stack ([Go, Definition 2.22]): an atlas is defined by the

morphism X → X given by the pull-back of the trivial bundle X ×G (see [Go, Example

2.25]).

Lemma D.1 (e.g. [Hei, Section 4]). The category of quasi-coherent sheaves on the Artin

stack X = [X/G] is equivalent to the category of G-equivariant quasi-coherent sheaves


on X. In particular, the ring O(X ) of regular functions on X is isomorphic to the ring

O(X)G of G-invariant regular functions on X.

A scheme V can be regarded as an Artin stack whose objects over B are morphisms

B → V ; the only morphism over B is the identity. A stack is said to be representable

if it is isomorphic to a stack arising from a scheme. For two Artin stacks X and Y , a

morphism φ : X → Y of stacks is said to be representable if for any morphism B → Y ,

the fiber product B ×Y X is representable. Informally speaking, a morphism B → Y can

be viewed as a “local chart” on Y , and the induced morphism B ×Y X → B is the “local

expression” of φ.

The following lemma tells us that we can obtain morphisms between quotient stacks

from equivariant morphisms of varieties.

Lemma D.2. Let X = [X/H ] and Y = [Y/G] be two quotient stacks. Let φ : X → Y

be a morphism equivariant with respect to an embedding τ : H → G. Then it induces a

representable morphism φ∗ : X → Y of Artin stacks. More precisely, for any morphism

u : B → Y from a scheme B which corresponds to an object (E, f) ∈ Y(B), the diagram

E ×G (G/H) X

B Y

uH

φ φ∗

u

is Cartesian. Here uH is a morphism corresponding to an H-bundle E → E × (G/H) →E ×G (G/H).

We give a proof for our convenience.

Proof. For an object (E, f) over B in X , the pair φ∗(E, f) := (E, φ ◦ f) is an object

over B in Y . This correspondence is clearly compatible with pull-backs, and defines a

morphism φ∗ : X → Y . It is not hard to see that the fiber product B ×Y X is isomorphic

to E ×G (G/H), with a notice that an H-sub-bundle of a G-bundle P → B is in one-to-

one correspondence with a section of the associated bundle P ×G (G/H). Thus φ∗ it is

representable. �

We call φ a presentation of the morphism φ∗.

Remark D.3. When τ is not an embedding, the morphism φ∗ is not representable in

general. For instance, the morphism (idpt)∗ : [pt/G] → [pt/e] = pt is not representable for

a non-trivial group G, where pt denotes the point scheme and e is the trivial group.

A property of morphisms of schemes that are local in nature on the target and stable

under base-change can be defined for representable morphisms of stacks. For instance, φ∗

is said to be an open embedding if φ : X = Y ×[Y/G] [X/G] → Y is an open embedding of

varieties.

Lemma D.4. Suppose that the G-action on X is free. Then the quotient stack X = [X/G]

is representable by the geometric quotient X/G.


Indeed, from the assumption every points of X are G-stable, and the geometric quotient

X/G exists (see, for instance, [Bri, Proposition 1.26]).

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