Olivier Guichard

Daniele Alessandrini

Antonin Guilloux

Elisha Falbel

Devant la commission d'examen composée de

Introduction en français

Une représentation du groupe fondamental d'une surface S de caractéristique d'Euler négative dans un groupe de Lie G peut être vue comme un G-fibré principal sur S, ce qui permet d'utiliser des outils topologiques et géométriques pour étudier ces représentations. Les fibrés obtenus à partir des représentations ont la propriété particulière d'avoir des changements de cartes localement constants. Ce type de fibrés est appelé un G-système local sur la surface S. La représentation correspondant à un G-système local est appelée représentation de monodromie. Habituellement, il est plus intéressant d'étudier l'espace de modules des représentations à conjugaison près (ou de manière équivalente les systèmes locaux à transformation de jauge près), que l'on appelle variété de caractères. Cependant, prendre le quotient par l'action de G par conjugaison résulte généralement en un espace non-Hausdorff. Il existe deux solutions à ce problème : prendre le quotient GIT qui nécessite des outils de géométrie algébrique, ou prendre le quotient Hausdorff à la place. Ces deux solutions coïncident lorsque G est un groupe réductif complexe, ce qui est l'un des cas couverts par le travail présenté ici. Un exemple classique de cette approche est l'espace de Teichmüller de la surface S qui peut être identifié à une composante connexe de l'espace de modules des PSL 2 (R)-systèmes locaux sur S. Il s'agit d'un exemple très particulier où une composante connexe entière de la variété de caractères est constituée uniquement de représentations fidèles et discrètes. Trouver des composantes connexes similaires dans la variété de caractères d'autres groupes de Lie est devenu un domaine très fructueux appelé théorie des espaces de Teichmüller supérieurs. Pour une exposition détaillée sur le sujet, voir [Wie18, [START_REF] Flamm | Proceedings of the young researchers workshop on positivity in lie groups[END_REF][START_REF] Guichard | Positivity and higher Teichmüller theory[END_REF][START_REF] Burger | Surface group representations with maximal Toledo invariant[END_REF].

Dans [START_REF] Fock | Moduli spaces of local systems and higher Teichmüller theory[END_REF], Vladimir Fock et Alexander Goncharov introduisent les variétés amassées pour étudier les systèmes locaux en utilisant une structure algébrique appelée algèbre amassée introduite par Sergei Fomin et Andrei Zelevinsky dans [START_REF] Fomin | Cluster algebras. I. Foundations[END_REF] quelques années auparavant. La théorie introduite par Fock et Goncharov décrit les G-systèmes locaux sur une surface ciliée, où G est un groupe de Lie semi-simple réel déployé. Les surfaces ciliées sont une classe très générique de surfaces hyperboliques contenant les surfaces épointées et des polygones idéaux, voir la Section 2.1 pour une définition précise. Les surfaces ciliées sont toujours non-compactes, et leurs groupes fondamentaux sont des groupes libres. Une propriété

INTRODUCTION EN FRANÇAIS

essentielle des surfaces ciliées est qu'elles admettent toutes des triangulations idéales. Une triangulation idéale ∆ d'une surface ciliée S est un ensemble maximal de classes d'homotopie d'arcs sur S deux à deux sans intersections et non homotopes, ne s'auto-intersectant pas, dont les extrémités sont des pointes. De manière similaire aux coordonnées de Penner sur l'espace de Teichmüller décoré, les coordonnées de Fock-Goncharov nécessitent des données additionelles au système local : un cadrage ou une décoration (voir les sections 2.2.2 et 2.2.3 pour des définitions précises). Soit G un groupe de Lie semi-simple et réel déployé, B un sous-groupe de Borel et U le radical unipotent de B.

Définition. Un cadrage d'un G-système local L est la donnée pour chaque pointe de S d'une classe de G/B (appelée drapeau) stabilisée par la monodromie de L autour d'une petite boucle autour de la ppointe. Une décoration d'un G-système local L est la donnée pour chaque pointe d'une classe de G/U (appelée drapeau décoré) stabilisée par la monodromie de L autour d'une petite boucle autour de la pointe.

Les travaux de Fock et Goncharov reposent sur la définition d'un atlas spécial sur l'espace de modules des systèmes locaux encadrés (resp. décorés) sur une surface ciliée, appelé X -coordonnées (resp. A-coordonnées). Cet atlas spécial a une carte associée à chaque triangulation idéale de la surface ciliée et les changements de cartes peuvent être calculés avec une suite de mutations élémentaires, chacune étant une transformation birationnelle. Deux triangulations quelconques sont reliées par une suite finie de flips (changement de diagonale dans un quadrilatère), de sorte qu'il suffit de calculer les changements de cartes pour deux triangulations qui diffèrent d'un flip pour pouvoir calculer les changements de cartes de n'importe quelle triangulation vers n'importe quelle autre. La combinatoire de la suite de mutations conduisant à un flip de la triangulation est décrite par un carquois inscrit sur la surface, et les formules de mutations sont données par la structure d'algèbre amassée. En particulier, ces transformations birationnelles préservent la positivité des coordonnées. Une représentation est dite positive si toutes ses coordonnées dans n'importe quelle triangulation (donc dans toutes) sont positives. Cette notion de positivité étend la définition donnée par George Lusztig dans [START_REF] Lusztig | Total positivity in reductive groups[END_REF]. Dans le cas G = PGL n (R), les coordonnées X sont des birapports et des triple rapports de diverses droites construites en utilisant le cadrage du système local. Dans le cas G = SL n (R), les coordonnées A sont des volumes de diverses bases construites en utilisant la décoration du système local. L'un des principaux résultats de leur théorie est l'identification entre la composante de Hitchin de la variétédes caractères et le sous-ensemble des représentations positives. Ils montrent également que les représentations positives sont fidèles et discrètes, et donc que les composantes de Hitchin sont des espaces de Teichmüller supérieurs.

La théorie des coordonnées de Fock-Goncharov s'applique aux groupes de Lie semisimples réels déployés G, et le cadrage doit être pris dans la variété des drapeaux complets G/B où B est le sous-groupe de Borel de G. Une autre classe connue d'espaces de Teichmüller supérieurs est l'espace des représentations maximales. Les représentations maximales ont été définies par Marc Burger, Alessandra Iozzi et Anna Wienhard dans [START_REF] Burger | Surface group representations with maximal Toledo invariant[END_REF] pour les surfaces non fermées. Il s'agit de représentations dans un groupe de Lie hermitien de type tubulaire non compact pour lequel le nombre de Toledo, qui est une généralisation du nombre d'Euler pour les représentations dans PSL 2 (R), prend sa valeur maximale. Dans [START_REF] Burger | Surface group representations with maximal Toledo invariant[END_REF], Burger-Iozzi-Wienhard montrent que les représentations maximales sont fidèles et discrètes, et qu'elles satisfont une propriété de positivité, mais le cadrage dans ce cas est à valeurs dans une variété de drapeaux partiels, comme l'espace des sous-espaces lagrangiens lorsque G = Sp 2n (R). Contrairement aux composantes de Hitchin qui sont contractiles, l'espace des représentations maximales n'est en général ni connexe ni simplement connexe. Pour les groupes de Lie qui sont à la fois réels déployés et hermitiens (c'est-à-dire Sp 2n (R)), l'ensemble des représentations maximales contient strictement la composante de Hitchin. Une grande classe de groupes de Lie hermitiens de type tubulaire peut être exprimée comme des groupes symplectiques à coefficients dans une algèbre involutive (voir [ABR + 22] ou Section 2.7.2 pour plus de détails). Dans le cas symplectique, le nombre de Toledo peut être calculé en utilisant l'indice de Kashiwara-Maslov, voir [START_REF] Alessandrini | Noncommutative coordinates for symplectic representations[END_REF]. Dans [START_REF] Alessandrini | Noncommutative coordinates for symplectic representations[END_REF], Daniele Alessandrini, Olivier Guichard, Eugen Rogozinnikov et Anna Wienhard ont construit des généralisations non-commutatives des coordonnées de Fock-Goncharov X et A pour les représentations symplectiques, et ont montré que le lieu de positivité est exactement l'ensemble des représentations maximales. Ces coordonnées généralisées sont à valeurs dans des anneaux de matrices carrées, et sont donc génériquement non-commutatives. Cependant, elles admettent une forte structure de type "amassée", comme des cartes birationelles (non-commutatives) entre les coordonnées associées aux triangulations différant d'un flip. Afin de disposer d'un cadre général pour étudier à la fois les représentations positives et les représentations maximales, Guichard-Wienhard ont introduit dans [GW18, GW22] la notion de Θ-positivité. Cette notion dépend de la donnée d'un sous-groupe parabolique P Θ d'un groupe de Lie semi-simple G qui est déterminé par le choix d'un sous-ensemble Θ de racines restreintes de G. Toutes les paires (G, P ) n'admettent pas une structure Θ-positive, et toutes les paires admettant une structure positive sont classifiées dans [START_REF] Guichard | Positivity and higher Teichmüller theory[END_REF]. Il existe quatre familles de structures Θ-positives. La première est celle où G est réel déployé et P = B est le sous-groupe de Borel de G (c'est-à-dire quand Θ est l'ensemble de toutes les racines), et les représentations Θ-positives sont exactement les représentations positives introduites par Fock-Goncharov. La seconde famille est celle où G est un groupe de Lie hermitien de type tubulaire et P est associé à la dernière racine du système de Dynkin de type B n , et dans ce cas les représentations Θ-positives coïncident avec les représentations maximales. Les deux "nouvelles" familles sont respectivement celle des groupes localement isomorphes à SO(p, q) avec p ̸ = q et celle qui contient une classe de groupes de Lie exceptionnels. Dans ces deux derniers cas, le sous-groupe parabolique est associé à un sous-ensemble strict de racines. Dans [START_REF] Guichard | Positivity and representations of surface groups[END_REF], Guichard-Labourie-Wienhard ont montré que les représentations Θ-positives sont P Θ -Anosov, donc fidèles et discrètes. La conjecture selon laquelle les représentations T heta-positives forment des espaces de Teichmüller supérieurs est encore ouverte dans le cas général au moment de l'écriture de ce manuscrit, mais elle a été démontrée pour une grande classe d'exemples, voir [START_REF] Guichard | Positivity and representations of surface groups[END_REF][START_REF] Beyrer | Positive surface group representations in po(p,q)[END_REF].

L'idée principale du travail présenté ici est d'étudier l'espace de modules d'un système local encadré ou décoré sur une surface ciliée, mais avec un cadrage ou une décoration associé à n'importe quel sous-groupe parabolique d'un groupe de Lie semi-simple ou réductif. En particulier, nous cherchons des coordonnées qui partagent certaines propriétés calculatoires avec les coordonnées de Fock-Goncharov, à savoir une description combinatoire qui permet le calcul de formules de mutation de flips à partir d'une suite de mutations élémentaires. Dans ce travail, nous nous concentrons sur un groupe de Lie de type A, à savoir SL n (R) (ou plutôt GL n (R), comme nous l'expliquerons ci-dessous). Pour des raisons techniques, nous devrons nous limiter aux sous-groupes paraboliques de GL n (R) dont le sous-groupe de Levi est de la forme GL d (R) k , avec n = kd. Cela correspond à un sous-groupe parabolique conjugué au sous-groupe des matrices triangulaires supérieures où tous les blocs diagonaux sont des carrés de taille d. Pour ce faire, nous remplaçons le corps R par une R-algèbre de dimension finie R. Dans tout ce travail, nous nous intéressons principalement au cas où R = M n (R) est une algèbre de matrices carrées à coefficients réels. Ce faisant, nous obtiendrons des coordonnées qui sont à valeurs dans une certaine algèbre de matrices carrées, et qui sont donc génériquement non-commutatives. Ceci s'accompagne de plusieurs difficultés techniques. La première de ces difficultés est la définition même des coordonnées de type X : l'une des définitions possibles du birapport (resp. du triple rapport) s'étend au cas non-commutatif, donnons d'abord ces définitions :

Définition. Soit L 1 , L 2 , L 3 , L 4 quatre droites dans R 2 , transverses deux à deux. Soit π i,j la restriction de la projection canonique π i,j : L j → R 2 /L i pour tous i, j ∈ {1, 2, 3, 4} distincts. Le birapport des quatre droites est alors le suivant

[L 1 : L 2 : L 3 : L 4 ] = π -1 4,1 • π 4,3 • π -1 2,3 • π 2,1 .
Définition. Soit (L 1 , P 1 ), (L 2 , P 2 ), (L 3 , P 3 ) trois drapeaux complets dans R 3 avec dim F i = 1 et dim P i = 2 pour i = 1, 2, 3, transverses deux à deux. Soit π i,j la restriction de la projection canonique π i,j : L j → R 2 /P i pour tous ß, j ∈ {1, 2, 3} distincts. Alors le triple rapport du triple des drapeaux est

r = π -1 3,1 • π 3,2 • π -1 1,2 • π 1,3 • π -1 2,3 • π 2,1 .
Dans ces deux définitions, l'application L 1 → L 1 est identifiée à un scalaire car sa matrice ne dépend pas du choix d'une base de L 1 . Cependant, cette identification n'est plus possible si la dimension de L 1 est plus grande. La généralisation correcte de X serait alors la classe de conjugaison des applications définies ci-dessus, mais nous perdrions les formules de mutation, car on ne peut pas ajouter ou multiplier les classes de conjugaison. Cependant, ces "coordonnées X généralisées" conservent une partie du comportement de leur contrepartie commutative. En faisant attention à ces difficultés techniques, nous les étudions dans la Section 3.1.2.

Lorsque l'on considère les drapeaux décorés, la situation est toutefois très similaire dans le cadre commutatif et dans le cadre non-commutatif. Le besoin d'une base pour exprimer les applications ci-dessus est couvert par la décoration. Les définitions du birapport et du rapport triple indiquent également de bons candidats pour les A-coordonnées noncommutatives. En effet, nous voulons que les A-coordonnées déterminent les X -coordonnées de la configuration de drapeaux, et dans le cas commutatif, les définitions ci-dessus donnent les formules habituelles concernant les X -coordonnées et les A-coordonnées, étant donné que les A-coordonnées sont définies comme les matrices des cartes π i,j dans les bases données par la décoration des drapeaux. Cependant, cette affirmation n'est valable qu'au rang 2. Ceci nous amène à la difficulté technique suivante découlant de la non-commutativité : il n'y a pas de déterminant pour les matrices à valeurs dans un anneau générique, donc pas de groupe spécial linéaire SL n . Puisque nous ne pouvons pas travailler avec SL n , nous devons travailler dans le groupe général linéaire GL n qui, même sur un anneau commutatif, n'est pas un groupe de Lie semi-simple, mais un groupe réductif. Par conséquent, les A-coordonnées non-commutatives que nous définirons dans les Sections 3.3 et 3.2 ne se réduiront pas directement aux coordonnées A de Fock-Goncharov dans le cas commutatif. Au lieu de cela, nous obtenons une généralisation des coordonnées de Fock-Goncharov aux GL n (R)-systèmes locaux. Une autre difficulté technique rencontrée est la nécessité de travailler avec des systèmes locaux tordus pour définir les A-coordonnées. Ces systèmes locaux tordus sont définis non pas sur S mais sur le fibré tangent unitaire T ′ S de S, avec pour condition d'avoir la monodromie autour de la fibre de T ′ S → S égale à -Id.

Maintenant que nous avons de bons candidats pour les A-coordonnées généralisées, plusieurs questions se posent. Premièrement, existe-t-il des relations entre les coordonnées définies par rapport à une triangulation fixée ? Deuxièmement, comment calculer les formules de mutations reliant les coordonnées correspondant à deux triangulations différentes ? Pour répondre à ces deux questions, nous utilisons un outil introduit en 2014 par Davide Gaiotto, Gregory W. Moore et Andrew Neitzke dans [START_REF] Gaiotto | Spectral networks[END_REF] appelé réseaux spectraux. Il s'agit d'un outil combinatoire utilisé pour étudier les systèmes locaux encadrés en permettant la définition d'une procédure appelée abélianisation (et son inverse appelée non-abélianisation).

Cette procédure transforme un fibré vectoriel plat de rang n sur une surface ciliée S en un fibré en droites plat sur un certain revêtement ramifié à n feuilles Σ n de S appelé surface spectrale. Il s'agit d'un outil très important en physique théorique pour étudier une certaine classe de théories quantiques des champs topologiques, mais nous n'utiliserons pas tout le potentiel des réseaux spectraux dans ce travail. Initialement, les réseaux spectraux étaient définis comme le graphe critique d'une différentielle quadratique holomorphe sur

INTRODUCTION EN FRANÇAIS

S relevée à la surface spectrale. Cependant, une description géométrique et combinatoire existe et sera plus pratique à utiliser dans notre étude. Pour cela, nous utilisons une description combinatoire de la surface spectrale introduite par Alexander Goncharov et Maxim Kontsevich dans [START_REF] Goncharov | Spectral description of noncommutative local systems on surfaces and non-commutative cluster varieties[END_REF] que nous rappelons dans la Section 2.3.1, et nous ajoutons la définition d'un réseau spectral adapté à cette surface spectrale dans la section 2.3.3. Ensuite, nous devons étendre la définition de la procédure de non-abélianisation pour travailler avec des systèmes locaux tordus. Cette partie est un travail commun avec Eugen Rogozinnikov dans [START_REF] Kineider | On partial abelianization of framed local systems[END_REF] dans le cas n = 2, et se généralise immédiatement à n ≥ 3. Cette extension est décrite dans la section 2.4. Le processus de non-abélianisation lui-même fonctionne exactement de la même manière que le fibré sur la surface spectrale soit un fibré en droites ou un fibré de rang supérieur. Une fois cela fait, nous décrivons une construction inverse au processus de non-abélianisation appelée abélianisation (partielle), qui transforme un fibré vectoriel plat encadré de rang nd sur la surface ciliée S en un fibré vectoriel plat de rang d sur la surface spectrale. Nous la décrivons plus précisément dans la section 2.5. Cela nous permet de décrire la topologie d'un sous-ensemble (en raison des hypothèses de généricité) ouvert de l'espace de modules des GL 2 (R)-systèmes locaux encadré sur une surface ciliée S, où R est une R-algèbre de dimension finie, par exemple l'anneau des matrices n × n à coefficients réels.

Théorème. L'espace de modules des GL 2 (R)-systèmes locaux encadrés tordus sur S qui sont ∆-génériques par rapport à une triangulation ∆ est homéomorphe à l'espace de modules des R × -systèmes locaux tordus sur Σ 2 qui est homéomorphe à (R × ) 1-4χ(S)+2p+ n i /R × où R × agit diagonalement par conjugaison sur (R × ) 1-4χ(S)+2p+ n i .

Cette construction nous permet également d'étudier les systèmes locaux symplectiques dans la Section 2.7 car une matrice dans le groupe symplectique s'écrit naturellement comme une matrice par bloc. Il s'agit d'un travail commun avec Eugen Rogozinnikov dans [START_REF] Kineider | On partial abelianization of framed local systems[END_REF]. Pour cela, nous devons traduire la donnée de la forme symplectique présente dans un système local symplectique sur S en une donnée supplémentaire dans le système local abélianisé. Comme corollaire, nous obtenons aussi une description topologique de l'espace de modules des systèmes locaux encadrés maximaux.

Théorème. Si A est hermitien, alors l'espace de modules des Sp 2 (A, σ)-systèmes locaux encadrés tordus maximaux sur S est homéomorphe à :

(A σ + ) -2χ(S)+p × (A × ) -2χ(S)+2p-1+ n i /A × où A × agit par conjugaison sur (A × ) -2χ(S)+2p-1+ n i et par congruence sur (A σ + ) -2χ(S)+p .

Dans le chapitre 3, nous nous concentrons sur la définition et l'étude d'une généralisation non-commutative des A-coordonnées de Fock-Goncharov. Nous commençons par étudier le cas particulier des GL 2 (R)-systèmes locaux, étant donné que de nombreuses difficultés techniques apparaissant au rang supérieur ne sont pas présentes en rang 2. Dans ce cas, nous montrons que l'algèbre non-commutative introduite par Arkady Berenstein et Vladimir Retakh dans [START_REF] Berenstein | Noncommutative marked surfaces[END_REF] a une représentation par des fonctions rationnelles à valeurs matricielles sur l'espace de modules des GL 2 (R)-systèmes locaux tordus, que nous appelons A-coordonnées. Comme pour les coordonnées commutatives de Fock-Goncharov, il existe une famille de fonctions associées à chaque triangulation de la surface, et il existe des formules de mutations qui expriment les fonctions associées à une triangulation comme des fonctions rationnelles non-commutatives de fonctions associées à une autre triangulation. Cependant, il y a une différence fondamentale avec les coordonnées commutatives de Fock-Goncharov : les fonctions associées à une triangulation fixée ont des relations entre elles, appelées relations triangulaires. Le terme "coordonnées" est donc un abus de langage.

Définition. Soit S une surface ciliée. Nous définissons la R-algèbre unitaire A S engendrée par les symboles x γ et x -1 γ pour tous les arcs idéaux γ sur S (avec la convention x γ = 1 si γ est trivial) avec les relations :

• ∀γ ∈ E(S), x -1 γ x γ = x γ x -1 γ = 1 • ∀f : P 3 → S triangle, x γ 1,3 x -1 γ 2,3 x γ 2,1 = x γ 1,2 x -1 γ 3,2 x γ 3,1 . • ∀f : P 4 → S quadrilateral, x γ 4,2 = x γ 4,3 x -1 γ 1,3 x γ 1,2 + x γ 4,1 x -1 γ 3,1 x γ 3,2 and x γ 2,4 = x γ 2,3 x -1 γ 1,3 x γ 1,4 + x γ 2,1 x -1 γ 3,1 x γ 3,4
Dans le cas R = M d (R), l'espace X des GL 2 (R)-systèmes locaux décorés tordus sur S est une variété algébrique et le sous-ensemble X ∆ des systèmes locaux ∆-génériques est un sous-ensemble ouvert et dense de X. Chaque A-coordonnée associée à ∆ peut être considérée comme une fonction rationnelle sur X ∆ à coefficients dans M d (R).

Théorème. Soit S une surface ciliée. La carte

ψ : A S → Rat(X, M d (R))
x γ → a γ est un homomorphisme d'algèbre.

Pour le démontrer, nous devons calculer à la fois les relations triangulaires et les formules de mutations. En rang 2, nous pouvons calculer les deux en utilisant les réseaux spectraux.

INTRODUCTION EN FRANÇAIS

, . . . , i k ∈ N tel que i 1 +• • •+i k = (k-1)n+1 d'un élément b i 1 ,...,i k ∈ R n qui engendre librement A (1) i 1 ∩ • • • ∩ A (k) i k .
Pour une configuration de drapeaux dans R n , la définition ne pose pas de problème, mais pour un système local tordu, ces bases devraient être des sections plates du système local tordu le long de petites boucles situées à l'intérieur de la surface. Cependant, ces petites boucles ont une monodromie -Id car nous travaillons avec des systèmes locaux tordus, il n'est donc possible de définir que ces sections qu'au signe près. Nous définissons une version légèrement modifiée des carquois habituels associés aux algèbres amassées commutatives, pour prendre en compte le fait qu'une extra-décoration dépend de la triangulation choisie, et doit donc muter en même temps que les coordonnées elles-mêmes. Nous ne décrivons que les mutations nécessaires à un flip, car il nous manque une interprétation géométrique des autres mutations pour les décrire.

Introduction

A representation of the fundamental group of a surface S with negative Euler characteristic into a Lie group G can be seen as a principal bundle over S, allowing topological and geometrical tools to be used to study those representations. The bundles obtained from representations have the special property that the changes of charts are locally constant. This kind of bundle is called a G-local system on the surface S. The representation corresponding to a G-local system is called the monodromy representation. Usually we are more interested in the moduli space representations up to conjugation (or equivalently local systems up to gauge transformation), which is called the character variety. However, taking the quotient by the action of G by conjugation results usually in a non-Hausdorff space. There are two workarounds to this problem: taking a GIT quotient which require algebraic geometry tools, or taking the Hausdorff quotient instead. Both these solutions coincides when G is a complex reductive group which is one of the case covered in the work presented here. A classical example of this approach is the Teichmüller space of the surface S which can be identified with a connected component in the moduli space of SL 2 (R)-local systems on S. This is a very special example where an entire connected component of the character variety is constituted of only discrete and faithful representations. Finding similar connected components in the character variety of other Lie groups has become a very fruitful domain called higher Teichmüller theory. For a detailed account on the subject, see [Wie18,FM22,GW18,BIW10].

In [START_REF] Fock | Moduli spaces of local systems and higher Teichmüller theory[END_REF], Vladimir Fock and Alexander Goncharov introduce cluster varieties to study local systems using an algebraic structure called cluster algebras introduced by Sergei Fomin and Andrei Zelevinsky in [START_REF] Fomin | Cluster algebras. I. Foundations[END_REF] a few years before. The theory introduced by Fock and Goncharov describes G-local systems over a ciliated surface, where G is a split-real semisimple Lie group. Ciliated surfaces are a very generic class of hyperbolic surfaces containing punctured surfaces and ideal polygons, see Section 2.1 for a precise definition. Ciliated surfaces are always non-compact, and their fundamental groups are free groups.

A key property of ciliated surfaces is that they all admit ideal triangulations. An ideal triangulation ∆ of a ciliated surface S is a maximal set of (homotopy classes of) pairwise nonintersecting and non-homotopic, not self-intersecting arcs on S with endpoints on punctures.

Similarly to Penner's coordinates on the decorated Teichmüller space, Fock-Goncharov

13

INTRODUCTION

coordinates require additional data to a local system: a framing or a decoration (see Section 2.2.2 and 2.2.3 for precise definitions). Let G be a semisimple split-real Lie group, B be a Borel subgroup and U be the unipotent radical of B.

Definition.

A framing of a G-local system L is the data for each puncture of a coset in G/B (called a flag) stabilized by the monodromy of L around a small loop around the puncture. A decoration of a G-local system L is the data for each puncture of a coset in G/U (called a decorated flag) stabilized by the monodromy of L around a small loop around the puncture.

The work of Fock and Goncharov revolves around defining a special atlas on an open dense subset of the moduli space of framed (resp. decorated) local systems on a ciliated surface, called X -coordinates (resp. A-coordinates). This special atlas has a chart associated to each triangulation of the ciliated surface and the changes of charts can be computed with a sequence of elementary mutations, each being a birational map. Any two triangulation are related by a finite sequence of flips, so it is enough to compute the change of charts for two triangulations that differ from a flip to be able to compute change of charts from any triangulation to any other. The combinatorics of the sequence of mutations leading to a flip of the triangulation is described by a quiver embedded on the surface, and the mutations formulas are given by the cluster algebra structure. In particular, these birational transformations preserve the positivity of the coordinates. A representation is said to be positive if all of its coordinates in any (hence all) triangulation are positive. This notion of positivity extends the definition given by George Lusztig in [START_REF] Lusztig | Total positivity in reductive groups[END_REF]. In the case G = PGL n (R), the X -coordinates are cross ratios and triple ratios of various lines constructed using the given framing of the local system. In the case G = SL n (R), the A-coordinates are volumes of various bases constructed using the given decoration of the local system. One of the main result of their theory is the identification between the Hitchin component and the subset of positive representations. They also show that positive representations are discrete and faithful, hence that Hitchin components are higher Teichmüller spaces.

The theory of Fock-Goncharov coordinates hold for split-real semisimple Lie groups G, and the framing has to be taken in the complete flag variety G/B where B is the Borel subgroup of G. Another known class of higher Teichmüller space is the space of maximal representations. Maximal representations were defined by Marc Burger, Alessandra Iozzi and Anna Wienhard in [START_REF] Burger | Surface group representations with maximal Toledo invariant[END_REF] for non-closed surfaces. They are representations in a non-compact hermitian Lie group of tube type for which the Toledo number, which is a generalization of the Euler number for representations into PSL 2 (R), takes its maximal value. In [START_REF] Burger | Surface group representations with maximal Toledo invariant[END_REF], Burger-Iozzi-Wienhard show that maximal representations are discrete and faithful, and that they satisfy a positivity property, but the framing in this case takes value in a partial flag variety, like the space of lagrangian subspaces when G = Sp 2n (R).

Unlike Hitchin components which are contractible, the space of maximal representations is in general neither connected nor simply connected. For Lie groups that are both split-real and Hermitian (i.e Sp 2n (R)), the set of maximal representations strictly contains the Hitchin component. A large class of hermitian Lie group of tube type can we expressed as symplectic groups over involutive algebras (see [ABR + 22] or Section 2.7.2 for more details). In the symplectic case, the Toledo number can be computed using the Kashiwara-Maslov index, see [START_REF] Alessandrini | Noncommutative coordinates for symplectic representations[END_REF]. In [START_REF] Alessandrini | Noncommutative coordinates for symplectic representations[END_REF], Daniele Alessandrini, Olivier Guichard, Eugen Rogozinnikov and Anna Wienhard constructed non-commutative generalizations of Fock-Goncharov X and A coordinates for symplectic representations, and showed that the positivity locus is exactly the set of maximal representations. These generalized coordinates take values in square matrix rings, hence are non-commutative. However, they admits strong "clusterlike" structure, like (non-commutative) birational maps between coordinates associated to triangulations differing from a flip.

In order to have a general framework to study both positive representations and maximal representations, Guichard-Wienhard introduced in [GW18, GW22] the notion of Θ-positivity. This notions depend on the data of a parabolic subgroup P Θ of a semisimple Lie group G which is determined by the choice of a subset Θ of restricted roots of G. Not every pair (G, P ) admits a Θ-positive structure, and all the pairs admitting a positive structure are classified in [START_REF] Guichard | Positivity and higher Teichmüller theory[END_REF]. There are four families of Θ-positive structure. The first one is when G is split-real and P = B is the Borel subgroup of G (i.e. when Θ is the set of all roots), and Θ-positive representations are exactly the positive representations introduced by Fock-Goncharov. The second family is G hermitian Lie group of tube type and P is the last root of the Dynkin system of type B n , and in this case the Θ-positive representations coincide with the maximal representations. The two "new" family are respectively for groups locally isomorphic to SO(p, q) with p ̸ = q and for some class of exceptional Lie groups. In both those last cases, the parabolic subgroup is associated to a strict subset of roots. In [START_REF] Guichard | Positivity and representations of surface groups[END_REF], Guichard-Labourie-Wienhard showed that Θ-positive representations are P Θ -Anosov, hence discrete and faithful. The conjecture that Θ-positive representations form higher Teichmüller spaces is still open in the general case at the time of writing, but has been shown for large class of examples, see [START_REF] Guichard | Positivity and representations of surface groups[END_REF][START_REF] Beyrer | Positive surface group representations in po(p,q)[END_REF].

The main idea behind the work presented here is to study the moduli space of framed or decorated local system on a ciliated surface, but with a framing/decoration associated to any parabolic subgroup of a semisimple or reductive Lie group. In particular, we are interested in finding coordinates that share some computational properties with Fock-Goncharov coordinates, namely a combinatorial description that allows the computation of flip mutation formulas from a sequence of elementary mutations. In this work we focus on A-type Lie group, namely SL n (R) (or rather GL n (R), as we will explain below). For technical reasons, we will need to restrict to parabolic subgroups of GL n (R) whose Levi

INTRODUCTION

subgroup is of the form GL d (R) k , with n = kd. This corresponds to a parabolic subgroup conjugated to the subgroup of blockwise upper-triangular matrices where all the diagonal blocks are square of size d. The way we do this is by replacing the field R by a finite dimensional R-algebra R. In all this work, we are mainly interested in the case R = M n (R) of square matrices with real coefficients. Doing so will result in coordinates which take values in some square matrix algebra, thus being generically non-commutative. This comes with several technical difficulties. The first of these difficulties is the very definition of cluster X -coordinates: one of the possible definition of the cross ratio (resp. the triple ratio) extend to the non-commutative case, let us first give those definitions.

Definition. Let L 1 , L 2 , L 3 , L 4 be four lines in R 2 , pairwise transverse. Let π i,j denote the restriction of the canonical projection π i,j : L j → R 2 /L i for all i, j ∈ {1, 2, 3, 4} distinct. Then the cross-ratio of the four lines is

[L 1 : L 2 : L 3 : L 4 ] = π -1 4,1 • π 4,3 • π -1 2,3 • π 2,1 .
Definition. Let (L 1 , P 1 ), (L 2 , P 2 ), (L 3 , P 3 ) be three complete flags in R 3 with dim F i = 1 and dim P i = 2, pairwise transverse. Let π i,j denote the restriction of the canonical projection π i,j : L j → R 2 /P i for all i, j ∈ {1, 2, 3} distinct. Then the triple-ratio of the triple of flags is

r = π -1 3,1 • π 3,2 • π -1 1,2 • π 1,3 • π -1 2,3 • π 2,1 .
In both those definitions, the map L 1 → L 1 is identified with a scalar because its (1 by 1) matrix does not depend on the choice of a basis of L 1 . However, this identification is no longer possible if the dimension of L 1 is higher. The correct generalization of X would then be the conjugacy class of the maps defined above, however we would lose the mutation formulas, as one cannot add or multiply conjugacy classes. Still, those "generalized X -coordinates" retain some of the behavior of their commutative counterpart. Being careful with those technical difficulties, we discuss those in Section 3.1.2.

When considering decorated flags however, the situation is very similar in the commutative setting and in the non-commutative setting. The need for a basis is covered by the decoration. The definitions of cross-ratio and triple-ratio also indicate good candidates for the non-commutative A-coordinates. Indeed, we want the A-coordinates to determine the X -coordinates of the configuration of flags, and in the commutative case the definitions above give the usual formulas relating X -coordinates and A-coordinates, given that the Acoordinates are defined as the matrices of the maps π i,j in the bases given by the decoration of the flags. [START_REF] Kineider | On partial abelianization of framed local systems[END_REF] in the case n = 2, and generalizes immediately to n ≥ 3. This extension is described in Section 2.4. The non-abelianization process itself works exactly the same whether the the bundle over the spectral surface is a line bundle or a higher rank bundle. Once this is done, we describe an inverse construction to the non-abelianization process called (partial) abelianization, which transform a framed flat rank nd vector bundle over the ciliated surface S into a flat rank d vector bundle over the spectral surface. We describe it more precisely in Section 2.5. This allow us to describe the topology of a subset (due to genericity assumptions) of the moduli space of framed GL 2 (R)-local system over a ciliated surface S, where R is a finite dimensional R-algebra, such as the ring of n × n matrices with real coefficients.

Theorem. The moduli space of framed (twisted) GL 2 (R)-local systems on S that are ∆generic with respect to a fixed triangulation ∆ is homeomorphic to the moduli space of (twisted) R × -local systems on Σ 2 which is homeomorphic to (R × ) 1-4χ(S)+2p+ n i /R × where R × acts diagonally by conjugation on (R × ) 1-4χ(S)+2p+ n i .

This construction also allow us to study symplectic local systems in Section 2.7 as a matrix in the symplectic group naturally writes as a two by two block matrix. This is a join

INTRODUCTION

work with Eugen Rogozinnikov in [START_REF] Kineider | On partial abelianization of framed local systems[END_REF]. For this, we need to translate the data of the symplectic form present in a symplectic local system on S into an additional data in the abelianized local system. As a corollary we also get a topological description of the moduli space of maximal framed local systems.

Theorem. If A is Hermitian, then the moduli space of framed (twisted) maximal Sp 2 (A, σ)local systems on S is homeomorphic to:

(A σ + ) -2χ(S)+p × (A × ) -2χ(S)+2p-1+ n i /A ×
where A × acts componentwisely by conjugation on (A × ) -2χ(S)+2p-1+ n i and by congruence on (A σ + ) -2χ(S)+p .

In Chapter 3, we focus on defining and studying a non-commutative generalization of Fock-Goncharov's A-coordinates. First we study the special case of GL 2 (R)-local systems, since a lot of technical difficulties appearing in higher rank are not present in rank 2. In this case, we show that the non-commutative algebra introduced by Arkady Berenstein and Vladimir Retakh in [START_REF] Berenstein | Noncommutative marked surfaces[END_REF] have a representation as matrix valued rational function on the moduli space of decorated twisted GL 2 (R)-local systems, which we call A-coordinates. Similarly to commutative Fock-Goncharov coordinates, there is a family of functions associated to each triangulation of the surface, and there are mutations formulas that express the functions associated to a triangulation as non-commutative rational functions of functions associated to another triangulation. However, there is a fundamental difference with commutative Fock-Goncharov coordinates: the functions associated to a fixed triangulations have relations between them, called triangle relations. The term "coordinates" is thus an an abuse of terminology.

Definition. Let S be a ciliated surface. We define the unitary R-algebra A S generated by the symbols x γ and x -1 γ for all homotopy class of arcs γ joining two punctures on S (with the convention x γ = 1 if γ is trivial) with the relations:

• ∀γ ∈ E(S), x -1 γ x γ = x γ x -1 γ = 1 • ∀f : P 3 → S triangle, x γ 1,3 x -1 γ 2,3 x γ 2,1 = x γ 1,2 x -1 γ 3,2 x γ 3,1
• ∀f : P 4 → S quadrilateral,

x γ 4,2 = x γ 4,3 x -1 γ 1,3 x γ 1,2 + x γ 4,1 x -1 γ 3,1 x γ 3,2 and x γ 2,4 = x γ 2,3 x -1 γ 1,3 x γ 1,4 + x γ 2,1 x -1 γ 3,1 x γ 3,4
We define the non-commutative A-coordinates associated to a decorated GL 2 (R)-local system L as follows: for each ideal arc γ on S (i.e. a homotopy class of arc joining two punctures) from p to q, let γ * L be the pullback of L| γ . This is a trivial flat R 2 -bundle over [0, 1] and the decoration of L induces two flat R-subbundles F p and F q of γ * L. We define

a γ : F p → R 2 /F q
to be the restriction of the canonical projection to F p , identified with its matrix in the bases induces by the decoration. In the case R = M d (R), the space X of decorated twisted GL 2 (R)-local systems on S is an algebraic variety and the subset X ∆ of ∆-generic local systems is an open dense subset of X. Each A-coordinate associated to ∆ can be seen as a rational function on X ∆ with coefficient in M d (R).

Theorem. Let S be a ciliated surface. The map

ψ : A S → Rat(X, M d (R)) x γ → a γ
is an algebra homomorphism.

To show this, we need to compute both the triangle relations and the mutations relations associated with flips of the triangulation. In rank 2, we can compute both with the use of spectral networks. The triangle relations appear naturally in the abelianization process: the A-coordinates are holonomies of certain paths on the abelianized local system, and the triangle relations correspond to holonomies of contractible paths, hence are trivial. For the mutation relation, we take advantage of the fact that the local system on the base surface S does not depend on the triangulation, while the spectral surface and the abelianized local system do. Then by doing abelianizations corresponding to two different triangulations and identifying the right terms, we can express the A-coordinates corresponding to one triangulation in terms of the A-coordinates corresponding to the other one. The same technique can be used to show that given a specific triangulation ∆, any other A-coordinates can be written as a non-commutative Laurent polynomial of coordinates associated to ∆. This property is known as (non-commutative) Laurent phenomenon and is the noncommutative generalization of a well known property of commutative cluster algebras.

Theorem. Let n ≥ 3 and let S be the closed disk with n punctures on the boundary. Let i, j ∈ {1, . . . , n}, i ̸ = j. Then for every triangulation ∆ of S and every decorated twisted GL 2 (R)-local system L that is ∆-generic and such that (F (i) , F (j) ) is generic, the A-coordinate a γ i,j is a non-commutative Laurent polynomial in the A-coordinates (a γ ) γ∈∆ associated to the triangulation ∆.

We also study in Section 3.1.4 and 3.1.5 a type of coordinates in-between the X -coordinates and the A-coordinates, which yield a representation of a subalgebra of A S introduced

INTRODUCTION

in [START_REF] Berenstein | Noncommutative marked surfaces[END_REF]. The A-coordinates presented here restrict to the non-commutative symplectic coordinates introduced in [AGRW22] when defined over a symplectic local system with a symplectic decoration.

We then generalize these coordinates to configurations of flags in R n in Section 3.2 and to twisted GL n (R)-coordinates in Section 3.3. The reason we split this in two different sections is because of a technical difficulty arising from twisted local systems. Indeed, the generalization we describe require additional data to the usual decorated flags. We call that additional data an extra-decoration.

Definition. Let (A (1) , . . . , A (k) ) be a k-tuple of flags in generic position (recall Section 1.2 for the definition of flags in generic position). An extra-decoration of the k-tuple of flags is the data of a decoration of each flag A (i) together with the data for every i 1 , . . .

, i k ∈ N such that i 1 + • • • + i k = (k -1)n + 1 of an element b i 1 ,...,i k ∈ R n that freely span A (1) i 1 ∩ • • • ∩ A (k) i k .
For a configuration of flags in R n this is easy to define, but for a twisted local system, these bases should be sections of the local system along small loops sitting in the interior of the surface. However these small loops have monodromy -Id because we are working with twisted local systems, so this is only possible to define those section up to sign. We define a slightly altered version of the usual quivers associated to commutative cluster algebras, to take into account the fact that an extra-decoration depends on the triangulation chosen, so should mutate alongside the coordinates themselves. We only describe the mutations necessary to a flip, as we lack a geometric interpretation of a general mutation to describe it.

This definition of non-commutative A-coordinates is slightly different than the one introduced by Goncharov-Kontsevich in [START_REF] Goncharov | Spectral description of noncommutative local systems on surfaces and non-commutative cluster varieties[END_REF]. Indeed, the Goncharov-Kontsevich coordinates can be written as quotients of two of the coordinates introduced in this work. In other words, the A-coordinates presented here gives factorization of Goncharov-Kontsevich coordinates.

Another difference is that Goncharov-Kontsevich coordinates only require a decoration of the local systems, whereas the coordinates presented here require this additional extradecoration. The combinatorics of the coordinates presented in this work are much closer to the combinatorics of a cluster algebra, with mutations dictated by an underlying quiver. As such, Goncharov-Kontsevich coordinates can be thought as "in-between" the A-coordinates presented here and non-commutative X -coordinates. It is worth noting that for n = 2 (i.e. for G = GL 2 (R)) the Goncharov-Kontsevich coordinates coincide with the A-coordinates introduced i this work, the differences between the two types of coordinates arising for n ≥ 3.

The work presented in Chapter 3 is a step toward the definition of (possibly non-commutative) cluster structures on moduli spaces of G-local systems with framing in partial flag varieties G/P for more general Lie groups and more general parabolic subgroups than what is already known. While there is no direct reduction from the coordinates presented here to coordinates on SO(p, q) or exceptional Lie groups with their parabolic subgroup P Θ (the two remaining classes of Θ-positive structures), the constructions present in this manuscript should help to understand the expected combinatorics of non-commutative cluster coordinates a general case.

INTRODUCTION

Chapter 1

Algebra and flags

The purpose of this first chapter is to collect basic definitions about flags in free modules over non-commutative algebras. The main example of such non-commutative algebras that we will be interested throughout this work is the algebra of square matrices over a commutative field. This chapter contains elementary linear algebra results that will serve as ground level results in the rest of this manuscript.

Algebra vocabulary

Let R be a finite dimensional unitary associative R-algebra and R × the group of invertible elements of R. For n ≥ 2 we consider R n as a right R-module. Let M n (R) be the ring of all n × n-matrices with entries in R, and GL n (R) be the group of all invertible matrices of M n (R). We see R n as the set of column vectors with coefficients in R, so that GL n (R) acts on R n by left multiplication.

The work presented here could be generalized to a larger class of ring R, but we are formulating it here with R a finite dimensional R-algebra so that R has a "nice" free module theory. In particular, under this assumption a free R-module M has a unique rank k, meaning that when there exists a non-negative integer k such that M ≃ R k , then this integer k (called the rank of M ) is unique. Moreover, if N is a free submodule of M , then the rank of N is necessarily lesser or equal to k, with equality if and only if

N = M . Definition 1.1.1. Let 0 ≤ k ≤ n. A k-dimensional subspace of R n is a R-submodule F isomorphic to R k that is a direct factor of R n , i.e. R n ≃ F ⊕ R n-k . A 1-dimensional
subspace of R n will sometimes be called an R-line. We denote the space of R-lines of R n by P(R n ).

Remark

1.1.2. If F is a i-dimensional subspace of R n , then the quotient R n /F is isomorphic to R n-i . However a R-submodule F satisfying F ≃ R k and R n /F ≃ R n-k for some 1 ≤ k ≤ n -1 is not necessarily a k-dimensional subspace.
We denote by x+F the image of an element x ∈ R n by the canonical projection R n → R n /F .

Definition 1.1.3. We make the following definitions:

• An n-tuple (x 1 , . . . , x n ) for x 1 , . . . , x n ∈ R n is called basis of R n if the map R n → R n (a 1 , . . . , a n ) → n i=1 x i a i is an isomorphism of right R-modules. • An element x ∈ R n is called regular if it freely spans an R-line of R n .
• Regular elements x 1 , . . . , x k ∈ R n for k ≤ n are called linearly independent if they freely span a k-dimensional subspace of R n , i.e. there exists x k+1 , . . . , x n such that

(x i ) 1≤i≤n is a basis of R n . • A k-dimensional subspace F and a (n-k)-dimensional subspace G are called transverse if F ⊕ G = R n as right R-modules. Remark 1.1.4. Let x ∈ R n . The map m x : R → R n a → xa
is an isomorphism of right R-module if and only if x is regular, and if x is not regular then the kernel of m x is a R-subalgebra of R.

In this work, we are mainly interested in the case when R is the algebra M d (K) of square matrices of size d ≥ 2 over a commutative field K, or a subalgebra of it.

= R or K = C then R × is dense in R and GL n (R) is dense in M n (R).

Configuration of flags

One of the reasons the notion of flags became central in the study of character varieties is because of its relations with Hitchin components. Indeed, studying the action of a split real Lie group on a space of (complete) flags proved to be useful to determine faithful and discreteness properties, see [START_REF] Fock | Moduli spaces of local systems and higher Teichmüller theory[END_REF][START_REF] Guichard | Composantes de Hitchin et représentations hyperconvexes de groupes de surface[END_REF]. The goal of this work is to work with partial flags instead of complete ones. A partial flag in R nd with subspaces of dimensions (0, d, 2d, . . . , (n -1)d, nd) is the same as a complete flag in M d (R) n , which lead to the following definitions.

Definition 1.2.1. A (complete) flag F in R n is a sequence F 0 ⊂ F 1 ⊂ • • • ⊂ F n-1 ⊂ F n where every F i is a i-dimensional subspace of R n . A decoration of a flag F is the data of (f 1 , . . . , f n ) such that f i ∈ F i /F i-1 freely spans F i /F i-1 for all 1 ≤ i ≤ n.
Definition 1.2.2. A k-tuple of flags (F (1) , . . . , F (k) ) is said to be in generic position if :

• For every i 1 , . . . , i k ∈ N such that i 1 + • • • + i k = (k -1)n + r with 0 ≤ r ≤ n, the R-submodule F (1) i 1 ∩ • • • ∩ F (k) i k ⊂ R n is a r-dimensional subspace of R n • For every i 1 , . . . , i k ∈ N such that i 1 + • • • + i k = (k -1)n + 1 and for all 1 ≤ m ≤ k,
the canonical projection π m : F

(1)

i 1 ∩ • • • ∩ F (k) i k → F (m) im /F (m)
im-1 is an isomorphism.

The group GL n (R) acts naturally on the space of k-tuple of flags in R n in generic position. We denote by Conf R,k (n) the quotient of this space by the GL n (R) action.

Kashiwara-Maslov map and the exchange relation

This section contains most of the generic linear algebra results that will play a fundamental role in the definition of both our non-commutative A-coordinates (Section 3.1.2 and 3.2) and the abelianization procedure (Section 2.5).

Let (A, B, C) be a triple of flags in generic position. For all i, j, k ∈ N such that i + j + k = 2n + 1, we define the following maps by restriction of the canonical projections

A i → A i /A i-1 , B j → B j /B j-1 and C k → C k /C k-1 : a A i,j,k : A i ∩ B j ∩ C k → A i /A i-1 a B i,j,k : A i ∩ B j ∩ C k → B j /B j-1 a C i,j,k : A i ∩ B j ∩ C k → C k /C k-1 Remark 1.3.1. If i = 1, then j = k = n and A i ∩ B j ∩ C k = A 1 .
Then the map a A 1,n,n : A 1 → A 1 is the identity. Notice also that if k = n, then i + j = n + 1 and the maps a A i,j,n : A i ∩ B j → A i /A i-1 and a B i,j,n : A i ∩ B j → B j /B j-1 do not depend on the flag C.

For i, j, k ∈ N such that i + j + k = 2n + 2, we define the 2-dimensional subspace P i,j,k = A i ∩ B j ∩ C k . There is a map

A i-1 ∩ B j ∩ C k ⊕ A i ∩ B j-1 ∩ C k ⊕ A i ∩ B j ∩ C k-1 → P i,j,k (x, y, z) → x + y + z
and we denote by K i,j,k its kernel. This space is endowed with three canonical projections

p A i,j,k : K i,j,k → A i-1 ∩ B j ∩ C k , p B i,j,k : K i,j,k → A i ∩ B j-1 ∩ C k , p C i,j,k : K i,j,k → A i ∩ B j ∩ C k-1 Lemma 1.3.2.
The maps p A i,j,k , p A i,j,k and p A i,j,k defined above are isomorphisms. In particular K i,j,k is a R-line.

Proof. We will show that p A i,j,k is an isomorphism, the other case being similar. First, note that the R lines

A i-1 ∩ B j ∩ C k , A i ∩ B j-1 ∩ C k and A i ∩ B j ∩ C k-1 have pairwise trivial intersection. Indeed, (A i-1 ∩ B j ∩ C k ) ∩ (A i ∩ B j-1 ∩ C k ) = A i-1 ∩ B j-1 ∩ C k = 0 because (i -1) + (j -1) + k = 2n
and the triple (A, B, C) is in generic position. Then we have

P i,j,k = A i ∩ B j-1 ∩ C k ⊕ A i ∩ B j ∩ C k-1
because both terms are free R-module of rank 2 and the right-hand side is a submodule of the left-hand side. So given

x ∈ A i-1 ∩ B j ∩ C k , there exists a unique pair (y, z) ∈ A i ∩ B j-1 ∩ C k ⊕ A i ∩ B j ∩ C k-1 such that x = y + z, i.e. (x, -y, -z) ∈ K i,j,k . So p A i,j,k (x, -y, -z) = x and p A i,j,k is an isomorphism. Proposition 1.3.3. Let i, j, k ∈ N such that i + j + k = 2n + 2. The following diagram anti-commutes, i.e. a C i-1,j,k • p A i,j,k = -a C i,j-1,k • p B i,j,k : A i-1 ∩ B j ∩ C k a C i-1,j,k ' ' K i,j,k p A i,j,k 7 7 p B i,j,k ' ' C k /C k-1 A i ∩ B j-1 ∩ C k a C i,j-1,k 7 7 Proof. Let (x, y, z) ∈ K i,j,k . Then a C i-1,j,k (p A i,j,k (x, y, z)) = a C i-1,j,k (x) = x + C k-1 and a C i,j-1,k (p B i,j,k (x, y, z)) = a C i-1,j,k (y) = y + C k-1 . Since x + y + z = 0 and z ∈ A i ∩ B j ∩ C k-1 ⊂ C k-1 , we have x + C k-1 = -y + C k-1 .
More generally for all i, j, k ∈ N such that i + j + k = 2n + 2, the following diagram anti-commutes, i.e. the monodromy around each cell is equal to -Id:

A i-1 ∩ B j ∩ C k a C i-1,j,k u u a B i-1,j,k ) ) C k /C k-1 B j /B j-1 K i,j,k p A i,j,k O O p B i,j,k u u p C i,j,k
) )

A i ∩ B j-1 ∩ C k a C i,j-1,k O O a A i,j-1,k
) )

A i ∩ B j ∩ C k-1 a B i,j,k-1 O O a A i,j,k-1 u u A i /A i-1
In particular, we have

(a B i-1,j,k ) -1 • a B i,j,k-1 • (a A i,j,k-1 ) -1 • a A i,j-1,k • (a C i,j-1,k ) -1 • a C i-1,j,k = -Id .
In the case n = 2 a flag is just a R-line and the map µ

B,C A := a A 2,2,1 (a B 2,2,1 ) -1 a B 1,2,2 : C 1 → R 2 /C 1 is called the Kashiwara-Maslov map of the triple of R-lines (A 1 , B 1 , C 1 ). Proposition 1.3.3 imply that µ B,C A = -µ C,B
A . When n = 2 we can also describe an additional relation between these isomorphisms for a configuration of 4 flags in generic position: Proposition 1.3.4. Let F 1 , F 2 , F 3 , F 4 be four R-lines in R 2 in generic position. For all i ̸ = j ∈ {1, 2, 3, 4} let a j,i : F i → R 2 /F j be the restriction of the canonical projection R 2 → R 2 /F j to F i . These maps are isomorphisms and we have the following relation:

a 2,4 = a 2,3 a -1 1,3 a 1,4 + a 2,1 a -1 3,1 a 3,4 .
Proof. These maps are isomorphisms because of the genericity condition. Let b ∈ F 4 freely spanning F 4 . Since F 1 ⊕ F 3 = R 2 by genericity again, there exists

x 1 ∈ F 1 and x 3 ∈ F 3 such that b = x 1 + x 3 . The left-hand term is then a 2,4 (b) = b + F 2 . The first term of the right-hand side is a 2,3 a -1 1,3 a 1,4 (b) = a 2,3 a -1 1,3 (b + F 1 ) = a 2,3 a -1 1,3 (x 1 + x 3 + F 1 ) = a 2,3 a -1 1,3 (x 3 + F 1 ) = a 2,3 (x 3 ) = x 3 + F 2
and similarly the second term is

a 2,3 a -1 1,3 a 1,4 (b) = x 1 + F 2 hence the relation.
Lastly, we describe an additional relation in the case n = 3. Proposition 1.3.5. Let (A, B, C, D) be four flags in R 3 in generic position. The following diagram anti-commutes:

R 3 /(A 2 ∩ B 2 ⊕ C 2 ∩ D 2 ) ) ) B 2 ∩ C 2 5 5 ) ) A 2 ∩ D 2 R 3 /(A 2 ∩ C 2 ⊕ B 2 ∩ D 2 ) 5 5
where all the maps are restrictions of the canonical projections. All these maps are isomorphisms. We call the top map

T + AD,BC : B 2 ∩ C 2 → A 2 ∩ D 2
and the bottom map

T - AD,BC : B 2 ∩ C 2 → A 2 ∩ D 2
Proof. Let b freely spanning B 2 ∩ C 2 . By genericity we have

A 2 ∩ B 2 ⊕ C 2 ∩ D 2 ⊕ A 2 ∩ D 2 = R 3 since A 2 ∩ B 2 ⊕ A 2 ∩ D 2 = A 2 because A 2 ∩ B 2 and A 2 ∩ D 2 are two transverse R-lines in A 2 which is a 2-dimensional subspace, and A 2 ∩ (C 2 ∩ D 2 ) is trivial by genericity. Similarly, we have A 2 ∩ C 2 ⊕ B 2 ∩ D 2 ⊕ A 2 ∩ D 2 = R 3 so there exists unique x ∈ A 2 ∩C 2 , y ∈ B 2 ∩D 2 , x ′ ∈ A 2 ∩B 2 , y ′ ∈ C 2 ∩D 2 and z, z ′ ∈ A 2 ∩D 2 such that b = x + y + z = x ′ + y ′ + z ′
We want to show that z + z ′ = 0. We have y

+ z ∈ D 2 but since y + z = b -x ∈ C 2 , we have y + z ∈ C 2 ∩ D 2 . Similarly x ′ + z ′ ∈ A 2 ∩ C 2 . So b = x + (y + z) = (x ′ + z ′ ) + y ′ are two decomposition of b in A 2 ∩ C 2 ⊕ C 2 ∩ D 2 , so they must coincide. This means that x = x ′ + z ′ , so b = x ′ + z ′ + y + z. So z + z ′ = b -x ′ -y ∈ B 2 , and z + z ′ ∈ A 2 ∩ D 2 which means that z + z ′ = 0 since A 2 ∩ D 2 ∩ B 2 = 0.
Chapter 2

Local systems and partial (non-)abelianization

In this chapter, we give all the definitions of framed and decorated local systems we need in this work. To understand the combinatorial properties of the non-commutative cluster coordinates we will define in Chapter 3, we start from a geometrical point of view. In 2014 Davide Gaiotto, Gregory W. Moore and Andrew Neitzke introduce in [START_REF] Gaiotto | Spectral networks[END_REF] a combinatorial tool called spectral networks to describe a procedure called abelianization and its inverse called non-abelianization. The purpose of these constructions is to transform a flat vector bundle over a surface into a flat line bundle over a ramified covering of the surface.

In this line bundle, many of the geometric invariant of the initial bundle can be seen. This is true in particular for the Fock-Goncharov coordinates of the initial bundle, which we find as monodromies of some special curves in the associated line bundle, see [START_REF] Gaiotto | Spectral networks and snakes[END_REF][START_REF] Hollands | Spectral networks and Fenchel-Nielsen coordinates[END_REF]. The main idea behind this chapter is that the abelianization procedure can be carried "partially", i.e. transforming a vector bundle of high rank over a surface into a vector bundle of lower rank over a ramified covering but not necessarily a line bundle. In this lower rank bundle, looking at the monodromies of the curves representing the Fock-Goncharov coordinates in the abelian case should naturally give rise to non-commutative analogs of Fock-Goncharov coordinates. However since we want to define non-commutative cluster A-coordinates, we need to work with twisted local system, as introduced in [START_REF] Fock | Moduli spaces of local systems and higher Teichmüller theory[END_REF]. Thus, the first step is to extend the construction of Gaiotto-Moore-Neitzke to twisted local system. This is a joint work with Eugen Rogozinnikov in [START_REF] Kineider | On partial abelianization of framed local systems[END_REF], together with a direct application of this construction to the study of symplectic local systems and maximal representations.

Ciliated surfaces, topological and combinatorial data

To study local system on surfaces, we first need to describe the kind of surfaces we want to consider. The construction of cluster coordinates on moduli spaces of local system in [START_REF] Fock | Moduli spaces of local systems and higher Teichmüller theory[END_REF] uses a very general class of surfaces called ciliated surfaces. Since our goal is to extend this construction we will naturally deal with the same kind of surfaces.

Let S a smooth compact orientable surface with boundary. Let P ⊂ S be a finite non-empty set such that every connected component of the boundary ∂S of S contains at least one point in P. We call the elements of P punctures. We split the set of all punctures in two subsets: the set of external punctures P ext = P ∩ ∂S and the set of internal punctures P int = P\P ext . Let S = S\P. We call a surface S obtained this way a ciliated surface. A ciliated surface will be called hyperbolic if it is not the sphere with one or two (internal) punctures nor the closed disk with one or two external punctures and no internal punctures. Every hyperbolic ciliated surface admits a hyperbolic metric of finite volume with totally geodesic boundary, but the choice of such a metric does not matter in this work. For every such hyperbolic structure, all the internal punctures are cusps and all boundary curves are (infinite) geodesics. Once equipped with a hyperbolic structure as above, the universal covering S ′ of S can be seen as a closed convex subset of the hyperbolic plane H 2 with totally geodesic boundary, which is invariant under the natural action of π 1 (S) on H 2 by the holonomy representation. Punctures of S are lifted to points of the ideal boundary of H 2 which we call punctures of S ′ and denote their set by

P ′ ⊆ ∂ ∞ S ′ ⊆ ∂ ∞ H 2 . Notice, if S does not have boundary, then S ′ is the entire H 2 .
Example 2.1.1. An important example of a ciliated surface is the k-gon for any integer k ≥ 3 which is a (closed) disk with k external punctures and no internal punctures. In the following, a 3-gon will be called a triangle and a 4-gon will ce called a quadrilateral.

Later on we will need to replace the base surface S with its unit tangent bundle (see Section 2.2.3). Let T S the tangent bundle of S and T S\ {0} the tangent bundle without the zero section. The group R * + of positive real numbers acts on T S\ {0} by scalar multiplication. With a slight abuse of terminology, we call unit tangent bundle the quotient T ′ S = (T S\ {0}) /R * + . This is a bundle over S of fiber S 1 . We will write an element of T ′ S as an ordered pair (x, v) with x ∈ S and v a non-zero vector in T x S, identified with the half-line it spans. Every smooth path γ : [0, 1] → S lifts to a path

T ′ γ = (γ, γ ′ ) : [0, 1] → T ′ S.
We fix an orientation on S. In the figures, the surface S will be oriented clockwise. For every internal puncture p ∈ P int we choose a neighborhood V p of p in S such that V p is diffeomorphic to an open disk with one internal puncture, and such for p ̸ = q two distinct punctures the neighborhoods V p and V q are disjoint. We fix a smooth oriented loop β p around p in V p such that β p is homotopic to p and the orientation of β p is such that p is on its right. We call T ′ β p = (β p , β ′ p ) the lift of β p to T ′ S. For every external puncture p ∈ P ext we choose a neighborhood V p of p in S such that V p is diffeomorphic to (x, y) ∈ R 2 |x 2 + y 2 < 1, x ≥ 0, (x, y) ̸ = (0, 0) (we call such a neighborhood a punctured half-disk ). We choose the neighborhoods V p to be pairwise disjoints. We also choose a smooth path β p inside V p going from one of the boundary components surrounding p to the other one, again oriented such that p is on its right. We also denote by T ′ β p its lift to T ′ S.

An ideal triangulation (which we will simply call triangulation) of S is a maximal set of (homotopy classes of) pairwise non-intersecting and non-homotopic, not self-intersecting arcs

γ i : [0, 1] → S i∈I such that ∀i ∈ I, ∀t ∈ [0, 1], γ i (t) ∈ P ⇔ t ∈ {0, 1}
. By maximality, the complement in S of a triangulation ∆ is a disjoint union of open disks which we call triangles of ∆. Note that the boundary components of S (i.e. the connected components of ∂S\P ext ) are arcs present in every triangulation of S. We call those external edges of the triangulation, and we call the other arcs internal edges. Note that the endpoints of edges of ∆ are punctures. For a triangle t of ∆ with vertices p, q, r ∈ P we will sometimes write t = (p, q, r) even though the data of the vertices does not in general determine the triangle t.

Remark 2.1.1. It is well known (see [START_REF] Hatcher | On triangulations of surfaces[END_REF]) that any two triangulations of S can be related by a finite sequence of flips. A flip is the change of triangulation obtained by changing the diagonal of a quadrilateral that is part of the triangulation.

Framed and decorated local systems

Local systems

Let X be a smooth manifold and let n ≥ 1. Definition 2.2.1. A GL n (R)-local system on X is a R n -bundle L over X with locally constant transitions functions, i.e. there exists an open covering X = i∈I U i such that

L| U i α i ≃ U i × R n
and for every i, j ∈ I such that U i ∩ U j ̸ = ∅ (up to refining the covering we can assume U i ∩ U j to be connected), there exists g i,j ∈ GL n (R) such that the following diagram commutes:

(U i ∩ U j ) × R n α i Id ×g i,j / / (U i ∩ U j ) × R n α j L| U i ∩U j Id / / L| U i ∩U j
Remark 2.2.2. This is the same as a locally constant sheaf on X of fiber R n . Given a local system L on X, we will denote by L| Y its restriction to a subset Y ⊂ X and by L x its fiber above x ∈ X. Definition 2.2.3. Let U be an open subset of X. A regular R k -subbundle or k-dimensional subbundle F of a GL n (R)-local system L over U is a subbundle of L| U such that for every p ∈ U there exists a neighborhood U p of p and a trivialization

Φ p : F| Up → U p × R n such that Φ p (F | Up ) = U p × G where G is an k-dimensional subspace in R n . A section v : U → F is regular if its span vR is a regular R 1 -subbundle of F| U . Definition 2.2.4. Let L 1 and L 2 be two GL n (R)-local systems over X. A morphism φ : L 1 → L 2 of
vector bundles is a morphism of local systems if there exists an open covering X = i∈I U i such that both L 1 and L 2 are trivial over U i and such that the family of linear maps (g(x) :

R n → R n ) x∈U i induced by U i × R n ≃ L 1 | U i φ -→ L 2 | U i ≃ U i × R n is locally constant.
The following results relate those geometrical objects to more algebraic objects, namely representations of groups. These are well-known results, the reader can find the proofs in [START_REF] Deligne | Equations Différentielles à Points Singuliers Réguliers[END_REF].

Proposition 2.2.5. A GL n (R)-local system on a simply connected manifold X is isomorphic to the trivial bundle X × GL n (R).

Proposition 2.2.6 (Riemann-Hilbert correspondence). The set of GL n (R)-local systems on X up to isomorphism is in 1:1 correspondence with the set of representations ρ : π 1 (X) → GL n (R) up to the action of GL n (R) by conjugation.

Definition 2.2.7. Let L be a GL n (R)-local system on X, and let γ : [0, 1] → X be a path on X. Let x = γ(0) and y = γ(1). The bundle γ * L is trivial since [0, 1] is simply connected, we fix an isomorphism γ * L ≃ [0, 1] × R n . Hence we can define an isomorphism of R-module

φ γ : L x → L y such that φ γ (v) = v ′ if and only if γ * (v) = (0, v ′′ ) and γ * (v ′ ) = (1, v ′′ ) for some v ′′ ∈ R n .
We call the map φ γ the parallel transport of L along γ. This map only depends on the homotopy class of γ. The parallel transport along a path will sometimes be called the holonomy of L along γ, and in the special case when the path is a loop we will instead call the parallel transport the monodromy of L around γ. Definition 2.2.8. Let L be a GL n (R)-local system on X, and let γ : [0, 1] → X be a path on X. Let x = γ(0) and y = γ(1). The bundle γ * L is trivial since [0, 1] is simply connected, we fix an isomorphism γ * L ≃ [0, 1] × R n . Hence we can define an isomorphism of R-module

φ γ : L x → L y such that φ γ (v) = v ′ if and only if γ * (v) = (0, v ′′ ) and γ * (v ′ ) = (1, v ′′ ) for some v ′′ ∈ R n .
We call the map φ γ the parallel transport of L along γ. This map only depends on the homotopy class of γ.

Framed local systems

The framework of Fock-Goncharov requires to work with additional data to a local system on a surface. We will add to the local system the data of a flag at each puncture to get a framed local system. This impose a monodromy condition on the local system around internal puncture.

Let S be an hyperbolic ciliated surface.

Definition 2.2.9. Let L a GL n (R)-local system on S. A framing of L is the data of for each puncture p of n + 1 subbundles

F (p) 0 ⊂ • • • ⊂ F (p) n of L| βp , with F (p) i being a i-dimensional subbundle of L| βp . The data of a GL n (R)-local system together with a framing is called a framed GL n (R)-local system on S.
Remark 2.2.10. For every puncture p, F (p) 0 is the zero section of L| βp and

F (p) n = L| βp .
Remark 2.2.11. For a GL n (R)-local system on a hyperbolic ciliated surface S to admit a framing, the monodromy around β p for every internal puncture p must be upper triangular in some basis.

Remark 2.2.12. Let L be a framed local system on S. Let ∆ be a triangulation of S and let t be a triangle of ∆. Since L| t is trivial we have in any fiber L x over x ∈ t a triple of flags given by the parallel transport of F (p) , F (q) and F (r) . Definition 2.2.13. Let L be a framed GL n (R)-local system on a ciliated surface S, and let ∆ be a triangulation of S. We say that the framing of L is ∆-generic if for every triangle t = (p, q, r) of ∆ and for any point x ∈ t, the triple of flags defined by parallel transport of F (p) , F (q) and F (r) to L x is in generic position.

Note that in a contractible subset of S like a triangle t in a triangulation ∆ of S, for every x, y ∈ t and F ⊂ L x all the parallel transports of F from x to y in t define the same subspace F ′ ⊂ L y . Given a triangulation, a framing of L can thus be extended to a triple of flags in every fiber of L.

Twisted and decorated local systems

The data of a framed local system is enough to carry the abelianization/non-abelianization constructions, but the moduli space of framed local system in the commutative setting carry X -coordinates, which are less well behaved in the non-commutative setting. To define A-coordinates, we need even more additional data, called a decoration: a basis of the graded of each flags of the framing. Very few local systems on a ciliated surface admit a decoration, hence we need to consider a slight modification of local systems called twisted local systems, introduced in [START_REF] Fock | Moduli spaces of local systems and higher Teichmüller theory[END_REF] and used in numerous work in the field like [START_REF] Guichard | Parametrizing positive representations[END_REF] or [START_REF] Goncharov | Spectral description of noncommutative local systems on surfaces and non-commutative cluster varieties[END_REF].

Definition 2.2.14. Let S be a ciliated surface and let T ′ S be its unit tangent bundle. A twisted GL n (R)-local system on S is a GL n (R)-local system on T ′ S such that the monodromy around the loop going once around the fiber of T ′ S → S is -Id.

We define a framing of a twisted local system the same way as for not twisted ones, except the flag subbundles are now taken above the lifted peripheral paths T ′ β p . More precisely, let L a twisted GL n (R)-local system on S. A framing of L is the data for each puncture

p of n + 1 subbundles F (p) 0 ⊂ • • • ⊂ F (p) n of L| T ′ βp such that for all 0 ≤ i ≤ n, F (p) i is a i-dimensional subbundle.
The data of a GL n (R)-local system together with a framing is called a framed twisted GL n (R)-local system on S.

Note that since the monodromy of a twisted local system L around a fiber of T ′ S → S is -Id, it preserves any subspace of a fiber of L. Thus as in the non-twisted case, given a triangulation of S we can define a triple of flags in every fiber of L.

Again, we define ∆-generic framing for twisted local system in a similar way to before. Let L be a framed twisted GL n (R)-local system on a ciliated surface S, and let ∆ be a triangulation of S. We say that the framing of L is ∆-generic if for every triangle t = (p, q, r) of ∆ and for any point x ∈ t, the triple of flags defined by parallel transport of F (p) , F (q) and F (r) to L v is in generic position, where v ∈ T ′

x S. The fundamental group of T ′ S is an extension of π 1 (S) by π 1 (S 1 ) = Z:

0 → Z → π 1 (T ′ S) → π 1 (S) → 0
and this sequence splits because π 1 (S) is a free group. We denote by π s 1 (S) the quotient of π 1 (T ′ S) by the subgroup 2Z ⊂ Z = π 1 (S 1 ).

Proposition 2.2.15. The set of twisted GL n (R)-local systems on S up to isomorphism is in one-to-one correspondence with the set of representations ρ : π s 1 (S) → GL n (R) such that ρ(δ) = -Id up to the action of GL n (R) by conjugation, where δ is the loop going once around the fiber of T ′ S → S in any direction.

Remark 2.2.16. If the ciliated surface S does not have any internal puncture, S embeds into T ′ S via the choice of any non-vanishing vector field agreeing with the paths T ′ β p for all internal puncture p. Thus, twisted GL n (R)-local system and GL n (R)-local system are equivalent on surfaces without internal punctures. Definition 2.2.17. Let L be a framed twisted GL n -local system on S. A decoration of L is the data for every puncture p and for every

1 ≤ i ≤ n of a flat section b (p) i of F (p) i /F (p) i-1 along T ′ β p that freely spans F (p) i /F (p)
i-1 . The data of a framed twisted GL n (R)-local system together with a decoration is called a decorated twisted GL n (R)-local system. Remark 2.2.18. For a twisted GL n (R)-local system on S to admit a decoration, the monodromy around every loop T ′ β p must be upper triangular unipotent in some basis.

Spectral data

In this section we introduce the necessary tools for the abelianization/non-abelianization construction, namely the spectral surface in Section 2.3.1 which is a type of ramified covering and the spectral network in Section 2.3.3. We also define a few special curves on the spectral surface that will be useful in Section 3.3 to define the cluster coordinates.

Ramified coverings

We first introduce some vocabulary about ramified coverings. We will only consider finite ramified covering.

Definition 2.3.1. Let X, X be two smooth manifolds and let n ≥ 2. A continuous surjective map π : X → X is called a n-fold (topological) ramified covering if there exists a nonempty finite set B ⊂ X such that π| π -1 (X\B) : π -1 (X\B) → X\B is a n-fold covering and every b ∈ B has strictly less than n preimages. The subset B above is called the branch locus of π, and a point b ∈ B is called a branch point. If a branch point b ∈ B has exactly n -1 preimages we call it a simple branch point, and if all the branch points are simple the ramified covering is called simple. Given a branch point

b ∈ B, a preimage b ∈ π -1 (b) is called a ramification point if in a neighborhood U of b the map π| U \ b : U \ b → π(U )\b is a k-fold covering with k ≥ 2.
In this section we will give a combinatorial construction of the spectral surface for bundles of rank n ≥ 2 introduced in [GK22] by A. Goncharov and M. Kontsevich. We recall the construction of the spectral cover given in [START_REF] Goncharov | Spectral description of noncommutative local systems on surfaces and non-commutative cluster varieties[END_REF] and introduce a few notations, and we will use this construction in Section 2.3.3 to construct a spectral network associated to this simple ramified n-covering. Let S a ciliated surface, endowed with a triangulation ∆. For every integer n ≥ 2 we define a bipartite graph Γ n on S by gluing for all triangles t of S elementary pieces Γ (t) n described as follows. We can fix for each triangle t of ∆ a diffeomorphism φ t between t and the euclidean triangle T in R 3 that is the convex hull of the points of coordinates (0, n + 1, n + 1), (n + 1, 0, n + 1) and (n + 1, n + 1, 0). This euclidean triangle T can be decomposed into smaller triangles as in Figure 3.1. We call this subdivision into smaller triangles a n + 1-subdivision of T. On each side of T there are n + 1 smaller upward triangles. There is a black vertex of Γ (t) n on every point of coordinates (a, b, c) with a + b + c = 2n + 2 and a, b, c positive integers (i.e. every vertex of smaller triangles except for the vertices of the big triangle T). There is a white vertex of Γ (t) n inside every downward small triangle. Then for every downward small triangle, there is an edge in Γ Then for every pair of triangles t and t ′ that share an edge e of the triangulation, we identify the corresponding black vertices of Γ for two adjacent triangles t and t ′ .

The white vertices are trivalent and the black vertices are either univalent, bivalent or trivalent: the univalent and bivalent black vertices will be called external and the trivalent black vertices will be called internal. The graph Γ n is embedded into S so every vertex inherits a cyclic order on the edges incident to it from the orientation of S. Let γ = (v 1 , . . . , v r ) be a path on Γ n and 1 < i < r. We say γ turns left (resp. turns right) at v i if the edges

(v i-1 , v i ) and (v i , v i+1 ) (resp. (v i , v i+1 ) and (v i-1 , v i ))
are consecutive in that order with respect to the cyclic ordering at v i . Since the graph Γ n is embedded into S, a path in Γ n will be identified with its image in S.

A zig-zag path on Γ n is a path (v 1 , . . . , v r ) that turns right at every white vertex and turns left at every black vertex. In particular, every oriented edge of Γ n belong to exactly one maximal zig-zag path. Since the graph Γ n is finite, a zig-zag path can be extended either to a cycle on Γ n , every extensions of this cycle being periodic, or to a path connecting two boundary components of S. From the construction of Γ n , we get the following proposition: Proposition 2.3.2. Let γ be a zig-zag path on Γ n . Then the image of γ in S is either:

• homeomorphic to an oriented circle S 1 , and S\γ has two connected component, one being homeomorphic to an open disk with one puncture. We denote this connected component S γ and the puncture inside it s γ . The puncture s γ is on the right of the path γ.

• a path without self-intersection, and S\γ has two connected component, one being homeomorphic to a punctured half-disk. We denote this connected component S γ and the puncture inside it s γ .

Let γ a zig-zag path on Γ n and S γ the closure in S of S γ . The boundary of S γ is the image of γ in S, in particular it does not contain any punctures so S γ is homeomorphic to either a closed disk with one puncture in its interior or a closed punctured half-disk.

Lemma 2.3.3. Let n ≥ 2. For every puncture s of S, there is exactly n zig-zag paths γ

(1) s , . . . , γ (n) s in Γ n such that s = s γ (i) s
for all 1 ≤ i ≤ n. Moreover, up to reordering, for all

1 ≤ i < j ≤ k we have γ (i) s ⊂ S γ (j) s .
Figure 3.4: The three zig-zag paths γ

(1)

s , γ (2)
s and γ

(3)

s in Γ (t)
3 that circle around the puncture s. The subset of t above a path γ

(i) s is contained in S γ (i) s .
We define Σ n as the topological surface obtained by gluing punctured disks to Γ n : along every zig-zag path γ in Γ n we glue a copy of S γ . We define π : Σ n → S as follows: the image of a point in Γ n is the immersion in S, and every cell S γ is endowed with a map S γ → S given by the inclusion. For all puncture s of S, we denote by s i the puncture inside S γ (i)

s in Σ n .
Proposition 2.3.4. The map π : Σ n → S is a is a surface, and a ramified n-fold covering with simple ramification points. The ramification points are the internal black vertices of Γ n , we denote the set of all ramification points in Σ n by B, and the set of branch points of S by B = π(B).

Proof. Every edge of Γ n is part of exactly two zig-zag paths so Σ n is a topological surface in the neighborhood of a point in the interior of every edge of Γ n and in the neighborhood of an external black vertex. Every white vertex is on the boundary of exactly three zig-zag paths, each cell S γ being glued to a pair of adjacent edges (see Figure 3.5). The neighborhood of a white vertex is then a surface. In all these case, we also see that the covering π is regular. In the neighborhood of every internal black vertex Σ n is also a surface, but the covering π is simply ramified at every internal black vertex (see Figure 3.5). Note that inside a triangle t of ∆ with vertex s, q, r, each white vertex v of Γ n is uniquely determined by the triple of integers

(i q , i r , i s ) such that v ∈ γ (iq) q ∩ γ (ir) r ∩ γ (is)
s . This triple satisfies i q + i r + i s = 2n + 1.

In the case n = 2, we can describe more precisely the ramified covering. Let S be an hyperbolic ciliated surface and let ∆ be a triangulation of S. We can endow S with a Euclidean structure with conical points by choosing for each triangle t of ∆ an orientation preserving diffeomorphism φ t : T → t where T is the Euclidean triangle in R 2 = C with vertices 1, j = e 2iπ 3 and j 2 . Then for each gluing of two triangles (not necessarily distinct) in S, glue the corresponding Euclidean triangles with the composition of a rotation and a translation. The conical points of this structure are exactly the points in P, meaning that this structure once restricted to S is smooth. Let B = {φ t (0) | t triangle of ∆} ⊂ S. There is one point of B in the interior of each triangle of T . With this data, we can construct a two-fold branched covering π : Σ 2 → S such that the branched points are precisely elements of B and Σ 2 has a Euclidean structure as follows. Let H be the Euclidean hexagon with vertices the sixth roots of unity in C. Then the map z → z 2 is a ramified covering from H to T that has exactly one ramification of order 2 at the point 0. Then take as many copies of H as there are triangles in T and for each gluing of two triangles (not necessarily distinct) in S, glue the corresponding Euclidean hexagons on both edges that are mapped to the glued edge in S with rotation and a translation (see Figure 3.7). The surface obtained is Σ 2 . This defines a two-fold ramified covering π : Σ 2 → S with ramification points at B, and the conical points of Σ 2 are a subset of π -1 (P). This means the map π restricted to Σ 2 = Σ 2 \ π -1 (P) is a smooth two-fold branched covering from Σ 2 to S, with simple ramifications on points of B. The lift ∆ * := π -1 (∆) of ∆ to Σ 2 induces a hexagonal tiling of Σ 2 such that in every hexagon there is exactly one element of π -1 (B).

Figure 3.7: The ramified two-fold covering of two glued triangles. The preimages of p are p 1 and p 2 , same for q, r, s. The branched points are the blue crosses. The two outer edges with an arrow are glued according to arrow orientation.

Remark 2.3.5. Since the map π is a local diffeomorphism on Σ 2 \ π -1 (B), it induces the tangent (differential) map dπ : T (Σ 2 \ π -1 (B)) → T (S \ B) that factorizes to unit tangent bundles T ′ (Σ 2 \ π -1 (B)) → T (S \ B) ′ .
In order to simplify the notation, we will sometime write π * : T Σ 2 → T S and π * :

T ′ Σ 2 → T ′ S instead of dπ.
Remark 2.3.6. The unit tangent bundle of H (resp. T) is canonically identified to H × S 1 (resp. T × S 1 ) as H (resp. T) is a subset of R 2 . With this identification, the preimages by dπ of (x, v) ∈ T ′ S are of the form (x 1 , v ′ ) and (x 2 , -v ′ ) where x 1 and x 2 are the preimages of x by π.

The following proposition describe the topology of the ramified covering Σ 2 : Proposition 2.3.7. Let S be a compact orientable surface with k ≥ 0 boundary components C 1 , . . . , C k and let P be a finite set of points (called punctures) of S such that for all i ∈ {1, . . . , k}, n i = #(C i ∩ P) > 0. Let k e (resp. k o ) be the number of components of ∂S with an even (resp. odd) number of punctures, such that k = k e + k o . Let p = #(P \ ∂S), let g be the genus of S and let S = S \ P. Then the two-fold ramified covering Σ 2 = Σ 2 \ π -1 (P) of S is a surface that verifies:

• Σ 2 is a compact orientable surface of genus g ′ = 1 2 2p + 2k e + 3k o + 8g -6 + k i=1 n i ,
• for each of the k e boundary components C of S with aneven number n of punctures, π -1 (C) is the union of two distinct boundary components in Σ 2 , each with n punctures,

• for each of the k o boundary components C of S with an odd number n of punctures, π -1 (C) is one boundary component in Σ 2 with 2n punctures,

• Σ 2 has 2p internal punctures.

Proof. First, note that the formula given above defines an integer because 3k o + n i is always even. It is clear from the construction that Σ 2 is compact and orientable, and that Σ 2 has 2p internal punctures. To compute the number of boundary components of Σ 2 , we will glue to each boundary component of S a disk with the corresponding number of puncture on the boundary to get a surface Ŝ with no boundary, only internal punctures. Since a disk with one (resp. two) puncture on the boundary does not admit an ideal triangulation, we glue a disk with one (resp. two) puncture on the boundary and one internal puncture instead. In the corresponding ramified covering Σ2 of Ŝ, we then remove the lifts of the interior of the glued disks to obtain Σ 2 . The result follows from the following lemma: Proof. The two cases with an internal puncture can be computed individually. Let S be a disk with n ≥ 3 punctures on the boundary. Let ∆ be a triangulation of S and Σ 2 the corresponding ramified covering. Let γ be a loop homotopic to the boundary of the disk going around all the n -2 branched points in S. Let x be the base point of γ, and x 1 , x 2 the lifts of x to Σ 2 . Let γ the lift of γ starting at x 1 . If γ is a loop then there are two lifts of the boundary of S to Σ 2 , and if γ is a path from x 1 to x 2 then the lift of the boundary of S is connected in Σ 2 . The loop γ is homotopic to the concatenations of loops γ 1 , . . . , γ ⌊ n-2 2 ⌋ , γ ′ based at x such that each γ i goes around two branched points in S and γ ′ is either trivial if n -2 is even or goes around one branched point if n -2 is odd. Then γ is the concatenation of the lifts γ1 , . . . , γ⌊ n-2 2 ⌋ , γ′ . Since the γi are loops based at p 1 and γ′ is either trivial or a path from p 1 to p 2 (depending on the parity of n), we get the result.

Lemma 2.3.8. If
The Euler characteristic of S is

χ(S) = 2 -2g -k = 2 -2g -k o -k e and the Euler characteristic of Σ 2 is χ(Σ 2 ) = 2 -2g ′ -k o -2k e .
The number of branched points is the same as the number of triangles in ∆, which is -2χ(S) + 2p + n i . Riemann-Hurwitz formula gives us:

χ(Σ 2 ) = 2 -2g ′ -k o -2k e = 2χ(S) --2χ(S) + 2p + k i=1 n i = 4χ(S) -2p - k i=1 n i = 8 -8g -2p -4k o -4k e - k i=1 n i
We can then solve for g ′ to get the result.

Remark 2.3.9. To precisely describe the topology of Σ n for n ≥ 3 we would need to understand first the topology of the ramified covering of a disk with external punctures. We haven't found a convenient description of these coverings yet.

We denote by θ : Σ 2 → Σ 2 the covering involution, i.e. the map swapping the two preimages of a regular point, and fixing ramification points. The following result is a direct consequence of the above proposition.

Corollary 2.3.10. The fundamental group π 1 (Σ 2 ) is a free group of rank

1 -χ(Σ 2 ) + 2p = 1 -4χ(S) + 4p + n i .
Let b ∈ Σ 2 be a ramification point of the covering π : Σ 2 → S. Let α 1 , . . . , α s : [0, 1] → S be free generators of the fundamental group π 1 (S, π(b)) that do not pass through other branch points. The fundamental group π 1 (Σ 2 , b) is the free group freely generated by the following collection of loops on Σ 2 :

• For every generator α i , there are two closed lifts γ 1 i and

γ 2 i = θ • γ 1 i on Σ 2 based at b (in total 2 -2χ(S) + 2p curves); • For every ramification point b ′ ̸ = b in Σ 2 ,
we fix a simple segment on S connecting π(b) and π(b ′ ) and take the lift of this segment on

Σ 2 . It is a closed loop ξ based at b (in total -2χ(S) + 2p -1 + n i curves). The fundamental group π 1 (T ′ Σ 2 , b) where b ∈ T ′ Σ 2 is a lift of b to T ′ Σ 2 is
generated by lifts of curves described above and the curve going once around the fiber of T ′ Σ 2 → Σ 2 at b.

Decorating loops and joining paths

We want to fix for every white vertex of Γ n a small loop going twice around it, in both S and Σ n . These loops will support sections of the local system which will be used to define the A-coordinates of a local system on S in Section 3.3. However for consistency reasons, white vertices of Γ n adjacent to a same external black vertex should be grouped together. This creates three different types of white vertices:

• White vertices that are not adjacent to any external black vertex, for which all three coordinates are at most n -1

• White vertices adjacent to exactly one external black vertex, for which exactly one of the three coordinates is equal to n. The white vertices adjacent to bivalent external black vertices are grouped two-by-two.

• White vertices adjacent to two external black vertices, for which two coordinates are equal to n and the last coordinate is 1. In this case the white vertex is on the path γ

(1) p for some puncture p. We call those peripheral white vertices.

We fix the following paths on Σ n , corresponding to each kind of white vertex in Γ n :

• For every white vertex v that is not adjacent to any external black vertex, let β v be a small loop going twice around v, with positive orientation.

• For every pair of white vertices v and v ′ adjacent to the same external black vertex b (and not adjacent to any other external black vertex), let β v = β v ′ be a loop going twice around the three vertices v, v ′ and b, with positive orientation.

• For every puncture p of S, let β p be a small smooth deformation of γ

(1) p in S. For every lift p i of p to Σ n , let β p i be the preimage of β p by π going around p i . For every white vertex v on γ

(1) p , we set β v = β p 1 .
We call all the paths T ′ β v and T ′ β p i the decorating curves of Σ n . Figure 3.8: All projections on S of the decorating curves on a quadrilateral for n = 3.

Remark 2.3.11. Notice that in the last case, the path β p is not a loop if the puncture p is external, and if β p is a loop, it goes only once around p and not twice as in the other cases. This is due to the fact we are working with twisted local systems on S and Σ n , see Section 2.2.3.

Remark 2.3.12. All decorating curves β v going twice around a vertex or a group of vertices are obtained by choosing a smooth loop βv : S 1 → Σ n going once around, and β v = βv • τ where τ : S 1 → S 1 is the two-fold cover of the circle over itself. In particular, the image of every β v in Σ n is diffeomorphic to a circle. This construction is equivalent to contracting each group of white vertices and external black vertices to a single white vertex, and taking a loop βv around that new white vertex v. Doing so will create a white vertex at every puncture. If this new white vertex v is a puncture we set β v = βv and if not, we set

β v = βv • τ .
The cover Σ n → S is regular in the neighborhood of punctures, white vertices and external black vertices, so by taking the images of the paths above by π : Σ n → S we obtain similar paths π(β v ) on S. Notice that for all puncture p and every

p i ∈ π -1 (p), π(β p i ) = β p .
For every white vertex v of Γ n and for every puncture p of S that is a vertex of the triangle containing v and such that v / ∈ γ

(1) p , let p i be the lift of p to Σ n such that v ∈ γ

(i) p . Fix a smooth path γ p,v : [0, 1] → Σ n without self-intersection such that T ′ γ p,v (0) ∈ T ′ β v and T ′ γ p,v (1) ∈ T ′ β p i , as in Figure 3.9. If v is peripheral, T ′ γ p,v intersects T ′ β v only once. If v is not peripheral, then T ′ γ p,v intersects T ′ β v twice, so there exists x 1 , x 2 ∈ S 1 such that T ′ γ p,v (1) = T ′ β v (x 1 ) = T ′ β v (x 2
), and x 1 , x 2 satisfy τ -1 (x 1 ) = τ -1 (x 2 ). We call those paths γ p,v joining paths.

For each white vertex v, we get a finite collection of points E(v) on the circle S 1 given by all x ∈ S 1 such that T ′ β v (x) = T ′ γ p,v (0) for some puncture p. This set E(v) inherit the cyclic ordering of S 1 . In the next sections, it will be convenient to view this cyclic ordering as a bijection σ :

E(v) → E(v) satisfying σ #E(v) = Id, which sends an element x ∈ E(v)
to the next element in the cyclic order. If v is not peripheral, #E(v) is even and we have

T ′ β v (x) = T ′ β v (σ #E(v)/2 (x)
) for all x ∈ E(v) because β v loops twice around v. We denote the union of all the images in Σ n of the sets β v (E(v)) for v a white vertex of Γ n by E ∆ (Σ n ), and its projection on S by E ∆ (S) = π(E ∆ (Σ n )).

Figure 3.9: The decorating curve β v around a non-peripheral white vertex v ∈ γ

(i) p ∩γ (j) q ∩γ (k)
r , together with the joining paths and the images of the points in E(v) on β v . The points in

E(v) are x 1 < • • • < x 6 < x 1 and satisfy β v (x i ) = β v (x i+3 ) for i = 1, 2, 3.

Spectral networks

Spectral networks are tools introduced by Davide Gaiotto, Gregory W. Moore and Andrew Neitzke in [START_REF] Gaiotto | Spectral networks[END_REF] to study local systems by breaking them down into a smaller rank local system on a ramified covering of the initial surface. In this section we give a combinatorial definition of finite spectral networks, which will be sufficient for our work. For a general account on spectral networks, see [START_REF] Gaiotto | Spectral networks[END_REF][START_REF] Gaiotto | Spectral networks and snakes[END_REF][START_REF] Hollands | Spectral networks and Fenchel-Nielsen coordinates[END_REF].

Let S be a ciliated surface, let n ≥ 2 and let π : Σ n → S be the ramified covering constructed in Section 2.3.1. We first recall the definition of a spectral network.

Definition 2.3.13. A small spectral network associated with the ramified covering π : Σ n → S is a finite set W of paths [-1, 1] → Σ n (called lines of the spectral network) satisfying:

• ∀α ∈ W, α(-1), α(1) ∈ π -1 (P), α(0) ∈ B and if t / ∈ {-1, 0, 1}, then α(t) / ∈ P ∪ B • ∀α ∈ W, ∀t ∈ [-1, 1], π(α(t)) = π(α(-t))
• ∀b ∈ B, there are exactly 3 lines α 1 , α 2 , α 3 ∈ W passing through b, and locally around b the lines look like so:

• ∀α, α ′ ∈ W with α ̸ = α ′ , α(] -1, 0[) ∩ α ′ (]0, 1[) ̸ = ∅.
Remark 2.3.14. Each line α of a spectral network goes through exactly one ramification point b of Σ n , and we say this ramification point b is associated to α, or that b is the branch point of α.

Remark 2.3.15. The general construction of a spectral network given in [START_REF] Gaiotto | Spectral networks[END_REF] describe more general spectral networks. Generically they have infinitely many lines and are dense on the surface. However we will only need the very small class of spectral networks defined above.

We can now construct a small spectral network on Σ n associated with the n-fold ramified covering π : Σ n → S. For this, we will describe the spectral network on the covering of a triangle of S, the complete spectral network on Σ n being the union of all the spectral networks on the coverings of the triangles. First, notice that every puncture p of S is unramified, i.e. |π -1 (p)| = n. Those n lifts p 1 , . . . , p n of p each lies in one of the punctured disks S γ (i) p defined in Lemma 2.3.3, and inherit the order given by the inclusion of these

punctured disks p 1 < • • • < p n , where p i < p j if γ (i) p ⊂ S γ (j) p .
Let t be a triangle of S and p a vertex of this triangle, and let 2 ≤ i ≤ n. There are i -1 internal black vertices on the boundary of S γ

= c 3 .c 2 .c 1 : [-1, 1] → Σ n , parameterized such that α(0) = v b .
Remark 2.3.16. Notice that every line of this spectral network goes from the i-th preimage p i of a puncture p to the (i -1)-th preimage p i-1 for some 2 ≤ i ≤ n.

Example 2.3.1. Note that in the case n = 2, every puncture of Σ 2 is either at the end of every line of the spectral network containing it, in which case we call it a sink , or at the beginning of every line of the spectral network containing it, in which case it is called a source. This way, every point of P have two preimages, one source and one sink. For each hexagon H of π -1 (∆), the three lines going through the ramification point in H are the three Euclidean segments going from the source to the sink for each of the three puncture in T.

Figure 3.11: Picture of the spectral network on each triangle of the triangulation in the case n = 2. Here the sources are p 2 , q 2 , r 2 and the sinks are p 1 , q 1 , r 1 .

Figure 3.12: The small spectral network on the lift of a triangle to Σ 3 . Proposition 2.3.17. The set W of all paths on Σ n constructed as above is a small spectral network associated to the n-fold ramified covering π : Σ n → S.

Proof. With the right coice of parametrization, every line p : [-1, 1] → Σ n defined above satisfy π(p(t)) = π(p(-t)) for t ∈ [-1, 1], π(p(-1)) = π(p(1)) is a puncture in S and p(0) is an internal black vertex, i.e. a ramification point. Every internal black vertex is part of three zig-zag paths, so it is on the boundary of three cells of the form S γ (i) s . There is three line of W going through every black vertex, with alternating directions (see Figure 3.13).

Figure 3.13: The intersection of three lines of the spectral network at an internal black vertex.

Each edge of Γ n incident to an internal black vertex is followed by exactly one line of W, and the edges incident to an external black vertex do not meet any lines of W. There are no intersection between lines of W inside a punctured disk of the form S γ (i) s , so the only intersections between lines of W except those at ramification points lies on white vertices of Γ n . That means all those intersections happen between lines having already crossed their ramification points, thus the set W is a small spectral network.

We can endow the neighborhood U b of a ramification point b ∈ B defined in Section 2.3.1 with an Euclidean structure such that the three lines of the spectral network going through b are straight lines with respect to this structure, so that locally the figure looks like the one in rank two (Figure 2.3.13). The Euclidean structure on U b defines a flat connection ∇ on T U b given by the restriction of the standard flat connection on R 2 . Since it is a bilinear map on the space of sections of T U b (denoted Γ(T U b )), this connection induces a flat connection (which we also call ∇) on the unit tangent bundle

∇ : Γ(T ′ U b ) × Γ(T ′ U b ) → Γ(T ′ U b ).
Remark 2.3.18. A consequence of Remark 2.3.6 is that when n = 2, we can assume the neighborhood U b of a ramification point b ∈ B to be the whole hexagon tile containing it in Σ 2 .

Partial non-abelianization

In this section we describe in detail the non-abelianization procedure introduced by Gaiotto-Moore-Neitzke in [START_REF] Gaiotto | Spectral networks[END_REF]: we want to construct from a framed GL n (R)-local system on S a R × -local system on Σ n . Since we will need to work with twisted local systems to define A-coordinates, we first need to extend the spectral network lifting rule to work with paths on the unit tangent bundle of a surface. This construction is a joint work with Eugen Rogozinnikov in [START_REF] Kineider | On partial abelianization of framed local systems[END_REF] in the case n = 2, and is presented here in the general case.

The spectral network map

Lifting paths to a ramified covering is not homotopy invariant: a contractible loop around a branch point b ∈ B is lifted as two paths on Σ that are not loops, thus not homotopic to the lift of the trivial loop. The goal of this section is to construct a path-lifting map denoted by SN , which depends on the spectral network W, from paths on T ′ S to paths on T ′ Σ such that SN is well-defined on homotopy classes.

We will use the symbol ≈ to represent homotopy (with fixed extremities) of paths.

Let S be a ciliated surface, ∆ an ideal triangulation of S and π : Σ n → S the n-fold branched covering constructed in Section 2.3.1. Let W be the spectral network adapted to this covering constructed in Section 2.3.3. Every path α :] -1, 1[→ Σ n of W is smooth. We can thus lift the paths of W to

T ′ α : ] -1, 1[ → T ′ Σ n t → (α(t), α(t)
) .

We will call this set of paths in T ′ Σ n a tangent spectral network and denote it T ′ W.

Let U b ⊂ Σ n be the neighborhood of a ramification point b endowed with an Euclidean structure. Note that the Euclidean structure on U b allows us to identify

T ′ U b with U b × S 1 .
In the following, a path γ on T ′ U b ≃ U b × S 1 will be written as a couple (x, v) where x is the projection of γ on U b ⊂ Σ n and v is the projection of γ on S 1 . Note that S 1 has a natural orientation given by the one on Σ n . For all θ ∈ S 1 , define s + θ to be the (homotopy class of the) path in S 1 going from θ to -θ following the orientation of S 1 , and s - θ going from θ to -θ in the opposite direction. For a path v on S 1 , we will denote -v the image of v under the involution θ → -θ. The path -v goes from -v(0) to -v(1). In particular, we have s - θ = -s + θ (where s + θ denote the path obtained by reversing the direction of s + θ ) and (-s + θ ).s

+ θ = δ + θ (resp. (-s - θ ).s - θ = δ - θ ) where δ + θ : t → θ + 2iπt (resp. δ - θ : t → θ -2πt
) is the path going once around S 1 . When the context is clear, we will omit the subscript describing the starting point of the paths s ± and δ ± . The paths δ ± satisfy δ -= δ + and if v is a path on S 1 from θ 1 to θ 2 , we write δ

± .v ≈ v.δ ± , meaning that δ ± θ 2 .v ≈ v.δ ± θ 1 .
Definition 2.4.1. Let X be a topological space. The path algebra of X (denoted Z[Path(X)]) is the free abelian group generated by homotopy classes of paths [0, 1] → X, together with a product given by concatenation of paths: if γ 1 (0) ̸ = γ 2 (1) then γ 1 .γ 2 = 0 and if γ 1 (0) = γ 2 (1) then γ 1 .γ 2 is the path obtained by going through γ 2 then γ 1 .

Remark 2.4.2. The path algebra of a topological space is in general a ring without a unit element.

Definition 2.4.3. Now let X be a smooth surface. Define the twisted path algebra of X as

TPA(X) = Z[Path(T ′ X)]/I
where I is the two-sided ideal generated by the elements e x,θ + δ x,θ for (x, θ) ∈ T ′ X, with

e x,θ : [0, 1] → T ′ X t → (x, θ) and δ x,θ : [0, 1] → T ′ X t → (x, θ + 2πt) .
Remark 2.4.4. Given any non-empty subset E ⊂ T ′ X, the subset

{γ 1 + • • • + γ r + I | for all 1 ≤ i ≤ r, endpoints of γ i are in E} ⊂ TPA(X)
is a subring of TPA(X) because composition of paths preserves the set of endpoints. We will denote TPA E (X) this subring.

We first describe the path lifting rule for a path on S, and then we will extend the construction to paths on T ′ S. Since there is a projection T ′ S → S, we want the lift to T ′ Σ n of a path γ on T ′ S to project onto the lift to Σ n of the projection x of γ to S. Let x be a path on S intersecting only once (and not at its endpoints) the spectral network W and not going through a branch point. Let α ∈ W be the path such that π(α) intersects x and let b ∈ B be the ramification point associated to α. Up to homotopy, we can assume the intersection between x and α to be in U b . Among the n standard lifts x 1 , . . . , x n of x to Σ n , two of them intersect once α in U b , one of them intersecting α(] -1, 0[) and the other intersecting α(]0, 1[). Suppose x 1 is the one intersecting α(] -1, 0[) and x 2 is the one intersecting α(]0, 1[).

We can then define a new path x ′ on Σ n as the concatenation of 5 paths x ′ 1 , . . . , x ′ 5 defined as follows:

• x ′ 1 is the part of x 1 from its starting point to the intersection point with α

• x ′ 2 is the part of α from the intersection with x 1 to the ramification point b

• x ′ 3 is a constant path at the ramification point b point (it will be useful in the next paragraph when we will consider the lifted spectral network T ′ W)

• x ′
4 is the part of α from b to the intersection with x 2

• x ′ 5 is the part of x 2 from the intersection with α to its endpoint.

The spectral lifts of x to Σ n is the collection x 1 , . . . , x n , x ′ .

We now extend the path lifting rule to paths on T ′ Σ n . Let γ : t → (x(t), v(t)) be a path on T ′ S such that the path x on S intersects only once the spectral network on the projection π(α) of a line α ∈ W at a time t 0 ∈ ]0, 1[. Let b be the ramification point associated to α. Up to homotopy, we can assume that the intersection of x and α is in U b . Let γ 1 = (x 1 , v 1 ), . . . , γ n = (x n , v n ) be the n standard lifts of γ to T ′ Σ n , with γ 1 and γ 2 being the two lifts such that x 1 and x 2 intersect the spectral network. Note that γ 1 and γ 2 do not intersect T ′ W in general, but x 1 and x 2 intersect α ∈ W. Let x ′ be the path on Σ n obtained Figure 4.14: The path x ′ added by the intersection of the lifts x 1 and x 2 of x with α with the construction above. We now want a continuous map v ′ : [0, 1] → S 1 which coincide with the standard lifts v 1 and v 2 when x ′ coincides with either x 1 or x 2 . Without loss of generality, suppose x is smooth at the intersection point with α and that the intersection is transverse. Then x 1 and x 2 are also smooth at their intersection points with α. We say the intersection of x 1 with α is positively oriented if ( ẋ1 (t 0 ), α(t 0 )) agrees with the orientation on Σ, negatively oriented if not.

Remark 2.4.5. The positivity of the intersection of a path (x, v) in T ′ Σ with a line of the spectral network is determined using the derivative of the underlying path x, and does not depend on the vector field v on x.

Let v ′ be the concatenation of 5 paths v ′ 1 , . . . , v ′ 5 defined as follows:

• v ′ 1 is the part of v 1 from its starting point to the intersection point with α

• v ′ 2 is obtained by parallel transport with respect to the flat connection ∇ on U p from the vector v 1 (t 0 ) along the path

x ′ 2 • v ′ 3 is the path s + v ′ 2 (0) in T ′ b Σ n ≃ S 1 if the intersection of x 1 with α is positively oriented, and s - v ′ 2 (0) if the intersection is negatively oriented. • v ′
4 is obtained by parallel transport with respect to ∇ from the vector v 2 (t

0 ) = -v 1 (t 0 ) along the path x ′ 4 • v ′
5 is the part of v 2 from the intersection with α to its endpoint Remark 2.4.6. The resulting path v ′ on S 1 is homotopic to

(-v 2 1 ).s ± v 1 (t 0 ) .v 1 1 where v 1 1 = v 1 | [0,t 0 ] and v 2 1 = v 1 | [t 0 ,1] .
Note that for all path w on S 1 from θ 0 to θ 1 , we have

s ± θ 1 .w ≈ (-w).s ± θ 0 so the path v ′ is homotopic to s ± v 1 (1) .v 1 .
Figure 4.15: The path (x ′ , v ′ ) added by the intersection with α. The intersection of x 1 with α is positively oriented if Σ n is oriented clockwise.

Let γ ′ = (x ′ , v ′ ) and let SN (γ) be the element

γ 1 + • • • + γ n + γ ′ ∈ TPA(Σ n ).
Let γ be a path in T ′ S. We can write γ as a concatenation of smaller paths γ 1 , . . . , γ r , each having a projection to S intersecting at most once the spectral network and for each of these small paths, apply the construction above to obtain SN (γ 1 ), . . . , SN (γ r ) (if γ i have a projection to S which does not intersect the spectral network, define SN (γ i ) to be the sum of the n standard lifts of γ i ). Define the lift of γ with respect to the spectral network W to be the product SN (γ) = SN (γ 1 ) . . . SN (γ r ) ∈ TPA(Σ).

Theorem 2.4.7. Let γ 1 and γ 2 be two homotopic paths in T ′ S. Then SN (γ 1 ) = SN (γ 2 ).

In particular, the map

SN : TPA(S) → TPA(Σ) γ → SN (γ)
is well-defined.

Remark 2.4.8. The map SN is not defined on the whole twisted path algebra of S as paths with endpoints on a line of the spectral network can not be lifted consistently, but we will never need to lift such paths. The subset of TPA(S) (resp. TPA(Σ n )) of elements where no term has an endpoint on W is a subring (see Remark 2.4.4), and with a slight abuse of notation we will still denote it TPA(S) (resp. TPA(Σ n )) instead of TPA(S) S\π(W) (resp.

TPA(Σ n ) Σn\W ).
The theorem is a consequence of the two following lemmas:

Lemma 2.4.9. Let γ = (x, v) be a path in T ′ S that intersects exactly twice the same line α of the spectral network and no other line of W, as in Figure 4.16. Then SN (γ) = γ 1 +• • •+γ n where γ 1 , . . . , γ n are the n standard lifts of γ to T ′ Σ n .

Proof. Let t 1 < t 2 be the two elements of the interval [0, 1] such that x(t 1 ) and x(t 2 ) are on α. Let (x (1) , v (1) ) = γ(t 1 ) and (x (2) , v (2) ) = γ(t 2 ), and let γ 1 = (x 1 , v 1 ) and γ 2 = (x 2 , v 2 )

Figure 4.16: A loop intersecting twice the same line of W the two standard lifts of γ such that x 1 and x 2 intersect α, γ 1 being the lift such that

x 1 intersects α(] -1, 0[). Then SN (γ) = γ 1 + γ 2 + • • • + γ n + γ ′ + γ ′′ where γ ′ = (x ′ , v ′ ) is such that x ′ follow α from x (1) 1 to x (1)
2 and γ ′′ = (x ′′ , v ′′ ) is such that x ′′ follows α from x

(2)

1 to x (2) 2 .
Figure 4.17: Spectral network lift of γ In order to prove the lemma, we need to show that the two paths γ ′ and γ ′′ added by the intersections with the spectral network cancel each other in TPA(Σ), i.e. that γ ′ + γ ′′ = 0. For this, we need to show that γ ′′ .γ ′ is homotopic to an odd power of δ x 1 (0),v 1 (0) . The paths x ′ and x ′′ are homotopic on Σ so the concatenation x ′′ .x ′ is trivial. What is left is to show that v ′′ .v ′ is homotopic to an odd power of δ + . Suppose the intersection of x 1 with α at x

(1) 1 is positive, the other case being symmetric. Then the intersection of x 1 with α at x

(2) 1 is negative. Then by remark 2.4.6, v ′ ≈ s + .v 1 and v ′′ ≈ s -.v 1 , so we have

v ′′ .v ′ ≈ v 1 .s -.s + .v 1 ≈ v 1 .δ + .v 1 ≈ δ + .
Lemma 2.4.10. Let m be a point in S in a small neighborhood of a branch point b but not on a line of W and θ ∈ T ′ m S. Let γ be a path homotopic to e m,θ in T ′ S that loops around the branch point b in S, intersecting exactly once each of the three lines of W going out of b, as in Figure 4.18. Then SN (γ) = e m 1 ,θ 1 + • • • + e mn,θn where (m 1 , θ 1 ), . . . , (m n , θ n ) are n lifts of (m, θ) in T ′ Σ n . Proof. Suppose the path γ is looping around b in the direction given by the orientation of Σ n , the other case being symmetric. Then all the intersections of the standard lifts of γ with the spectral network in Σ n are positive. By applying the spectral network lifting rule to γ, we get n + 6 paths: the n standard lifts γ 1 = (x 1 , v 1 ), . . . , γ n = (x n , v n ) (with x 1 and x 2 intersecting W as before), and 6 additional paths γ ′ 1 , . . . , γ ′ 6 shown in Figure 4.19. Let α 1 , α 2 and α 3 be the three line of W intersected by γ, in that order. Let γ 1 be the standard lift of γ intersecting α 1 before the branch point, and let (m 1 , θ 1 ) be its starting point and (m 2 , θ 2 ) be its endpoint. We will label the spectral network lifts γ ′ i = (x ′ i , v ′ i ) of γ as follows:

• γ ′ 1 follows γ 1 until the intersection with α 3 , then α 3 , then γ 2 until its end

• γ ′ 2 follows γ 2 until the intersection with α 2 , then α 2 , then γ 1 until its end

• γ ′ 3 follows γ 2 until the intersection with α 2 , then α 2 , then γ 1 until the intersection with α 3 , then α 3 , then γ 2 until its end • γ ′ 4 follows γ 1 until the intersection with α 1 , then α 1 , then γ 2 until its end

• γ ′ 5 follows γ 1 until the intersection with α 1 , then α 1 , then γ 2 until the intersection with α 2 , then α 2 , then γ 1 until the intersection with α 3 , then α 3 , then γ 1 until its end • γ ′ 6 follows γ 1 until the intersection with α 1 , then α 1 , then γ 2 until the intersection with α 2 , then α 2 , then γ 1 until its end.

The paths x ′ 1 , x ′ 4 and x ′ 5 are homotopic to the trivial path e m 1 , x ′ 6 is homotopic to x 1 , x ′ 2 is homotopic to e m 2 and x ′ 3 is homotopic to x 2 . Since γ is homotopic to e m,θ and x is looping around b in the direction given by the orientation of Σ, we have v 1 ≈ s - θ 1 and v 2 ≈ s - θ 2 . Using the same reasoning, we get the following:

v ′ 1 ≈ s + .v 1 ≈ e θ 1 v ′ 2 ≈ s + .v 2 ≈ e θ 2 v ′ 3 ≈ s + .s + .v 2 ≈ δ + .v 2 v ′ 4 ≈ s + .v 1 ≈ e θ 1 v ′ 5 ≈ s + .s + .s + .v 1 ≈ δ + θ 1 v ′ 3 ≈ s + .s + .v 1 ≈ δ + .v 1
So in TPA(Σ), we have:

γ 2 + γ ′ 3 = 0 γ 1 + γ ′ 6 = 0 γ ′ 1 + γ ′ 5 = 0 γ ′ 4 = e m 1 ,θ 1 γ ′ 2 = e m 2 ,θ 2 so SN (γ) = γ 1 + γ 2 + γ ′ 1 + γ ′ 2 + γ ′ 3 + γ ′ 4 + γ ′ 5 + γ ′ 6 = e m 2 ,θ 2 + e m 2 ,θ 2 .

Twisted path algebras and representations

Let X be an hyperbolic ciliated surface and n ≥ 1. If L is a twisted GL n (R)-local system on X and γ is a path on T ′ X, the flat connection defines a holonomy map m γ from L γ(0) to L γ(1) . Moreover, by definition of a twisted local system the path δ x,θ induces the linear map -

Id on L x,θ . Thus, if γ = γ 1 + • • • + γ r + I ∈ TPA(X)
where all the γ i ∈ Path(T ′ X) have the same extremities, the holonomy map

m γ = m γ 1 + • • • + m γr : L γ(0) → L γ(1)
is well-defined (if there is more than one term in γ the holonomy map m γ may not be an isomorphism). However, if γ 1 and γ 2 do not have the same extremities, it is not possible to associate an element of M n (R) to γ 1 + γ 2 , which is a problem we need to solve in order to consider representations of TPA(X). To make a link between twisted local systems and representations of TPA(S), we first need to modify the ring M n (R) to solve this issue of endpoints. Since multiplication in TPA(X) is zero for paths whose extremities do not match, we need a ring with the same behavior.

Definition 2.4.11. Let A be a unital ring and E ⊂ X any non-empty subset. Let A E be the ring A (E×E) of finite formal sums of elements of the form a (p,q) , a ∈ A, p, q ∈ E, endowed with the multiplication defined as follows:

• ∀a, b ∈ A, ∀x, y, z ∈ E, a (x,y) .b (y,z) = (a.b) (x,z) • ∀a, b ∈ A, ∀x, y, z, t ∈ E, y ̸ = z, a (x,y) .b (z,t) = 0
The elements of this ring are copies of elements of A indexed by pairs of points in E, thought as "endpoints" of these elements. The sum of two elements is a formal sum except when the indices match, it then agrees with the sum in A. The multiplication of two elements is made so it agrees with the composition of paths: multiplication of two elements with "non-composable" indices is zero and multiplication with "composable" indices agrees with the one on A, and the index of the result is the composition of the indices. Given two rings A and B, a ring homomorphism φ : A E → B E is called a graded ring homomorphism if for all a ∈ A and for all x, y ∈ E, φ(a x,y ) = b x,y for some b ∈ B.

The ring A E contains many isomorphic copies of A as subrings: for all x ∈ E,

A x := a (x,x) | a ∈ A
is a subring of A E isomorphic to A. Note however that the ring A E is not unital if E is infinite, but contains many idempotent elements.

Let TPA ∆ (S) = TPA E ∆ (S) (S) be the subring of TPA(S) of paths with endpoints in E ∆ (S) described in Remark 2.4.4, where E ∆ (S) is defined in Section 2.3.2. Similarly, let

TPA ∆ (Σ n ) = TPA E ∆ (Σn) (Σ n ).
For any unital ring A, let A ∆,S = A E ∆ (S) and A ∆,Σn = A E ∆ (Σn) . In the following, X will denote either the surface S or its ramified covering Σ n . Since E ∆ (S) and E ∆ (Σ) are finite, TPA ∆ (X) and A ∆,X are unital, the units elements being respectively

x∈E ∆ (X) e x and x∈E ∆ (X) 1 (x,x) .
There is then a diagonal embedding

A E ∆ (X) → A ∆,X (a x ) x∈E ∆ (X) → x∈E ∆ (X) (a x ) (x,x)
.

Remark 2.4.12. For every x ∈ E ∆ (X), there is an injective group homomorphism

π s 1 (X, x) → TPA x (X) × ⊂ TPA ∆ (X)
where TPA x (X) × denotes the group of invertible elements of TPA x (X).

Two elements a, b ∈ A ∆,X are said conjugated if there exists an invertible element u in A E ∆ (X) such that b = u.a.u -1 . This is an equivalence relation.

Proposition 2.4.13. Let R be a finite dimensional R-algebra. Let ∆ be a triangulation of a hyperbolic ciliated surface S and let X be either the surface S or its ramified covering Σ n .

There is a 1:1 correspondence between the set of twisted GL n (R)-local systems on X up to isomorphism and the set of graded ring homomorphisms TPA ∆ (X) → M n (R) ∆,X up to the action of GL n (R) E ∆ (X) by conjugation.

Proof. Given a twisted GL n (R)-local system L on X, for all x ∈ E ∆ (X) choose a basis of the fiber of L over x. The map

φ : TPA ∆ (X) → M n (R) ∆,X γ → (Hol L (γ)) (t(γ),s(γ))
is a ring homomorphism, where s(γ) (resp. t(γ)) is the source (resp. the target) of γ (which are in E ∆ (X)), and Hol L (γ) is the holonomy of γ in L in the corresponding bases. The conjugacy class of φ does not depend on the choices of the bases.

Conversely, let φ be the conjugacy class of a representation TPA ∆ (X) → M n (R) ∆,X , and let x ∈ E ∆ (X). Then TPA x (X) contains an isomorphic copy of π s 1 (X, x) and the restriction of φ to π s 1 (X, x) yield a representation π s 1 (X, x) → GL n (A) mapping δ ± x to -Id, which define a unique isomorphism class of twisted GL n (A)-local system by Proposition 2.2.15, having holonomies described by φ. By construction, those are inverse.

Partial non-abelianization of twisted local systems

In Section 2.4.1, we constructed an algebra homomorphism SN : TPA(S) → TPA(Σ n ). This homomorphism restricts to a graded ring homomorphism

SN : TPA ∆ (S) → TPA ∆ (Σ n )
as mentioned in Remark 2.4.4. Let γ ∈ TPA ∆ (S) be a path from p to q, p, q ∈ E ∆ (S), and let p 1 , . . . , p n be the lifts of p to Σ n labeled as in Section 2.3.1, and q 1 , . . . , q n the lifts of q. Then SN (γ) = 1≤i,j≤n γ j,i where γ j,i is the sum of all terms of SN (γ) from p i to q j (γ j,i may be 0). Instead of a formal sum, it will be more convenient to see SN (γ) as a n by n matrix with coefficients in TPA ∆ ( Σ n ). The definition of the multiplication on TPA ∆ (Σ n ) makes it so the map:

SN n : TPA ∆ (S) → M n (TPA ∆ (Σ n )) γ →    γ 1,1 . . . γ 1,n . . . . . . . . . γ n,1 . . . γ n,n   
is a ring homomorphism. We also have a ring homomorphism π R :

M n (R ∆,Σn ) → M n (R) ∆,S
such that for all p, q ∈ P and for all (a (i,j)

) 1≤i,j≤n ∈ R n 2 , π R    a (1,1) q 1 ,p 1 . . . a (1,n) q 1 ,pn . . . . . . . . . a (n,1) qn,p 1 . . . a (n,n) qn,pn    =    a (1,1) . . . a (1,n) . . . . . . . . . a (n,1) . . . a (n,n)    q,p
.

Note that we can always write an element of M n (R ∆,Σn ) as the sum of elements of the form (possibly with some coefficients equal to 0), with a (i,j) ∈ R for all 1 ≤ i, j ≤ n. Lastly given any ring homomorphism φ : R 1 → R 2 between two rings R 1 and R 2 , we will denote by M n (φ) : M n (R 1 ) → M n (R 2 ) the morphism obtained by applying φ to each entry of the matrix.

   a (1,1) q 1 ,p 1 . . . a (1,n) q 1 ,
Proposition 2.4.14. Let E be a twisted R × -local system over Σ n and let φ : TPA ∆ (Σ n ) → R ∆,Σn the corresponding ring homomorphism given by Proposition 2.4.13. Then the ring homomorphism

ψ = π R • M n (φ) • SN n : TPA ∆ (S) → M n (R) ∆,S
corresponds to a twisted GL n (R)-local system on S, together with a ∆-generic framing.

Proof. Let E be a twisted R × -local system on Σ n , and let L be the twisted GL n (R)-local system obtained on S. We need to show that L admits a flat section on any peripheral curve β p on S corresponding to an internal puncture, i.e. that the monodromy along β p is upper triangular in some basis. Let p ∈ P int and p 1 , . . . , p n the lifts of p to Σ n , labeled as in Section 2.3.1. Let q ∈ E ∆ (S) ∩ β p and q 1 , . . . , q n the lifts of q to Σ n , q i ∈ β p i . We will assume β p is a loop based on q. The fiber L q of L over q can be identified with the direct sum E q 1 ⊕ • • • ⊕ E qn of the fibers of E over q 1 to q n . Every line of the spectral network W crossed by β p on S lifts to a line from p i to p i-1 for some 2 ≤ i ≤ n on Σ n . This means that the lifts added by the spectral network all go from q i to q j with j < i, so the image of β p via SN : TPA ∆ (S) → M n (TPA ∆ (Σ)) is upper triangular. Then ψ(β p ) ∈ M n (R) ∆,S which is the monodromy of β p is also upper triangular. For each

1 ≤ k ≤ n, the k-dimensional subspace F (p) k = E q 1 ⊕ • • • ⊕ E q k ⊂
L q is preserved by the peripheral monodromy which means that the parallel transport of F (p) along β p defines a framing F (p) ⊂ L βp around p. This framing is ∆-generic because for every triangle t = (p, q, r) of ∆ and every triple i, j, k ∈ N such that i + j + k = 2n + 1, the map

F (p) i ∩ F (q) j ∩ F (r) → F (p) i /F (p) i-1
is the holonomy of the path γ p,v on Σ n defined in Section 2.3.2, thus is an isomorphism.

The twisted GL n (R)-local system L on S obtained from a twisted R × -local system E on Σ n via this construction is called the partial non-abelianization of E. In the next part, we define an inverse construction.

Partial abelianization

Partial abelianization of generic framed local systems

Let S be a ciliated surface and let ∆ be an ideal triangulation of S. Let n ≥ 2. Let π : Σ n → S be the ramified n-fold covering constructed in section 2.3.1. Let L be a ∆-generic framed twisted GL n (R)-local system.

We construct the line bundle E on Σ n as follows. Let p be a puncture of S and let 1 ≤ i ≤ n. Let S γ (i) p be the punctured disk defined in Lemma 2.3.3. On S γ (i) p the line bundle E is the pullback of the bundle F p i /F p i-1 by π. Let t = (p, q, r) be a triangle of ∆. Let v be a white vertex of Γ (t) n of coordinates (i, j, k), and let U i,j,k be a small contractible neighborhood of v together with the three edges incident to v in Σ n that does not contain any black vertex of Γ n (see Figure 5.20).

Figure 5.20: The neighborhood U i,j,k of a white vertex of coordinates (i, j, k) in Σ n .

The line bundle E on T ′ U i,j,k is the pullback of the subbundle

F (p) i ∩ F (q) j ∩ F (r)
k via π. On the intersection of T ′ U i,j,k with T ′ S γ (i) p , the transition function is given by the isomorphism

F (p) i ∩ F (q) j ∩ F (r) k → F p i /F p i-1
which is well defined in every point of the intersection

T ′ U i,j,k ∩ T ′ S γ (i) p .
This defines a line bundle E on Σ n minus the set of all black vertex of Γ n . To extend E to Σ n , we need to check that the monodromy of E along a smooth loop without self-intersection going around a black vertex of Γ n ⊂ Σ n is -Id. When the black vertex v is external this is immediate because the line bundle E is trivial around v. When the black vertex v is internal, the monodromy of such a loop around v is -Id as computed in Proposition 3.2.4.

This construction of E has locally constant transition function so it defines a twisted R × local-system E on Σ n , which we call the partial abelianization of L. By construction, the partial abelianization and the partial non-abelianization processes are inverse.

Proposition 2.5.1. Let L be a ∆-generic framed twisted GL n (R)-local system, let E be the partial abelianization of L. Let L ′ be the partial non-abelianization of E and let E ′ be the partial abelianization of L ′ . Then L ≃ L ′ and E ≃ E ′ .

TOPOLOGY OF THE MODULI SPACE OF FRAMED GL 2 (R)-TWISTED LOCAL SYSTEMS63

Partial abelianization of generic decorated local systems

Applying the construction of Section 2.5.1 to a decorated twisted GL n (R)-local system, we get an R × -local system E → T ′ Σ n . Moreover, the existence of sections b (p) i over T ′ β p for all p ∈ P and for all 1 ≤ i ≤ n induces that the holonomies of E around internal punctures of Σ n are all trivial, i.e. the bundle E is trivial over the punctured disk around p bordered by β p . That means that the local system E → T ′ Σ n can be uniquely extended to the local system over T ′ Σ n . For simplicity, slightly abusing the notation, we will write E → T ′ Σ n this new local-system.

A decoration of L provides additionally a parallel section of E| T ′ βp . We call the set of all those parallel sections a decoration of the twisted R × -local system E.

Theorem 2.5.2. The partial abelianization and partial non-abelianization processes define a bijection between the set of decorated twisted R × -local systems on Σ n with trivial monodromy around punctures up to isomorphism and the set of decorated ∆-generic twisted local systems on S up to isomorphism.

Topology of the moduli space of framed GL 2 (R)-twisted local systems

In this section, we focus on the special case n = 2. We describe the topology of the moduli space of framed twisted GL 2 (R)-local systems on S that are ∆-generic with respect to a fixed triangulation ∆.

As we have seen, framed twisted GL 2 (R)-local systems on S that are ∆-generic with respect to a fixed triangulation ∆ are in 1:1-correspondence with twisted R × -local systems on Σ 2 . Since Σ 2 has punctures, the space of twisted and non-twisted R × -local systems are homeomorphic. So we obtain the following theorem, using the same notations as in Proposition 2.3.7:

Theorem 2.6.1. The moduli space of framed (twisted) GL 2 (R)-local systems on S that are ∆-generic with respect to a fixed triangulation ∆ is homeomorphic to the moduli space of (twisted) R × -local systems on Σ 2 which is homeomorphic to (R × ) 1-4χ(S)+2p+ n i /R × where R × acts diagonally by conjugation on (R × ) 1-4χ(S)+2p+ n i .

Remark 2.6.2. In [START_REF] Guichard | Parametrizing positive representations[END_REF] the authors prove the same result using different techniques.

They define local systems on some appropriate graphs over S and parametrize them using coordinates that are similar to Fock-Goncharov's GL n -cluster X -coordinates [START_REF] Fock | Moduli spaces of local systems and higher Teichmüller theory[END_REF].

Symplectic local systems

Involutive algebras are an important class of non-commutative algebras. Over involutive algebras, generalizations of many classical groups can be constructed (e.g. orthogonal groups, symplectic groups). In this chapter which is a joint work with Eugen Rogozinnikov, we define algebras with anti-involutions and symplectic groups over such algebras that were introduced and studied in [ABR + 22]. Further, we introduce framed twisted symplectic local system and characterize them in terms of partial abelianization introduced before. As a result of this construction, we describe the topology of the moduli space of framed or decorated twisted maximal symplectic local systems.

Involutive algebras

Let A be a unital associative, possibly non-commutative finite-dimensional R-algebra.

Definition 2.7.1. An anti-involution on A is a R-linear map σ :

A → A such that • σ(ab) = σ(b)σ(a); • σ 2 = Id.
An involutive R-algebra is a pair (A, σ), where A is a R-algebra and σ is an anti-involution on A.

Example 2.7.1. The set of n × n matrices with real or complex coefficients endowed with the transposition is an involutive algebra.

Definition 2.7.2. Two elements a, a ′ ∈ A are called congruent, if there exists b ∈ A × such that a ′ = σ(b)ab. We call the action of A × on A by b → (a → σ(b)ab) the action by congruence.

Definition 2.7.3. An element a ∈ A is called σ-symmetric if σ(a) = a. An element a ∈ A is called σ-anti-symmetric if σ(a) = -a. We denote

A σ := Fix A (σ) = {a ∈ A | σ(a) = a}, A -σ := Fix A (-σ) = {a ∈ A | σ(a) = -a}.
Definition 2.7.4. The closed subgroup

U (A,σ) = {a ∈ A × | σ(a)a = 1}
of A × is called the unitary group of A. It is a Lie group whose Lie algebra agrees with A -σ .

Remark 2.7.10. In [ABR + 22] is shown that, if A is a Hermitian algebra, then Sp 2 (A, σ) is a Hermitian Lie group of tube type.

Let (x, y) be a basis of A 2 . We say that this basis is isotropic if ω(x, x) = ω(y, y) = 0. We say that this basis is symplectic if furthermore ω(x, y) = 1.

Let x ∈ A 2 be a regular isotropic element (recall the definition of a regular element in Section 1.1). We call the set xA := {xa | a ∈ A} an isotropic A-line. The space of all isotropic A-lines is denoted by Is(ω).

Symplectic local systems

We consider a twisted GL 2 (A)-local system L → T ′ S. We say that L is a twisted Sp 2 (A, σ)local system (or just twisted symplectic local system) if the transition functions between trivializations of L preserve the standard symplectic 2-form on A 2 , i.e. belong to Sp 2 (A, σ).

We then get a well defined symplectic form ω : L × L → A on T ′ S.

A framing of a parabolic twisted symplectic local system is called isotropic if the A-line defining the framing in a neighborhood of every puncture is isotropic with respect to the field of the form ω. A decoration ((b

(p) 1 ) p∈P , (b (p) 2 ) p∈P ) of a twisted symplectic local system is called symplectic if ω(b (p) 1 , b (p) 1 ) = 0 and ω(b (p) 1 , b (p) 2 ) = 1. Remark 2.7.11. Since b (p) 2 ∈ L/b (p)
1 A, we need to check that symplectic decorations are well defined. Notice that if ω(b

(p) 1 , b (p) 1 ) = 0, then the expression ω(b (p) 1 , b (p) 2 ) is well-defined. Indeed, let b(p) 2 and ( b(p) 2 ) ′ be two lifts of b (p) 2 to A 2 . Then ( b(p) 2 ) ′ = b(p) 2 + b (p)
1 a for some a ∈ A. Further,

ω(b (p) 1 , ( b(p) 2 ) ′ ) = ω(b (p) 1 , b(p) 2 + b (p) 1 a) = ω(b (p) 1 , b(p) 1 ) =: ω(b (p) 1 , b (p)
2 ). 2 becomes uniquely defined. A framed twisted symplectic local system is a twisted symplectic local system with an isotropic framing. A decorated twisted symplectic local system is a twisted symplectic local system with a symplectic decoration.

It is always enough to choose b

Remark 2.7.12. For a twisted symplectic local system to admit a framing (resp. a decoration), the monodromy around every internal puncture must stabilize (resp. fix) an isotropic A-line.

Remark 2.7.13. Notice, that since ω is a parallel form of even degree, the parallel transport of ω around the fiber of T ′ S is trivial.

Let π : Σ 2 → S be the ramified two-fold covering defined in Section 2.3.1. Let E → T ′ Σ 2 be an A × -local system over the spectral covering Σ 2 of S that is obtained by the partial abelianization procedure.

Let θ : Σ 2 → Σ 2 be the covering involution. Slightly abusing the notation, we also denote

θ = θ * : T ′ Σ 2 → T ′ Σ 2 .
Remark 2.7.14. Notice that θ does not have fixed points in T ′ Σ 2 , even though it has fixed points in Σ 2 .

We consider the pull-back of E with respect to θ and denote it by E ′ := θ * E. To simplify the notation, we will identify E ′ p and E θ(p) for all p ∈ Σ 2 . We denote by P γ : E γ(0) → E γ(1) ,

P ′ γ = P θ•γ : E θ(γ(0)) → E θ(γ(1)
) the parallel transport along γ : [0, 1] → Σ 2 in E and E ′ . We denote by P S α : L α(0) → L α(1) the parallel transport along α : [0, 1] → S in L.

Definition 2.7.15. Let V and V ′ be two right A-modules. A map b :

V × V ′ → A is called an A-sesquilinear pairing between V and V ′ if it is additive in each argument and if for all v ∈ V , v ′ ∈ V ′ , and for all a, a ′ ∈ A, b(va, v ′ a ′ ) = σ(a)b(v, v ′ )a ′ . An A-sesquilinear paring b is non-degenerate if for every regular v ∈ V there exists v ′ ∈ V ′ such that b(v, v ′ ) ∈ A × and for every regular v ′ ∈ V ′ there exists v ∈ V such that b(v, v ′ ) ∈ A × .
We denote by B(E, E ′ ) → T ′ Σ 2 the vector bundle of all A-sesquilinear parings between E and

E ′ . A section β ∈ Γ(T ′ Σ 2 , B(E, E ′ )) is called parallel if β γ(0) (x, y) = β γ(1) (P γ (x), P ′ γ (y)) = β γ(1) (P γ (x), P θ•γ (y))
for every γ : [0, 1] → T ′ Σ 2 and for every

x ∈ E γ(0) , y ∈ E ′ γ(0) = E θ(γ(0)) . Remark 2.7.16. Notice that if β ∈ Γ(T ′ Σ 2 , B(E, E ′ )) is parallel and β p is non-degenerate for one p ∈ T ′ Σ 2 , then β p is non-degenerate for all p ∈ T ′ Σ 2 .
Theorem 2.7.17. Let L be a framed twisted GL 2 (A)-local system on S and let E be the partial abelianization of L over Σ 2 . Then L is an Sp 2 (A, σ)-local system if and only if there exists a non-degenerate parallel section β ∈ Γ(T ′ Σ 2 , B(E, E ′ )) such that β p (x, y) = -σ(β θ(p) (y, x)) for every p ∈ T ′ Σ 2 , for every x ∈ E p and for every y ∈ E θ(p) .

Proof. (⇒) Assume, L is an Sp 2 (A, σ)-local system. That means, there exists a field of standard symplectic forms ω on L → T ′ S, such that for every α : [0, 1] → T ′ S and for every v, w ∈ L α(0) , ω α(0) (v, w) = ω α(1) (P S α (v), P S α (w)).

Let γ : [0, 1] → T ′ Σ 2 be a smooth path such that γ(0), γ(1) do not project to points on lines of the spectral network on Σ 2 , and x ∈ E γ(0) and y ∈ E θ(γ(0)) regular elements. We consider

γ ′ = θ • γ and α = π • γ = π • γ ′ . Moreover, (π * (x), π * (y)) is an isotropic basis of L α(0) . We can define β γ(0) (x, y) := ω α(0) (π * (x), π * (y)).
Since ω is non-degenerate and skew-Hermitian, β is non-degenerate and sesquilinear pairing. Moreover, we have

β γ(0) (x, y) = -σ(β θ(γ(0)) (y, x)) because ω α(0) (π * (x), π * (y)) = -σ(ω α(0) (π * (y), π * (x))).
If γ does not intersect lines of the spectral network, then β along γ is parallel because in this case P S α = P γ ⊕ P σ•γ If γ is a small segment intersecting a line of spectral network, then

ω α(1) (P S α (π * (x)), P S α (π * (y))) = ω α(1) (π * (P γ (x)) + π * (P γ (x)), π * (P θ•γ (y)))
where γ is a lift of α going along a line of spectral network from γ(0) to θ(γ(1)). But elements P γ (x), P θ•γ (y) ∈ E θ(γ(1)) , therefore, ω(π * (P γ (x)), π * (P θ•γ (y))) = 0. So

β γ(0) (x, y) = ω α(0) (π * (x), π * (y)) = ω α(1) (P S α (π * (x)), P S α (π * (y))) = ω α(1) (π * (P γ (x)), π * (P θ•γ (y))) = β α(1) (P γ (x), P θ•γ (y)),
i.e. β is parallel and extends also along lines of the spectral network on Σ 2 . Finally, let p ∈ T ′ Σ 2 . Let x ∈ E p and y ∈ E θ(p) regular elements. Then (π * (x), π * (y)) is an isotropic basis of L π(p) , i.e. β(x, y) = ω(π * (x), π * (y)) ∈ A × . So the pairing β is non-degenerate.

(⇐) Assume, there exists a non-degenerate parallel sesquilinear pairing β. Let p ∈ T ′ Σ 2 that does not project to a point on a line of the spectral network on Σ 2 . We define for every x ∈ E p , y ∈ E θ(p) : ω π(p) (π * (x), π * (y)) := β p (x, y).

Because (π * (x), π * (y)) is a basis of V π(p) , ω extends by sesquilinearity on V π(p) if we assume

ω π(p) (π * (x), π * (x ′ )) = ω π(p) (π * (y), π * (y ′ )) = 0
for all x, x ′ ∈ E p and y, y ′ ∈ E θ(p) . Since β is non-degenerate, ω is non-degenerate as well.

Since β p (x, y) = -σ(β θ(p) (y, x)), we get

ω π(p) (π * (y), π * (x)) = β θ(p) (y, x) = -σ(β p (x, y)) = -σ(ω π(p) (π * (x), π * (y))).
Further, ω is parallel. Indeed, let α : [0, 1] → T ′ S be a path such that the projections of α(0) and α(1) to S are not on the lines of the spectral network. Let x, y ∈ L α(0) . Let

α 1 , α 2 := θ • α 1 are two standard lifts of α to T ′ Σ 2 . Then x = π * (x 1 ) + π * (x 2 ) and y = π * (y 1 ) + π * (y 2 ) where x 1 , y 1 ∈ E α 1 (0) and x 2 , y 2 ∈ E α 2 (0)
. If the projection of α to Σ 2 does not intersect the spectral network, then the projection T ′ Σ 2 → T ′ S and the parallel transport along α and α 1 , α 2 commute. So ω is parallel because β is parallel.

Assume now that the projection of α intersects the spectral network once. We denote by α 3 the additional lift of α along the spectral network. Without loss of generality, assume α 3 (0) = α 1 (0) and α 3 (1) = α 2 (1). Notice that the path θ • (α 3 .α 1 ).α 3 .α 1 is homotopic to the fiber of T ′ Σ 2 → Σ 2 . Therefore, P θ•α 1 .α 3 = -P θ•α 3 .α 1 . Therefore, ω α(1) (P S α (x), P S α (y)) = ω α(1) (P S α (x), P S α (y)) = ω α(1) (P S α (π * (x 1 )) + P S α (π * (x 2 )), P S α (π * (y 1 )) + P S α (π * (y 2 ))) = ω α(1) (π * (P α 1 (x 1 ) + P α 3 (x 1 ) + P α 2 (x 2 )), π * (P α 1 (y 1 ) + P α 3 (y 1 ) + P α 2 (y 2 )))

= ω α(1) (π * (P α 1 (x 1 ), π * (P α 3 (y 1 ) + P α 2 (y 2 ))))

+ ω α(1) (π * (P α 3 (x 1 ) + P α 2 (x 2 )), π * (P α 1 (y 1 ))) = β α 1 (1) (P α 1 (x 1 ), P α 3 (y 1 ) + P α 2 (y 2 )) + β α 2 (1) (P α 3 (x 1 ) + P α 2 (x 2 ), P α 1 (y 1 )) = β α 1 (1) (P α 1 (x 1 ), P α 3 (y 1 )) + β α 1 (1) (P α 1 (x 1 ), P α 2 (y 2 )) + β α 2 (1) (P α 3 (x 1 ), P α 1 (y 1 )) + β α 2 (1) (P α 2 (x 2 ), P α 1 (y 1 )) = β α 1 (1) (P α 1 (x 1 ), P α 2 (y 2 )) + β α 2 (1) (P α 2 (x 2 ), P α 1 (y 1 )) + β α 1 (0) (x 1 , P (θ•α 3 ).α 1 (y 1 ) + P (θ•α 1 ).α 3 (y 1 )) = β α 1 (1) (P α 1 (x 1 ), P α 2 (y 2 )) + β α 2 (1) (P α 2 (x 2 ), P α 1 (y 1 )) + β α 1 (0) (x 1 , P θ•α 3 .α 1 (y 1 ) + P θ•α 1 .α 3 (y 1 )) = β α 1 (1) (P α 1 (x 1 ), P α 2 (y 2 )) + β α 2 (1) (P α 2 (x 2 ), P α 1 (y 1 )) = β α 1 (0) (x 1 , y 2 ) + β α 2 (0) (x 2 , y 1 ) = ω α(0) (x, y).
So ω is parallel and extends also along lines of the spectral network on S.

Finally, let p ∈ Σ 2 and x ∈ E p , y ∈ E θ(p) such that β p (x, y) = 1, then ω(π * (x), π * (y)) = 1. So ω is a field of standard symplectic forms.

Topology of the moduli space of framed twisted symplectic local systems

We keep the same notation as in Proposition 2.3.7. Our goal in this section is to prove the following theorem:

Theorem 2.7.18. The moduli space of framed (twisted) Sp 2 (A, σ)-local systems on S that are ∆-generic with respect to a fixed triangulation ∆ is homeomorphic to:

((A σ ) × ) -2χ(S)+2p-1+ n i × (A × ) 1-χ(S)+p /A ×
where the group A × acts componentwisely by conjugation on (A × ) 1-χ(S)+p and by congruence on ((A σ ) × ) -2χ(S)+2p-1+ n i .

Proof. We use the 1:1-correspondence between framed twisted Sp 2 (A, σ)-local systems on S that are transverse to a fixed triangulation ∆ and twisted A × -local systems on Σ 2 equipped with a non-degenerate parallel pairing β as in Theorem 2.7.17.

Let b ∈ T ′ Σ 2 such that it projects to a ramification point b ∈ Σ 2 . Let α 1 , . . . , α s : [0, 1] → S are free generators of the fundamental group π 1 (S, π(b)). Let γ 1 i , γ 2 i are closed lifts of α i to

T ′ Σ 2 such that θ • γ 1 i = γ 2 i and
β b(x, P s + (x)) = β θ( b) (P s + (x), P s + (P s + (x))) = β θ( b) (P s + (x), -x) = -β θ( b) (P s + (x), x).
So we obtain:

β b(x, P s + (x)) = -β θ( b) (P s + (x), x) = -σ(β θ( b) (P s + (x), x)) =: a 0 ∈ A σ .
Let now γ be a loop based at b and

a 0 = β b(x, P s + (x)) = β b(P γ (x), P θ•γ P s + (x)).
For every x ∈ E b, P γ (x) = xa γ where a γ ∈ A × . Let

P θ•γ P s + (x) = P s + (x)a ′ γ for a ′ γ ∈ A × . Then a 0 = σ(a γ )β b(x, P s + (x))a ′ γ = σ(a γ )a 0 a ′ γ , a ′ γ = a -1 0 σ(a -1 γ )a 0 .
Let γ and s -.(θ • γ).s + are different generators of π 1 (T ′ Σ 2 , b) (this corresponds to curves γ 1 i and γ 2 i of Lemma 2.3.10 case (1) lifted to T ′ Σ 2 ). In particular, they are not homotopic. Then a γ and a 0 determine uniquely a andπ(b ′ ) where b ′ is another ramification point on Σ 2 . Let b := γ(0) and b′ := γ(1). In this case, ξ b′ := ξ := s - θ( b) .θ(γ).s + b′ .γ and s -.(θ • ξ).s + are homotopic in T ′ Σ 2 . Therefore, a ξ = a -1 0 σ(a ξ )a 0 , i.e. a 0 a ξ ∈ A σ . Moreover, an easy calculation shows that a 0 a ξ = β b′ (y, P s + b′ y) where y = P γ (x).

′ γ . Let γ : [0, 1] → T ′ Σ 2 and θ • γ : [0, 1] → T ′ Σ 2 are two lifts to T ′ Σ 2 of a segment in S connecting π(b)
Chapter 3

Cluster A-coordinates

In this chapter we present an attempt at the construction of a non-commutative generalization of the Fock-Goncharov cluster A-coordinates introduced in [FG06] in the type A, i.e. for SL n (R)-local systems. Note however that many Lie groups does not admit a definition over a non-commutative algebra: this is the case in particular of SL n for which there is no analog over a non-commutative ring, thus forcing us to deal with GL n instead. We cannot yet express the mutation formulas for any mutation, we only describe those which are necessary to describe a flip in the triangulation chosen. We lack a geometric interpretation of a general mutation to extend the formula to any mutation in a given quiver. Still, the coordinates presented here have some convenient characteristics to justify the name of cluster A-coordinates. First, they admit a nice description using (slightly altered) quivers which make their computation easy when changing the underlying triangulation. Second, they coincide with the non-commutative algebra introduced by Arkady Berenstein and Vladimir Retakh in [START_REF] Berenstein | Noncommutative marked surfaces[END_REF] in the case n = 2, while also providing a quiver combinatorial description of the mutation formulas.

The coordinates introduced here are closely related to the cluster coordinates introduced by Alexander Goncharov and Maxim Kontsevich in [START_REF] Goncharov | Spectral description of noncommutative local systems on surfaces and non-commutative cluster varieties[END_REF]. The Goncharov-Kontsevich coordinates are products of two of the coordinates presented here. Another way to put it is to say that the coordinates presented here describe a factorization of each of the Goncharov-Kontsevich. It is worth noting that both coordinates systems coincide with the algebra defined by Berenstein-Retakh in the case n = 2. These two different definition of "cluster A-coordinates" both have their advantages and drawbacks. The Goncharov-Kontsevich coordinates describe the moduli space that is the most natural generalization of the Amoduli space introduced by Fock-Goncharov in [START_REF] Fock | Moduli spaces of local systems and higher Teichmüller theory[END_REF], namely the moduli space of twisted decorated GL n (R)-local systems on an hyperbolic ciliated surface. However, they present some differences with the commutative Fock-Goncharov coordinates: the combinatorial description of the mutations sequences uses a bipartite graph instead of a quiver, and the mutation formulas do not coincide with the usual commutative ones when the ring R is commutative. A goal of the work presented here is to give a solution to both of those problems, and to see what is the trade-off to have coordinates that fits on a quiver with mutations formulas as close to the commutative ones as possible.

Rank 2

In rank 2 most of the technical difficulties arising from our definition of non-commutative cluster coordinates do not appear. We show in Section 3.1.2 that the non-commutative cluster algebra A S introduced in [BR18] can be realized as the ring of non-commutative rational functions on the moduli space of decorated twisted GL 2 (R)-local systems over an hyperbolic ciliated surface S. Furthermore, the mutation formulas can be explicitly computed using spectral networks, which allow us to give a geometric and topological proof of an algebraic property of this algebra: the non-commutative Laurent phenomenon. We also give in Section 3.1.5 a representation of a subalgebra Q S of A S introduced in [BR18], and give a presentation of the subalgebra Q S .

The rank 2 cluster algebra

Arkady Berenstein and Vladimir Retakh introduced in [BR18] a non-commutative algebra associated to any hyperbolic ciliated surface that have some properties that can be thought as generalizations of properties of (commutative) cluster algebras. This algebra arose from the study of non-commutative Plücker coordinates constructed with a non-commutative algebra tool called quasideterminants introduced in [START_REF] Gelfand | Quasideterminants[END_REF]. In this section we recall some definitions and properties of these algebras introduced in [START_REF] Berenstein | Noncommutative marked surfaces[END_REF], and in Section 3.1.2 we will give a geometric representation of these non-commutative algebras.

Let S be an hyperbolic ciliated surface, endowed with a hyperbolic metric with totally geodesic boundary. In this section, an arc of S is a (homotopy class) of path γ : [0, 1] → S such that γ(0), γ(1) ∈ P and γ(]0, 1[) ⊂ S. We will denote by E(S) the set of all arcs of S. Two arcs γ 1 and γ 2 are composable if γ 1 (1) = γ 2 (0).

Given two hyperbolic ciliated surfaces S 1 and S 2 with puncture set P 1 and P 2 respectively, a map f : S 1 → S 2 is a morphism of ciliated surface if it extends to an orientation-preserving local homeomorphism f : S 1 → S 2 such that f -1 (P 2 ) = P 1 .

For n ≥ 3, we will call a n-gon a closed disk with n external punctures, and will denote it by P n . The punctures of a n-gon are cyclically ordered from 1 to n. A triangle of S is a morphism P 3 → S of ciliated surface and a quadrilateral of S is a morphism P 4 → S of ciliated surface. Given a morphism f : P n → S, we will denote by γ i,j the image of the arc from the puncture j to the puncture i in P n by f . Definition 3.1.1. We define the unitary R-algebra A S generated by the symbols x γ and x -1 γ for all arcs γ ∈ E(S) (with the convention x γ = 1 if γ is trivial) with the relations:

• ∀γ ∈ E(S), x -1 γ x γ = x γ x -1 γ = 1 • ∀f : P 3 → S triangle, x γ 1,3 x -1 γ 2,3 x γ 2,1 = x γ 1,2 x -1 γ 3,2 x γ 3,1 • ∀f : P 4 → S quadrilateral, x γ 4,2 = x γ 4,3 x -1 γ 1,3 x γ 1,2 + x γ 4,1 x -1 γ 3,1 x γ 3,2 and x γ 2,4 = x γ 2,3 x -1 γ 1,3 x γ 1,4 + x γ 2,1 x -1 γ 3,1 x γ 3,4
For ∆ a triangulation of S, we define A ∆ the subalgebra of A S generated by the set of all x γ for γ ∈ E(S) and x -1 γ for γ ∈ ∆. When S is the n-gon, we will sometimes write A n instead of A Pn .

The study of the subalgebra A ∆ for a fixed triangulation ∆ of S is considerably easier than the study of A S . Given a group G and a commutative field K, the group algebra of G denoted by KG is the vector space freely spanned by the elements of G, together with the multiplication obtained by extending the multiplication on G by linearity. Any element x ∈ KG can be written as x = w∈G λ w w where all λ w ∈ K but a finite number are zero, and Supp(x) is the finite set of w ∈ G such that λ w ̸ = 0. The following definition is used in [START_REF] Baumslag | One-relator groups: an overview[END_REF] and will be used later in Theorem 3.1.12.

Definition 3.1.2. A cyclically pinched 1-relator group is a finitely generated group G that writes as an amalgamated product

G = F 1 * w 1 =w 2 F 2
where F 1 and F 2 are finitely generated free groups and w i ∈ F i is a non-trivial element for i = 1, 2.

Theorem 3.1.3. Let S be an hyperbolic ciliated surface and let ∆ be a triangulation of S.

Then the algebra A ∆ is isomorphic to the group algebra RG of G, where G is :

• A finitely generated free group if S has at least one boundary component or if S is a sphere with 3 internal punctures

• A cyclically pinched 1-relator torsion free group otherwise.

Proof. See [START_REF] Berenstein | Noncommutative marked surfaces[END_REF], Theorem 3.26.

Coordinates on decorated GL 2 -local systems

In this section we show that the algebra introduced in 3.1.1 can be realized as noncommutative rational functions on the moduli space of decorated twisted GL 2 (R)-local systems on an hyperbolic ciliated surface S. Additionally, many algebraic computations can be done with purely topological/geometrical tools, namely spectral networks.

Let S be an hyperbolic ciliated surface and let ∆ be a triangulation of S. Let Σ = Σ 2 and π : Σ → S be the ramified covering constructed in Section 2.3.1. Then for every arc γ of ∆ from p ∈ P to q ∈ P we can bend it so that T ′ γ intersect T ′ β p and T ′ β q as in Figure 1.1, i.e. β p and γ are tangent at their intersection point.

Let E ∆ (S) ⊂ T ′ S be the set of intersection points between ∆ and the lifted decoration curves T ′ β p . This means that now each edge of the triangulation ∆ is endowed with two special points (one for each extremity) lying on the peripheral curves associated to its endpoints. For every edge γ ∈ ∆ of the triangulation, let τ γ be the path in T ′ S with extremities in E ∆ (S) obtained by restricting γ to the part in between the two special points on it. Note that since this is applied to all the oriented arcs of the triangulation, the chosen representative for τ γ and τ γ are such that T ′ (τ γ .τ γ ) is homotopic to a fiber T ′ S → S (see Figure 1.1).

Figure 1.1: The bending of an edge of the triangulation. In red are the peripheral decoration, in blue and green are the oriented edges of the triangulation, the crosses are the points in E ∆ (S) and the thicker part of the edges in between the peripheral curves are the paths τ γ and τ γ

We also apply the same construction in Σ to equip each edge of the hexagonal tiling π -1 (∆) with two special points, and denote E ∆ (Σ) the set of all special points in T ′ Σ. Note that both E ∆ (Σ) and E ∆ (S) are finite sets, and that the restriction of dπ :

T ′ Σ → T ′ S is 2:1 from E ∆ (Σ) to E ∆ (S).
Since the points in E ∆ (Σ) lie on peripheral curves associated with punctures, they inherit the source/sink naming from the puncture.

Let L be a decorated twisted GL 2 (R)-local system on the surface S, and assume L is ∆-generic. Let γ ∈ ∆ be an edge from p ∈ P to q ∈ P. We can trivialize the GL 2 (R)-local system L over T ′ τ γ and the 1-dimensional subbundles F (p) 1

and F

(q) 1 are transverse. The natural projection a γ :

F (p) 1 → L/F (q) 1
is an isomorphism, and we can identify it with its (1 by 1) matrix in the bases b

(p)
1 and b (q) 2 . We thus obtain a family (a γ ) γ∈∆ of elements of R × which we call non-commutative A-coordinates of L. The name "coordinates" is a slight abuse, since they are not independent. Proposition 3.1.4. For every oriented triangle (γ 1 , γ 2 , γ 3 ) of ∆, we have

a γ 3 a -1 γ 2 a γ 1 = a γ 1 a -1 γ 2 a γ 3 (1.1)
Proof. This is a direct corollary of Proposition 1.3.3. The minus sign appearing in the relation from Proposition 1.3.3 is canceled out by the minus sign appearing from the fact the local system is twisted. Indeed, the path τ γ 3 τ γ 2 τ γ 1 τ γ 1 τ γ 2 τ γ 3 is homotopic to a loop going once around the fiber of T ′ S → S.

The coordinates of a decorated twisted GL 2 (R)-local system are the holonomies of its abelianized system E along the lifts of the arcs τ γ , γ ∈ ∆. Indeed, for each puncture p ∈ P, the two lifts of p to Σ are a sink p 1 and a source p 2 . In the neighborhood of p 1 , the bundle E is the pullback of

F (p) 1
and in the neighborhood of p 2 the bundle is the pullback of L/F (p)

1 . Now for an arc γ ∈ ∆ from p ∈ P to q ∈ P, the lifts of τ γ to Σ join a sink and a source. Denote γ 1 the lift from p 1 to q 2 and γ 2 the lift from p 2 to q 1 . Then a γ is the holonomy of E along T ′ τ γ 1 and a γ is thus the holonomy of E along T ′ τ γ 2 .

Let G be the graph embedded in T ′ Σ with vertices the points of E ∆ (Σ) and edges the arcs τ γ , γ ∈ π -1 (∆) oriented from sink to source and portions of lifts of peripheral curves connecting two different points of E ∆ (Σ). To each oriented edge of this graph a coordinate is associated, and we assign to the edges with reversed orientation the inverse of this coordinate. The monodromy of E around a peripheral curve is trivial because L is decorated, so given a path on G, its holonomy in E is well-defined and only depends on the edges of the form τ γ of the path. Then the triangle relation (1.1) imply that the monodromy of the abelianized system E restricted to the graph G is trivial around every hexagonal tile. Note that since Σ is obtained by gluing the hexagons back to G, the data of a family (a γ ) γ∈∆ of invertible elements of R satisfying the relations 1.1 give rise to a twisted local system on Σ.

We now want to compute the "change of charts" induced by a flip in the triangulation. Let ∆ 1 and ∆ 2 be two triangulations differing only by one flip. Let p 1 , p 2 , p 3 , p 4 be the four (not necessarily distinct) punctures at the vertices of the quadrilateral supporting the flip, in the cyclic order such that ∆ 1 \ ∆ 2 = {γ 1,3 , γ 3,1 } and ∆ 2 \ ∆ 1 = {γ 2,4 , γ 4,2 } where γ i,j is the arc of the quadrilateral going from p j to p i . Using the path-lifting map constructed in Section 2.4.1, we can compute the relations between the A-coordinates associated to ∆ 1 and the A-coordinates associated to ∆ 2 . Proposition 3.1.5. Let L a decorated twisted GL 2 (R)-local system that is both ∆ 1 -generic and ∆ 2 -generic. Then its A-coordinates with respect to ∆ 1 and ∆ 2 satisfy the following exchange relations:

a γ 2,4 = a γ 2,1 a -1 γ 3,1 a γ 3,4 + a γ 2,3 a -1 γ 1,3 a γ 1,4 a γ 4,2 = a γ 4,1 a -1 γ 3,1 a γ 3,2 + a γ 4,3 a -1 γ 1,3 a γ 1,2 Remark 3.1.6
. This way we retrieve with a geometric argument the relation given in Proposition 1.3.4.

Proof. For i ∈ {1, 2, 3, 4}, let p ′ i , p ′′ i be the two lifts of p i to Σ where p ′ i is the sink and

p ′′ i is the source. Let s ∈ E ∆ 1 (S) ∩ E ∆ 2 (S)
be the intersection of T ′ β p 2 and γ 2,1 and let t ∈ E ∆ 1 (S) ∩ E ∆ 2 (S) be the intersection of T ′ β p 4 and γ 3,4 . Let δ be a path in T ′ S from s to t as in Figure 1.2. Let Σ 1 and W 1 be the ramified covering and the spectral network associated to the triangulation ∆ 1 and Σ 2 , W 2 the ones associated to ∆ 2 . The corresponding path-lifting maps will be denoted SN 1 and SN 2 , and the corresponding partial abelianizations of L will be denoted E 1 and E 2 .

First, let's lift δ to Σ 2 using SN 2 . Let s 1 , s 2 be the lifts of s to Σ 2 , s 1 being the sink and s 2 the source. Similarly, let t 1 , t 2 the lifts of t, t 1 being the sink and t 2 the source. We get

SN 2 (δ) = δ 1 + δ 2 + δ ′ 1 + δ ′ 2 + δ ′ 3
where δ 1 is a standard lift from s 1 to t 2 , δ 2 is a standard lift from s 2 to t 1 , δ ′ 1 is a lift from s 2 to t 2 added by the spectral network, δ ′ 2 is a lift from s 2 to t 1 added by the spectral network and δ ′ 3 is a lift from s 1 to t 1 added by the spectral network (see Figure 1.3). The path δ 1 is the only lift going from s 1 to t 2 , and its holonomy in E 2 in the corresponding bases is a γ 4,2 since it is homotopic to τ γ 4,2 precomposed with a piece of β p 2 and postcomposed with a piece of β p 4 , both of which have trivial holonomies. Since L is the partial non-abelianization of E 2 , this means that the map

F (p 2 ) 1 → L/F (p 4 ) 1
obtained by trivializing L along δ is exactly a γ 4,2 .

Figure 1.3: All the lifts of δ to Σ 2 using SN 2 . Now we will lift δ to Σ 1 using SN 1 . Let s 1 , s 2 be the lifts of s to Σ 1 , s 1 being the sink and s 2 the source. Similarly, let t 1 , t 2 the lifts of t, t 1 being the sink and t 2 the source. We get

SN 1 (δ) = δ 1 + δ 2 + δ ′ 1 + δ ′ 2 + δ ′ 3
where δ 1 is a standard lift from s 1 to t 2 , δ 2 is a standard lift from s 2 to t 1 , δ ′ 1 is a lift from s 1 to t 1 added by the spectral network, δ ′ 2 is a lift from s 1 to t 2 added by the spectral network and δ ′ 3 is a lift from s 2 to t 2 added by the spectral network (see Figure 1.4). The paths going from s 1 to t 2 are δ 1 and δ ′ 2 , and their holonomies in E 1 in the corresponding bases are respectively a γ 4,3 a -1 γ 1,3 a γ 1,2 and a γ 4,1 a -1 γ 3,1 a γ 3,2 . These are obtained by retracting the paths on the graph Γ, as the oriented edges of Γ have holonomies given by the A-coordinates. Since L is also the partial non-abelianization of E 1 , this means that the map

F (p 2 ) 1 → L/F (p 4 ) 1
obtained by trivializing L along δ must be equal to the holonomy of δ 1 + δ ′ 2 , which give the formula:

a γ 4,2 = a γ 4,1 a -1 γ 3,1 a γ 3,2 + a γ 4,3 a -1 γ 1,3 a γ 1,2
The formula for a γ 2,4 is obtained similarly.

The proposition above yield a geometric realization of the non-commutative algebra A S introduced in [START_REF] Berenstein | Noncommutative marked surfaces[END_REF]. Let S be an hyperbolic ciliated surface and let ∆ be a triangulation of S. We now focus on the case R = M d (R). In this case, the space X of decorated twisted GL 2 (R)-local systems on S is an algebraic variety and the subset X ∆ of ∆-generic local Figure 1.4: All the lifts of δ to Σ 1 using SN 1 .

systems is an open dense subset of X. Each A-coordinate associated to ∆ can be seen as a rational function on X ∆ with coefficient in M d (R).

Corollary 3.1.7. Let S be an hyperbolic ciliated surface. The map

ψ : A S → Rat(X, M d (R)) x γ → a γ is an algebra homomorphism.
Using the same type of arguments as above, we can give a topological/geometrical proof of the Laurent phenomenon for the cluster algebra of a polygon:

Theorem 3.1.8. Let n ≥ 3 and let S be the closed disk with n punctures on the boundary. Let i, j ∈ {1, . . . , n}, i ̸ = j. Then for every triangulation ∆ of S and every decorated twisted GL 2 (R)-local system L that is ∆-generic and such that (F (i) , F (j) ) is generic, the A-coordinate a γ i,j is a non-commutative Laurent polynomial in the A-coordinates (a γ ) γ∈∆ associated to the triangulation ∆.

Proof. All the edges of the form γ i,i+1 , with i ∈ P ordered cyclically, belong to every triangulation of S so the result is immediate if j ∈ {i -1, i + 1}. Now let i, j ∈ {1, . . . , n}, i ̸ = j ± 1. Let ∆ 0 be a triangulation of S containing the edges γ i,j , γ i,j-1 and γ i-1,j . Such a triangulation always exists when i ̸ = j ± 1. Let s ∈ E ∆ 0 (S) ∩ E ∆ (S) be the intersection of β j and γ j-1,j and let t ∈ E ∆ 0 (S) ∩ E ∆ (S) be the intersection of β i and γ i-1,i . Let δ be the path from s to t drawn in Figure 1.5.

As we have seen in the proof of the flip relation (Proposition 3.1.5) and keeping the same notations, in the spectral network lift of δ with respect to the triangulation ∆ 0 the only term from s 1 to t 2 has the holonomy a γ i,j in the partial abelianization of L with respect to ∆ 0 . This means that the map F (j) → L/F (i) obtained by trivializing L on δ is a γ i,j .

Figure 1.5: The path δ in the triangulation ∆ 0 . Only the quadrilateral (i -1, i, j -1, j) is drawn.

Let E be the partial abelianization of L with respect to ∆. In spectral network lift of δ with respect to the triangulation ∆, let δ ′ = δ ′ 1 + • • • + δ ′ r be the sum of all paths from s 1 to t 2 . Each δ ′ k has a holonomy in E that is a monomial in the coordinates (a ±1 γ ) γ∈∆ as it retracts on the graph G. Since L is the partial non-abelianization of E, the map F (j) → L/F (i) obtained by trivializing L on δ is equal to the sum of the holonomies of the δ ′ k in E, so it is a Laurent polynomial in the A-coordinates (a γ ) γ∈∆ .

Using these A-coordinates, we can describe precisely the changes on the abelianized R × -local system on Σ = Σ 2 induced by a flip in the triangulation. We use the same notations as in Proposition 3.1.5. Let L be a framed twisted GL 2 (R)-local system on S that is transverse with respect to two triangulations ∆ 1 and ∆ 2 . Let E 1 (resp. E 2 ) be the R × -local system on Σ obtained by abelianizing L with respect to ∆ 1 (resp. ∆ 2 ). These changes on the abelianized local system are supported in the lift C Q of the quadrilateral Q surrounding the flip, which is homeomorphic to a cylinder with four punctures on each boundary components in Σ. Let γ be a loop on T ′ Σ. If γ crosses only one of the two boundary component of C Q then the monodromies of γ in E 1 and E 2 are equal. Suppose γ crosses exactly once each of the two boundary components of C Q . Let γ Q be the loop going around C Q with the same orientation as the boundary of C Q containing the sinks lifts of p 2 and p 4 (we refer to this boundary as the positive one, and the other one as negative).

Remark 3.1.9. We think of the holonomy of γ Q in E as a generalization in the noncommutative setting of Fock-Goncharov's X -coordinate of the quadrilateral Q. If R = R, the holonomy of γ Q is the cross-ratio of the four lines in R 2 given by the framing of L.

Up to homotopy, we can assume γ is going through at least one point

x 0 ∈ E ∆ 1 (Σ) ∩ E ∆ 2 (Σ)
on one of the eight external edges of the hexagon tiling of Q. We also choose a representative of γ Q based at x 0 . Let b be a basis of the fiber of E 1 over x 0 . Since x 0 is not in the interior of the cylinder supporting the flip in Σ, the fibers of E 1 and E 2 over x 0 are the same. Let Y 1 ∈ R × (resp. Y 2 ) be the holonomy of γ in E 1 (resp. E 2 ), and let X be the holonomy of γ Q in both E 1 and E 2 . Proposition 3.1.10. If the part of γ inside C Q goes from the positive boundary to the negative boundary, then

Y 2 = Y 1 (1 + X).
If the part of γ inside C Q goes from the negative boundary to the positive boundary, then

Y 2 = Y 1 (1 + X -1 ) -1
Remark 3.1.11. The element 1 + X -1 ∈ R is invertible because of the transversality of L with respect to ∆ 2 .

We have shown that a representation A ∆ → R give rise to a ∆-generic decorated twisted GL 2 (R)-local system on S. We now want to show a partial converse result: that there is no additional relations satisfied by the A-coordinates. For this, we will focus on R = M d (R).

A group G is called residually finite if for all g ∈ G\ {1}, there exists a finite index normal subgroup H not containing g, or equivalently if there exists a finite group G g and a morphism φ g : G → G g such that the image of g by φ g is non-trivial. All the finitely generated free groups are residually finite. Indeed, a rank n free group F n = ⟨x 1 , . . . , x n ⟩ embeds into

F 2 = ⟨a, b⟩ via the map F n → F 2 x i → ab i .
In turn, F 2 embeds into SL n (Z) ⊂ Isom + (H 2 ) via the group homomorphism defined by a → 1 2 0 1 and b → 1 0 2 1 .

Given any non-trivial M ∈ SL 2 (Z), there exists an integer N ≥ 2 such that the image of M in the group SL n (Z/N Z) is non-trivial. This hows that SL n (Z) is residually finite, hence F n is too.

The following theorem is due to Baumslag in [START_REF] Baumslag | One-relator groups: an overview[END_REF].

Theorem 3.1.12. A cyclically pinched 1-relator group is residually finite.

Let S be an hyperbolic ciliated surface and let ∆ be a triangulation of S. We now focus on the case R = M d (R). In this case, the space X of decorated twisted GL 2 (R)-local systems on S is an algebraic variety and the subset X ∆ of ∆-generic local systems is an open Zariski-dense subset of X. Each A-coordinate associated to ∆ can be seen as a rational function on

X ∆ with coefficient in M d (R). Let ψ d : A ∆ → Rat(X ∆ , M d (R))
be the representation defined in Proposition 3.1.7.

Theorem 3.1.13. The family of representations (ψ

d : A ∆ → Rat(X ∆ , M d (R))) d≥2 is asymptotically injective, i.e. d≥2 Ker(ψ d ) = {0} .
Proof. Combining the Theorem 3.1.3 and 3.1.12, the algebra A ∆ is isomorphic to RG where G is a residually finite group. Let x ∈ A ∆ \ {0}. We can write

x = w∈Supp(x) λ w w.
Since G is residually finite, for all w, w ′ ∈ Supp(x) distinct there exists a finite group G ww ′-1 and a morphism

φ ww ′-1 : G → G ww ′-1 such that φ ww ′-1 (ww ′-1 ) ̸ = 1. Let G = w,w ′ ∈Supp(x),w̸ =w ′ G ww ′-1 and let φ = w,w ′ ∈Supp(x),w̸ =w ′ φ ww ′-1
be the morphism associated. It satisfies φ| Supp(x) is injective because for any w, w ′ ∈ Supp(x) distinct, φ(ww ′-1 ) ̸ = 1 so φ(w) ̸ = φ(w ′ ). The map φ :

RG = A ∆ → R G satisfies φ(x) ̸ = 0.
The algebra R G acts on itself by left multiplication, providing an embedding

R G → End(R G). Since G is finite, the algebra R G is finite dimensional so End(R G) is isomorphic to M d (R) where d = |G|. Since φ(x) ̸ = 0, φ(x) acts non trivially on R G, hence is mapped to an non-zero element of M d (R). This yield a representation f : A ∆ → M d (R) such that f (x) ̸ = 0, so x / ∈ Ker(φ d ).

A-coordinates for symplectic local systems

Recall the definition of symplectic local systems given in Section 2.7. Since Sp 2 (A, σ) is a subgroup of GL 2 (A), the A-coordinates defined in section 3.1.2, a twisted symplectic local system have well-defined A-coordinates, and because of the additional structure of symplectic local systems, they satisfy additional relations. The following proposition is immediate:

Proposition 3.1.14. Let S be an hyperbolic ciliated surface and let ∆ be a triangulation of S. Let L → S be a ∆-transverse decorated symplectic local system. Let γ be an arc of the triangulation ∆ from p ∈ P to q ∈ P. Then a γ = ω(b

(q) 1 , b (p)
1 ). In particular a γ = -σ(a γ ).

Proof. By definition of non-commutative

A-coordinates, b (p) 1 ∈ F (p) 1 projects to b (q) 2 a γ ∈ L/F (q) 1 , i.e. for some lift b(q) 2 ∈ A 2 of b (q) 2 , b (p) 1 = b(q) 2 a γ + b (q)
1 r for some r ∈ R. Therefore, ω(b

(q) 1 , b (p) 1 ) = ω(b (q) 1 , b(q) 2 a γ + b (q) 1 r) = ω(b (q) 1 , b(q) 2 )a γ = a γ .
From Proposition 1.1 follows:

Corollary 3.1.15. For each oriented triangle t = (γ 1 , γ 2 , γ 3 ) of ∆, we have

β t := a γ 3 a -1 γ 2 a γ 1 ∈ A σ .
If A is Hermitian, the signature of β t agrees with the Kashiwara-Maslov index of t. Furthermore if β t ∈ A σ + for all oriented triangles t of ∆, then the decorated local system is maximal.

A subalgebra of A S

In [START_REF] Berenstein | Noncommutative marked surfaces[END_REF], Berenstein and Retakh define a subalgebra Q n of A n and give a representation of both A n and Q n in terms of quasideterminants. Our goal in this section is to define a subalgebra Q S of A S that coincides with Q n when S = P n , and to give a presentation of Q S similar to the presentation of Q n given in [BR18], Theorem 2.14. We follow a similar approach to the original proof of Berenstein and Retakh. For this, we write the algebra A S as a quotient of an algebra defined similarly to A n , except over an "infinite polygon", namely the universal cover of S in H 2 .

Let S be an hyperbolic ciliated surface, endowed with a hyperbolic metric with totally geodesic boundary. Let π : S → S be the universal cover of S, seen as a closed convex totally geodesic subspace of H. We have S = Γ\ S where π 1 (Σ) ≃ Γ ⊂ PSL 2 (R) acts properly discontinuously on S ⊂ H. We endow P 1 R with the canonical cyclic order, and we see P 1 R = ∂H as the boundary of the hyperbolic plane H. Let Λ = π -1 (P) ⊂ P 1 R be the Farey set of S, i.e. the preimage of all punctures of S. A pair (p, q) ∈ Λ 2 with p ̸ = q is identified with the oriented geodesic in H from p to q. The surface S admits (infinite) ideal triangulations, in particular any pullback of a triangulation of S.

Definition 3.1.16. Let Q S be the subalgebra of A S spanned by the elements y α,α ′ = x -1 α x α ′ with α and α ′ composable arcs on S. Given a triangulation ∆ of S, we denote by Q ∆ the subalgebra of Q S spanned by the elements y α,α ′ = x -1 α x α ′ with α and α ′ composable arcs in ∆. Definition 3.1.17. We define the unitary R-algebra A Λ generated by the symbols x p,q and x -1 p,q for p, q ∈ Λ (with the convention x p,p = 1 for all p ∈ Λ) with the relations:

• ∀p, q ∈ Λ, x -1 p,q x p,q = x p,q x -1 p,q = 1

• ∀p, q, r ∈ Λ, x p,q x -1 r,q x r,p = x p,r x -1 q,r x q,p • ∀(p, q, r, s) cyclically ordered, x p,r = x p,q x -1 s,q x s,r + x p,s x -1 q,s x q,r For ∆ a triangulation of S, we define A ∆ the subalgebra of A Λ generated by the set of all x p,q for p, q ∈ Λ and x -1 p,q for (p, q) ∈ ∆. We define Q Λ the subalgebra of A Λ generated by the elements y r p,q = x -1 r,p x r,q for p, q, r ∈ Λ, r ̸ = p, q.

Definition 3.1.18. A filtered set is a nonempty set I together with a (partial) order ⩽, with the additional property that every pair of elements has an upper bound.

Let (I, ⩽) a filtered set. Let (A i ) i∈I be a family of unitary algebras and let (f i,j ) i,j∈I with f i,j : A i → A j algebra homomorphisms such that f ii = Id A i and f i,j • f k,i = f k,j . We call ((A i ) i∈I , (f ij ) i,j∈I ) a direct system. We denote by (lim

-→
A i , (f i ) i∈I ) the direct limit of the system, which is defined by the following universal property: for all algebra B and for all families of algebra homomorphisms (g i ) i∈I such that for all i ⩽ j ∈ I, g i = g j • f ij , there exists a unique homomorphism φ : lim -→ A i → B such that the following diagram commutes:

A i f i ! ! f ij / / g i $ $ A j f j } } g j y y lim -→ A i φ B This direct limit is isomorphic to i∈I A i / ∼,
where the equivalence relation ∼ is defined by (i, a i ) ∼ (j, a j ) ⇔ ∃k ∈ I, i ⩽ k, j ⩽ k, f i,k (a i ) = f j,k (a j ). In particular, for all a ∈ lim -→ A i , there exists i ∈ I and a i ∈ A i such that a = f i (a i ).

Let P be the set of all finite subsets of Λ. Endowed with the inclusion, it is a filtered set. Let lim -→ A P be the direct limit of the system ((A P ) P ∈P , (j P,P ′ ) P ⊂P ′ ∈P ), where j P,P ′ : A P → A P ′ is the algebra homomorphism induced by the inclusion P ⊂ P ′ (i.e. ∀p, q ∈ P, j P,P ′ (x p,q ) = x p,q ). Theorem 3.1.19. The inclusions A P → A Λ for P ∈ P induce an algebra isomorphism

lim -→ A P ≃ A Λ .
Proof. Let P ∈ P. Let j P : A P → lim -→ A P be the direct limit map associated to P , and let f P : A P → A Λ be the algebra homomorphism induced by the inclusion P ⊂ Λ. This algebra homomorphism is well defined because Σ contains all the triangles and the quadrilaterals of P . The morphisms f P then satisfy f P = f P ′ • j P,P ′ for all P ⊂ P ′ ∈ P. By universal property, there exists an algebra homomorphism φ : lim -→ A P → A Λ such that φ • j P = f P for all P ∈ P. For p, q ∈ Λ, we define ψ(x p,q ) = j P (x p,q ) ∈ lim -→ A P , where P ∈ P contains p and q. This definition does not depends on the choice of P by the property of direct limits. Moreover, for all triangle or quadrilateral of S, there exists P ∈ P containing it. So ψ : A Λ → lim -→ A P is a well defined algebra homomorphism. By construction, the morphisms φ and ψ are inverse one of each other.

A P j P # # j P,P ′ / / f P % % A P ′ j P ′ { { f P ′ y y lim -→ A P φ A Λ ψ O O
The following lemmas recall a few elementary properties of direct limits, they are stated here without proof.

Lemma 3.1.20. Let (I, ⩽) be a filtered set. Let ((A i ) i∈I , (f ij ) i,j∈I ) be a direct system and let (B i ) i∈I a family of subalgebras

B i ⊂ A i such that f ij (B i ) ⊂ B j . Then the maps B i → lim -→
A j induce an identification of the limit lim

-→ B i of the system ((B i ) i∈I , (f ij | B i ) i,j∈I ) with a subalgebra of lim -→ A i .
Corollary 3.1.21. For all triangulations ∆ of S, the subalgebra A ∆ of A Λ is identified with the direct limit lim -→ A ∆ P with P ∈ P which admits a triangulation ∆ P such that ∆ P ⊂ ∆. We then have an isomorphism A ∆ ≃ RF ∞ , where F ∞ is the free group freely spanned by a countable infinite set of generators. The subalgebra Q Λ of A Λ is identified with the direct limit lim -→ Q P with P ∈ P. Lemma 3.1.22. Let P be a polygon. The subalgebra Q P of A P is generated by the elements y r p,q with p, q, r ∈ P distinct, with the relations:

Lemma 3.1.23. Let 0

I i A i Q i 0 m i π i
be a family of short exact sequences indexed by i ∈ I, with ((A i ) i∈I , (f ij ) i,j∈I ) a direct system such that f ij • m i (I i ) ⊂ m j (I j ). Let g ij = f ij | I i : I i → I j and h ij : Q i → Q j the map induced by π j • f ij . Then we have a short exact sequence 0 lim

-→ I i lim -→ A i lim -→ Q i 0.
Corollary 3.1.24. The subalgebra Q Λ of A Λ is generated by the elements y r p,q with p, q, r ∈ Λ distinct, with the relations:

1. ∀p, q, r ∈ Λ distinct, y r p,q y r q,p = 1 2. ∀p, q, r ∈ Λ distinct, y r p,q y p q,r y q r,p = 1 3. ∀p, q, r, s ∈ Λ distinct, y s p,q y s q,r y s r,p = 1 4. ∀(p, q, r, s) ∈ Λ distinct cyclically ordered, y q p,s = y r p,q y p q,s + y r p,s

Given two algebras A and B, we denote by A * B their free product.

Proposition 3.1.25. Let f : Λ → Λ be a map without fixed points. Let F Λ be the free group generated by c i , i ∈ Λ and let RF Λ be its group algebra. Then the map:

A Λ → (RF Λ ) * Q Λ x ij → c i * y i f (i),j
is an algebra isomorphism.

Proof. Let φ :

A Λ → (RF Λ ) * Q Λ x ij → c i * y i f (i),j
. Firs we show that φ is well defined. Let i, j, k ∈ Λ.

We have

φ(x ij x -1 kj x ki ) = c i * y i f (i),j * (c k * y k f (k),j ) -1 * c k * y k f (k),i = c i * y i f (i),j * (y k f (k),j ) -1 * c -1 k * c k * y k f (k),i = c i * y i f (i),j * y k j,f (k) * y k f (k),i = c i * y i f (i),j * y k j,i = c i * y i f (i),j * y i j,k * y j k,i = c i * y i f (i),k * y j k,i φ(x ik x -1 jk x ji ) = c i * y i f (i),k * (c j * y j f (j),k ) -1 * c j * y j f (j),i = c i * y i f (i),k * (y j f (j),k ) -1 * c -1 j * c j * y j f (j),i = c i * y i f (i),k * y j k,f (j) * y j f (j),i = c i * y i f (i),k * y j k,i .
Let (i, j, k, ℓ) ∈ Λ cyclically ordered.

φ(x jk x -1 ik x iℓ + x ji x -1 ki x kℓ ) = c j * y j f (j),k * (c i * y i f (i),k ) -1 * c i * y i f (i),ℓ + c j * y j f (j),i * (c k * y k f (k),i ) -1 * c k * y k f (k),ℓ = c j * y j f (j),k * y i k,f (i) * y i f (i),ℓ + c j * y j f (j),i * y k i,f (k) * y k f (k),ℓ = c j * y j f (j),k * y i k,ℓ + c j * y j f (j),i * y k i,ℓ = c j * y j f (j),k * (y i k,ℓ + y j k,i * y k i,ℓ ) = c j * y j f (j),k * y j k,ℓ = c j * y j f (j),ℓ = φ(x jℓ )
So φ is well defined. Let ψ 1 :

RF Λ → A Λ c i → x i,f (i) and let ψ 2 : Q Λ → A Λ be the canonical inclusion. Let ψ = ψ 1 * ψ 2 : (QF Λ ) * Q Λ → A Λ .
We now show that φ and ψ are reciprocal inverse. We have

(ψ • φ)(x ij ) = ψ(c i * y i f (i),j ) = x i,f (i) y i f (i),j = x ij . Moreover, (φ • ψ)(c i ) = φ(x i,f (i) ) = c i * y i f (i),f (i) = c i and (φ • ψ)(y k ij ) = φ(y k ij ) = (c k * y k f (i),i ) -1 * c k * y k f (i),j = y k i,f (i) * y k f (i),j = y k ij .
So φ is an isomorphism with inverse ψ.

Corollary 3.1.26. With the same notation as above,

A Λ → Q Λ x ij → y i f (i),j
is a surjective algebra homomorphism, and its kernel is spanned by the elements x f (i),i -1 with i ∈ Λ.

Let A S and Q S be the algebras defined in Section 3.1.1. For all p, q ∈ Λ distinct, we denote by (p, q) the oriented geodesic of S from p to q and π(p, q) the image of this geodesic in S. Let ∆ be a triangulation of S and let ∆ be the lift of ∆ to S. Since π maps a polygon of S to a polygon of S, π induces a surjective algebra homomorphism π * : A Λ → A S defined by π * (x p,q ) = x π(p,q) . The action of Γ = π 1 (S) is orientation-preserving, so it preserves the cyclic order on Λ. An element γ ∈ Γ induces an algebra automorphism A Λ given by γ • x p,q = x γ(p),γ(q) . Proposition 3.1.27. The kernel of π * : A Λ → A S is generated by the elements (γ • x) -x with γ ∈ Γ and x ∈ A Λ . In particular, we have an isomorphism

A S ≃ A Λ / ((γ • x) -x)
Proof. For all x ∈ A Λ and γ ∈ Γ, we have (γ •x)-x ∈ ker π * . Let φ : A Λ / ((γ • x) -x) → A Σ be the morphism induced by π * . Let

ψ : A Λ → A Λ / ((γ • x) -x) x α → x α
, where α = π( α) is an arc of S.

This defines an algebra homomorphism: on one hand if α ′ is another lift of α by π, then there exists γ ∈ Γ such that α ′ = γ • α so x α = x α ′ and the definition of ψ(x α ) does not depend on the choice of the lift of α. On the other hand, if P is a n-gon of S, then there exists a n-gon P of S such that π( P ) = P . This allows to deduce that ψ preserves the relations of A Λ , hence defines an algebra homomorphism. By construction, φ and ψ are inverse, so φ is an isomorphism.

Theorem 3.1.28. Let I ⊂ Q Λ be the ideal spanned by the elements γ • y -y, γ ∈ Γ, y ∈ Q Λ and y r γ(p),q -y r p,q , p, q, r ∈ Λ distinct with γ ∈ Stab Γ (r).

Then Q S ≃ Q Λ /I.
Proof. The kernel of the morphism φ : Q Λ → Q S y r p,q → y π(p,r),π(r,q) satisfies I ⊂ ker φ, so we get a morphism φ : Moreover as shown in Corollary 3.1.26, the map

Q Λ /I → Q S y r p,q → y π(p,
ψ ′ : A Λ → Q Λ /I x p,q → y p f (p),q
is an algebra homomorphism. We now show that the ideal ((γ•x)-x) is contained in its kernel. Let p, q ∈ Λ and let γ ∈ Γ. We have y

γ(p) f (γ(p)),γ(q) = γ • y p γ -1 (f (γ(p))),q and γ -1 (f (γ(p))) ∈ Stab Γ (p) • f (p), so there exists γ ′ ∈ Stab Γ (p) such that γ -1 (f (γ(p))) = γ ′ (f (p)). Then γ • y p γ -1 (f (γ(p))),q = γ • y p γ ′ (f (p))
,q , and since γ • y p γ ′ (f (p)),q = ←y r p,q so ψ ′ (x p,q ) = ψ ′ (γ • x p,q ). So ψ ′ is an algebra homomorphism A Λ /((γ • x) -x) → Q Λ /I, whose composition with the isomorphism A S → A Λ /((γ • x) -x) from Proposition 3.1.27 yield a morphism ψ :

A S → Q Λ /I x α → y p f (p),q with α = π(p, q)

. We now show that ψ| Q Σ = φ -1 .

(ψ • φ)(y r p,q ) = ψ(y π(p,r),π(r,q) ) = ψ(x -1 π(r,p) x π(r,q) ) = y r f (r),p -1 y r f (r),q = y r p,f (r) y r f (r),q = y r p,q

Let α, α ′ be two arcs of S with α(1) = α ′ (0), and let (p, r) and (r, q) be lifts of α and α ′ respectively, sharing an endpoint r ∈ Λ. We have

(φ • ψ)(y α,α ′ ) = φ(ψ(x -1 α x α ′ )) = φ(y r f (r),p -1 y r f (r),q ) = φ(y r p,f (r) y r f (r),q ) = φ(y r p,q ) = y α,α ′ So φ : Q Λ /I → Q S is an isomorphism.
Corollary 3.1.29. The algebra Q S is generated by the elements y α,α ′ with α, α ′ composable arcs of S with the following relations:

1. y α,α = 1 for all α arc of S 2. y α,α ′ y α ′ ,α = 1 for all α, α ′ composable arcs of S 3. For all triangle (α, α ′ , α ′′ ) of S: y α,α ′ y α ′′ ,α y α ′ ,α ′′ = 1 4. For all α, α ′ , α ′′ such that α(1) = α ′ (1) = α ′′ (1): y α,α ′ y α ′ ,α ′′ y α ′′ ,α = 1 5. ∀f : P 4 → S quadrilateral, y α 12 ,α 24 = y α 13 ,α 32 y α 21 ,α 14 + y α 13 ,α 34 Proof. Let Q be the algebra defined with the presentation above. We have a morphism φ :

Q Λ → Q y r p,q → y π(p,r),π(r,q) because the relations of Q Λ given by Corollary 3.1.24 are satisfied by φ. This morphism induces φ : Q Λ /I → Q. By composing it with the isomorphism of Theorem 3.1.28, we get a morphism φ ′ : Q Σ → Q. The relations of Q are satisfied in Q Σ , so we have a morphism

ψ : Q → Q Σ y α,α ′ → y α,α ′
and ψ and φ ′ are inverse.

Coordinates on semi-decorated GL 2 -local systems

We now give a representation of the subalgebra Q S as coordinates on GL 2 -local systems on S with a new type of additional data in between a decoration and a framing. Definition 3.1.30. Let S be an hyperbolic ciliated surface and let L be a twisted GL 2 (R)local system on S. A semi-decoration of L is the data of a framing (F (p) ) p∈P together with a regular section b Let L be a ∆-generic semi-decorated GL 2 (R)-local system on S. Let α, α ′ ∈ ∆ be composable edges. Let p ∈ P be the start of α ′ and q ∈ P be the end of α. The map

a -1 α • a α ′ : F (p) 1 → F (q) 1
is well defined. We denote by q α,α ′ its (1 by 1) matrix in the respective bases of F (p) 1

and

F (q) 1
given by the semi-decoration. We call those elements the Q-coordinates of L.

Let R = M d (R). Let S be an hyperbolic ciliated surface and let ∆ be a triangulation of S.

The space Q of semi-decorated twisted GL 2 (R)-local systems on S is an algebraic variety and the subset Q ∆ of ∆-generic local systems is an open Zariski-dense subset of Q. Each Q-coordinate associated to ∆ can be seen as a rational function on Q ∆ with coefficient in M d (R). As a corollary of 3.1.7, we get:

Corollary 3.1.32. Let S be an hyperbolic ciliated surface and let ∆ be a triangulation of S. The map

ψ Q : Q ∆ → Rat(Q ∆ , M d (R)) y α,α ′ → q α,α ′ is an algebra homomorphism.

Coordinates on extra-decorated configurations of flags

We want extend the A-coordinates defined in Section 3.1 for GL 2 (R)-local systems to GL n (R)local systems. Because of some technical difficulties related to twisted local system, it will be more convenient to introduce first those A-coordinates in the case of a configuration of flags where those technical difficulties do not arise. The mutations formulas of the coordinates defined in this section are meant to be the closest possible to both the commutative case, with a quiver to break down the mutation of a flip into smaller mutations, and the noncommutative rank 2 case described above. For this, it is natural to introduce a new kind of decoration on a configuration of flags, which we call an extra-decoration since it contains a decoration in the sense introduced in Section 1.2 plus additional data. This new kind of decoration depends on the choice of a triangulation, so the mutation rules must include a way to "mutate" an extra-decoration associated to a triangulation to one associated to another triangulation. The coordinates introduced can be expressed in terms of quasideterminants (see Section 3.2.7). It should be noted however that these coordinates fail to satisfy the non-commutative Laurent phenomenon in general, however other situations in which noncommutative Laurent phenomenon arise in small rank also fail to satisfy it in higher rank. These coordinates also fail to satisfy the pentagon relation because the extra-decoration itself does not satisfy it, so the coordinates that depends on this extra-decoration cannot satisfy it either.

Extra-decorated configurations of flags

Definition 3.2.1. Let (A (1) , . . . , A (k) ) be a k-tuple of flags and let ∆ be a triangulation of the k-gon. If for all triangle (i, j, k) of ∆, (A (i) , A (j) , A (k) ) is a triple of flag in generic position, the k-tuple (A (1) , . . . , A (k) ) is called ∆-generic. The group GL n (R) acts naturally on the space of ∆-generic k-tuple of flags in R n . We denote by Conf ∆ R,k (n) the quotient of this space by the GL n (R) action. Definition 3.2.2. Let (A (1) , . . . , A (k) ) be a k-tuple of flags in generic position (recall Section 1.2 for the definition of flags in generic position). An extra-decoration of the k-tuple of flags is the data of a decoration of each flag A (i) together with the data for every i

1 , . . . , i k ∈ N such that i 1 + • • • + i k = (k -1)n + 1 of an element b i 1 ,...,i k ∈ R n that freely span A (1) i 1 ∩ • • • ∩ A (k) i k .
Definition 3.2.3. Let ∆ be a triangulation of the k-gon. Let (A (1) , . . . , A (k) ) be a ∆-generic k-tuple of flags. A ∆-extra-decoration of (A (1) , . . . , A (k) ) is the data for each triangle (i, j, k) of ∆ of an extra-decoration of the triple (A (i) , A (j) , A (k) ). The data of a ∆-generic k-tuple of flags together with a ∆-extra-decoration is called a ∆-extra-decorated k-tuple of flags. The group GL n (R) acts naturally on the space of ∆-extra-decorated k-tuple of flags in R n . We denote by DecConf ∆ k (n) the quotient of this space by the GL n (R) action.

Coordinates on the configuration space of extra-decorated triple or quadruple of flags

Let (A, B, C) be a triple of flags in generic position. Recall from Section 1.3 that we defined for all i, j, k ∈ N such that i + j + k = 2n + 1 the following maps by restriction of the canonical projections

R n → A i /A i-1 , R n → B j /B j-1 and R n → C k /C k-1 : a A i,j,k : A i ∩ B j ∩ C k → A i /A i-1 a B i,j,k : A i ∩ B j ∩ C k → B j /B j-1 a C i,j,k : A i ∩ B j ∩ C k → C k /C k-1
Given an extra-decoration of the triple of flags (A, B, C), we identify these linear maps with their matrices in the corresponding bases. We call these elements of R × the (noncommutative) A-coordinates of the extra-decorated triple of flags.

A reformulation of Proposition 1.3.3 is the following: Proposition 3.2.4. The A-coordinates satisfy the following relations, called triangle relations:

∀i, j, k ∈ N s.t. i+j+k = 2n+2, a A i,j,k-1 . a B i,j,k-1 -1 .a B i-1,j,k = -a A i,j-1,k . a C i,j-1,k -1 .a C i-1,j,k

Quiver and mutations

Let P be a polygon and ∆ a triangulation of P . Let Q be the A n-1 -type quiver on P associated to the triangulation ∆, see Figure 2.6. More precisely, we see any triangle t as the Euclidean triangle in R 3 with vertices of coordinates (n, 0, 0), (0, n, 0) and (0, 0, n).

The A n-1 -type quiver of the triangle is the quiver with vertex the points of t with integer coordinates except for the vertices of t itself, and a vertex of coordinates (a, b, c) has outgoing arrows to (a + 1, b -1, c), (a, b + 1, c -1) and (a -1, b, c + 1) and ongoing arrows from vertex (a + 1, b, c -1), (a -1, b + 1, c) and (a, b -1, c + 1), provided those vertex belong to t and that the edge is not contained in an edge of t. The quiver Q is planar, and every vertex has the same number of edges going in and going out of it. Note that these two properties are preserved during the mutation sequence that leads to a flip in the triangulation. To handle the fact we are working with GL n and not SL n or PGL n -and our coordinates being non-commutative-we need to make a few adjustments to the usual mutation rules.

Figure 2.6: From left to right, the quivers of type A 1 , A 2 , A 3 on a triangle.

First, we want to allow pairs of vertices v and v ′ with arrows both from v to v ′ and from v ′ to v, although only in some precise situations.

Definition 3.2.5. Let Q = (Q 0 , Q 1 ) be a cyclic quiver, i.e. a quiver with n vertices

Q 0 = {v 1 , . . . , v n } and n -1 arrows Q 1 = {(v i , v i+1 )|1 ≤ i ≤ n} forming a cycle, with v n+1 = v 1 . An quiver!internal oriented triangulation of Q is a set of arrows Q ′ 1 such that (Q 0 , Q 1 ∪ Q 1 ∪ Q ′ 1 )
is the graph of a polygon with an oriented triangulation, where Q 1 denote the set of reversed edges of Q 1 . See Figure 2.7 for an example of a cyclic quiver with an internal oriented triangulation. • Each vertex of Q has as many arrows going in and going out of it. We define the valency of a vertex v, denoted by val(v), to be the number of arrows going to v if v is not frozen and this number plus one if v is frozen

• There are two types of arrows in Q, we call them either plain or dashed

• At each vertex of Q, there exists a pairing between arrows going in and arrows going out, such that each pair of paired arrows is part of an oriented triangle for which the orientation agrees with the orientation on P (in the following, an oriented cycle in a quiver will always suppose matching orientation with P ), and there are no edges inside such oriented triangles

• If an arrow v → v ′ is dashed, then there is another dashed arrow v ′ → v and those are the only arrows between v and v ′

• Each oriented cycle of plain arrows of Q has an internal oriented triangulation of dashed arrows, and every dashed arrow belongs to one such triangulation

• If there is a plain arrow v → v ′ , then there is no arrow v ′ → v.

• There is at most one arrow from a vertex v to another vertex v ′ When we say "arrow" without specifying if its dashed or plain it means any arrow of the quiver regardless of their nature, same for triangles and oriented cycles.

A notable difference with the usual quiver of cluster algebras is that we allow pairs of opposite arrows between two vertices, but only inside the oriented plain cycles. The set of all plain arrows in a triangulated quiver is a quiver in the usual sense. The dashed arrows are here to carry an additional information: a triangulation of the oriented cycles of plain arrows.

Proposition 3.2.7. Let P be a polygon, let ∆ be a triangulation of P and let n ≥ 1. The A n -type quiver associated with (P, ∆) is a triangulated quiver (without any dashed arrow). At every vertex v of a triangulated quiver, we will assign r variables a 1 (v), . . . , a r (v) ∈ R × where r is the valency of v. Notice there is exactly r oriented triangles t 1 , . . . , t r containing v in Q if v is not frozen. If v is frozen, then there is r -1 oriented triangles t 1 , . . . , t r-1 containing v in Q, and we set t r to be the boundary containing v. The labeling on t 1 , . . . , t r is set to agree with the cyclic ordering induced by the orientation of P . To get consistent mutation formulas, we will think of those r variables a 1 (v), . . . , a r (v) as "in between" the oriented triangles containing v. More precisely, we choose a cyclic order on the set of those r variables together with the oriented triangles containing v such that a

1 (v) < t 1 < a 2 (v) < • • • < t r-1 < a r (v) < t r < a 1 (v).
We call the choice of those variable together with the cyclic order an affectation of Q, and we will denote by V the set of all the variables.

Around every oriented triangle t, the situation looks like the one in Figure 2.10.

Notice that in the case when one (or more) of the arrow of the oriented triangle t is dashed, then there are variables located "in between" the two opposite dashed arrows, see Figure 2.9.

Figure 2.9: An affectation on a quiver containing a triangulated oriented quadrilateral.

Definition 3.2.8. Let Q be a triangulated quiver, together with an affectation V. Let t be an oriented triangle of Q. The affectation around t gives 6 variables a 1 , a 2 , b 1 , b 2 , c 1 , c 2 as in Figure 2.10. We define two elements which we call triangle elements at a:

R + t,a = b 1 a -1 2 c 1 and R - t,a = c 2 a -1 1 b 2 .
We define similarly R ± t,b and R ± t,c . An affectation of a triangulated quiver Q is said to satisfy the triangle relations at an oriented triangle t if the variables satisfy the following relation, which we call triangle relation at t:

R + t,x = -R - t,x
where x is any of the three vertices of t. If the affectation satisfies the triangle relation for every oriented triangle of Q, it is said to satisfy the triangle relations. Remark 3.2.9. At every oriented triangle of Q there is three equivalent triangles relations, one for each vertex. Also note that the triangle elements are always invertible elements of R.

Definition 3.2.10. A non-commutative seed is the data of:

• A triangulated quiver Q

• An affectation V of Q satisfying the triangle relations. We call those elements of R the A-coordinates of the seed.

To alleviate the notations, we will just refer to non-commutative seeds as seeds.

Definition 3.2.11. Let (Q, V) be a seed. Let x be a vertex of Q of valency 2. We say that x is mutable (or that the ▷◁-mutation at x is admissible) if none of the arrows incident to x are dashed and if the elements

x ′ 1 = d 2 x -1 1 a 2 + c 1 x -1 2 b 1 x ′ 2 = b 2 x -1 1 c 2 + a 1 x -1 2 d 1 are invertible in R,
where the variables a i , b i , y i , z i are defined as in Figure 2.11. If x is mutable, the ▷◁-mutation of (Q, V) at x is a new seed (Q ′ , V ′ ) = µ ▷◁,x (Q, V) defined as follows:

• The new quiver Q ′ is obtained by reversing the direction of the arrows incident to x and adding the four plain arrows needed to complete the triangles created. If this process creates a pair of plain arrows between two vertex with opposite directions, then if this pair lies inside an oriented plain cycle of the quiver we change the pair of arrows to be dashed, and if not we remove the pair of arrows.

• The two A-coordinates x 1 , x 2 at the vertex x are replaced by the new variables x ′ 1 and x ′ 2 defined above. The new cyclic ordering at the vertex x is described on Figure 2.11.

The result of an admissible ▷◁-mutation is still a seed, i.e. satisfies the triangle relations: only two triangles are modified by the mutation, and using the same notation as in Figure 2.11 the two new triangle relations are:

R + t ′ 1 ,a = x ′ 1 a -1 2 c 2 = (d 2 x -1 1 a 2 + c 1 x -1 2 b 1 )a -1 2 c 2 = d 2 x -1 1 c 2 + c 1 x -1 2 b 1 a -1 2 c 2 = R + t 2 ,x + c 1 (R - t 1 ,b ) -1 c 2 = -R - t 2 ,x -c 1 (R + t 1 ,b ) -1 c 2 = -c 1 x -1 2 d 1 -c 1 a -1 1 b 2 x -1 1 c 2 = -c 1 a -1 1 (b 2 x -1 1 c 2 + a 1 x -1 2 d 1 ) = -c 1 a -1 1 x ′ 2 = -R - t ′ 1 ,a
and a similar computation shows

R + t ′ 2 ,b = -R - t ′ 2 ,b .
Figure 2.11: The situation before and after the ▷◁-mutation at the vertex x.

Remark 3.2.12. We can express the new triangle elements in term of the previous triangle elements and the unchanged A-coordinates. Using the same notations as in Figure 2.11, we

Since the number of coordinates associated to a vertex depends on its valency and the valency can change during the sequence of mutation of a flip, we need a second type of mutation that changes the valency of a vertex.

Definition 3.2.15. Let (Q, V) be a seed. Let a and d two vertex of Q with a pair of dashed arrows between them. Let b, c ∈ Q 0 such that the oriented triangle containing a → d

(resp. d → a) is t 1 = c → a → d → c (resp. t 2 = a → b → d → a), see Figure 2.12. The -mutation of (Q, V) at the pair of arrows a ↔ d is a new seed (Q ′ , V ′ ) = µ ,a↔d (Q, V) defined as follows:
• The pair of dashed arrow is part of an internal triangulation of a plain oriented cycle.

The new quiver Q ′ is obtained by doing a flip of this edge in the internal triangulation:

Q ′ 1 = (Q 1 \ {a → d, d → a}) ∪ {b → c, c → b} .
The new arrows are also dashed.

• The valency of a and d is decreased by 1 so the variables a 3 and d 3 in between the removed dashed arrows are also removed. We then need to add two new variables in between the new pair of dashed arrows. The new added variable are:

b 3 = R + t 1 ,c = a 1 c -1 2 d 3 and c 3 = R + t 2 ,b = d 1 b -1 2 a 3 .
See Figure 2.12.

Figure 2.12: The situation before and after the -mutation in an oriented quadrilateral of Q.

Coordinates on configurations of flags

Let (A, B, C) be an extra-decorated triple of flags in R n . Let Q be the standard A n-1 -type quiver in a triangle (see Figure 2.6). The quiver Q is a triangulated quiver on a disk with 3 external punctures (with clockwise orientation). We label those three punctures p A , p B , p C (in that order) and think of the flags A, B, C as associated to their corresponding puncture. Every vertex of Q is determined by a triple of non-negative integers (i, j, k) such that i + j + k = 2n, where i (resp. j, k) is the distance of the vertex to the puncture p A (resp. p B , p C ). This labeling is different from the one used in Section 3.2.3, a vertex of coordinates (a, b, c) with Section 3.2.3 notations will have coordinates (i, j, k) = (n -a, n -b, n -c) in this new notation. We denote the vertex with coordinates (i, j, k) by v i,j,k . The two vertices closest to p A are v 1,n-1,n and v 1,n,n-1 . Frozen vertices of Q are the vertices for which one of the three coordinates is n.

Every non-frozen vertex v i,j,k of Q has valency three, and the three oriented triangles containing v i,j,k are:

t i,j,k,A = v i,j,k → v i-1,j+1,k → v i-1,j,k+1 → v i,j,k t i,j,k,B = v i,j,k → v i,j-1,k+1 → v i+1,j-1,k → v i,j,k t i,j,k,C = v i,j,k → v i+1,j,k-1 → v i,j+1,k-1 → v i,j,k
We assign to v i,j,k the coordinates a A i+1,j,k , a B i,j+1,k , a C i,j,k+1 constructed in Section 3.2.2 with the following cyclic order:

a A i+1,j,k < t i,j,k,C < a B i,j+1,k < t i,j,k,A < a C i,j,k+1 < t i,j,k,B < a A i+1,j,k
To remember this, t i,j,k,A is the triangle the closest to p A and a A i,j,k+1 is the coordinate the furthest from p A , with a similar statement for the other triangles/coordinates. Every frozen vertex v i,j,k of Q has valency two and is part of one oriented triangle t of Q. Suppose v i,j,k is on the boundary between p A and p B (which means that k = n and i + j = n), the other cases being symmetric. Let b A,B be the side of t between p A and p B . We assign to v i,j,n the variables a A i+1,j,n and a B i,j+1,n with the following cyclic ordering:

a A i+1,j,n < t < a B i,j+1,n < b A,B < a A i+1,j,n .
As an immediate corollary from Proposition 3.2.4, we have:

Proposition 3.2.18. The set of variables described above is an affectation of Q and satisfy the triangle relations.

Let P be a k-gon (k ≥ 3) and ∆ be a triangulation of P . Let Q be the quiver obtained by gluing together the A n-1 -type quiver of each triangle of ∆ along internal edges of ∆. The vertices on the internal edges of ∆ are no longer frozen. Let (A (1) , . . . , A (k) ) ∈ DecConf ∆ k (n). Inside each triangle (i, j, k) of ∆, the triple (A (i) , A (j) , A (k) ) is extra-decorated. The affectation described above is compatible with the gluing of the quivers along edges of ∆. Indeed, for every pair of triangles (t 1 , t 2 ) sharing an edge e, the variables affected to the vertices on e are identical in Q t 1 and Q t 2 , where Q t 1 (resp. Q t 2 ) is the quiver inside t 1 (resp. t 2 ). For every vertex v on e, the oriented triangle in Q t 1 (resp. Q t 2 ) replaces the boundary of the triangle t 1 (resp. t 2 ) regarding the cyclic order of the variables and oriented triangles at v. Remark 3.2.19. Let (Q, V) be the seed described above. The complement of Q in P has two types of connected components:

• interior of oriented triangles, which carry triangle relations • interior of cycles with opposite orientation to P (including the regions containing an edge of P ) which carries variables, all corresponding to maps with the same starting space. In this case we will say this common starting space is associated to this region. The four regions surrounding the two oriented triangle containing a vertex of valency 2 (as in Figure 2.11) are associated to four R-lines spanning a 2-dimensional subspace.

The five regions surrounding a pair of vertices with a pair of dashed arrows are associated to five R-lines obtained from intersections of four 2-dimensional subspaces spanning a 3-dimensional subspace.

As a corollary to Proposition 1.3.4, doing a ▷◁-mutation at a mutable vertex yield a new set of coordinates on the same extra-decorated configuration of flags. However, doing a -mutation replaces an anti-oriented region with another one. This means that for two triangulations ∆ 1 and ∆ 2 differing from a flip, we need a map from one ∆ 1 -extra-decoration of a configuration of flags to a ∆ 2 -extra-decoration of the same configuration of flags. For this, we use the isomorphism T + AD,BC defined in 1.3.5. This isomorphism transports the vector spanning the R-line associated to the region in-between a pair of dashed arrows to a vector spanning the R-line associated to the region in between the new pair of dashed arrows after the -mutation (See Figure 2.14).

Remark 3.2.20. A configuration of k ∆-extra-decorated flags is equivalent to a ∆-extradecorated local system (see Section 3.3 for the definition) on a disk with k external punctures and no internal puncture. Since on surfaces with no internal punctures twisted and nontwisted local systems are equivalent, we will show that the set of A-coordinates determines the configuration as a corollary of the fact that the A-coordinates of a local system can reconstruct it.

Figure 2.13: The flip mutation sequence for an A 1 -type quiver.

Flip mutation sequence

Let P be a quadrilateral with a triangulation ∆, let n ≥ 2 and let Q be the A n-1 -type quiver on P . We now describe a sequence of mutations that transform Q into the A n-1 -type quiver Q ′ associated with the other triangulation ∆ ′ of P . The sequence of mutations is the same as the standard one described in [START_REF] Fock | Moduli spaces of local systems and higher Teichmüller theory[END_REF] Section 10.3, except that at every step we need to apply all the possible -mutations. We now describe the mutation sequence more precisely. Let (A, B, D, C) be the vertices of P , such that the internal edge of ∆ is (B, C). The frozen vertices of Q lies on the edges of P , thus are uniquely determined by a quadruple (a, b, d, c) of integers such that only two of these integers are non-zero, and the remaining two add to n. The two non-zero coordinates indicate the edge of P the vertex is on, and their value indicate its location on the edge. For example, with n = 4, the frozen vertex of coordinates (1, 3, 0, 0) is the vertex on the edge AB that is the closest to A. For all pair of positive integers (k, ℓ) with k + ℓ = n, there are k × ℓ mutable vertices of Q that lies in the interior the rectangle embedded in P whose vertices are (k, ℓ, 0, 0), (k, 0, 0, ℓ), (0, ℓ, k, 0) and (0, 0, k, ℓ). We denote this set of vertices of Q by V k,ℓ . For 1 ≤ k ≤ n -1, the k-th step is doing the ▷◁-mutation at every vertex of V k,n-k , then doing once every possible -mutation. The flip mutation sequence is the result of the n -1 steps. See Figures 2.13, 2.14 and 2.15.

Pentagon relation

Let P be a pentagon and ∆ a triangulation of P . Let Q be the A n-1 -type quiver associated to (P, ∆). The pentagon relation states that after having done the mutations corresponding to five flips (without doing twice the same flip in a row) in Q, the quiver Q and the triangulation ∆ are back to their initial state. However, given a quintuplet (A, B, C, D, E) of flags in generic position and a ∆-extra-decoration the A-coordinates associated to this configuration of flags is not back in their initial values after such a sequence of mutation. The reason for this is that the extra-decoration itself changes during the sequence of mutation that lead to a flip in the triangulation, namely during every -mutation. In fact, after the five flips of the pentagon relation, the basis of a subspace A i ∩ B j ∩ C k with i + j + k = 2n + 1 is in general not mapped to a basis of the same subspace. This is easily seen for n = 3: after doing the flips corresponding to the pentagon relation as in Figure 2.16, the vector spanning A 2 ∩ C 2 is mapped to a vector spanning A 2 ∩ D 2 and conversely the vector spanning A 2 ∩ D 2 is mapped to a vector spanning A 2 ∩ C 2 . In general, the pentagon flip sequence acts non-trivially on the extra-decoration of a fixed configuration of flags in generic position.

Quasideterminants

In this section we show that the A-coordinates of an extra-decorated configuration of flags in R n are quasideterminants. A general introduction to quasideterminants can be found in [START_REF] Gelfand | Quasideterminants[END_REF]. To get this representation of A-coordinates as quasideterminants, it is necessary to fix a basis of the ambient space. In this section, R n is endowed with its canonical basis (e 1 , . . . , e n ). Definition 3.2.21. Let M ∈ GL n (R) and let 1 ≤ i, j ≤ n. If the (j, i)-th coefficient of M -1 is invertible, we call its inverse the (i, j)-th quasideterminant of M . Otherwise, we say the quasideterminant does not exists. Given a matrix M of size n × n with coefficients in R and 1 ≤ i, j ≤ n, we denote by |M | i,j the (i, j)-quasideterminant of M if it exists.

Let P be a k-gon, k ≥ 2. Let ∆ be a triangulation of P and let (A (1) , . . . , A (k) ) ∈ DecConf ∆ k (n). To realize A-coordinates as quasideterminants we need to add a technical hypothesis on the configuration (A (1) , . . . , A (k) ): we suppose (A (1) , . . . , A (k) , F opp ) is in generic position, where F opp is the opposite standard flag, i.e. (i) j-1 for all 1 ≤ j ≤ n. For all 1 ≤ j ≤ n, choose a lift a (i) j of a (i) j to A j ⊂ R n (note that along the path π(T ′ γ p,v ) to transport the i-dimensional subbundle F (p) i along T ′ β v . Since the monodromy around the loop going once around v is -Id, this defines a subbundle of L| T ′ βv . If v and v ′ are grouped together, i.e. β v = β v ′ , then the paths T ′ γ p,v and T ′ γ p,v ′ are homotopic so the respective parallel transport of F (i) p induce the same subbundle over T ′ β v . Definition 3.3.1. Let L be a decorated twisted GL n (R)-local system on S and let ∆ be a triangulation of S. Let v be a white vertex of Γ n . Let p, q, r be the punctures at the vertex of the triangle t of ∆ containing v. Let 1 < i, j, k < n such that v ∈ γ A ∆-extra-decoration of a twisted GL n (R)-local system depends on the graph Γ n , which depends on the triangulation ∆.

A-coordinates

Let S be a ciliated surface, ∆ a triangulation of S, n ≥ 2 and Γ n the associated bipartite graph. Let L be a ∆-extra-decorated twisted GL n (R)-local system on S. Let E be the abelianized R × -local system on Σ n constructed in Section 2.5. The decoration and ∆-extradecoration of L induce along all the decorating curves of Σ n a flat section of E. Indeed by construction, E is isomorphic to F v in the neighborhood of a non-peripheral white vertex v in Σ n , and isomorphic to F (p) i /F (p) i-1 in the neighborhood of the i-th lift p i of a puncture p. Let v be a white vertex of Γ n , p, q, r the punctures at the vertices of the triangle containing v and (i, j, k) the coordinates of v in the triangle (p, q, r). Without loss of generality suppose i > 1. Let γ p,v be a joining path from T ′ β p i to T ′ β v . The fiber of E above the end point T ′ γ p,v (1) has a single basis given by the decoration of L, namely b (p) i . If the vertex v is peripheral, for instance if v is on γ

(1) q , then the fiber of E above T ′ β v also have a single basis given by the decoration, namely b If the vertex v is not peripheral, then the fiber of E above the starting point T ′ γ p,v (0) have two bases given by b v (x 1 ) and b v (x 2 ), where x 1 , x 2 ∈ E(v) are such that T ′ γ p,v (0) = T ′ β v (x 1 ) = T ′ β v (x 2 ). They satisfy b v (x 1 ) + b v (x 2 ) = 0. Let a This pair satisfies a (p) i,j,k (x 1 ) + a (p) i,j,k (x 2 ) = 0 but there is no canonical choice of a "positive" coordinate.

Note that on a white vertex v of coordinates (i, j, k) in the triangle (p, q, r) the map σ : E(v) → E(v) defined in Section 2.3.2 induce a cyclic order on the A-coordinates associated to (p, v), (q, v) and (r, v). Again, we see this cyclic order on the coordinates as a map σ such that σ #E(v) = Id. For instance if x 1 ∈ E(v) is such that T ′ γ p,v (1) = x 1 and σ(x 1 ) = x 2 = T ′ γ q,v (1) then σ(a (p) i,j,k (x 1 )) = a (q) i,j,k (x 2 ). Proposition 3.3.2. Let S be a ciliated surface, ∆ a triangulation of S, n ≥ 2 and t = (p, q, r) a triangle of ∆. Let L be a ∆-extra-decorated twisted GL n (R)-local system on S. Let i, j, k ∈≤ n + 1 s.t. i + j + k = 2n + 2 and let a (q) i-1,j,k ∈ a (q) i-1,j,k , a Proof. These coordinates correspond to holonomies of paths in the R × -twisted local system E on Σ n . The left-hand term correspond to the holonomy along a contractible loop in T ′ Σ n , thus is trivial.

Quiver and twisted affectations

Let Q be a triangulated quiver on S. To account for the A-coordinates being defined up to sign, we slightly alter the definition of an affectation of Q. For this, the cycles in the quiver that are oriented with the opposite orientation to S will carry additional variables. Unfortunately, dealing with this sign indeterminacy greatly complexify the notations, even though the situation is very similar to the case of configurations of flags exposed in Section 3.2.3. All the following definitions are adaptations of the definitions in Section 3.2.3 to the case where many of the coordinates are defined "up to sign". Definition 3.3.3. Let S be a marked surface. An triangulated quiver on S is a quiver Q embedded in S such that:

• Q has no trivial loops

COORDINATES ON EXTRA-DECORATED LOCAL SYSTEMS: THE GENERAL CASE111

• The (possibly empty) subset of vertices of Q lying on the boundary of S are called frozen.

• Each vertex of Q has as many arrows going in and going out of it. We define the valency of a vertex v, denoted by val(v), to be the number of arrows going to v if v is not frozen and this number plus one if v is frozen

• There are two types of arrows in Q, they are called either plain or dashed

• At each vertex of Q, there exists a pairing between arrows going in and arrows going out, such that each pair of arrows is part of an oriented triangle for which the orientation agrees with the orientation on S (in the following, an oriented cycle in an embedded quiver will always suppose matching orientation with the surface), and there are no edges inside such oriented triangles Remark 3.3.5. Let Q be a triangulated quiver on S. Similarly to Remark 3.2.19, the complement of Q in S has three types of connected components:

• If an arrow v → v ′ is
• interior of oriented triangles, which carry triangle relations

• interior of cycles with opposite orientation to S (including the regions that contain portions of the boundary of S without punctures), which will carry coordinates with sign indeterminacy

• Punctured disks or half-disks on the boundary of S, which will carry coordinates with well-defined signs A twisted affectation of Q is an affectation of Q, except the variables a 1 (v), . . . , a r (v) ∈ R × affected to a vertex v of Q are replaced by sets a 1 , . . . , a r of cardinal either 1 or 2. Moreover, we add the data of the sign indeterminacy: for each anti-cycle c = (v 1 , . . . , v r ) of Q and for each vertex v k of c there is exactly one coordinate a i k (v k ) in the interior of c, i.e. such that t k < a i k (v k ) < t k+1 where t k (resp. t k+1 ) is the triangle of Q containing (v k-1 , v k ) (resp.

(v k , v k+1 )). We set for each vertex v k of c a bijection σ k : a i k (v k ) → a i k-1 (v k-1 ) such that σ r = τ k , where τ k is either Id :

a i k (v k ) → a i k (v k ) if #a i k (v k ) = 1 or the transposition if #a i k (v k ) = 2
, and where σ r is a slight abuse of notation made by omitting the subscript k of the corresponding variable.

Remark 3.3.6. For any consecutive coordinates a → b inside an anti-cycle the product σ(a)a -1 does not depend on the choice of a ∈ a. Indeed, the choice of a ∈ a will change the sign of both a and σ(a) ∈ b. To simplify the notation in the formulas, we will write ba -1 instead of σ(a)a -1 . Similarly, we will write ab -1 instead of aσ(a) -1 . These products are equal to Goncharov-Kontsevich coordinates defined in [START_REF] Goncharov | Spectral description of noncommutative local systems on surfaces and non-commutative cluster varieties[END_REF].

Remark 3.3.7. When defining the map σ, it is sufficient to define the image of one element inside every coordinate. Indeed, either in every coordinate there is at most two elements, and knowing the image of one element is enough to determine a bijection between sets of cardinal 1 or 2.

Definition 3.3.8. Let Q be a triangulated quiver on S and V a twisted affectation of Q.

We say V satisfies the triangle relations if for every oriented triangle t of Q, we have

(c 1 b -1 2 )(a 1 c -1 2 )(b 1 a -1 2 ) = 1
where the variable are defined as in Figure 3.17.

A twisted seed is the data of a couple (Q, V) where Q is a triangulated quiver on S and V is a twisted affectation on Q satisfying the triangle relations.

Figure 3.17: The variable around an oriented triangle of Q.

Definition 3.3.9. Let (Q, V) be a twisted seed on S. Let x be a non-frozen vertex of Q of valency 2 and let a, b, c, d be the surrounding vertices as in Figure 3.18. Let b 1 ∈ b 1 and c 2 ∈ c 2 . We say that x is mutable (or that the ▷◁-mutation at x is admissible) if the elements

x ′ 1 (b 1 ) = c 1 x -1 2 b 1 + d 2 x -1 1 σ(b 1 ) x ′ 2 (c 2 ) = b 2 x -1 1 c 2 + a 1 x -1 2 σ(c 2 )

COORDINATES ON EXTRA-DECORATED LOCAL SYSTEMS: THE GENERAL CASE113

are invertible in R, were the variables a i , b i , c i , d i , x i are defined as in Figure 3.18. If x is mutable, we set

x ′ 1 = x ′ 1 (b 1 ) | b 1 ∈ b 1 x ′ 2 = x ′ 2 (c 2 ) | c 2 ∈ c 2
Note that if there are two elements in b 1 (resp. c 2 ), then there are also two elements in x ′ 1 (resp. x ′ 2 ).

Figure 3.18: The situation before and after the ▷◁-mutation at the vertex x.

Remark 3.3.10. Each of the element in x 1 (resp. x 2 ) depends on the choice of an element in b 1 (resp. c 2 ). However the coordinate x ′ 1 (resp. x ′ 2 ) which is either a singleton or a pair depends only on b 1 (resp. c 2 ). Definition 3.3.11. Let (Q, V) be a twisted seed on S. Let x be a non-frozen vertex of Q of valency 2. If x is mutable, the ▷◁-mutation of (Q, V) at x is a new seed (Q ′ , V ′ ) = µ ▷◁,x (Q, V) defined as follows:

• The new quiver Q ′ is obtained by reversing the direction of the arrows incident to x and adding the four plain arrows needed to complete the triangles created. If this process creates a pair of plain arrows between two vertex with opposite directions, then if this pair lies inside an oriented plain cycle of the quiver we change the pair of arrows to be dashed, and if not we remove the pair of arrows.

• The two A-coordinates x 1 , x 2 at the vertex x are replaced by the new variables x ′ 1 and x ′ 2 defined above. The new cyclic ordering at the vertex x is described on Figure 3.18.

• The new map σ ′ is defined as follows: The -mutation of (Q, V) at the pair of arrows a ↔ c is a new seed (Q ′ , V ′ ) = µ ,a↔d (Q, V) defined as follows:

σ ′ (a 1 ) = σ 2 (a 1 ) ∈ c 1 , a 1 ∈ a 1 σ ′ (d 2 ) = σ 2 (d 2 ) ∈ b 2 , d 2 ∈ d 2 σ ′ (b 1 ) = x ′ 1 (b 1 ), σ ′ (x ′ 1 (b 1 )) = σ(b 1 ) σ ′ (c 2 ) = x ′ 2 (c 2 ), σ ′ (x ′ 2 (c 2 )) =
• The pair of dashed arrow is part of an internal triangulation of a plain oriented cycle of length 4. The new quiver Q ′ is obtained by doing a flip of this edge in the internal triangulation:

Q ′ 1 = (Q 1 \ {a → d, d → a}) ∪ {b → c, c → b} .
The new arrows are also dashed.

• The valency of a and c is decreased by 1 so the variables a 3 and d 3 in between the removed dashed arrows are also removed. We then need to add two new variables in between the new pair of dashed arrows, see Figure 3.19. The new added variable are:

c 3 (a 3 ) = d 1 b -1 2 a 3 = a 2 b -1 1 σ(a 3 ), a 3 ∈ a 3 b 3 (d 3 ) = a 1 c -1 2 d 3 = d 2 c -1 1 σ(d 3 ), d 3 ∈ d 3
• The new map σ ′ is defined as follows: Proposition 3.3.13. If (Q, V) is a twisted seed on S, then the pair (Q ′ , V ′ ) obtained after any admissible ▷◁-mutation or any -mutation is a twisted seed on S.

Twisted affectation associated to an extra-decorated twisted local system

Let S be a ciliated surface and let ∆ be a triangulation of S. Let n ≥ 2 and let Q be the A n-1 -type quiver on (S, ∆). Let t be a triangle of ∆ and let (p, q, r) be the punctures at the vertices of t. Let L be a ∆-extra-decorated twisted local system on S.

Inside t, every vertex of Q is determined by a triple of positive integers (i, j, k) such that i + j + k = 2n and i, j, k ≤ n, where i (resp. j, k) is the distance of the vertex to the puncture p (resp. q, r). We denote the vertex with coordinates (i, j, k) by v i,j,k . The two vertices closest to p are then v 1,n-1,n and v 1,n,n-1 .

Every non-frozen vertex v i,j,k of Q that is not on an edge of t has valency three, and the three oriented triangles containing v i,j,k are:

t p i,j,k = v i,j,k → v i-1,j+1,k → v i-1,j,k+1 → v i,j,k t q i,j,k = v i,j,k → v i,j-1,k+1 → v i+1,j-1,k → v i,j,k t r i,j,k = v i,j,k → v i+1,j,k-1 → v i,j+1,k-1 → v i,j,k
We assign to v i,j,k the coordinates a (p) i+1,j,k , a

i,j+1,k , a

i,j,k+1 with the following cyclic order:

a (p)
i+1,j,k < t r i,j,k < a (q) i,j+1,k < t p i,j,k < a (r) i,j,k+1 < t q i,j,k < a (p) i+1,j,k

To remember this, t p i,j,k is the triangle the closest to p and a (p) i,j,k+1 is the coordinate the farthest from p, with a similar statement for the other triangles/coordinates. If one of the edges of t is a boundary component then the vertices of Q that lies on this edge are frozen. Suppose v i,j,k is on the boundary between p and q (which means that k = n and i + j = n), the other cases being symmetric. Then v i,j,n is part of exactly one oriented triangle t r i,j,n of Q. Let b p,q be the boundary between p and q. We assign to v i,j,n the variables a It remains to assign variables to non-frozen vertices that lie on an edge of t, which is then an edge shared by two triangles t = (p, q, r) and t ′ = (q, p, s), so the cyclic order on the punctures is (p, s, q, r). Such a vertex v has coordinates (i, j, n) in t with i + j = n and coordinates (j, i, n) in t ′ . The two oriented triangles of Q containing v are t r i,j,n in t and t ′s j,i,n in t ′ . We assign to v the variables a i,j+1,n (t) = a (q) j+1,i,n (t ′ ) with the following cyclic ordering: a (p) i+1,j,n (t) < t r i,j,n < a (q) i,j+1,n < t ′s j,i,n < a (p) i+1,j,n (t).

As an immediate corollary from Proposition 3.3.2, we have:

Proposition 3.3.14. The set of variables described above is an affectation of Q and satisfy the triangle relations.

Remark 3.3.15. The coordinates defined by Goncharov and Kontsevich in [START_REF] Goncharov | Spectral description of noncommutative local systems on surfaces and non-commutative cluster varieties[END_REF] are products of two of the A-coordinates described here, as in Remark 3.3.6. As such, they do not require an extra-decoration but only a decoration, and the sign indeterminacy doesn't appear. However, the mutations formulas described in [START_REF] Goncharov | Spectral description of noncommutative local systems on surfaces and non-commutative cluster varieties[END_REF] are closer to the mutations formulas for Q-coordinates described in Section 3.1.5.

Index

Figure 3

 3 Figure 3.1: From left to right, a 3-subdivision, a 4-subdivision and a 5-subdivision of a triangle.

n

  between the white vertex inside it and the three black vertices at the vertices of the downward small triangle, see Figure3.2.

Figure4

  Figure 3.2: The graphs Γ (t) 2 , Γ (t) 3 and Γ (t) 4 restricted to a triangle t of ∆.

  )n on e, see Figure3.3.

Figure 3

 3 Figure 3.3: The gluing of Γ (t) 3 and Γ (t ′ ) 3

Figure 3 . 5 :

 35 Figure 3.5: The neighborhood of a white vertex, an internal black vertex and an external black vertex, together with the different cells glued in the neighborhood of those vertices.

Figure 3 .

 3 Figure 3.10: All decorating curves and the paths joining them in the lift of a triangle to Σ 3 .

  (i) p ∩ t and there are i -1 white vertices on the boundary of S γ (i-1) p ∩ t. Each of those black vertices is connected to exactly one of those white vertices via an edge of Γ (t) n , and those black vertices are ramification points. On Σ n , we define for each of those black vertices a line of the spectral network as follows: let v b be an internal black vertex on the boundary of S γ (i) p ∩ t and v w the white vertex on the boundary of S γ (i-1) p ∩ t associated to v b . Let c 1 a path from p i to v b in S γ (i) p ∩ t, let c 2 be the path from v b to v w obtained by following the edge of the graph Γ (t) n , and let c 3 be the path from v w to p i-1 in S γ (i-1) p ∩ t. The line of the spectral network is the concatenation α

Figure 4 .

 4 Figure 4.18: A small loop around a branch point.

Figure 4 .

 4 Figure 4.19: All 6 paths added by intersections with the spectral network, together with the standard lifts γ 1 and γ 2 . On the upper left picture are the paths homotopic to trivial paths, and on the other are the remaining lifts, grouped as pairs of paths canceling each other in TPA(Σ).

  γ 1 i is based at b. Notice, that then γ 2 i is based at θ( b). Let s + b be as before the path from b to θ( b) going along the fiber at b in the positive direction and s - θ( b) := s + b the path from θ( b) to b going along the fiber at b in the negative direction. If the context is clear, we just write s + or s -to simplify the notation. Let x ∈ E b. Then on one hand: β b(x, P s + (x)) = -σ(β θ( b) (P s + (x), x)). On the other hand, since β is parallel:

Figure 1 . 2 :

 12 Figure 1.2: The path δ on S with triangulation ∆ 1 on the left and with triangulation ∆ 2 on the right.

  r),π(r,q) . Let f : Λ → Λ without fixed point such that for all γ ∈ Γ and for all p ∈ Λ, f (γ(p)) ∈ γ.Stab Γ (p) • f (p). One can construct such a function by choosing for each puncture p of S a lift p 0 ∈ Λ, then choosing f (p 0 ) ∈ Λ and extending f to the orbit of p by Γ with the choice of an element of Stab Γ (p) for each point in the orbit of p. Indeed, if γ(p) = γ ′ (p), then γ ′-1 γ ∈ Stab Γ (p) so γ ′ .Stab Γ (p) = γ.Stab Γ (p).

  p ∈ P.Remark 3.1.31. For a twisted GL 2 (R)-local system on S to admit a semi-decoration, its monodromies around each internal punctures must fix a R-line in R 2 . One can obtain a semi-decorated local system from a decorated one by forgetting about the sections (b

Figure 2

 2 Figure 2.7: A cyclic quiver with 6 vertices with an internal oriented triangulation.

Figure 2 .

 2 Figure 2.8: A triangulated quiver with one oriented plain cycle of length 4.

Figure 2 .

 2 Figure 2.10: An affectation around an oriented triangle of Q.

Figure 2 .

 2 Figure 2.14: The flip mutation sequence for an A 2 -type quiver. The anti-oriented cycles are labeled with the R-line associated to them.

Figure 2 .

 2 Figure2.15: The flip mutation sequence for an A 3 -type quiver. The first two groups of mutations correspond to the first step, the third and fourth groups of mutations correspond to the second step and the last group of mutations correspond to the third step.

Figure 2 .

 2 Figure 2.16: The sequence of flips corresponding to the pentagon relation.

F opp i =

 i Re n ⊕ • • • ⊕ Re i+1 . Let 1 ≤ i ≤ k. Let (a (i) 1 , . . . , a (i)n ) be the decoration of the flag A (i) , i.e. a

r

  (i.e. the coordinates of v in the triangle t are (i, j, k)). The three subbundles F defined along π(T ′ β v ), so we can define the 1-dimensional subbundle F π(T ′ βv) which we denote by F v or F i,j,k . A ∆-extra-decoration of L with respect to the graph Γ n is the data for each white vertex v of Γ n of a flat section b v of F v that freely spans F v . The data of a decorated twisted GL n (R)-local system on S together with a ∆-extra-decoration is called a ∆-extra-decorated twisted GL n (R)-local system on S.

  ,k be the holonomy in E of the path T ′ γ p,v in the bases b (q) 1 and b (p) i . The A-coordinate associated to the couple (p, v)

  ,k (x r ) be the holonomy in E of the path T ′ γ p,v in the bases b v (x r ) and b (p) i for r = 1, 2. The A-coordinate associated to the couple (p, v) is the paira (p) i,j,k = a (p) i,j,k (x 1 ), a(p)i,j,k (x 2 ) .

  -1,k . The A-coordinates satisfy the following relations, called triangle relations:

  dashed, then there is another dashed arrow v ′ → v and those are the only arrows between v and v ′ • Each oriented cycle of plain arrows of Q has an internal oriented triangulation of dashed arrows, and every dashed arrow belongs to one such triangulation • If there is a plain arrow v → v ′ , then there is no arrow v ′ → v • There is at most one arrow from a vertex v to another vertex v ′ Definition 3.3.4. Let Q be a triangulated quiver on S. The complement of Q in S is a disjoint union of open disks, open punctured half-disks and open punctured disks, all with an oriented boundary. We call a cycle in Q with opposite orientation to the one on S which is the boundary of a disk without puncture an anti-cycle of Q.

  σ(c 2 ) and σ ′ coincides with σ for any coordinate y /∈ {a 1 , d 2 , b 1 , c 2 , x ′ 1 , x ′ 1 }.Definition 3.3.12. Let (Q, V) be a seed on S. Let a and d two vertex of Q with a pair of dashed arrows between them. Let b, c ∈ Q 0 such that the oriented triangle containingd → a (resp. a → d) is t 1 = d → a → b → d (resp. t 2 = a → d → c → a), see Figure3.19.

σ

  Figure 3.19: The situation before and after the -mutation in an oriented quadrilateral of Q.

  ,n with the following cyclic ordering:a (p) i+1,j,n < t r i,j,n < a (q) i,j+1,n < b A,B < a (p)i+1,j,n .

  ,n (t) = a (p) j,i+1,n (t ′ ) and a (q)

  Les relations triangulaires apparaissent naturellement dans le processus d'abélianisation : les A-coordonnées sont des holonomies de certains chemins dans le système local abélianisé, et les relations triangulaires correspondent à des holonomies de chemins contractiles, et sont donc triviales. Pour les formules de mutation, nous profitons du fait que le système local sur la surface de base S ne dépend pas de la triangulation, alors que la surface spectrale et le système local abélianisé en dépendent. Ensuite, en effectuant des abélianisations correspondant à deux triangulations différentes et en identifiant les bons termes, nous pouvons exprimer les A-coordonnées correspondant à une triangulation en termes de Acoordonnées correspondant à l'autre triangulation. La même technique peut être utilisée pour montrer qu'étant donné une triangulation spécifique ∆, toutes les autres coordonnées A peuvent être écrites comme un polynôme de Laurent non-commutatif des coordonnées associées à ∆. Cette propriété est connue sous le nom de phénomène de Laurent (noncommutatif) et est la généralisation non-commutative d'une propriété bien connue des algèbres amassées commutatives.

Théorème. Soit n ≥ 3 et S le disque fermé avec n pointes sur le bord. Soit i, j ∈ {1, . . . , n}, i ̸ = j. Alors, pour toute triangulation ∆ de S et tout GL 2 (R)-système local tordu décoré L qui est ∆-générique et tel que (F (i) , F (j) ) est générique, la coordonnée a γ i,j est un polynôme de Laurent non-commutatif dans les coordonnées (a γ ) γ∈∆ associées à la triangulation ∆.

Nous étudions également dans les sections 3.1.4 et 3.1.5 un type de coordonnées intermédiaire entre les X -coordonnées et les A-coordonnées, qui donnent une représentation d'une sousalgèbre de A S introduite dans

[START_REF] Berenstein | Noncommutative marked surfaces[END_REF]

. Les A-coordonnées présentées ici se restreignent aux coordonnées symplectiques non-commutatives introduites dans [AGRW22] lorsqu'elles sont définies sur un système local symplectique avec une décoration symplectique.

Nous généralisons ensuite ces coordonnées à des configurations de drapeaux dans R n dans la Section 3.2 et à des GL n (R)-systèmes locaux tordus dans la Section 3.3. La raison pour laquelle nous avons divisé ceci en deux sections différentes est une difficulté technique provenant des systèmes locaux tordus. En effet, la généralisation que nous décrivons nécessite des données supplémentaires aux drapeaux décorés habituels. Nous appelons ces données supplémentaires un extra-décoration.

Definition. Soit (A (1) , . . . , A (k) ) un k-uplet de drapeaux en position générique. Une extradécoration de l'ensemble k de drapeaux est constitué des données d'une décoration de chaque drapeau A (i) ainsi que des données pour chaque i 1

  However this statement only holds in rank 2. This lead us to the next technical difficulty arising from non-commutativity: there is no determinant for matrices with values in a general ring, hence no special linear group SL n . Since we can't work with SL n , we need do work in the general linear group GL n which even over a commutative ring is not a semi-simple Lie group, but a reductive one. As a consequence, the non-commutative A-coordinates we will define in Section 3.3 and 3.2 will not directly reduce to Fock-Goncharov's A-coordinates in the commutative case. Instead, what we get is a generalization of Fock-Goncharov coordinates to GL n (R)-local systems. Another technical difficulty arising is the need to work with twisted local systems to define A-coordinates. These twisted local systems are defined on the unit tangent bundle T ′ S of S, with the extra condition of having monodromy around the fiber of T ′ S → S equal to -Id. Now that we have good candidates for generalized A-coordinates, several questions arise. First, are there relations between the coordinates defined with respect to a fixed triangulation? Second, how to compute the mutations formulas relating the coordinates with respect to two different triangulations? To answer both questions, we make use of a tool introduced in 2014 by Davide Gaiotto, Gregory W. Moore and Andrew Neitzke in[START_REF] Gaiotto | Spectral networks[END_REF] called spectral networks. This is a combinatorial tool used to study framed local systems by allowing the definition of a procedure called abelianization. This procedure transform a flat rank n vector bundle over a ciliated surface S into a line bundle over a certain ramified n-covering Σ n of S called the spectral surface. This is a tool of great importance in theoretical physics to study a certain class of topological quantum field theories, but we won't make use of all the potential of spectral networks in this work. Initially, spectral network were defined as the critical graph of a holomorphic quadratic differential on S lifted to the spectral surface.

	However, a
	geometric and combinatorial description exists and will be more convenient to work with in
	our study. For this, we use a combinatorial description of the spectral surface introduced by
	Alexander Goncharov and Maxim Kontsevich in [GK22] which we recall in Section 2.3.1, and
	we add the definition to a spectral network adapted to this spectral surface in Section 2.3.3.
	Then, we need to extend the definition of the non-abelianization procedure to work with
	twisted local systems. This part is a joint work with Eugen Rogozinnikov in

  The right R-module R n is isomorphic to the right R-module M nd,d (K) of matrices with nd rows and d columns with coefficients in K as a column of n square matrices of size d with coefficients in K is equivalent to a nd×d matrix with coefficients in K. We may consider an element e of R n as a d-tuple (e 1 , . . . , e d ) of vectors in K nd . The group R × is then GL d (K) and the group GL n (R) is identified with GL nd (K). An R-submodule of R n spanned by k elements e 1 , . . . , e k is a

	k-dimensional subspace if and only if the family of kd vectors (e 1 1 , . . . , e 1 d , e 2 1 , . . . , e k d ) is of rank kd because the vector space in K nd spanned by (e 1 1 , . . . , e 1 d , e 2 1 , . . . , e k d ) is then kd-dimensional and admits a supplement that is (n -k)d-dimensional. The family (e 1 1 , . . . , e 1 d , e 2 1 , . . . , e k d ) is
	then a basis of the R-module it spans. A regular element is a family e = (e 1 , . . . , e d ) of rank
	d. Note that if K

∀p, q, r ∈ P pairwise distinct, y r p,q y r q,p = 1

∀p, q, r ∈ P pairwise distinct, y r p,q y p q,r y q r,p = 1

∀p, q, r, s ∈ P pairwise distinct, y s p,q y s q,r y s r,p = 1

∀(p, q, r, s) ∈ P pairwise distinct cyclically ordered, y q p,s = y r p,q y p q,s + y r p,sProof. See [BR18], Theorem 2.14.

ramified covering, 14 rank, 3 regular element, 4 residually finite group, 58 seed, 73 twisted, 88 signature, 42 sink, 26 source, 26 spectral lifts, 29 spectral network, 24 line, 24 symplectic basis, 42 decoration, 43 group, 42 transverse subspaces, 4 triangle, 11, 50 triangle elements, 72 triangle relation, 69, 72, 86 twisted path algebra, 28 twisted symplectic local system, 43 decorated, 43 framed, 43 unit tangent bundle, 10 unitary group, 41 vertex mutable, 73 peripheral white, 22 valency, 71, 86

Cette définition des coordonnées A non-commutatives est légèrement différente de celle introduite par Goncharov-Kontsevich dans [GK22]. En effet, les coordonnées de Goncharov-Kontsevich peuvent être écrites comme des quotients de deux des coordonnées introduites dans ce travail. En d'autres termes, les A-coordonnées présentées ici donnent une factorisation des coordonnées de Goncharov-Kontsevich. Une autre différence est que les coordonnées de Goncharov-Kontsevich ne nécessitent qu'une décoration (au sens usuel) des systèmes locaux, alors que les coordonnées présentées ici nécessitent cette extra-décoration. La combinatoire des coordonnées présentées dans ce travail est beaucoup plus proche de la combinatoire d'une algèbre amassée, avec des mutations dictées par un carquois sous-jacent. En tant que telles, les coordonnées de Goncharov-Kontsevich peuvent être considérées comme "intermédiaires" entre les A-coordonnées présentées ici et les X -coordonnées non-commutatives. Il est intéressant de noter que pour n = 2 (c'est-à-dire pour G = GL 2 (R)) les coordonnées de Goncharov-Kontsevich coïncident avec les A-coordonnées introduites ici, les différences entre les deux types de coordonnées apparaissant pour n ≥ 3.

Le travail présenté dans le chapitre 3 est un pas vers une définition de structures amassées (éventuellement non-commutatives) sur les espaces de modules des G-systèmes locaux encadrés dans les variétés de drapeaux partiels G/P pour des groupes de Lie plus généraux et des sous-groupes paraboliques plus généraux que ce qui est déjà connu. Bien qu'il n'y ait pas de réduction directe des coordonnées présentées ici aux coordonnées sur SO(p, q) ou sur les groupes de Lie exceptionnels avec leur sous-groupe parabolique P Θ (les deux classes restantes de structures Θ-positives), les constructions présentées dans ce manuscrit devraient aider à comprendre la combinatoire attendue des coordonnées amassées non-commutatives dans un cas général.

Definition 2.7.5. Let (A, σ) be an R-algebra with an anti-involution. We define two set of squares:

A σ + := a 2 | a ∈ (A σ ) × , A σ ≥0 := a 2 | a ∈ A σ . Remark 2.7.6. Since the algebra A is unital, we always have the canonical copy of R in A, namely R • 1 where 1 is the unit of A. We will always identify R • 1 with R. Moreover, since σ is linear, for all k ∈ R, σ(k • 1) = kσ(1) = k • 1, i.e. R • 1 ⊆ A σ and R >0 • 1 ⊆ A σ + . Definition 2.7.7. A unital associative finite dimensional R-algebra with an anti-involution (A, σ) is called hermitian if for all x, y ∈ A σ , x 2 + y 2 = 0 implies x = y = 0.

Remark 2.7.8. In [ABR + 22] it is shown that, if (A, σ) is a Hermitian algebra, then A σ

+ is an open proper convex cone in A σ , where proper means that the set does not contain (affine) lines.

If (A, σ) is Hermitian, for an element a ∈ A σ the signature can be defined, which is a bounded function sgn : A σ → Z that is invariant under congruence by elements of A × . The elements of maximal signature are precisely the elements of A σ + . For more details about the signature see [ABR + 22].

≥0 is the set of positive symmetric matrices and A σ + is the set of positive definite symmetric matrices. The signature of a symmetric matrix with p positive eigenvalues and q negative eigenvalues (and any multiplicity of 0 as eigenvalue) is p -q.

Symplectic groups over non-commutative algebras

Let A be a unital associative finite dimensional R-algebra with an anti-involution σ. We consider A 2 as a right A-module over A.

Definition 2.7.9. Let ω(x, y) := σ(x) T Ωy with Ω = 0 1 -1 0 . The group

is the symplectic group Sp 2 over (A, σ). The form ω is called the standard symplectic form on A 2 .

We have

We can also determine the Lie algebra sp 2 (A, σ) of Sp 2 (A, σ):

So the symplectic local system provides us elements a i ∈ A × corresponding to P γ 1 i , a 0 ∈ A σ and a 0 a ξ ∈ A σ for every ξ as in (2) of Lemma 2.3.10 (lifted to T ′ Σ 2 ). These elements are well-defined up to a common conjugation of all a i and common congruence of all a 0 and a 0 a ξ by an element of A × . Conversely, if elements a i , a 0 , a ξ as above are given, then a twisted A × -local systems on Σ 2 equipped with a non-degenerate parallel pairing β can be reconstructed uniquely. Equivalent local system correspond to a common conjugation of all a i and common congruence of all a 0 and a 0 a ξ by an element of A × .

Symplectic local system over Hermitian algebras

Let A be a Hermitian algebra. Let A, B, C be a triple of generic isotropic A-lines. The Kashiwara-Maslov index of the triple (A, B, C) is the signature of the element ω(x, µ B,C A (x)) ∈ (A σ ) × for a regular x ∈ A where µ B,C A is the Kashiwara-Maslov map defined in Section 1.3. In fact, this signature does not depend on x ∈ A, and it is invariant under cyclic permutations of the triple (A, B, C) and it changes the sign by transposition of the elements of the triple.

Let L be a framed symplectic twisted local system on S. Let t ⊂ S be a triangle of the triangulation ∆ that is incident to punctures p, q, r and the orientation of the triangle agrees with the orientation of the triple (p, q, r). Let H = π -1 (t) ⊂ Σ 2 be the hexagon that covers t. Let b be the ramification point in H, let b be a lift of b in T ′ H and let s + be a path in T ′ H going from b to θ( b) along the fiber in the positive direction.

The following proposition is immediate:

agrees with the signature of the element β b(x, P s + (x)) ∈ A σ for a regular x ∈ E b.

Definition 2.7.20. Let L be a framed twisted Sp 2 (A, σ)-local systems on S. We say that L is maximal if for some triangulation ∆ of S, for all triangle t of ∆ the Kashiwara-Maslov index of the triple of isotropic A-lines associated to t is maximal.

The following proposition is proven in [BIW10]:

Proposition 2.7.21. Twisted maximal Sp 2 (A, σ)-local systems on S are ∆-generic with respect to any triangulation ∆ of S.

Theorem 2.7.22. If A is Hermitian, then the moduli space of framed (twisted) maximal Sp 2 (A, σ)-local systems on S is homeomorphic to:

where A × acts componentwisely by conjugation on (A × ) -2χ(S)+2p-1+ n i and by congruence on (A σ + ) -2χ(S)+p .

have:

and

The other triangle elements are obtained using similar formulas.

Proposition 3.2.13. Let (Q, V) be a seed, let x be a mutable vertex of Q and let

Proof. Let x ′′ 1 be the variable of µ ▷◁,x (Q ′ , V ′ ) at the location of x 1 . We have:

To conclude, we need a general identity on non-commutative algebras:

Lemma 3.2.14. Let x ∈ R × such that 1 + x ∈ R × . Then we have:

Proof.

Again, the result of a -mutation is a seed, i.e. satisfies the triangle relations. The new triangle relations are:

and similarly we have

Proposition 3.2.16. Let (Q, V) be a seed, let a, d be a pair of vertices of Q with a pair of dashed arrows between them and let (Q ′ , V ′ ) = µ ,a↔d (Q, V). We use the same notations as in Figure 2.12. Then µ ,b↔c (Q ′ , V ′ ) = (Q, Ṽ), where Ṽ is the same affectation as V except for ã3 = -a 3 and d3 = -d 3 .

Proof. The coordinate ã3 is by definition equal to R + t 3 ,d = c 1 d -1 2 b 3 . By replacing b 3 by R + t 1 ,c , we have:

Remark 3.2.17. The -mutation not being involutive is a consequence of the arbitrary choice of R + instead of R -in the definition of the new variables. There is no canonical choice that would solve this problem, however when we will deal with twisted local systems (see Section 2.2.3) these coordinates will be defined up to sign anyway.

). Denote by M (i) the matrix whose columns are (a

r the r × r submatrix of M (i) with the first r rows and the first r columns. Since the (p, q)-quasideterminant is invariant under columns operations except for adding a multiple of the q-th column to another, the quasideterminant m

r | r,r does not depend on the choice of the lifts a (i) j . Note that the fact that (A (i) , F opp ) is generic ensure the submatrix M (i) r is invertible, however its (p, r)-quasideterminants may not exist. Still, generically m (i) r exists for all 1 ≤ r ≤ n and is invertible. Thus, we can consider a modified basis (ã

. Let M (i) be the matrix whose columns are (ã

Let (i, j, k) be a triangle of ∆, and let p, q, r such that p + q + r = 2n + 1. Let b p,q,r be the basis of

r given by the extra-decoration. Then the matrix of a A (i) p,q,r in the bases b p,q,r and

p-1 |b p,q,r p,p .

Coordinates on extra-decorated local systems: the general case

In this section we extend the construction of A-coordinates defined in Section 3.2 to twisted local systems on ciliated surfaces. Everything works similarly, except for one thing: the data of an extra-decoration on a twisted local system induce a technical sign difficulty. Indeed, because of the definition of the partial abelianization procedure, it is natural to assign the additional bases needed to an extra-decoration to the white vertices of the graph Γ n . This is the purpose of the decorating curves introduced in Section 2.3.2. However, since the local system is twisted, the monodromy of a loop going once around a white vertex of Γ n is -Id so a section of a line bundle over such a loop is only defined up to sign. We deal with this difficulty by instead taking a pair of sections summing to zero. Most A-coordinates become then a pair of coordinates with opposite signs. Then being careful with the cyclic ordering around the white vertices, we can still compute the mutations for those coordinates.

In all this section, all triangulations of an hyperbolic ciliated surface S will be without self-folded triangles, i.e. triangles with two edges identified together in S.

Decoration and extra-decoration

Let k ≥ 2 and let S be a ciliated surface. Let L be a framed twisted GL n (R)-local system on S, and let ∆ be a triangulation of S. Recall the decorating paths defined in Section 2.3.2. Let v be a white vertex of Γ n and let i such that v ∈ γ (i)

p . We use parallel transport

Further considerations

Here is a non-exhaustive list of directions in which expand the work presented in this manuscript, as well as question arising naturally:

• Is it possible to have a geometric interpretation of the A-coordinates obtained by mutations of the quiver that are not part of the flip sequence? Without such interpretation, we would need another approach to define a generalization of Berenstein-Retakh algebra for coordinates on GL n (R)-local systems.

• Is there similar non-commutative coordinates on local systems for other Lie groups and/or other parabolic subgroups? A particularly interesting case of this would be for G = SO(p, q), p ̸ = q with a framing which takes values in the flag variety G/P Θ , where P Θ is the parabolic subgroup such that (G, P Θ ) admits a Θ-positive structure. Such coordinates would mix both commutative and non-commutative behaviors.

• Is there an additional structure on the space of A-coordinates, such as the symplectic structure on the space of Fock-Goncharov A-coordinates or the non-commutative 2-form on the space of Goncharov-Kontsevich coordinates?

• Is it possible to describe the elementary mutations as "mutations" of the spectral surface and of the spectral network, such that the intermediary A-coordinates arising in the middle of the flip mutation sequence would be also holonomies of certain paths in the abelianized local system?

• Does the partial abelianization procedure also hold for local systems whose group is not of type A? Generalizations of spectral networks and abelianization already exists for such groups, see [START_REF] Ionita | Spectral networks and non-abelianization[END_REF].