Wilson lines and their Laurent positivity

For a marked surface Σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma $$\end{document} and a semisimple algebraic group G of adjoint type, we study the Wilson line morphism g[c]:PG,Σ→G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{[c]}:{\mathcal {P} }_{G,\Sigma } \rightarrow G$$\end{document} associated with the homotopy class of an arc c connecting boundary intervals of Σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma $$\end{document}, which is the comparison element of pinnings via parallel-transport. The matrix coefficients of the Wilson lines give a generating set of the function algebra O(PG,Σ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}({\mathcal {P} }_{G,\Sigma })$$\end{document} when Σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma $$\end{document} has no punctures. The Wilson lines have the multiplicative nature with respect to the gluing morphisms introduced by Goncharov–Shen [18], hence can be decomposed into triangular pieces with respect to a given ideal triangulation of Σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma $$\end{document}. We show that the matrix coefficients cf,vV(g[c])\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_{f,v}^V(g_{[c]})$$\end{document} give Laurent polynomials with positive integral coefficients in the Goncharov–Shen coordinate system associated with any decorated triangulation of Σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma $$\end{document}, for suitable f and v.


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. Wilson 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 A G,Σ and X G,Σ of the moduli space of local systems, each of which admits a natural cluster structure. Here G is a simply-connected semisimple algebraic group, and G = G/Z( G) is its adjoint group. The cluster structures of these moduli spaces are distinguished collections of open embeddings of algebraic tori accompanied with weighted quivers, related by two kinds of cluster transformations. The collection of weighted quivers is shared by A G,Σ and X G,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 A n , by Le [Le16] for type B n , C n , D n (and further investigated Date: September 20, 2022. in [IIO19]), and by Goncharov-Shen [GS19] for all semisimple gauge groups, generalizing all the works mentioned above and giving a uniform construction.
In [GS19], Goncharov-Shen introduced a new moduli space P G,Σ closely related to the moduli space X G,Σ , which possesses the frozen coordinates that are missed in the latter. When ∂Σ = ∅, we have P G,Σ = X G,Σ , and otherwise the former includes additional data called the pinnings assigned to boundary intervals. The supplement of frozen coordinates turns out to be crucial in the quantum geometry of moduli spaces: for example, it is manifestly needed in the relation with the quantized enveloping algebra in their work. The data of pinnings also allow one to glue the G-local systems along boundary intervals in an unambiguous way, which leads to a gluing morphism q E 1 ,E 2 : P G,Σ → P G,Σ ′ .
Here Σ ′ is obtained from Σ by gluing two boundary intervals E 1 and E 2 of Σ.
1.1. The Wilson lines. Using the data of pinnings, we introduce a new class of G-valued morphisms g [c] : P G,Σ → G, which we call the Wilson line along the homotopy classes [c] of a curve connecting two boundary intervals called an arc class. 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. Our aim in this paper is a detailed study of these morphisms. Here are main features: Multiplicativity: We will see that the Wilson lines have the multiplicative nature for the gluing morphisms. If we have two arc classes [c 1 ] : E 1 → E 2 and [c 2 ] : E ′ 2 → E 3 on Σ, then by gluing the boundary intervals E 2 and E ′ 2 we obtain another marked surface Σ ′ equipped with an arc class [c] := [c 1 ] * [c 2 ], 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 to regular functions on P G,Σ . Moreover, we will see in Section 3.4 that the function algebra O(P G,Σ ) is generated by these matrix coefficients when Σ has no punctures. Therefore Wilson lines provide enough functions to study the function algebra O(P G,Σ ). Universal Laurent property: Shen [She20] proved that the algebra O(P G,Σ ) of regular functions on this moduli stack is isomorphic to the cluster Poisson algebra O cl (P G,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 O cl (P G,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).
We remark here that the essential notion of Wilson lines has been appeared in many related works including [FG06, GMN14, GS15, SS17, GS19, CS20] (mainly as a tool for the computation of Wilson loops), while our work would be the first on its systematic study in the setting of the moduli space P G,Σ . Via their coordinate expressions as we discuss below, the Wilson lines (loops) have been recognized as related to the spectral networks [GMN14] and certain integrable systems [SS17].
1.2. Laurent positivity of Wilson lines. Our goal in this paper is a detailed study of the Laurent expressions of the matrix coefficients of Wilson lines in cluster charts on P G,Σ . Moreover, it will turn out that a special class of matrix coefficients give rise to Laurent polynomials with non-negative coefficients.
Fock-Goncharov's snake formula. Coordinate expressions of Wilson loops (or the trace functions) have been studied by several authors. In the A 1 case, a combinatorial 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) = x 1/2 0 0 x −1/2 ∈ P GL 2 , which are multiplied according to the turning pattern after substituting the cross ratio coordinates into x.
In the A n case, Fock-Goncharov [FG06] gave a similar formula called the snake formula, which expresses the Wilson loops in the cluster coordinates associated with ideal triangulations (called the special coordinate systems). In particular, the trace functions are positive Laurent polynomials (with fractional powers) in any special coordinate systems.
Generalizations of the snake formula. Generalizing the special coordinate systems, Goncharov-Shen [GS19] gave a uniform construction of coordinate systems on P G,Σ associated with decorated triangulations 2 . Let us call them theGoncharov-Shen coordinate systems (GS coordinate systems for short). The special coordinate systems in the type A n case are special instances of the GS coordinate systems, where the choice of reduced words are the "standard" one (see (4.8)). Unlike the special coordinate systems, however, a general GS coordinate system no more have the cyclic symmetry on each triangle. The data of "directions" of coordinates is encoded in the data of decorated triangulations, as well as the choice of reduced words on each triangle.
Locally, a natural generalization of the snake formula is given by the evaluation map [FG06] parametrizing the double Bruhat cells of G. We will see that the "basic" Wilson lines b L , b R on the configuration space Conf 3 P G ,which models the moduli space on the triangle, can be expressed using the evaluation maps. Since the multiplicativity allows one to decompose Wilson lines into those on triangles, one can write the Wilson lines as a product of evaluation maps when the direction of GS coordinates agree with the direction that the arc class traverses on each triangle. This is basically the same strategy as Fock-Goncharov [FG06], but manipulations 2 Actually, they described more coordinate systems geometrically: those along flips of ideal triangulations and along rotations of dots. We do not investigate these additional coordinate systems in this paper.
in the recently-innovated moduli space P G,Σ makes the computation much clearer, thanks to the nice properties of the gluing morphism [GS19].
Transformations by cyclic shifts. In general, we need to transform the evalutation maps in the expression of Wilson lines by the cyclic shift automorphism on Conf 3 P G , in order to match the directions of a given GS coordinate system with the direction of the arc class on each triangle. The cyclic shifts are known to be written as a composite of cluster transformations, which is computable in nature but rather a complicated rational transformation. While the matrix coefficients of g [c] are at least guaranteed to be Laurent polynomials as we discussed above, it is therefore non-trivial whether their coefficients are non-negative integers.
Let us further clarify the problem which we will deal with. A function f ∈ O(P G,Σ ) is said to be GS-universally positive Laurent 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 X P GL n+1 ,Σ in [FG06]. Moreover, a morphism F : P G,Σ → G is said to be GS-universally positive Laurent if for any finite-dimensional representation V of G, there exists a basis B of V such that is GS-universally positive Laurent for all v ∈ B and f ∈ F, where F is the basis of V * dual to B. Our result is the following: Theorem 1 (Theorem 5.2). 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] : E in → E out , the Wilson line g [c] : P G,Σ → G is a GS-universally positive Laurent morphism.
Here we briefly comment on the proof of Theorem 1. By the construction of the GS coordinate system on P G,Σ associated with a decorated triangulation, the Laurent positivity of a regular function on P G,Σ can be deduced from the Laurent positivity of its pull-back via the gluing morphism q ∆ : T ∈t(∆) P G,T → P G,Σ associated with the underlying ideal triangulation ∆. In other words, we can investigate the Laurent positivity of a regular function on P G,Σ by a local argument on triangles. Indeed, a key to the proof of Theorem 1 is a construction of a basis F pos,T of O(P G,T ) consisting of those GS-universally positive Laurent, 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 F pos of the coordinate ring O(U + * ) of the unipotent cell U + * of G. In particular, the invariance of F pos,T under the cyclic shift on P G,T comes from the invariance of F pos 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 (see Theorem 5.7) is obtained from the theory of categorification of O(U + 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 the GS-universally positive Laurent property with universal positive Laurent property, we have the notion of universally positive Laurent morphisms. Then, it would be natural to expect the following: Conjecture 3. For any arc class [c] : E in → E out , the Wilson line g [c] : P G,Σ → G is a universally positive Laurent morphism. Moreover, the trace function tr V (ρ |γ| ) ∈ O(P G,Σ ) is universally positive Laurent.
Indeed, it is known that this conjecture on the trace functions holds true for type A 1 case [FG06]. In our continuing work [IOS22] with Linhui Shen, it is shown that the generalized minors of Wilson lines are cluster monomials. In particular, they are known to be universally positive Laurent [GHKK18].
on unipotent cells and double Bruhat cells in Section 4, we study we study the coordinate expressions of the Wilson lines and prove Theorem 1 in Section 5. In the course of the proof, we construct a basis of O(P G,T ) for a triangle T , which consists of those GS-universally positive Laurent, which is invariant under the cyclic shift. Some basic notions on the cluster varieties, weighted quivers and their amalgamation procedure are recollected in Appendix B.
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 Goncharov. 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 KAK-ENHI 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).

Configurations of pinnings
Denote by G m = Spec C[t, t −1 ] the multiplicative group scheme over C. For an algebraic torus T over C, let X * (T ) := Hom(T, G m ) be the lattice of characters, X * (T ) := Hom(G m , T ) the lattice of cocharacters, and −, − the natural pairing For t ∈ T and µ ∈ X * (T ), the evaluation of µ at t is denoted by t µ .
2.1. Notations from Lie theory. In this subsection, we briefly recall basic terminologies in Lie theory. See [Jan] for the details.
Let G be a simply-connected connected simple algebraic group over C. Let B + be a Borel subgroup of G and H a maximal torus (a.k.a. Cartan subgroup) contained in B + , respectively. Let U + be the unipotent radical of B + . Let • X * ( H) be the weight lattice and X * ( H) the coweight lattice; • Φ ⊂ X * ( H) the root system of ( G, H); • Φ + ⊂ Φ the set of positive roots consisting of the H-weights of the Lie algebra of U + ; The sub-lattice generated by α s for s ∈ S is called the root lattice. For s ∈ S, we have a pair of root homomorphisms x s , y s : for h ∈ H. After a suitable normalization, we obtain a homomorphism ϕ s : SL 2 → G such that The group G = G/Z( G) is called the adjoint group, where Z( G) denotes the center of G. Then B + := B + /Z( G) is a Borel subgroup of G and H := H/Z( G) is a Cartan subgroup of G. Moreover the unipotent radical of B + is isomorphic to U + through the natural map G → G, which we again denote by U + . Then we have B + = HU + . The natural map H → H induces Z-module homomorphisms where ̟ ∨ s ∈ X * (H) is the s-th fundamental coweight defined by ̟ ∨ s , α t = δ st ; we tacitly use the same notations for the elements related by these maps. The above mentioned one-parameter subgroups x s , y s descend to the homomorphisms x s , y s : A 1 → G with the same notation. There exists an anti-involution T : G → G, g → g T of the algebraic group G given by x s (t) T = y s (t) and h T = h for s ∈ S, t ∈ A 1 , h ∈ H. This is called the transpose in G. Let Definition 2.1. In G, define E s := x s (1) ∈ U + and F s := y s (1) ∈ U − for each s ∈ S. Let H s : G m → H be the one-parameter subgroup given by H s (a) = ̟ ∨ s (a).
The elements r s := r s H ∈ W ( G) have order 2, and give rise to a Coxeter generating set for W ( G) with the following presentation: where m st ∈ Z is given by the following table C st C ts : 0 1 2 3 m st : 2 3 4 6 .
For a reduced word s = (s 1 , . . . , s ℓ ) of w ∈ W ( G), let us write w := r s 1 . . . r s ℓ ∈ N G ( 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 N G ( H) on H. The action of r s is given by For w ∈ W ( G), write the length of w as l(w). Let w 0 ∈ W ( G) be the longest element of W ( G), and set s G := w 0 2 ∈ N G ( H). It turns out that s G ∈ Z( G), and s 2 G = 1 (cf. [FG06,§2]). We define an involution S → S, s → s * by We note that the Weyl group W (G) := N G (H)/H of G is naturally isomorphic to the Weyl group W ( G) of G, and we will frequently regard w as an element of N G (H) by abuse of notation. Remark that s G = w 0 2 = 1 in G.
Irreducible modules and matrix coefficients. Set X * ( H) + := s∈S Z ≥0 ̟ s ⊂ X * ( H) and There exists a unique non-degenerate symmetric bilinear form ( , is called a generalized minor. The * -involutions. We conclude this subsection by recalling an involution on G associated with a certain Dynkin diagram automorphism (cf. [GS18, (2)]).
2.2. The configuration space Conf k P G . Let G be an adjoint group. Here we introduce the configuration space Conf k P G based on [GS19], which models the moduli space P G,Π for a k-gon Π.
Definition 2.3. The homogeneous spaces A G := G/U + and B G := G/B + are called the principal affine space and the flag variety, respectively. An element of A G (resp. B G ) is called a decorated flag (resp. flag). We have a canonical projection π : A G → B G .
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 → A 1 is a non-degenerate character. See [GS15, Section 1.1.1] for a detailed discussion. The basepoint of A G is denoted by [U + ]. The flag variety B G will be identified with the set of connected maximal solvable subgroups of G via g.B + → gB + g −1 .
The Cartan subgroup H acts on A G from the right by g.[U + ].h := gh.[U + ] for g ∈ G and h ∈ H, which makes the projection π : A G → B G a principal H-bundle.
A pair (B 1 , B 2 ) of flags is said to be generic if there exists g ∈ G such that Using the Bruhat decomposition G = w∈W (G) U + HwU + , it can be verified that the G-orbit of any pair (A 1 , are called the h-invariant and the w-distance of the pair (A 1 , A 2 ), respectively. Note that the w-distance only depends on the underlying pair (π(A 1 ), π(A 2 )) of flags, and the pair is generic if and only if w(A 1 , A 2 ) = w 0 . The following lemma justifies the name "w-distance" and provides us a fundamental technique to define Goncharov-Shen coordinates.
Then the followings hold.
Corollary 2.5. Let (B l , B r ) be a pair of flags with w(B l , B r ) = w. Every reduced word s = (s 1 , . . . , s p ) of w gives rise to a unique chain of flags B l = B 0 , B 1 , . . . , B p = B r such that w(B k−1 , B k ) = r s k .
Next we define an enhanced configuration space by adding extra data called pinnings.
Definition 2.6 (pinnings). A pinning is a pair p = ( B 1 , B 2 ) ∈ A G × B G of a decorated flag and a flag such that the underlying pair (B 1 , B 2 ) ∈ B G × B G is generic, where B 1 := π( B 1 ). We say that p is a pinning over (B 1 , B 2 ).
An important feature is that the set P G of pinnings is a principal G-space, and in particular P G is an affine variety. In this paper, we fix the basepoint to be p std := ([U + ], B − ), so that any pinning can be writen as g.p std for a unique g ∈ G. The right H-action of A G induces a right H-action on P G , which is given by (g.p std ).h = gh.p std for g ∈ G and h ∈ H. Each fiber of the projection is a principal H-space. For p = g.p std , we define the opposite pinning to be p * := gw 0 .p std . We have (g.p std .h) * = g.p * std .w 0 (h) for g ∈ G and h ∈ H.
Remark 2.7. We have the following equivalent descriptions of a pinning. See [GS19] for details.
(1) A pair p = ( B 1 , B 2 ) ∈ A G × A G of decorated flags such that h( B 1 , B 2 ) = e and the underlying pair of flags is generic (i.e., w( B 1 , B 2 ) = w 0 ). The opposite pinning is given by p * = ( B 2 , B 1 ).
(2) A data p = (B, B op ; (ξ + s (t)) s∈S , (ξ − s (t)) s∈S ), where (B, B op ) is a pair of opposite Borel subgroups of G and (ξ + s (t)) s , (ξ − s (t)) s are one-parameter subgroups determined by a fundamental system for the root data with respect to the maximal torus B ∩ B op . The opposite pinning is given by For k ∈ Z ≥2 , we consider the configuration space where B i ∈ B G , and p i,i+1 is a pinning over (B i , B i+1 ) for cyclic indices i ∈ Z k . Here we use the notation for a quotient stack. See Appendix C. By Lemma C.5, Conf k P G is in fact a geometric quotient, whose points are G-orbits of the data (B 1 , . . . , B k ; p 12 , . . . , p k−1,k , p k,1 ). We will sometimes write an element of Conf k P G (i.e. a G-orbit) as [p 12 , . . . , p k−1,k , p k,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 − (p i−1,i ) = π + (p i,i+1 ) for i ∈ Z k .

Wilson lines on the moduli space P G,Σ
In this section, we first recall the definition of the moduli space P G,Σ for a marked surface Σ. We give an explicit description of the structure of P G,Σ as a quotient stack as an algebraic basis for the arguments in the subsequent sections. Then we introduce the Wilson line and Wilson loop morphisms on the stack P G,Σ and 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 P G,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.
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 e int (∆) ⊂ 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] 3 . More generally, one can consider an ideal cell decomposition: it is the isotopy class 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 manifold M is a principal G-bundle over M equipped with a flat connection. Let Σ be the compact oriented surface obtained from Σ by removing 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. Let L be a G-local system on Σ. A framing of L is a flat section β of the associated bundle L B := L × G B G on (a small neighborhood of) S ∪ P.
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 L P := L × G P G on the set ∂Σ \ (S ∪ P) such that for each boundary interval E ∈ B, the corresponding section p E is a pinning over (β + E , β − E ). Here the last sentence means that p E is projected to the pair (β + E , β − E ) via the bundle map where the former map is induced by the projection (2.4), and the latter is the evaluation at the points m ± E . Since L P is a principal G-bundle, a pinning of (L, β) determines a trivialization of L near each boundary interval.
If the marked surface Σ has empty boundary, we have P G,Σ ∼ = X G,Σ . In general we have a map P G,Σ → X G,Σ forgetting pinnings, which turns out to be a dominant morphism with respect to their Betti structures. The image X 0 G,Σ consists of the G-local systems with generic framings. For each boundary interval E, we have a natural action α E : P G,Σ × H → P G,Σ given by the rescaling of the pinning p E . Here recall that the set of pinnings over a given pair of flags is a principal H-space. Thus the dominant map P G,Σ → X G,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 P G,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.
3.1.1. The moduli space P G,Σ as a quotient stack. For simplicity, consider a connected marked surface Σ. Fix a basepoint x ∈ Σ. A rigidified framed G-local system with pinnings consists of a triple (L, β; p) together with a choice of s ∈ L x . The group G acts on the isomorphism classes of rigidified framed G-local systems pinnings (L, β, p; s) by fixing (L, β, p) and by s → s.g for g ∈ G.
In order to parametrize the isomorphisms classes of such rigidified objects, let us prepare some notations: • For each puncture a ∈ P, let γ a ∈ π 1 (Σ, x) denote a based loop freely homotopic to the boundary component C a . • Enumerate the connected components of ∂Σ \ P as ∂ 1 , . . . , ∂ b , and let δ k ∈ π 1 (Σ, x) be a based loop freely homotopic to ∂ k and following its orientation for k = 1, . . . , b.
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, . . . , N k .
In the pictures, the location of distinguished marked points is indicated by dashed lines. See Figure 2. We will use the notation ǫ E, Figure 2. Some of the curves in the defining data of the atlas of P G,Σ .
Notice that given a rigidified framed G-local system, the flat section of L B on C a gives an element of B G via the parallel-transport along the path from C a to x surrounded by the loop γ a and the isomorphism B G ∼ − → L x , g.B → s.g −1 . Similarly, the flat section of L P defined on a boundary interval E gives an element of P G via the parallel transport along the path ǫ E . Let m consisting of triples (ρ, λ, φ) ∈ Hom(π 1 (Σ, x), G) × (B G ) M × (P G ) B which satisfy the following conditions: • ρ(γ a ).λ a = λ a for all a ∈ P.
Hence the quotient stack P G,Σ is independent of the choice of distinguished marked points. When no confusion can occur, we simply write P G,Σ = P by forgetting the P G -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 P G,Σ;Ξ → P G,Σ;Ξ ′ . When Ξ = ∅, the stack P G,Σ;Ξ is still representable.
Disconnected case. When the marked surface Σ has N connected components, we consider a rigidification of a framed G-local system (with pinnings) on each connected component. Then the atlas P G,Σ is defined to be the direct product of those for the connected components, on which G N acts. The moduli stack P G,Σ is defined as the quotient stack for this G N -action.
3.2. Gluing morphisms. An advantage of considering the moduli space P G,Σ , rather than X G,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 E 1 and E 2 . Identifying the intervals E 1 and E 2 , we get a new marked surface Σ ′ . Let E denote the common image of E 1 and E 2 , which is a new interior edge. On the level of moduli spaces, given (L, β; p), note that the pinning p Eν assigned to the boundary interval E ν determines a trivialization of L near E ν for ν = 1, 2, since P G is a principal G-space. Then there is a unique isomorphism beween the restrictions of (L, β) on Σ to neighborhoods of E 1 and E 2 which identify the pinnings p E 1 and p * E 2 . In this way we get a framed G-local system with pinnings q E 1 ,E 2 (L, β; p) on Σ ′ . Note that the result is unchanged under the transformation which induces an open embedding q E 1 ,E 2 : The gluing operation is clearly associative. In particular, given an ideal triangulation ∆ of Σ, we can decompose the moduli space P G,Σ into a product of the configuration spaces Conf 3 P G 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(∆) P G,T from the right via α E 1 ,E op 2 for each glued pair (E 1 , E 2 ) of edges.
Theorem 3.7 ([GS19, Theorem 2.13]). Let ∆ be an ideal triangulation of the marked surface Σ. Then we have the gluing morphism which induces an open embedding q ∆ : The image of q ∆ is denoted by P ∆ G,Σ ⊂ P G,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 q E 1 ,E 2 : P of the gluing morphism (3.2) for some atlases for later use. For simplicity, we assume that the resulting marked surface Σ ′ is connected. Then we distinguish the two cases: (1) Σ has two connected components Σ 1 and Σ 2 containing E 1 and E 2 , respectively, or (2) Σ is also connected. For example, the gluing morphism in Theorem 3.7 is obtained by succesively applying the gluings of the first type.
(1) The disconnected case: In this case, we have the van Kampen isomorphism π 1 (Σ ′ , x) ∼ = π 1 (Σ 1 , x) * π 1 (Σ 2 , x) by choosing their common basepoints on the new edge E. We also For simplicity, we assume that the distinguished marked points on the boundary components containing E 1 and E 2 are identified under the gluing. The other cases are then obtained by composing the coordinate transformations given in Lemma 3.6.
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 Σ 1 as that for Σ.
(2) The connected case: In this case, we have an inclusion π 1 (Σ) = π 1 (Σ ′ \ E) → π 1 (Σ ′ ) and rankπ 1 (Σ ′ ) = rankπ 1 (Σ) + 1. Choose the basepoint x on the new edge E. Then When E 1 and E 2 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 3.6.
given above descends to a morphism P G,Σ → P G,Σ ′ , which agrees with the topological definition of the gluing morphism (3.2).
Proof. The morphism q E 1 ,E 2 is clearly G-equivariant, and hence descends to a morphism P G,Σ → P G,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 g 1 w 0 g −1 2 on the triple (ρ 2 , λ 2 , φ 2 ) by rescaling the rigidification. After such rescaling, the pinning assigned to the boundary interval E 2 gives g 1 w 0 g −1 2 .(φ 2 ) E 2 = g 1 w 0 .p std , which is the opposite of the pinning (φ 1 ) E 1 . Thus the gluing condition matches with the one explained in the beginning of this subsection.

Wilson lines and Wilson loops.
We are going to introduce the Wilson line morphisms (Wilson lines for short) on P G,Σ for a marked surface with non-empty boundary.
Let E in , E out ∈ B be two boundary intervals, and c a path from E in to E out 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] ∈ P G,Σ , choose a local trivialization of L on a vicinity of E in so that the flat section p E in of L P associated to E in corresponds to p std . This local trivialization can be extended along the path c until it reaches E out . Then the flat section p Eout determines a pinning under this trivialization, which is written as g.p * std for a unique element g = g c ([L, β; p]) ∈ G. It depends only on the homotopy class [c] of c relative to E in and E out : we call such a homotopy class [c] an arc class, and write [c] : E in → E out in the sequel. Then we have a map which we call the Wilson line along the arc class [c] : The Wilson lines can be defined as morphisms P G,Σ → G, as follows. Fix a basepoint x ∈ Σ and the collection {m k } of distinguished marked points, and consider the corresponding Here G acts on the first factor of G × G trivially, and by the left multiplication on the second. Then the induced morphism g [c] : P G,Σ → G of varieties agrees with the topological definition given above.
Proof. Observe that [G × G/G] = G by Lemma C.2. Therefore g [c] induces a morphism g [c] : P G,Σ → G of Artin stacks by Lemma C.3. The action on the target amounts to forget the second factor.
Consider the rigidified framed G-local system with pinnings (L, β, p; s) corresponding to a given point of P G,Σ . The rigidification s determines a local trivialization of L near x, and the section p E in (resp. p Eout ) gives the element g E in .p std (resp. g Eout .p std ) via the parallel-transport along the path ǫ E in (resp. ǫ Eout ) under this local trivialization. Moreover, notice that the section p Eout gives ρ(γ x )g Eout .p std via the parallel-transport along the path γ x * ǫ Eout . The local trivialization of L near E in so that p E in corresponds to p std is given by the rigidification s.g −1 E in . The latter trivialization can be continued along the path (1) As the proof indicates, the second component of the presentation morphism g [c] is introduced in order to make it G-equivariant, rather than G-invariant (note that when G acts on some variety X trivially, then [X/G] = X as a stack). As we shall see in Remark 3.17, P G,Σ is a variety if Σ has no punctures, hence g [c] : P G,Σ → G can be directly defined.
(2) The Wilson line g [c] along an arc class [c] : E in → E out can be defined as a morphism since it do not refer to the pinnings other than p E in and p Eout . Here recall (3.1).
The Wilson lines g [c] have the following multiplicative property with respect to the gluing of marked surfaces. Let Σ be a (possibly disconnected) marked surface, and consider two arc 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 (2) Connected case: In this case, consider the based loop α 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 ρ s The set of conjugacy classes in π 1 (Σ, x) is identified with the setπ(Σ) := [S 1 , 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 π 1 (Σ, x) →π(Σ), [γ] → |γ|. It is a classical fact that the conjugacy class  (Σ, x), G)/G] denotes the moduli stack of G-local systems and the first morphism is induced by the projection P G,Σ → Hom(π 1 (Σ, x), G), (ρ, λ, φ) → ρ. The second morphism is induced by the G-equivariant morphism Hom(π 1 (Σ, x), G) → G given by the evaluation at the based loop [γ x ] ∈ π 1 (Σ, x) presenting the free loop |γ|. Let ρ |γ| : P G,Σ → G denote the composite of these morphisms on atlases.
Proposition 3.12. Let Σ be a marked surface, [c] : E 1 → E 2 an arc class. Let Σ ′ be the marked surface obtained from Σ by gluing the boundary intervals E 1 and E 2 , and |γ| ∈π(Σ ′ ) be the free loop arising from [c]. Then we have the following commutative diagram of morphisms of stacks: where the right vertical morphism is the canonical projection.
Proof. From the presentation of the gluing morphism given in Section 3.2, we have where g (1) [c] denotes the first component of the morphism g [c] defined in Proposition 3.9. In other words, we have the commutative diagram and thus we get the desired assertion.
Remark 3.13 (Twisted Wilson lines). Let Π 1 (Σ, B) 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] ∈ P G,Σ defines a functor ([L, β; p])w 0 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 Π 1 (Σ, B) can be represented by a loop γ based at x ∈ E, and the conjugacy class of the twisted Wilson line g tw [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.
3.4. Generation of O(P G,Σ ) by matrix coefficients of (twisted) Wilson lines. We are going to obtain an explicit presentation of the function algebra O(P G,Σ ) by using the (twisted) Wilson lines when Σ has no punctures. In the contrary case ∂Σ = ∅, the description of the function algebra O(P G,Σ ) = O(X G,Σ ) as an O(Loc G,Σ )-module has been already obtained in [FG06, Section 12.5].
Choose a generating set j as in Section 3.1.1. Then we get the atlas , which consists of triples (ρ, λ, φ) satisfying certain conditions described in Lemma 3.4.
Assume that Σ has no punctures, and choose one boundary interval, say, Here ρ E 0 (γ) := g −1 0 ρ(γ)g 0 is the monodromy along γ for the local trivialization given by the pinning φ E 0 for γ ∈ S, and g E 0 ,E : The group G acts on the last factor of G 2g+b × G B\{E 0 } × G by left multiplication, and trivially on the other factors. Then it descends to an embedding of Artin stacks. Note from Remark 3.13 that ρ E 0 (γ) for γ ∈ S can be regarded as the twisted Wilson line along the based loop γ x at x ∈ E 0 . We can take their matrix coefficients, not only their traces.  Lemma 3.14. The Wilson line g Since j+1,j , we have N k for k = 2, . . . , b, and g (1) := 1. See Figure 5. Therefore the embedding (3.5) gives rise to another embedding (3.7) Theorem 3.15. The image of the embedding Φ E 0 is the closed subvariety which consists of the tuples (3.7) satisfying the following conditions: Proof. It is clear from the previous discussion and the multiplicative property of the twisted Wilson lines g tw [c] = g [c] w 0 that the image of Φ E 0 satisfies the conditions. Conversely, given a tuple (3.7) which satisfies the conditions, we can reconstruct the G-orbit of a triple (ρ, λ, φ) ∈ P , as follows. We first get the monodromy homomorphism ρ E 0 normalized at the boundary interval E 0 , and the pinning φ E 0 = p std . The other pinnings are given by determined by the formula (3.6). The collection λ of the underlying flags is given by Each consecutive pair of flags is generic, since Corollary 3.16. When Σ has no punctures, we have where I G,Σ is the radical of the ideal I G,Σ generated by the two relations described in Theorem 3.15. In particular, the function algebra O(P G,Σ ) is generated by the matrix coefficients of (twisted) Wilson lines.
Remark 3.17. When Σ has no punctures, one can see that the variety P is affine via the embedding Φ E 0 , on which G acts freely. Hence the moduli space P Remark 3.18. When Σ has punctures and non-empty boundary, we have where p is the number of punctures, G := {(g, B) ∈ G × B G | g ∈ B} denotes the Grothendieck-Springer resolution, to which the pair (ρ E 0 (γ a ), λ a ) for a ∈ P belongs. The ideal I G,Σ is generated by the monodromy relation where I G,T = g 3,2 w 0 g 2,1 w 0 g 1,3 w 0 = 1 . As we have seen in Corollary 3.25, the images of the Wilson lines g j,j−1 are in fact restricted to the double Bruhat cell B + * . This can be seen as g 3,2 = (w 0 g −1 1,3 w 0 )w 0 (w 0 g −1 2,1 w 0 ) ∈ B − w 0 B − and the cyclic symmetry. Example 3.20. When Σ = Q is a quadrilateral, we have where I G,Q = g 4,3 w 0 g 3,2 w 0 g 2,1 w 0 g 1,4 w 0 = 1 . Let us consider the Wilson line g 1,3 on P G,Q shown in the left of Figure 6. Letting g 3,1 := w 0 g −1 1,3 w 0 = (g * 1,3 ) T , we get the relations g 1,3 w 0 g 3,2 w 0 g 2,1 w 0 = 1, g 3,1 w 0 g 1,4 w 0 g 4,3 w 0 = 1.
Then similarly to the previous example, we get g 1,3 ∈ B − w 0 B − and g 3,1 ∈ B − w 0 B − , the latter being equivalent to g 1,3 ∈ B + w 0 B + . Thus we get g 1, Example 3.21 (partially generic cases). The restriction g 1,3 ∈ G w 0 ,w 0 in the previous example can be viewed as a consequence of the genericity condition for the consecutive flags. Let us consider the moduli space P G,Q:Ξ with Ξ = {E 1 , E 2 , E 3 } and Ξ = {E 1 , E 3 }, which are schematically shown in the middle and in the right in Figure 6, respectively. In these cases we have less Wilson lines and less restrictions for the values of g 1,3 : it can take an arbitrary value in B + w 0 B + and in G, respectively. In particular, we have P G,Q;{E 1 ,E 3 } ∼ = G. The configuration of flags is parametrized as 3.5. Decomposition formula for Wilson lines. We are going to give a certain explicit representative of an element of Conf 3 P G . 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.
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 3.6.
The coordinate transformation f m ′ • f −1 m is given by the cyclic shift which is an isomorphism.
An explicit computation of the cyclic shift S 3 in terms of the standard configuration is given in Section 5.
3.5.2. The standard configuration and basic Wilson lines. Now let us more look into the configuration space Conf 3 P G , which models the moduli space P G,T as we have seen just above. Let We call the representative in the right-hand side or the parametrization C 3 itself the standard configuration.
Proof. Since C 3 is clearly a morphism of varieties, it suffices to prove that it is bijective. Let (B 1 , B 2 , B 3 ; p 12 , p 23 , p 31 ) be an arbitrary configuration. Using the genericity condition for the pairs (B 1 , B 2 ), (B 2 , B 3 ) and (B 3 , B 1 ), we can write [B 1 , B 2 , B 3 ] = [B + , B − , u + .B − ] for some u + ∈ U + * . Using an element of B + ∩ B − = H, we can further translate the configuration so that p 12 = p std . Note that a representative of (B 1 , B 2 , B 3 ; p 12 , p 23 , p 31 ) satisfying these conditions is unique.
Since p 31 is now a pinning over the pair (u + .B − , B + ), there exists h 2 ∈ H such that p 31 = u + h 2 w 0 .p std . Let us write p 23 = g.p std for some g ∈ G. Since p 23 is a pinning over the pair (B − , u + .B − ), we have g.B + = B − and hence g = b − w 0 for some b − ∈ B − . We also have g.B − = u + .B − = φ ′ (u + )w 0 .B − , where the latter is the very definition of the map φ ′ . It implies that b − = φ ′ (u + )h 1 for some h 1 ∈ H. Thus we get the desired parametrization.

Thus we get an induced isomophism
of the rings of regular functions. Note that we can represent a configuration C ∈ Conf 3 P G in the following two ways: • C = [p std , p 23 , p 31 ], where the first component is normalized, where the last component is normalized. Such representatives are unique since the set of pinnings is a principal G-space.
Definition 3.24. Define the elements b L = b L (C), b R = b R (C) ∈ G ("left", "right") by the condition The resulting maps b L , b R : Conf 3 P G → G are called the basic Wilson lines.
We remark here that these functions already appeared in [GS15, Section 6.2]. The following is a direct consequence of Lemma 3.23: are morphisms of varieties, which are explicitly given by Remark 3.26. The definition of the basic Wilson lines b L and b R can be rephrased as follows.
Let us write a configuration as C = [g 1 .p std , g 2 .p std , g 3 .p std ] ∈ Conf 3 P G . Then we have and their regularity is also clear from this presentation.
The H 3 -action. Recall the right H 3 -action on Conf 3 P G given by [p 12 , p 23 , p 31 ].(k 1 , k 2 , k 3 ) = [p 12 .k 1 , p 23 .k 2 , p 31 .k 3 ] for (k 1 , k 2 , k 3 ) ∈ H 3 . It is expressed in the standard configuration by for (h 1 , h 2 , u + ) ∈ H × H × U + * . By this action, the functions b L and b R are rescaled as (3.14) The following is the geometric rephrasing: Figure 7. Two intersection patterns of c ∩ T ν Lemma 3.27. Let us use the notation in (3.9). Then via the isomorphism Let us briefly summarize what we have seen. Given an ideal triangulation ∆ of a marked surface Σ, we get the gluing morphism q ∆ : P ∆ G,Σ = T ∈t(∆) P G,T → P G,Σ (recall Theorem 3.7). It gives a decomposition of the moduli space P G,Σ into triangle pieces. Moreover, if we choose a marked point m T of each triangle T , then we get isomorphisms f m T : P G,T ∼ − → Conf 3 P G by (3.8), which enables us to study the moduli spaces in terms of the unipotent cells via the standard configuration (Lemma 3.23). In this paper, the choice of m T is indicated by the symbol * in the figures (e.g. Figure 8 below). We call the data ∆ * := (∆, (m T ) T ∈t(∆) ) a dotted triangulation.
3.5.3. Decomposition formula in the polygon case. 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). Note that for two boundary intervals E in , E out of Π, there is a unique arc class of the form [c] : E in → E out in this case.
Take an ideal triangulation ∆ of Π. Choose a representative c so that the intersection with ∆ is minimal. Let T 1 , . . . , T M be the triangles of ∆ which c traverses in this order. Note that for each ν = 1, . . . , M , the intersection c∩T ν is either one of the two patterns shown in Figure 7. The turning pattern of c with respect to ∆ is encoded in the sequence τ ∆ ([c]) = (τ ν ) M ν=1 ∈ {L, R} N , where τ ν = L (resp. τ ν = R) if c ∩ T ν is the left (resp. right) pattern in Figure 7. For our purpose, it is enough to consider the case T 1 ∪ · · · ∪ T M = Π. An example for k = 6 is shown in Figure 8.  Let us consider the restriction of the Wilson line g [c] to the substack P ∆ G,Π . Let q ∆ : M ν=1 P G,Tν → P ∆ G,Π be the gluing morphism, and define Here recall the isomorphisms (3.8). Then we have the following: Let π c := ̟| Π c;∆ : Π c;∆ → Σ, which is a covering map over its image. It induces a map π * c : P G,Σ → P G,Π c;∆ ;{ E in , Eout} via pull-back. Here recall Definition 3.3. From the definitions and Remark 3.10, we have: Lemma 3.30. The following diagram commutes: Combined with Proposition 3.29, we would obtain a decomposition formula for Wilson lines. We are going to write it down explicitly.
Fix a dotted triangulation ∆ * of Σ. Let m T denote the dot assigned to a triangle T ∈ t(∆). Note that ∆ * determines a dotted triangulation ∆ lift * of the polygon Π c;∆ over ∆ c by lifting the dots, which may not agree with the "canonical" dotted triangulation ∆ can * For ν = 1, . . . , M , let T ν := π c ( T ν ) ∈ t(∆) denote the projected image of the ν-th triangle, which do not need to be distinct. Finally, set f nν := f m Tν • pr Tν : T ∈t(∆) P G,T → Conf 3 P G .
See Example 3.32 and Figure 9 for an example.
Proof. Since the pull-back via the covering map π c commutes with the gluing morphisms, we have the commutative diagram where π * c := M ν=1 (π c | Tν ) * , and the right vertical map is the composite of the gluing morphism and the projection forgetting the pinnings except for those assigned to E in and E out . From the definition of t ν and f nν , the following diagram commutes: Combining together, we get as desired.
Note that by Proposition 3.12, the decomposition formula given in Theorem 3.31 also gives a formula for the presentation morphisms ρ |γ| of Wilson lines mod AdG. . Under the projection π c : Π c;∆ → Σ, we have T = π c ( T 1 ) = π c ( T 3 ), T ′ = π c ( T 2 ) = π c ( T 4 ). By comparing the two pictures in the right, we have t 1 = 0, t 2 = 0, t 3 = −1, t 4 = 1. Thus we have Of course, we could have chosen another triangulation ∆ ′ obtained from ∆ by the flip along the edge E 2 . In this case we have τ ∆ ′ ([c]) = (L, R) and we need no cyclic shifts to express g [c] .

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 configuration space Conf 3 P G . A necessary background on the cluster algebra is reviewed in Appendix B.
Notation 4.1. For a torus T = G N m equipped with a coordinate system X = (X k ) N k=1 and a map f : T → V to a variety V , we occasionally write f = f (X) as in the usual calculus.  We call these parametrizations Lusztig parametrizations. These maps induce injective Calgebra homomorphisms When w = w 0 is the longest element, we have U ± w 0 = U ± * . Goncharov-Shen potentials. Let us consider the configuration space It has a parametrization The map β −1 3 pulls-back the additive characters χ s : U + → A 1 , χ s (u + ) := ∆ ̟s,rs̟s (u + ) for s ∈ S to give a function which we call the Goncharov-Shen potential. Note that u + ∈ U + w if and only if w(B 2 , B 3 ) = w * , when we write β 3 (u + ) = (A 1 , B 2 , B 3 ). Here W (G) → W (G), w → w * is an involution given by The following relation will be used later: Lemma 4.3. For a reduced word s = (s 1 , . . . , s l ) of w, let u + = x s 1 (t 1 ) . . . x s l (t l ) ∈ U + w be the corresponding Lusztig parametrization. Then we have [Wil13].

Coweight parametrizations on double Bruhat cells. The coweight parametrizations on double Bruhat cells are introduced in [FG06] and further investigated in
Let G be an adjoint group. For each u, v ∈ W (G), the double Bruhat cell is defined to be G u,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 construction 4 .
Let us write B + v := G e,v and B − u := G u,e . First consider B + v . Let s = (s 1 , . . . , s l ) be a reduced word for v. Then the evaluation map ev + s : where x = (x k ) n+l k=1 and the symbol − → k=1,...,l means that we multiply the elements successively from the left to the right, namely − → k=1,...,l g k := g 1 . . . g l . Similarly in the case v = e, we take a reduced word s for u and define ev − s : G n+l m → B − u by replacing each E with F. We call the variables x = (x k ) k the coweight parameters.
The following indexing for the coweight parameters x will turn out to be useful: for a reduced word s = (s 1 , . . . , s l ) of an element of W (G), let k(s, i) denote the i-th number k such that s k = s. Let n s (s) be the number of s which appear in the word s. If we relabel the variables as   = (1, 2, 3, 1, 2, 1) is given by  Figure 10. Amalgamation of the quivers J + (1, 2, 3) and J + (1) for type A 3 produces the quiver J + (1, 2, 3, 1).
For a reduced word s of w ∈ W (G) and ǫ ∈ {+, −}, each variable x s i of the coweight parametrization ev ǫ s is assigned to the vertex v s i of the weighted quiver J ǫ (s). See Appendix B. The group multiplication corresponds to an appropriate amalgamation of quivers. For example, the multiplication considered in Example 4.4 corresponds to the quiver amalgamation shown in Figure 10. The pair S ǫ (s) := (J ǫ (s), (x s i ) (s,i)∈I(s) ) forms an X-seed in the ambient field F = K(B ǫ w ).
Theorem 4.5 (Fock-Goncharov [FG06], Williams [Wil13]). For an element w ∈ W (G) and ǫ ∈ {+, −}, the seeds S ǫ (s) associated with reduced words s of w are mutation-equivalent to each other. Hence the collection(S ǫ (s)) s is a cluster Poisson atlas (Definition B.2) on the double Bruhat cell B ǫ w .
The following lemma directly follows from the definition of the Dynkin involution and Lemma 2.2, which will be useful in the sequel.
Lemma 4.6. We have the following relations: Since the map g → w 0 −1 g −1 w 0 is an anti-homomorphism, we get the following: Corollary 4.7. For a reduced word s = (s 1 , . . . , s N ) of w 0 ∈ W (G), let s * op := (s * N , . . . , s * 1 ). Then we have where w(B k−1 2 , B k 2 ) = s * k for k = 1, . . . , N . Suppose that the triple (B 1 , B 2 , B 3 ) is "sufficiently generic" so that each pair (B 1 , B k 2 ) is generic for k = 0, . . . , N . Let B 1 , B ′ 1 be two lifts of B 1 determined by the pinnings p 12 , p * 31 , respectively. Now we define: Here as before, k(s, i) denotes the i-th number k such that s k = s in s. Using the embedding G m ֒→ A 1 , we regard X ( s i ) as a G m -valued rational function on Conf 3 P G . Let G be the simply-connected group which covers G and take a lift where B 1 is an arbitrary lift of B 1 ∈ A G to A G and Λ s : Note that P s,k does not depend on the choice of the lifts B 1 , B 2 and it gives a well-defined G m -valued regular function on Conf 3 P G . See also Lemma 4.17. Let β s k := r s N . . . r s k+1 (α ∨ s k ) be a sequence of coroots associated with s. For each s ∈ S, there exists a unique k = k(s) such that β s k = α ∨ s . Then we set Definition 4.8. The rational functions X ( s i ) (s ∈ S, i = −∞, 0, 1, . . . , n s (s)) are called the Goncharov-Shen coordinates (GS coordinates for short) on Conf 3 P G , associated with the reduced word s. When we want to emphasize the dependence on the reduced word s, we write X ( s i ) =: X s ( s i ) .
Conversely, we can construct an embedding Lemma 4.9 ([GS19, Lemma 9.2]). Let (k 1 , k 2 , k 3 ) ∈ H 3 and denote the action of (k 1 , k 2 , k 3 ) on Conf 3 P G described in Lemma 3.27 by α k 1 ,k 2 ,k 3 : Conf 3 P G → Conf 3 P G . Then for s ∈ S, we have Proof. The first three equalities are given in [GS19, Lemma 9.2]. The last one straightforwardly follows from the definition of the H 3 -action and W s .  (1) Note that the frozen coordinates X ( s 0 ) , X ( s n s (s) ) and X ( s −∞ ) depend only on one of the three pinnings. Moreover, observe from Lemma 4.9 that these frozen coordinates are uniquely distinguished by their degrees with respect to the H 3 -action among the GS coordinates associated with s.
(2) On the other hand, the unfrozen coordinates X ( s i ) for (s, i) ∈ I uf (s) only depend on the underlying flags (B 1 , B 2 , B 3 ). Hence we have the following birational charts for the configuration spaces with some of the pinnings dropped: is the isomorphism induced by (4.5).
Below we give a proof of this theorem based on the standard configuration (Lemma 3.23). Let us write (4.9) Then from Corollary 3.25, we have We are going to compute the functions u + (X) and h 2 (X).
In the following, we use the short-hand notations x s [i j] (t) := x s i (t i ) . . . x s j (t j ) and y s [i j] (t) := y s i (t i ) . . . y s j (t j ) for a reduced word s = (s 1 , . . . , s N ) of w 0 ∈ W (G) and 1 ≤ i < j ≤ N .
Lemma 4.14. For a configuration C = C 3 (h 1 , h 2 , u + ) ∈ Conf 3 P G and its representative as in Lemma 3.23, write u be the GS coordinates of C associated with s. Then we have the followings: Proof. To check that the right-hand side of the first statement indeed gives B k 2 , let us compute where we used the relation x s (t) = y s (t −1 )α ∨ s (t)r −1 s y s (t −1 ). Then the uniqueness statement of Corollary 2.5 and an induction on k implies B k 2 = x s [1 k] (t)B − . To prove the second statement, we compute With a notice that Ad −1 h 2 (x s (t k(s,n s (s)) )) = x s (h −αs 2 t k(s,n s (s)) ), we get = h αs 2 t −1 k(s,n s (s)) . Hence from Lemma 4.14 we get h αs 2 = t k(s,n s (s)) X ( s n s (s) ) = X s , which implies h 2 = s∈S H s (X s ).
Proof of Theorem 4.13. From the definition of the one-parameter subgroup x u , we have the relation H s (a)x u (b)H s (a) −1 = x u (a δsu b). In particular x s (t) = H s (t)E s H s (t) −1 . Combining with Lemmas 4.14 and 4.15, we get where each t k = t k (X) is a monomial given by (4.10). Since H s commutes with E u for s = u, we can obtain the following expression by the relabeling as in (4.2): where t k(s,n s (s)+1) := 1 for s ∈ S. Note that it already has the form of the coweight parametrization ev + s (x), where the parameter x = (x s i ) s∈S, i=0,...,n s (s) is computed as follows: . . , n s (s) − 1), t −1 k(s,n s (s)) X s = X ( s n s (s) ) (i = n s (s)), where we used Lemma 4.14 for the second steps. Thus we have ψ * s b L (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:

4.5.
Primary coordinates in the standard configuration. Let s = (s 1 , . . . , s N ) be a reduced word of w 0 ∈ W (G). The following computation of the primary coordinates P s,k in terms of the standard configuration will be used in Section 5.
Lemma 4.17. For a configuration C = C 3 (h 1 , h 2 , u + ) ∈ Conf 3 P G and its representative as in Lemma 3.23, write u + = x s 1 (t 1 ) . . . x s N (t N ) = x s [1 N ] (t) using the Lusztig coordinates associated with s. Then we have the following: 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 h 1 .
Proof. Since the second component in the representative of C is φ ′ (u + )h 1 w 0 .p std , the decorated flag B 2 must be a lift of Such an element is written as h 1 w 0 .[U + ] for some lift of h 1 to G, which proves the first statement of (1). Set In order to show the second statement of (1), it suffices to see that ) , B (k−1) ) = 1 for k = 1, . . . , N . The statement (i) is immediate from the definition. In Conf 2 A G , we have Moreover we have (4.11) Thus we get For the computation of P s,k (C 3 (h 1 , h 2 , u + )), we may take a lift of the first flag B 1 as [U + ]. Then, Corollary 4.18. For s ∈ S, we have Here recall that k(s) is determined by Proof. By the definition of X ( s −∞ ) and Lemma 4.17 (2), the desired statement follows from the following calculation: Lemma 4.19. Write t k(s,i) := X ( s 0 ) . . . X ( s i−1 ) for (i, s) ∈ I(s) (recall the notation in (4.10)). Then we have the following: 4.6. Goncharov-Shen coordinates on P G,Σ via amalgamation. Triangle case. Let Σ = T be a triangle. Recall that we have the isomorphism (3.8) determined by choosing a distinguished marked point m, or a dot. Let us choose a reduced word s of w 0 . Then we define the GS coordinates on P G,T to be X (T,m,s) denote the GS coordinates on Conf 3 P G associated with the reduced word s. The accompanying quiver Q T,m,s is defined to be the quiver J + (s) placed on T so that for s ∈ S, • the vertices v s 0 lie on the edge E, • the vertices v s n s (s T ) lie on the edge E R , and • the vertices y s lie on the edge E L .
Here we use the notation in (3.9) for the edges. See Figure 11 for an example. Here the isotopy class of the embedding of the quiver J + (s) into the triangle T relative to the boundary intervals is which has the degree −α s for the action α E .
General case. Let us consider 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 5 . Let m T denote the marked point corresponding to the dot on a triangle T . • A choice s ∆ = (s T ) T of reduced words s T of the longest element w 0 ∈ W (G), one for each triangle T of ∆ * .
We call ∆ * the underlying dotted triangulation of ∆. Let q ∆ : P ∆ G,Σ = T ∈t(∆) P G,T → P ∆ G,Σ be the gluing morphism with respect to the underlying triangulation ∆. For T ∈ t(∆), let pr T : P ∆ G,Σ → P G,T be the projection. Then the GS coordinate system on P ∆ G,Σ ⊂ P G,Σ associated with ∆ is the collection of rational functions, which are characterized as follows: • For T ∈ t(∆) and (s, i) ∈ I uf (s), • For an interior edge E ∈ e(∆) and s ∈ S, Here T L (resp, T R ) is the triangle that lie on the left (resp. right) of E, and E L ⊂ ∂T L (resp. E R ⊂ ∂T R ) is the edge that projects to E. • For a boundary edge E ∈ e(∆), Here T is the unique triangle that contain E.
One can verify that the right-hand sides of these characterizing equations are indeed H ∆invariant.
Correspondingly, for an interior edge E ∈ e(∆) shared by two triangles T L and T R as above, the two seeds S (T L ,m T L ,s T L ) and S (T R ,m T R ,s T R ) are amalgamated according to the gluing data (4.12) See Definition B.3 for a detail on the amalgamation procedure. See Figures 12 and 13 for examples. Applying this procedure for each interior edge, we get a weighted quiver Q ∆ drawn on the surface Σ. In the light of Theorem 3.7, the collection X ∆ of functions provide an open embedding ψ ∆ : G I ∆ m → P G,Σ , whose image is contained in P ∆ G,Σ and the index set is given by Thus the pair S(∆) := (X ∆ , Q ∆ ) forms a seed in the ambient field K(P G,Σ ). * * Figure 12. The quiver on T L E L ∪ E R T R with g = A 3 and s T L = s T L = (1, 2, 3, 1, 2, 1). We glue the vertices as (v 1 3 ) ′ = y ′′ 1 , (v 2 2 ) ′ = y ′′ 2 and (v 3 1 ) ′ = y ′′ 3 . * * 2 2 2 2 2 2 2 2 2 2 Figure 13. The quiver on T L E L ∪ E R T R with g = C 3 and s T L = s T R = (1, 2, 3) 3 . We glue the vertices as (v s 3 ) ′ = y ′′ s for s = 1, 2, 3. Comparison with the cluster Poisson algebra. Let S G,Σ denote the cluster Poisson structure on P G,Σ which includes the cluster Poisson atlas (S(∆)) ∆ . Then Theorem 4.21 tells us that our moduli space P G,Σ is birationally isomorphic to the cluster Poisson variety X S G,Σ , and hence their fields of rational functions are isomorphic. Slightly abusing the notation, let us denote the cluster Poisson algebra by O cl (P G,Σ ) := O(X S G,Σ ). Shen proved the following stronger result: Theorem 4.22 (Shen [She20]). We have an isomorphism O cl (P G,Σ ) ∼ = O(P G,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: 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 P G,Σ;Ξ of Ξ-generic framed G-local systems with Ξ-pinnings (recall Definition 3.3 and (3.1)). For a decorated triangulation ∆, set (4.13) Then by Remark 4.11, we have an open embedding G I Ξ (∆) m → P G,Σ;Ξ such that the projection P G,Σ → P G,Σ;Ξ forgetting the pinnings except for those assigned to the boundary intervals in Ξ is expressed as a coordinate projection. In other words, the moduli space P G,Σ;Ξ also has a canonical cluster Poisson atlas so that the projections P G,Σ → P G,Σ;Ξ is a cluster projection.

Laurent positivity of Wilson lines and Wilson loops
5.1. 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). We say that f ∈ O(P G,Σ ) is GS-universally positive Laurent if it is expressed as a Laurent polynomial with non-negative integral coefficients in the GS coordinate system associated with any decorated triangulation ∆. A morphism F : P G,Σ → G is called a GS-universally positive Laurent morphism if for any finite-dimensional representation V of G, there exists a basis B of V such that c V f,v • F ∈ O(P G,Σ ) is GS-universally positive Laurent for all v ∈ B and f ∈ F, where F is the basis of V * dual to B (see (2.2)).
We should remark that the definition of GS-universally positive Laurent morphism does not change if we replace "any finite-dimensional representation V " in its definition 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 5.2. 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] : E in → E out , the Wilson line g [c] : P G,Σ → G is a GS-universally positive Laurent morphism.
Combining with Proposition 3.12, we immediately get the following: Corollary 5.3. 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 tr V (ρ |γ| ) := tr V •ρ |γ| ∈ O(P G,Σ ) is GS-universally positive Laurent.
with λ 1 , λ 2 ∈ X * (H) + such that −λ 1 + λ 2 = ξ. Note that ∆ + w 0 ,ξ is a well-defined element. The description of the cyclic shift S 3 on Conf 3 P G (Lemma 3.22) in terms of the standard configuration is important in the sequel. In the description, we use the Berenstein-Fomin-Zelevinsky twist automorphism η w 0 [BFZ96, BZ97] (we call it the twist automorphism for short) defined by This is a regular automorphism of U + * . The properties of η w 0 are collected in Appendix A.
Lemma 5.5. 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 * w 0 : O(U + * ) → O(U + * ) as a set, and (M) ∆ + w 0 ,ξ · F ∈ F for any ξ ∈ X * (H) and F ∈ F. Then the basis F T of O(P G,T ) given by does not depend on the marked point m of T .
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(P G,T ) with an appropriate positivity.
(P2) Recall the notation (4.1). For any reduced word s = (s 1 , . . . , s N ) of w 0 , we have In the following, we write the weight of G λ (b) (and G up λ (b ′ )) as wt b. Remark 5.8. In (P1), either c λ , we can interchange the roles of B(λ) and B up (λ).
In the following, the notations , where the right-hand side is the pull-back of the GS coordinate on Conf 3 P G associated with s. By Lemma 5.5, given any decorated triangulation ∆ of T , we may regard F pos,T as f * m ( F pos,3 ), where m is the dot of ∆ and F pos,3 := {(C * 3 ) −1 (e µ 1 ⊗ e ν 2 ⊗ F ) | µ, ν ∈ X * (H), F ∈ F pos }. Therefore, it suffices to show that F pos,3 is expressed as a Laurent polynomial with non-negative integral coefficients in terms of the GS coordinates on Conf 3 P G associated with s. Recall the map ψ s in (4.7) and the maps h 1 , h 2 , u + in (4.9), whose explicit descriptions are given in Lemma 4.19. For µ, ν ∈ X * (H) and F ∈ F pos , we have By Lemma 4.19 and the property (P2) of F pos , the right-hand side is a Laurent polynomial with non-negative integral coefficients in {X s ( s i ) } (s,i)∈I∞(s) . This completes the proof.
Remark 5.11. In the proof of Theorem 5.10, we do not use the property (P1) of F pos .
Lemma 5.12. We have F • * ∈ F pos for all F ∈ F pos .
Recall the basic Wilson lines b L : Conf 3 P G → B + * and b R : Conf 3 P G → B − * from Definition 3.24. For a dot m of T , we set where the last maps ι are the inclusion maps.
Theorem 5.13. For λ ∈ X * (H) is written as a Z ≥0 -linear combination of elements of F pos,T . In particular, it is GS-universally positive Laurent, and g m,τ is a GS-universally positive Laurent morphism.
Proof. We have The remaining statements immediately follow from Remark 5.1 and Theorem 5.10.

5.3.
A proof of Theorem 5.2. 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 3.7, where ∆ is the underlying triangulation of ∆ * . Fix an arc class [c] : E in → E out . For our purpose, it suffices to show that is GS-universally positive Laurent for any λ ∈ X * (H) + and b, b ′ ∈ B(λ) (see Remark 5.1).
Let q ∆ : P ∆ G,Σ = T ∈t(∆) P G,T → P ∆ G,Σ be the gluing map in Theorem 3.7, and pr T : P ∆ G,Σ → P G,T the projection for T ∈ t(∆). Recall that for T ∈ t(∆), (s, i) ∈ I uf (s), and for E ∈ e(∆) and s ∈ S. Here T L (resp. T R ) is the triangle containing E and lies on the left (resp. right) side with respect to the orientation of E in the first case, and T is the unique triangle containing E in the second case. By the correspondence above, it suffices to show that q * ∆ (c λ ) is expressed as a Laurent polynomial with non-negative integral coefficients in any GS coordinate system on T ∈t(∆) P G,T .
Henceforth, we follow the notation in the beginning of Section 3.5.4. For ν = 1, . . . , M , denote by m ν the dot on T ν which is associated with the turning pattern (τ 1 , . . . , τ M ) of c. Moreover, we have the commutative diagram where π * c := M ν=1 (π c | Tν ) * (see the proof of Theorem 3.31). Then, by (3.20), we have ) for any dot m on T ν , any (s, i) ∈ I ∞ (s), and any reduced word s of w 0 . Thus q * ∆ (c λ ) is expressed as a Laurent polynomial with non-negative integral coefficients in the GS coordinate system on T ∈t(∆) P G,T , which completes the proof of Theorem 5.2.
Proof. Let us write u The second equality immediately follows from the first one.

Appendix B. 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 the gluing morphism (Theorem 3.7) via Goncharov-Shen coordinates. We also recall the construction of weighted quivers from reduced words, following [FG06] and [GS19].
B.1. Weighted quivers and the cluster Poisson varieties. We use the conventions for weighted quivers in [IIO19]. Recall that a weighted quiver Q = (I, I 0 , σ, d) is defined by the following data: • I 0 ⊂ I are finite sets.
>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 σ ij := #{arrows from i to j} − #{arrows from j to i}.
Here we have "half " arrows when σ ij ∈ Z/2 (shown by dashed arrows in figures). The quiver has no loops nor 2-cycles by definition. The subset I 0 is called the frozen set, and mutations will be allowed only at the vertices in the complement I uf := I \ I 0 . The ciral dual of Q is defined by Q op := (I, I 0 , −σ, d).
We define the exchange matrix ε = (ε ij ) i,j∈I of Q to be ε ij := d i σ ij gcd(d i , d j ) −1 . Since we can reconstruct the skew-symmetric matrix σ from the pair (ε, d), we sometimes write Q = (I, I 0 , ε, d). The following is a reformulation of the matrix mutation (see e.g., [FG09,(12)]) in terms of the weighted quiver: Definition B.1. For k ∈ I uf , let Q ′ = (I, I 0 , σ ′ , d) be the weighted quiver given by The operation µ k : Q → 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 variables with coefficients in C. An (X -)seed is a pair (Q, X), where X = (X i ) i∈I is a tuple of algebraically independent elements of F and Q is a weighted quiver. For k ∈ I \ I 0 , let (Q ′ , X ′ ) be another seed where Q ′ = µ k (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): The operation µ k : (Q, X) → (Q ′ , X ′ ) is called the seed mutation at k. It is not hard to see that seed mutations are involutive: µ k µ k = 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 I 0 setwise). Let T I uf be the regular |I uf |-valent tree, each of whose edge is labeled by an index in I uf so that two edges sharing a vertex have different labels. An assignment S = (S (t) ) t∈T I uf of a seed S (t) = (Q (t) , X (t) ) to each vertex t of T I uf is called a seed pattern if for two vertices t, t ′ sharing an edge labeled by k ∈ I uf , the corresponding seeds are related as S (t ′ ) = µ k S (t) .
The cluster Poisson variety X S = t∈T I uf X (t) is defined by patching the coordinate tori X (t) = G I m corresponding to seeds S (t) by the rational transformations given by the formula (B.1) whenever t and t ′ shares an edge labeled by k ∈ I uf . The cluster Poisson variety has a natural Poisson structure given by {X The ring O(X S ) = t∈T I uf 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 + (X S ) : is called a universally positive Laurent polynomial.
Definition B.2. A cluster Poisson atlas on a variety (or scheme, stack) V over C is a collection (S α ) α∈A of seeds (here A is an index set) in the field K(V ) of rational functions on V such that • each seed S α = (Q α , X α ) gives rise to a birational isomorphism X α : V G I m which admits an open embedding ψ α : G I m ֒→ V as a birational inverse; • the seeds S α for α ∈ A are mutation-equivalent to each other.
From the second condition, the collection (S α ) α∈A can be extended to a unique seed pattern S = (S (t) ) t∈T I uf . In particular we get a birational isomorphism V ∼ = X S . 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 X S ֒→ V when (S α ) α∈A S.
A rational function on V can be regarded as a rational function on X S , and we can ask whether it is a universally (positive) Laurent polynomial.
Definition B.3. Let Q = (I, I 0 , σ, d), Q ′ = (I ′ , I ′ 0 , σ ′ , d ′ ) be two weighted quivers. Fix two subsets F ⊂ I 0 , 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, J 0 , τ, c) defined as follows • The entry τ ij is given by: Here we can choose any subset J 0 of I 0 ∪ φ I ′ 0 such that σ ij is integral unless (i, j) ∈ J 0 × J 0 . In this paper, we consider the minimal J 0 given by 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 = (Y i ) i∈J is defined by Then it is not hard to check that the amalgamation of seeds commutes with the mutation at any vertex k ∈ (I \ I 0 ) ⊔ (I ′ \ I ′ 0 ). Thus for two seed patterns S and S ′ and a gluing data as above, we have a dominant morphism α φ : X S × X S ′ → X S * φ S ′ .
Here the seed pattern S * φ S ′ is obtained by the amalgamation of the seeds S (t) and S ′ (t) for t ∈ T I uf .
B.3. Weighted quivers from reduced words. Let us fix a finite dimensional complex semisimple Lie algebra g. Let C(g) = (C st ) s,t∈S be the associated Cartan matrix. For s ∈ S, we define a weighted quiver J + (s) = (J(s), J 0 (s), σ(s), d(s)) as follows. Note that other entries are determined by the skew-symmetricity. (1) Type A 3 : S = {1, 2, 3} and the Cartan matrix is given by The elementary quivers J + (1), J + (2) and J + (3) are given as follows: (2) Type C 3 : S = {1, 2, 3} and the Cartan matrix is given by 6 The elementary quivers J + (1), J + (2), J + (3) are given as follows. Note that the vertices with the same Dynkin label are drawn on the same level in the pictures.
For a reduced word s = (s 1 . . . s l ) of u ∈ W (g) and ǫ ∈ {+, −}, we construct a weighted quiver J ǫ (s) = J ǫ (s 1 , . . . , s l ) by amalgamating the elementary quivers J ǫ (s 1 ), · · · , J ǫ (s l ) in the following way: for k = 1, . . . , l − 1, amalgamate J ǫ (s k ) and J ǫ (s k+1 ) by setting the gluing data in Definition B.3 as F := J(s k ) \ {s L k }, F ′ := J(s k+1 ) \ {s R k+1 }, φ : F → F ′ , s R k → s k , s k+1 → s L k+1 , t → t for t = s k , s k+1 . Note that the Dynkin labelings δ(s k ) 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 + (s 1 ) * · · · * J + (s l ), let v s i be the (i + 1)-st vertex with Dynkin label s from the left, for s ∈ S and i = 0, . . . , n s (s). Here n s (s) is the number of s which appear in the word s. We also use the labelling v s i =: v k(s,i) for s ∈ S and i = 1, . . . , n s (s), where k(s, i) ∈ {1, . . . , l} denotes the i-th number k such that s k = s in Example B.6. Here are some examples of the quiver J + (s).
Thus we get k(1) = 1, k(2) = 4, = k(3) = 9, and the quiver J + ((1, 2, 3) 3 ) is given by Appendix C. 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 P G,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 C.5); 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 = Spec C 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 (Spec C). 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 C.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 Ginvariant 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. Here is an easy example: Lemma C.2. Let V be a scheme over C and G an affine algebraic group. Consider the G-action on V × G given by the trivial action on the first factor and left multiplication on the second. Then the quotient stack [V × G/G] is representable by V .
The following lemma tells us that we can obtain morphisms between quotient stacks from equivariant morphisms of varieties. Lemma C.3. 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 Here u H 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 C.4. When τ is not an embedding, the morphism φ * is not representable in general. Lemma C.5. Suppose that X is an affine algebraic variety, G is reductive, and 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]).