A duality map for the quantum symplectic double

Transcription

1 A dualty map for the quantum symplectc double Dylan G.L. Allegrett arxv: v2 [math.qa] 14 Nov 2018 Abstract We consder a cluster varety called the symplectc double, defned for an orented dsk wth fntely many marked ponts on ts boundary. We construct a canoncal map from the tropcal ntegral ponts of ths cluster varety nto ts quantzed algebra of ratonal functons. As a specal case, we obtan a soluton to Fock and Goncharov s dualty conjectures for quantum cluster varetes assocated to a dsk wth marked ponts. Ths extends the author s prevous work wth Km on quantum cluster varetes assocated to punctured surfaces. Contents 1 Introducton The cluster Posson varety The symplectc double Extenson of prevous results Organzaton Quantum cluster varetes Seeds and mutatons The quantum dlogarthm and quantum tor The quantum double constructon The classcal lmt The dsk case Quantum cluster algebras General theory of quantum cluster algebras Quantum F-polynomals The sken algebra and lamnatons Defnton of the sken algebra Relaton to quantum cluster algebras Lamnatons

2 5 Dualty map for the quantum Posson varety Realzaton of the cluster Posson varety Constructon of the map Propertes Dualty map for the quantum symplectc double Realzaton of the symplectc double Constructon of the map Propertes A Dervaton of mutaton formulas 33 Acknowledgements 36 References 36 1 Introducton 1.1 The cluster Posson varety Cluster varetes are geometrc objects ntroduced and studed by Fock and Goncharov n a seres of recent papers [6, 7, 8, 9]. They arse naturally as modul spaces of geometrc structures on a compact orented surface wth fntely many marked ponts on ts boundary. In the orgnal paper on the subject, Fock and Goncharov defned a verson of the modul space of PGL 2 (C)-local systems on such a surface S [6]. Ths space s denoted X PGL2,S(C). It s bratonally equvalent to a partcular knd of cluster varety called a cluster Posson varety [6]. One of the man deas from the work of Fock and Goncharov s that the algebra of regular functons on the cluster Posson varety possesses a canoncal bass wth many specal propertes. To construct ths canoncal bass, Fock and Goncharov consdered a space denoted A SL2,S(Z t ). Ths space can be understood as a tropcalzaton of a certan modul space of SL 2 (C)-local systems assocated to the surface S. In the case where S s a surface wthout marked ponts, Fock and Goncharov constructed a canoncal map I A : A SL2,S(Z t ) Q(X PGL2,S) from ths space nto the feld of ratonal functons on X PGL2,S(C), and they argued that the mage of ths map s a vector space bass for the algebra of regular functons [6]. Ths canoncal bass constructon s closely related to Lusztg s canoncal bases n the theory quantum groups [16], as well as the work of Musker, Schffler, and Wllams [19] and Gross, Hackng, Keel, and Kontsevch [14] on canoncal bases for cluster algebras. It has also found applcatons n mathematcal physcs where t was used by Gaotto, Moore, and Netzke to calculate BPS degeneraces n certan four-dmensonal supersymmetrc quantum feld theores [13]. 2

3 An mportant feature of the cluster Posson varety s that ths object carres a canoncal Posson structure and can be canoncally quantzed [8, 9]. In other words, there s a famly of noncommutatve algebras X q PGL 2,S dependng on a parameter q such that Xq PGL 2,S concdes wth the commutatve algebra Q(X PGL2,S) when q = 1. In recent jont work wth Km [3], the author showed that Fock and Goncharov s canoncal bass constructon extends naturally to the quantum settng. Theorem 1.1 ([3]). When S s a surface wthout marked ponts, there exsts a natural map I q A : A SL 2,S(Z t ) X q PGL 2,S whch reduces to I A n the classcal lmt q = 1. In fact, the man result of [3] was a much more detaled statement whch establshed a number of propertes of I q A conjectured n [6, 8]. The map Iq A s expected to be mportant n physcs where t s related to the protected spn character ntroduced by Gaotto, Moore, and Netzke [13]. 1.2 The symplectc double The man goal of the present paper s to extend the results descrbed above to another cluster varety called the cluster symplectc varety or symplectc double. Ths cluster varety was ntroduced by Fock and Goncharov and plays a key role n ther work on quantzaton of cluster varetes [9]. As before, one can assocate, to a compact orented surface S wth fntely many marked ponts on ts boundary, a certan modul space D PGL2,S(C) of local systems, and ths modul space s bratonally equvalent to the symplectc double [10, 2]. The man result of [1] was the constructon of a dual space D PGL2,S(Z t ). Ths space can be understood as a tropcalzaton of D PGL2,S(C). Just as the tropcal space A SL2,S(Z t ) parametrzes a canoncal bass for the algebra of regular functons on the cluster Posson varety, the tropcal space D PGL2,S(Z t ) parametrzes a canoncal collecton of functons on the symplectc double. More precsely, there s a canoncal map I D : D PGL2,S(Z t ) Q(D PGL2,S) from ths space nto the feld of ratonal functons on D PGL2,S(C), and there s an explct formula expressng ths map n terms specal polynomals called F-polynomals from the work of Fomn and Zelevnsky [12]. An mportant and surprsng feature of ths constructon s the fact that the functon obtaned by applyng the map I D to a pont of D PGL2,S(Z t ) s n general not a regular functon on the symplectc double. The natural geometrc defnton of the map I D produces ratonal ratherthanregularfunctons. Inpartcular, themapi D doesnotprovde acanoncal bass for the coordnate rng, and the relaton between canoncal bases and the map I D s more subtle than orgnally envsoned by Fock and Goncharov. It would be nterestng to understand ths relatonshp more precsely and compare wth other approaches to canoncal 3

4 bases such as [14] and [19]. Another mportant open problem s to relate these canoncal functons to the physcal deas of Gaotto, Moore, and Netzke [13]. As mentoned above, the map I A appears n ther work on BPS states and the protected spn character, but an nterpretaton of the map I D n ther framework s not yet avalable. Lke the cluster Posson varety, the symplectc double admts a natural deformaton quantzaton [9]. In other words, there s a famly of noncommutatve algebras D q PGL 2,S dependng on a parameter q such that we recover the functon feld Q(D PGL2,S) when q = 1. The man result of the present paper s a canoncal q-deformaton of the map I D n the mportant specal case where S s a dsk wth fntely many marked ponts on ts boundary. Theorem 1.2. When S s a dsk wth fntely many marked ponts, there exsts a natural map I q D : D PGL 2,S(Z t ) D q PGL 2,S whch reduces to I D n the classcal lmt q = 1. The exstence of the map I q D was essentally conjectured by Fock and Goncharov n Conjecture 3.6 of [9]. However, ths conjecture was formulated before t was understood that the classcal map I D naturally maps nto the feld of ratonal functons. In lght of [1], we know I q D should map nto the quantzed algebra of ratonal functons, rather than the quantzed algebra of regular functons as predcted n [9]. Naturally, one would lke to generalze Theorem 1.2 to any surface S. It should be possble to do ths by combnng the technques of the present paper wth those of [3], although some parts of the constructon wll lkely be more complcated. We leave ths as a problem for future research. 1.3 Extenson of prevous results The cluster symplectc varety and cluster Posson varety are closely related. For example, there s a natural projecton π : D PGL2,S(C) X PGL2,S(C). In fact, for each q, we have a map π q : X q PGL 2,S Dq PGL 2,S such that ths map concdes wth the pullback of π when q = 1. One also has a natural embeddng ϕ : A 0 SL 2,S (Zt ) D PGL2,S(Z t ) where A 0 SL 2,S (Zt ) s a verson of the tropcal space A SL2,S(Z t ). By restrctng I q D to the subspace A0 SL 2,S (Zt ) of D PGL2,S(Z t ), we obtan a map I q A, whch fts nto the followng commutatve dagram: A 0 SL 2,S (Zt ) ϕ D PGL2,S(Z t ) I q A I q D X q PGL 2,S π q D q PGL 2,S. In fact, we wll see n Theorem 5.7 that ths map I q A satsfes several propertes conjectured n [6, 8]. Thus we have an extenson of the constructon of [3], whch s vald only when S 4

5 s a surface wthout marked ponts, to the case where S s a dsk wth fntely many marked ponts. Interestngly, our constructon of the map I q A n the dsk case uses several tools that were not present n [3]. These nclude the theory of quantum cluster algebras [4], the noton of quantum F-polynomal from [21], and the work of Muller on sken algebras [17]. The orgnal constructon of I q A n [3] was nstead based on the quantum trace map ntroduced by Bonahon and Wong [5]. Recent results of Lê [15] suggest that these two approaches are n fact equvalent. 1.4 Organzaton The rest of ths paper s organzed as follows. In Secton 2, we defne the cluster Posson varety, the cluster symplectc varety, and ther q-deformatons usng the quantum dlogarthm functon. In Secton 3, we revew the tools we need from the theory of quantum cluster algebras. In Secton 4, we defne the sken algebra and revew the results of Muller [17], whch relate sken algebras to quantum cluster algebras. In Secton 5, we construct the map I q A and prove a number of conjectured propertes ofths map. In Secton 6, we defne the mapi q D and dscuss ts relaton to I q A. Appendx A gves a detaled dervaton of the mutaton formulas used to defne the quantum symplectc double and may be of ndependent nterest to some readers. 2 Quantum cluster varetes 2.1 Seeds and mutatons Cluster varetes are geometrc objects defned usng combnatoral deas from Fomn and Zelevnsky s theory of cluster algebras [11]. One of the man concepts needed to defne cluster varetes s the noton of a seed. Defnton 2.1. A seed = (Λ,{e } I,{e j } j J,(, )) s a quadruple where 1. Λ s a lattce wth bass {e } I. 2. {e j } j J s a subset of the bass. 3. (, ) s a Z-valued skew-symmetrc blnear form on Λ. A bass vector e j wth j J s sad to be mutable, whle a bass vector e wth I J s sad to be frozen. Note that f we are gven a seed, we can form a skew-symmetrc nteger matrx wth entres ε j = (e,e j ) (, j I). Thesecondmanconceptthatweneedtodefnecluster varetessthenotonofmutaton. For any nteger n, let us wrte [n] + = max(0,n). 5

6 Defnton 2.2. Let = (Λ,{e } I,{e j } j J,(, )) be a seed and e k (k J) a mutable bass vector. Then we defne a new seed = (Λ,{e } I,{e j } j J,(, ) ) called the seed obtaned by mutaton n the drecton of e k. It s gven by Λ = Λ, (, ) = (, ), and { e = e k f = k e +[ε k ] + e k f k. It s straghtforward to calculate the change of the matrx ε j under a mutaton of seeds. Proposton 2.3. A mutaton n the drecton k changes the matrx ε j to the matrx { ε j = ε j f k {,j} ε j + ε k ε kj +ε k ε kj f k {,j}. 2 We wll denote by the mutaton equvalence class of a seed, that s, the collecton of all seeds obtaned from by applyng sequences of mutatons. 2.2 The quantum dlogarthm and quantum tor To defne quantum cluster varetes, we employ the followng specal functon. Defnton 2.4. The quantum dlogarthm s the formal nfnte product Ψ q (x) = (1+q 2k 1 x) 1. k=1 We wll study the quantum dlogarthm as a functon on the followng algebra of q- commutng varables. Defnton 2.5. Let Λ be a lattce equpped wth a Z-valued skew-symmetrc blnear form (, ). Then the quantum torus algebra s the noncommutatve algebra over Q[q,q 1 ] generated by varables Y v (v Λ) subject to the relatons q (v 1,v 2 ) Y v1 Y v2 = Y v1 +v 2. Ths defntonallowsustoassocatetoanyseed = (Λ,{e } I,{e j } j J,(, )), aquantum torus algebra X q. The set {e j} j J provdes a set of generators X j ±1 gven by X j = Y ej for ths algebra. They obey the commutaton relatons X X j = q 2ε j X j X. Ths algebra X q satsfes the Ore condton from rng theory, so we can form ts noncommutatve fracton feld X q. In addton to assocatng a quantum torus algebra to every seed, we use the quantum dlogarthm to construct a natural map X q X q whenever two seeds and are related by a mutaton. 6

7 Defnton The automorphsm µ k : Xq X q s gven by conjugaton wth Ψ q (X k ): µ k = Ad Ψ q (X k ). 2. The somorphsm µ k : Xq X q s nduced by the natural lattce map Λ Λ. 3. The mutaton map µ q k : Xq X q s the composton µ q k = µ k µ k. Note that conjugaton by Ψ q (X k ) produces a pror a formal power seres. We wll see below that ths constructon n fact provdes a map X q X q of skew felds. 2.3 The quantum double constructon As explaned n [9], t s natural to embed the above constructon n a larger one. If = (Λ,{e } I,{e j } j J,(, )) s any seed, then we can form the double Λ D = Λ D, of the lattce Λ gven by the formula Λ D = Λ Λ where Λ = Hom(Λ,Z). The bass {e } for Λ provdes a dual bass {f } for Λ, and hence we have a bass {e,f } for Λ D. Moreover, there s a natural skew-symmetrc blnear form (, ) D on Λ D gven by the formula ((v 1,ϕ 1 ),(v 2,ϕ 2 )) D = (v 1,v 2 )+ϕ 2 (v 1 ) ϕ 1 (v 2 ). We can apply the constructon of Defnton 2.5 to these data to get a quantum torus algebra whch we denote D q. If we let X and B denote the generators assocated to the bass elements e and f, respectvely, then we have the commutaton relatons X X j = q 2ε j X j X, B B j = B j B, X B j = q 2δ j B j X. We wll wrte D q for the (noncommutatve) fracton feld of D q. The followng notatons wll be mportant n the sequel: B + k = B (e k,e ), B k = B (e k,e ), X = X B (e,e j ) j. (e k,e )<0 j (e k,e )>0 One can check usng the above relatons that the elements X k and X k commute. Just as before, we have a natural map D q D q whenever and are two seeds related by a mutaton. Defnton The automorphsm µ k : D q D q s gven by µ k = Ad Ψ q (X k )/Ψ q ( X k ). 7

8 2. The somorphsm µ k : D q D q s nduced by the natural lattce map Λ D, Λ D,. 3. The mutaton map µ q k : D q D q s the composton µ q k = µ k µ k. NotcethatthegeneratorsX spanasubalgebraofd q whchssomorphctothequantum torus algebra X q defned prevously. Moreover, snce X k and X k commute, the restrcton of µ q k to ths subalgebra concdes wth the mutaton map from Defnton 2.6. Although conjugaton by Ψ q (X k )/Ψ q ( X k ) produces a pror a formal power seres, ths constructon n fact provdes a map D q D q of skew felds. Indeed, one has the followng formulas for the values of the map µ q k on generators. Theorem 2.8. The map µ q k s gven on generators by the formulas { µ q (qx k B + k (B k ) = +B k )B 1 k (1+q 1 X k ) 1 f = k f k and B X εk 1 µ q p=0 (1+q 2p+1 X k ) f ε k 0 and k k (X ) = X X ε k εk 1 k p=0 (X k +q 2p+1 ) 1 f ε k 0 and k X 1 k f = k. The proof of ths theorem can be found n Appendx A. 2.4 The classcal lmt Wehavenowseenhowtoassocate, toanyseed, anoncommutatve algebrad q. Inaddton, we have seen how to assocate, to any par of seeds and related by a mutaton, an somorphsm D q D q of the correspondng fracton felds. Note that f we set q = 1, then D q becomes the Laurent polynomal rng Q[B ±1,X ±1 ] J. Its spectrum s a splt algebrac torus, whch we denote by D. The formulas of Theorem 2.8 specalze to and µ k B = µ kx = X k j ε B ε kj kj>0 j (1+X k )B k B + j ε kj<0 B ε kj j { X 1 k X (1+X sgn(ε k) k ) ε k f = k f k f = k f k where {B j,x j } j J are the coordnates on D. These formulas defne bratonal maps D D of tor. Defnton 2.9. The symplectc double D = D s the scheme over Q obtaned by glung the splt algebrac tor D for all seeds n the mutaton equvalence class of usng the bratonal maps above. 8

9 In exactly the same way, the algebra X q becomes the Laurent polynomal rng Q[X ±1 ] J when q = 1. Its spectrum s a splt algebrac torus denoted X. As before, we get a bratonal map X X whenever two seeds and are related by a mutaton. Defnton The cluster Posson varety X = X s the scheme over Q obtaned by glung the splt algebrac tor X for all seeds n the mutaton equvalence class of usng the bratonal maps above. As the names suggest, the cluster Posson varety has a natural Posson structure, and the symplectc double carres a natural symplectc form wth a compatble Posson structure. The cluster Posson varety embeds nto the symplectc double as a Lagrangan subspace [9]. For generc q, we can use the formulas of Theorem 2.8 to glue the algebras X q and D q : X q = X q /dentfcatons, D q = D q /dentfcatons. The sets X q and D q nhert natural algebra structures and n the classcal lmt q = 1 are dentfed wth the functon felds of the cluster Posson varety and symplectc double, respectvely. For q 1, we thnk of X q and D q as the functon felds of correspondng quantum cluster varetes. 2.5 The dsk case From now on, we wll wrte S for a compact orented dsk wth fntely many marked ponts on ts boundary. As we wll see n ths secton, there are quantum cluster varetes naturally assocated to such a dsk. Defnton An deal trangulaton of S s a trangulaton whose vertces are the marked ponts on the boundary. The dsk S admts an deal trangulaton f and only f t has at least three marked ponts on ts boundary. From now on, we wll always assume ths s the case. The llustraton below shows an example of an deal trangulaton of a dsk wth fve marked ponts. An edge of an deal trangulaton s sad to be external f t les along the boundary of S, connectng adjacent marked ponts, and s sad to be nternal otherwse. For a gven deal 9

10 trangulaton T, we wll wrte I = I T for the set of edges of T and J = J T for the set of nternal edges. Defnton Let T be an deal trangulaton of S. Then we defne a skew-symmetrc matrx ε j (, j I) by 1 f, j share a vertex and s mmedately clockwse to j ε j = 1 f, j share a vertex and j s mmedately clockwse to 0 otherwse. Gven an deal trangulaton T, we can consder the assocated lattce Λ = Z[I]. Ths lattce has a bass {e } gven by e = {} for I and a Z-valued skew-symmetrc blnear form (, ) gven on bass elements by (e,e j ) = ε j. Thus we assocate a seed to any deal trangulaton. Defnton If k s an nternal edge of an deal trangulaton T, then a flp at k s the transformaton of T that removes k and replaces t by the unque dfferent edge that, together wth the remanng edges, forms an deal trangulaton: It s a fact that any two deal trangulatons are related by a sequence of flps. It s therefore natural to ask how the matrx ε j changes when we perform a flp at some edge k. To answer ths queston, frst note that there s a natural bjecton between the edges of an deal trangulaton and the edges of the trangulaton obtaned by a flp at some edge. If we use ths bjecton to dentfy edges of the flpped trangulaton wth the set I, then we have the followng result. Proposton A flp at an edge k of an deal trangulaton changes the matrx ε j to the matrx { ε ε j f k {,j} j = ε j + ε k ε kj +ε k ε kj f k {,j}. 2 10

11 Thus we see that the new matrx ε j that we get after performng a flp at the edge k s the same as the matrx that we get by mutatng the seed assocated to T. It follows that the quantum cluster varetes defned above can be canoncally assocated to a dsk wth marked ponts so that each seed corresponds to an deal trangulaton of the dsk. Defnton We wll wrte D q PGL 2,S for the algebra Dq assocated to a dsk S n ths way and X q PGL 2,S for the algebra Xq assocated to the dsk. The cluster varetes consdered here are related to confguratons of flags for the group PGL 2 [6, 10, 2]. Ths accounts for the notaton used n the above defnton. 3 Quantum cluster algebras 3.1 General theory of quantum cluster algebras Here we revew the theory of quantum cluster algebras, followng [4, 21]. Throughout ths secton, m and n wll be postve ntegers wth m n. Defnton 3.1. Let k {1,...,n}. We say that an m n matrx B = (b j) s obtaned from an m n matrx B = (b j ) by matrx mutaton n the drecton k f the entres of B are gven by b j = In ths case, we wrte µ k (B) = B. { b j f k {,j} b j + b k b kj +b k b kj f k {,j}. 2 Defnton 3.2. Let B = (b j ) be an m n nteger matrx, and let Λ = (λ j ) be a skewsymmetrc m m nteger matrx. We say that the par (Λ,B) s compatble f for each j {1,...,n} and {1,...,m}, we have m b kj λ k = δ j d j k=1 for some postve ntegers d j (j {1,...,n}). Equvalently, the product B t Λ equals the n m matrx (D 0) where D s the n n dagonal matrx wth dagonal entres d 1,...,d n. Let k {1,...,n} and choose a sgn ǫ {±1}. Denote by E ǫ the m m matrx wth entres gven by δ j f j k e j = 1 f = j = k max(0, ǫb k ) f j = k and set Λ = E t ǫλe ǫ. 11

12 Proposton 3.3 ([4], Proposton 3.4). The matrx Λ s skew-symmetrc and ndependent of the sgn ǫ. Moreover, (Λ,µ k (B)) s a compatble par. Defnton 3.4 ([4], Defnton 3.5). Let (Λ,B) be a compatble par and let k {1,...,n}. We say that the par (Λ,µ k (B)) s obtaned from (Λ,B) by mutaton n the drecton k and wrte µ k (Λ,B) = (Λ,µ k (B)). Let Lbealattceofrankmequpped wthaskew-symmetrc blnear formλ : L L Z. Let ω be a formal varable. We can assocate to these data a quantum torus algebra T. It s generated over Z[ω,ω 1 ] by varables A v (v Λ) subject to the commutaton relatons A v 1 A v 2 = ω Λ(v 1,v 2 ) A v 1+v 2. In the lterature on quantum cluster algebras, ths quantum torus algebra s typcally called a based quantum torus, and the parameter s denoted q 1/2, rather than ω. (See [4, 17, 21] for example.) Ths quantum torus algebra has a noncommutatve fracton feld whch we denote F. Defnton 3.5. A torc frame n F s a mappng M : Z m F {0} of the form M(v) = φ(a η(v) ) where φ s an automorphsm of F and η : Z m L s an somorphsm of lattces. Note that the mage M(Z m ) of a torc frame s a bass for an somorphc copy of T n F. We have the relatons M(v 1 )M(v 2 ) = ω Λ M(v 1,v 2 ) M(v 1 +v 2 ), M(v 1 )M(v 2 ) = ω 2Λ M(v 1,v 2 ) M(v 2 )M(v 1 ), M(v) 1 = M( v), M(0) = 1 where the form Λ M on Z m s obtaned from Λ usng the somorphsm η. Defnton 3.6. A quantum seed s a par (M,B) where M s a torc frame n F and B s an m n nteger matrx such that (Λ M,B) s a compatble par. Defnton 3.7. Let (M,B) be a quantum seed and wrte B = (b j ). For any ndex k {1,...,n} and any sgn ǫ {±1}, we defne a mappng M : Z m F {0} by the formulas v k ( ) M vk (v) = M(E ǫ v +ǫpb k ), p p=0 M ( v) = M (v) 1 where v = (v 1,...,v m ) Z m s such that v k 0 and b k denotes the kth column of B. Here the t-bnomal coeffcent s gven by ( ) r = (tr t r )...(t r p+1 t r+p 1 ). p (t p t p )...(t t 1 ) t 12 ω d k

13 Proposton 3.8 ([4], Proposton 4.7). The mappng M satsfes the followng propertes: 1. The mappng M s a torc frame whch s ndependent of the sgn ǫ. 2. The par (Λ M,µ k (B)) s compatble and obtaned from the par (Λ M,B) by mutaton n the drecton k. 3. The par (M,µ k (B)) s a quantum seed. Defnton 3.9. Let (M,B) be a quantum seed, and let k {1,...,n}. Let M be the mappng from Defnton 3.7, and let B = µ k (B). Then we say that the quantum seed (M,B ) s obtaned from (M,B) by mutaton n the drecton k. Proposton 3.10 ([4], Proposton 4.9). Let (M, B) be a quantum seed, and suppose that (M,B ) s obtaned from (M,B) by mutaton n the drecton k. Then M (e k ) = M ( e k + and M (e ) = M(e ) for k. m ) ( [b k ] + e +M ek + =1 m ) [ b k ] + e Defnton Denoteby T n ann-regular tree wth edges labeled by thenumbers 1,...,n n such a way that the n edges emanatng from any vertex have dstnct labels. A quantum cluster pattern s an assgnment of a quantum seed Σ t = (M t,b t ) to each vertex t T n so that f t and t are vertces connected by an edge labeled k, then Σ t s obtaned from Σ t by a mutaton n the drecton k. Gven a quantum cluster pattern, let us defne A ;t = M t (e ). For {n+1,...,m}, we have A ;t = A ;t for all t, t T n, so we may omt one of the subscrpts and wrte A = A ;t for all t T n. Let S = {A ;t : {1,...,n},t T n }. Defnton 3.12 ([4], Defnton 4.12). Gven a quantum cluster pattern t (M t,b t ), the assocated quantum cluster algebra A s the Z[ω ±1,A ±1 n+1,...,a±1 m ]-subalgebra of the ambent skew-feld F generated by elements of S. 3.2 Quantum F-polynomals One of the mportant tools that we apply n our constructon of the map I q D s the noton of a quantum F-polynomal from [21]. Ths extends Fomn and Zelevnsky s noton of F- polynomal [12] to the noncommutatve settng and allows us to express a generator A j;t of a quantum cluster algebra n terms of the generators assocated wth an ntal seed. Theorem 3.13 ([21], Theorem 5.3). Let (M 0,B 0 ) be an ntal quantum seed n a quantum cluster algebra A and wrte B 0 = (b j ). Then there exsts an nteger λ j;t Z and a polynomal F j;t n the varables ( ) Y j = M 0 b j e 13 =1

14 wth coeffcents n Z[ω,ω 1 ] such that the cluster varable A j;t A s gven by A j;t = ω λ j;t F j;t M 0 (g j;t ) where g j;t s an nteger vector called the extended g-vector of A j;t. The polynomal F j;t appearng n the theorem s known as a quantum F-polynomal. In ths paper, we restrct attenton to quantum cluster algebras where the matrx B arses from a trangulaton of a dsk. For such algebras, we have the followng refnement of Theorem Corollary Let A be a quantum cluster algebra assocated to a dsk, and suppose the matrx D appearng n the compatblty condton of Defnton 3.2 s four tmes the dentty. Then there exsts a polynomal F j;t n the varables Y 1,...,Y n wth coeffcents n Z 0 [ω 4,ω 4 ] such that the cluster varable A j;t A s gven by A j;t = F j;t M 0 (g j;t ). Proof. In the dsk case, t s known that each classcal F-polynomal has nonzero constant term (for example by [18]). Hence, by [21], Theorem 6.1, we have λ j;t = 0 n Theorem By [21], Theorem 7.4, we know that the coeffcents of F j;t are Laurent polynomals n ω 4 wth postve ntegral coeffcents. 4 The sken algebra and lamnatons 4.1 Defnton of the sken algebra The quantum cluster algebras that we consder n ths paper all arse from the sken algebra of the dsk S. Here we wll revew the the general theory of sken algebras, followng [17]. Defnton 4.1. By a curve n S, we mean an mmerson C S of a compact, connected, one-dmensonal manfold C wth (possbly empty) boundary nto S. We requre that any boundary ponts of C map to the marked ponts on S and no pont n the nteror of C maps to a marked pont. By a homotopy of two curves α and β, we mean a homotopy of α and β wthn the class of such curves. Two curves are sad to be homotopc f they can be related by homotopy and orentaton-reversal. Defnton 4.2. A multcurve s an unordered fnte set of curves whch may contan duplcates. Two multcurves are homotopc f there s a bjecton between ther consttuent curves whch takes each curve to a homotopc one. Defnton 4.3. A framed lnk n S s a multcurve such that each ntersecton of strands s transverse and 1. At each crossng, there s an orderng of the strands. 14

15 2. At each marked pont, there s an equvalence relaton on the strands and an orderng on equvalence classes of strands. By a homotopy of framed lnks, we mean a homotopy through the class of multcurves wth transverse ntersectons where the crossng data are not changed. When drawng pctures of framed lnks, we ndcate the orderng of strands at a transverse ntersecton or marked pont by makng one strand pass over the other: (In the second of these pctures, the dotted lne ndcates a porton of S contanng a marked pont.) If two strands are dentfed by the equvalence relaton at a marked pont, we ndcate ths as n the followng pcture: Defnton 4.4. A multcurve wth transverse ntersectons s sad to be smple f t has no nteror ntersectons and no contractble curves. Note that any smple multcurve can be regarded as a framed lnk wth the smultaneous orderng at each endpont. Defnton 4.5. Let us wrte K(S) for the free Z[ω,ω 1 ]-module generated by equvalence classes of framed lnks n S. The sken module Sk ω (S) s defned as the quotent of K(S) by the followng local relatons. In each of these expressons, we depct the porton of a framed lnk over a small dsk n S. The framed lnks appearng n a gven relaton are assumed to be dentcal to each other outsde of the small dsk. In the last two relatons, the dotted lne segment represents a porton of S. In these pctures, there may be addtonal undrawn curves endng at marked ponts, provded ther order wth respect to the drawn curves and each other does not change. = ω 2 + ω 2 = (ω 4 +ω 4 ) ω = = ω 1 = = = 0 If K s a framed lnk n S, then the class of K n Sk ω (S) wll be denoted [K]. 15

16 Suppose K and L are two lnks such that the unon of the underlyng multcurves has transverse ntersectons. Then the superposton K L s the framed lnk whose underlyng multcurve s the unon of the underlyng multcurves of K and L, wth each strand of K crossng over each strand of L and all other crossngs ordered as n K and L. Proposton 4.6 ([17], Proposton 3.5). [K L] depends only on the homotopy classes of K and L. Defnton 4.7. For any framed lnks K and L, choose homotopc lnks K and L such that the unon of the multcurves underlyng K and L s transverse. Then the superposton product s defned by [K][L] := [K L ]. Ths extends to a product on Sk ω (S) by blnearty. Note that the superposton product s well defned and ndependent of the choce of K and L by Proposton Relaton to quantum cluster algebras We conclude ths secton by descrbng the relaton, frst dscovered by Muller n[17], between sken algebras and quantum cluster algebras. As usual, we take S to be a dsk wth fntely many marked ponts on ts boundary. Defnton 4.8. Let T be an deal trangulaton of S. For any I and j J, we defne 1 f, j share a vertex and s mmedately clockwse to j b j = 1 f, j share a vertex and j s mmedately clockwse to 0 otherwse. These are entres of a skew-symmetrc I J matrx whch we denote B T. Defnton 4.9. Let T be an deal trangulaton of S. For, j I, we defne 1 f, j share a vertex and s clockwse to j λ j = 1 f, j share a vertex and j s clockwse to 0 otherwse. These are entres of a skew-symmetrc I I matrx whch we denote Λ T. We wll use the same notaton Λ T for the assocated skew-symmetrc blnear form Z I Z I Z. Note that the word clockwse n the defnton of Λ T does not necessarly mean mmedately clockwse. Ths defnton makes sense because we work on a dsk and therefore cannot be both clockwse and counterclockwse to j. 16

17 Proposton 4.10 ([17], Proposton7.8). The matrces Λ T andb T satsfy thecompatblty condton b kj λ k = 4δ j. k Defnton Let F denote the skew-feld of fractons of the sken algebra Sk ω (S). For any deal trangulaton T and any vector v = (v 1,...,v m ) Z I 0, we wll wrte Tv for the smple multcurve havng v -many curves homotopc to I and no other components. The correspondng class [T v ] s called a monomal n the trangulaton T. More generally, we wrte [T u u ] = ω Λ T(u,u ) [T u ] 1 [T u ]. Ths s well defned and provdes a map M T : Z I F {0} gven by M T (v) = [T v ]. Wth these defntons, the par (B T,M T ) s a quantum seed and Λ T s the compatblty matrx assocated to the torc frame M T. Proposton 4.12 ([17], Theorem 7.9). Let T be an deal trangulaton of S, and let T be the deal trangulaton obtaned from T by performng a flp of the edge k. Then the quantum seed (B T,M T ) s obtaned from (B T,M T ) by a mutaton n the drecton k. It follows from Proposton 4.12 that there s a quantum cluster algebra A canoncally assocated to the dsk S. Ths algebra s generated by the cluster varables nsde of the skew-feld of fractons of the sken algebra. 4.3 Lamnatons In order to construct the maps I q A and Iq D, we need to defne the spaces A0 SL 2,S (Zt ) and D PGL2,S(Z t ). As explaned n the ntroducton, these can be understood as tropcalzatons of a certan modul spaces assocated to the groups SL 2 and PGL 2. Further explanaton can be found n [1, 2, 7, 10]. Defnton Let S be a dsk wth fntely many marked ponts on ts boundary. A pont of A 0 SL 2,S (Zt ) s defned as a collecton of edges and mutually nonntersectng dagonals of the polygon S wth nonzero ntegral weghts, subject to the followng condtons: 1. The weght of a dagonal s postve. 2. The total weght of the dagonals and edges ncdent to a gven vertex s zero. Defnton 4.14 ([1], Defnton 4.1). If S s an orented dsk wth fntely many marked ponts on ts boundary, we wll wrte S for the same dsk equpped wth the opposte orentaton. A pont of D PGL2,S(Z t ) s defned as a collecton of edges and nonntersectng dagonals of the polygons S and S wth postve ntegral weghts, subject to the followng condtons: 17

18 1. The total weght of the dagonals and edges ncdent to a gven vertex of S s the same as the total weght of the dagonals and edges ncdent to the correspondng vertex of S. 2. The collecton does not nclude an edge of S together wth the correspondng edge of S. Theabovedefntonsareequvalent tothedefntonsn[7,1]. In[7], thespacea 0 SL 2,S (Zt ) was understood as a space of measured lamnatons on S. In ths context, a measured lamnaton s a fnte collecton of homotopy classes of curves on a surface wth ntegral weghts and subject to certan condtons and equvalence relatons. Smlarly, n[1]the spaced PGL2,S(Z t )was understoodasaspace ofmeasured lamnatons on the surface S D obtaned by glung S and S along correspondng boundary edges and deletng the mages of the marked ponts. In ths case, a measured lamnaton conssts of fntely many homotopy classes of closed loopson S D. These loopsareequpped wth postve ntegral weghts and are agan subject to certan condtons and equvalence relatons. In partcular, we requre that no loop s retractble to a sngle puncture. 5 Dualty map for the quantum Posson varety 5.1 Realzaton of the cluster Posson varety We have assocated a quantum cluster algebra A to the dsk S. Let F be the ambent skew-feld of ths algebra A. Defnton 5.1. Let T be an deal trangulaton of S. For any j J, we defne an element X j = X j;t of F by the formula ( ) X j = M T ε js e s where the e s are standard bass vectors. In addton, we wll use the notaton q = ω 4. As the notaton suggests, these elements are related to the generators of the quantum Posson varety. Proposton 5.2. The elements X j (j J) satsfy the commutaton relatons X X j = q 2ε j X j X. Proof. By the propertes of torc frames, we have X X j = ω 2Λ T( s ε se s, t ε jte t) X j X. s 18

19 Now the compatblty condton n the defnton of a quantum seed mples ( Λ T ε s e s, ) ε jt e t = ε s ε jt Λ T (e s,e t ) s t Therefore X X j = ω 8ε j X j X = q 2ε j X j X. s,t = ε s ε jt λ st s,t = ε s b tj λ ts s t = 4ε j. In the followng result, T denotes the trangulaton obtaned from T by a flp of the edge k. Proposton 5.3. For each, let X = X ;T. Then X εk 1 p=0 (1+q 2p+1 X k ) f ε k 0 and k X = X X ε k εk 1 k p=0 (X k +q 2p+1 ) 1 f ε k 0 and k X 1 k f = k. Proof. Accordng to Lemma 5.4 of [21], a mutaton n the drecton k transforms X to the new element X bk 1 p=0 (1+ω 2( d kp d k ) 2 X k ) f b k 0 and k X = X X b k bk 1 k p=0 (X k +ω 2( d kp d k ) 2 ) 1 f b k 0 and k f = k X 1 k where d k s the number appearng n Defnton 3.2. The statement follows f we substtute d k = 4 nto ths formula. 5.2 Constructon of the map We wll now defne the map I q A. To do ths, let l be a pont of A0 SL 2,S (Zt ), represented by a collecton of curves on S. By the defnton of the space A 0 SL 2,S (Zt ), these curves are nonntersectng, so there exsts an deal trangulaton T l of S such that each of these curves concdes wth an edge of T l. Let w = (w 1,...,w m ) be the ntegral vector whose th component w s the weght of the curve correspondng to the edge of T l. Defnton 5.4. We wll wrte I q A (l) for the element of F gven by I q A (l) = M T l (w). 19

20 It s clear from the defnton of the torc frames that ths s ndependent of the choce of T l. Our goal s to show that ths defnton provdes a map I q A : A0 SL 2,S (Zt ) X q PGL 2,S. To prove ths, we wll need a techncal lemma on the g-vectors from Secton 3.2. Lemma 5.5. Fx an deal trangulaton T of S, and let c 1,...,c 2k be noncontractble curves connectng marked ponts. Assume these curves form a closed path on S n whch c and c +1 share a common endpont for = 1,...,2k where we number the ndces modulo 2k. Let g c be the g-vector assocated to c. Then the alternatng sum s = 2k =1 ( 1) g c s an ntegral lnear combnaton of expressons of the form s ε se s. Proof. Consder the trangle n T that ncludes p I J as one of ts edges and label the other edges as follows: p q r We can defne a vector assocated to ths edge p as y p = e p + e q e r. Fx an edge j J. For any endpont v of j, there s a collecton of edges of T that start at v and le n the counterclockwse drecton from j. Consder an arc j that goes dagonally across j, ntersectng each of these edges transversally and termnatng on the boundary. An example s llustrated below. 0 j j 1 Gven such an arc j, let E j be the set of all edges n T that j crosses. Then we can form the sum p j = y 0 + x +y 1 E j where 0 and 1 are the edges on whch j termnates and we have wrtten x = s ε se s. One can check that ths expresson p j equals 2e j. Now let c be a curve connectng marked ponts. It follows from the results of [18] that the g-vector g c s an alternatng sum of standard bass vectors correspondng to a path n T. In fact, the entre sum appearng n the statement of the lemma can be wrtten as an alternatng sum s = 2N l=1 ( 1)l e jl of standard bass vectors 20

21 assocated to the edges j 1,...,j 2N of a closed path n T. We assume that these edges are ordered consecutvely. To each edge j of ths path, we assocate the arc j descrbed above. We have 2s = 2 2N l=1 ( 1) l e jl = 2N l=1 ( 1) l p jl and all of the y-terms cancel on the rght hand sde. Thus ths expresson s lnear combnaton of the x k. By constructon, these x k are assocated to the ponts of ntersecton between edges of T and a certan closed path on S. Each edge ntersects ths path n an even number of ponts, so the vector 2s s n fact a lnear combnaton of the x k wth even coeffcents. Dvdng by 2 yelds the desred result. Theorem 5.6. Let T be an deal trangulaton of S, and let X j = X j;t be defned as above. Then for any l A 0 SL 2,S (Zt ), the element I q A (l) s a Laurent polynomal n the X j wth coeffcents n Z 0 [q,q 1 ]. Proof. By applyng the relatons from Defnton 4.5, we can wrte I q A (l) = m ωα M Tl (e ) w where α = <j Λ T l (e,e j )w w j. Then Corollary 3.14 mples =1 I q A (l) = m ωα (F M T (g )) w. =1 Each F n ths last lne denotes a polynomal n the expressons M T ( j ε kje j ) wth coeffcents n Z 0 [ω 4,ω 4 ]. The g are ntegral vectors. We have ( ) Λ T ε k e,e j = ε k λ j = b k λ j = 4δ kj so that ( ) ( ) M T ε k e MT (e j ) = ω 8δ kj M T (e j )M T ε k e. It follows that we can rearrange the factors n the last expresson for I q A (l) to get m I q A (l) = ωα Q M T (g ) w where Q s a polynomal n the expressons M T ( j ε kje j ) wth coeffcents n the semrng Z 0 [ω 4,ω 4 ]. By Lemma 6.2 n [21], we know that Λ Tl (e,e j ) = Λ T (g,g j ), and so ( ) M T w g = ω =1 <j Λ T(g,g j )w w j m m = ω α M T (g ) w. =1 21 =1 M T (g ) w

22 Hence I q A (l) = Q M ( ) T w g. The vector w g s a sum of vectors of the type appearng n the statement of Lemma 5.5, so ths lemma mples that the second factor s a monomal n the X j wth coeffcent n Z 0 [ω 4,ω 4 ]. Ths completes the proof. By the propertes establshed n Propostons 5.2, and 5.3, we can regard the Laurent polynomal of Theorem 5.6 as an element of the algebra X q PGL 2,S. Thus we have constructed a canoncal map I q A : A0 SL 2,S (Zt ) X q PGL 2,S as desred. 5.3 Propertes Fx an deal trangulaton T of S, and let X j be the elements of F defned above. We wll now prove several propertes of the map I q A conjectured n [6, 8]. Theorem 5.7. The map I q A : A0 SL 2,S (Zt ) X q PGL 2,S satsfes the followng propertes: 1. Each I q A (l) s a Laurent polynomal n the varables X j wth coeffcents n Z 0 [q,q 1 ]. 2. The expresson I q A (l) agrees wth the Laurent polynomal I A(l) when q = Let bethecanoncalnvolutve antautomorphsmofx q thatfxeseachx andsendsq to q 1. Then I q A (l) = Iq A (l). 4. The hghest term of I q A (l) s q <j ε ja a j X a Xan n where X a X an n s the hghest term of the classcal expresson I A (l). 5. For any l, l A 0 SL 2,S (Zt ), we have I q A (l)iq A (l ) = l A 0 SL 2,S (Zt ) c q (l,l ;l )I q A (l ) where c q (l,l ;l ) Z[q,q 1 ] and only fntely many terms are nonzero. Proof. 1. Ths was proved n Theorem Ths follows mmedately from the defnton of I A gven n [7]. 3. If K s any framed lnk n S, let K be the framed lnk wth the same underlyng multcurve and the order of each crossng reversed. By Proposton 3.11 of [17] the map sendng [K] to [K ] and ω to ω := ω 1 extends to an nvolutve antautomorphsm of the sken algebra Sk ω (S). It extends further to an antautomorphsm of the fracton feld F by the rule (xy 1 ) = (y ) 1 x. Ths operaton s compatble wth n the sense that X j = X j = X j 22

23 and q = q 1 = q. It s easy to check that preserves M Tl (w) = ω <j Λ T l (e,e j ) M Tl (e 1 ) w 1...M Tl (e n ) wm. It follows that I q A (l) s -nvarant. 4. By part 1, we can wrte I q A (l) = c 1,..., n X Xn n ( 1,..., n) supp(l) for some fnte subset supp(l) Z n where the coeffcent c 1,..., n Z 0 [q,q 1 ] s nonzero for all ( 1,..., n ) supp(l). Consder the coeffcent c = c a1,...,a n. We can expand ths coeffcent as c = s c sq s. By settng q = 1, we see that s c s = 1. It follows that we must have c s = 1 for some s, and all other terms must vansh. Thus c = q s, and the leadng term of I q A (l) has the form q s X a X an n for some nteger s. By part 3, we know that the expresson I q A (l), and hence ts leadng term, s -nvarant. Ths means We have q s X an n...xa 1 1 = q s X a Xan n. X an n...xa 1 1 = q 2 n j=2 ε 1ja 1 a j X a 1 1 (X an n...xa 2 2 ) = q 2 n j=2 ε 1ja 1 a j q 2 n j=3 ε 2ja 2 a j X a 1 1 Xa 2 2 (Xan n...xa 3 3 ) =... = q 2 <j ε ja a j X a X an n, so -nvarance of the leadng term s equvalent to q s 2 <j ε ja a j X a Xan n = qs X a Xan n. Equatng coeffcents, we see that s = <j ε ja a j. Thus we see that the hghest term equals q <j ε ja a j X a Xan n. 5. Let l, l A 0 SL 2,S (Zt ). Then there exst deal trangulatons T l and T l of S and ntegral vectors w and w such that I q A (l) = M T l (w) and I q A (l ) = M Tl (w ). For any ntegral vector v = (v 1,...,v m ), we can consder the vector v + whose th component equals v f v 0 and zero otherwse. We can lkewse consder the vector v whose th component equals v f v 0 and zero otherwse. Then there exsts an nteger N such that I q A (l)iq A (l ) = M Tl (w)m Tl (w ) = ω N M Tl (w )M Tl (w + )M Tl (w +)M Tl (w ) = ω N M Tl (w )[T w + l ][T w + l ]M Tl (w ). 23

24 By applyng the sken relatons, we can wrte the product [T w + l ][T w + ] as [T w + l ][T w + l ] = k d ω [K ] where the K are dstnct smple multcurves on S and d ω Z[ω,ω 1 ] are nonzero. Therefore =1 l I q A (l)iq A (l ) = k ω N d ω M T (w )M T (v )M T (w ) =1 where T s an deal trangulaton of S such that each each curve of K concdes wth an edge of T. Here we have wrtten v for the unque ntegral vector such that [K ] = M T (v ). Each of the products ω N d ω M T (w )M T (v )M T (w ) n ths last expresson can be wrtten as c ω (l,l ;l )I q A (l ) for some l A 0 SL 2,S (Zt ) and some c ω (l,l ;l ) Z[ω,ω 1 ]. Hence I q A (l)iq A (l ) = k c ω (l,l ;l )I q A (l ). =1 The left hand sde of ths last equaton s a Laurent polynomal wth coeffcents n Z[q,q 1 ]. Let us wrte [I q A (l )] H for the hghest term of I q A (l ). We can mpose a lexcographc total orderng on the set of all commutatve Laurent monomals n X 1,...,X n so that [I 1 A (l )] H s ndeed the hghest term of I 1 A (l ) wth respect to ths total orderng. These hghest terms are dstnct, so we may assume [I 1 A (l 1)] H > [I 1 A (l 2)] H > > [I 1 A (l k)] H. Consder the expresson c ω (l,l ;l 1 )[I q A (l 1)] H. It cannot cancel wth any other term n the sum, so we must have c ω (l,l ;l 1 ) Z[q,q 1 ]. It follows that the sum k c ω (l,l ;l )I q A (l ) =2 s a Laurent polynomal wth coeffcents n Z[q,q 1 ]. Argung as before, we see that c ω (l,l ;l 2 ) Z[q,q 1 ]. Contnung n ths way, we see that all c ω (l,l ;l ) le n Z[q,q 1 ]. Ths completes the proof. In [6, 8], Fock and Goncharov conjectured n addton that the Laurent polynomals c q (l,l,l ) have postve coeffcents. Ths property should be closely related to Conjecture 4.20 of [20]. However, t does not obvously follow from the sken relatons as these may nvolve negatve coeffcents. 24

25 6 Dualty map for the quantum symplectc double 6.1 Realzaton of the symplectc double We have assocated a quantum cluster algebra A to the dsk S. One can lkewse assocate a quantum cluster algebra A to the dsk S obtaned from S by reversng ts orentaton. If T s any deal trangulaton of S, then there s a correspondng trangulaton T of S. We wll wrte Λ T and M T, respectvely, for the compatblty matrx and torc frame correspondng to the trangulaton T. We wll wrte e for the standard bass vector n Z m assocated to the edge of T or the correspondng edge of T. Note that we have Λ T (e,e j ) = Λ T (e,e j ) for all and j. Let F be the ambent skew-feld of the quantum cluster algebra A, and let F be the skew-feld of the quantum cluster algebra A. Each of these skew-felds s a two-sded module over the rng Z[ω,ω 1 ], and we wll consder the followng elements of the tensor product F Z[ω,ω 1 ] F. Defnton 6.1. Let T be an deal trangulaton of S. For any I and any j J, we defne elements B = B ;T and X j = X j;t of F Z[ω,ω 1 ] F by the formulas ( ) X j = M T ε js e s 1, In addton, we wll use the notaton q = ω 4. s B = M T ( e ) M T (e ). Let us denote by G the subalgebra of F Z[ω,ω 1 ]F consstng of Laurent polynomals n thevarablesb for I whosecoeffcentsareratonalexpressons nthex j wthcoeffcents n Z[ω,ω 1 ]. It s easy to check that the B for I J are central elements of ths algebra, and n what follows t wll be mportant to consder the quotent F D = G/I where I s the two-sded deal of G generated by B 1 for I J. The followng result relates the elements X j and B j to the generators appearng n the defnton of the quantum symplectc double. Proposton 6.2. The elements B j and X j (j J) satsfy the followng commutaton relatons: X X j = q 2ε j X j X, B B j = B j B, X B j = q 2δ j B j X. Proof. The proof of the frst relaton s dentcal to the proof of Proposton 5.2. To prove the second relaton, observe that B B j = M T ( e )M T ( e j ) M T (e )M T (e j ) = ω 2(Λ T(e,e j )+Λ T (e,e j )) M T ( e j )M T ( e ) M T (e j )M T (e ) = ω 2(Λ T(e,e j )+Λ T (e,e j )) B j B 25

26 and Λ T (e,e j )+Λ T (e,e j ) = 0. Therefore B B j = B j B. Fnally, we have ( ) X B j = M T ε s e s MT ( e j ) M T (e j ) s = ω 2Λ T( s ε se s, e ( ) j ) M T ( e j )M T ε s e s MT (e j ) = ω 2Λ T( s ε se s, e j ) B j X. By the compatblty condton, we have ( ) Λ T ε s e s, e j = ε s Λ T (e s,e j ) s s s = s b s λ sj = 4δ j. Therefore X B j = ω 8δ j B j X = q 2δ j B j X. In the followng result, T denotes the trangulaton obtaned from T by a flp of the edge k. Proposton 6.3. For each, let B = B ;T. Then Proof. By Proposton 3.10, we have { B = (qx k B + k +B k )B 1 k (1+q 1 X k ) 1 f = k B f k. ( M T (e k ) = M T ek + ) ( [ε k ] + e +MT ek + ) [ ε k ] + e ( = M T ε k e e k + ) ( [ ε k ] + e +MT ek + ) [ ε k ] + e ( ) ( ) = ω α M T ε k e MT ( e k )M T [ ε k ] + e ( ) +ω β M T ( e k )M T [ ε k ] + e where we have wrtten ( α = Λ T ε k e e k, ) ( ) [ ε k ] + e +ΛT ε k e, e k 26

27 and β = Λ T ( ek, [ ε k ] + e ). Factorng, we obtan M T (e k ) = ( ( ) ) ( ) 1+ω α β M T ε k e ω β M T ( e k )M T [ ε k ] + e. A straghtforward calculaton usng the compatblty condton shows α β = 4. Substtutng ths nto the last expresson, we see that ( ) ( ) M T (e k ) 1 = (1+q 1 X k )ω β (M( e k ) 1) M T [ ε k ] + e 1. On the other hand, we have 1 M (T ) (e ( k) = 1 M T ek + ( = M T ε k e [ε k ] + e + ( +M T [ ε k ] + e + ) ( [ε k ] + e +1 MT ek + [ ε k ] + e ) ) ( [ ε k ] + e MT ek + ) ( [ ε k ] + e MT ek + [ε k ] + e ) [ ε k ] + e ). By the propertes of torc frames, ths equals ( ) ( ) ( ) ( ) ω γ M T ε k e MT [ε k ] + e MT [ ε k ] + e MT [ε k ] + e MT ( e k ) ( ) ( ) ( ) +ω δ M T [ ε k ] + e MT [ ε k ] + e MT [ ε k ] + e MT ( e k ) for certan exponents γ and δ. Factorng, we obtan ( ( ) ( ) ( ) ω γ δ M T ε k e MT [ε k ] + e MT [ε k ] + e ( ) ( ) ) ( ) +M T [ ε k ] + e MT [ ε k ] + e ω δ M T [ ε k ] + e MT ( e k ) ( ( ) = (ω γ δ X k B + k +B k )ωδ (1 M T ( e k )) M T [ ε k ] + e 1 ). A straghtforward calculaton shows that δ = β and γ δ = 4. Substtutng these nto the last expresson, we see that ( 1 M (T ) (e ( ) k) = (qx k B + k +B k )ωβ (1 M T ( e k )) M T [ ε k ] + e 1 ). 27

28 Fnally, by the defnton of B k, we have B k = M T ( e k) M (T ) (e k) = (1 M (T ) (e k))(m T (e k ) 1) 1 ( ) ( ) = (qx k B + k +B k )ωβ (1 M T ( e k )) M T [ ε k ] + e 1 ( ) ( ) 1 M T [ ε k ] + e 1 (M T ( e k ) 1) 1 ω β (1+q 1 X k ) 1 = (qx k B + k +B k )B 1 k (1+q 1 X k ) 1. For k, we clearly have B = B. Ths completes the proof. By Proposton 5.3, there s a smlar statement descrbng the transformaton of the X j under a flp of the trangulaton. 6.2 Constructon of the map We are now ready to defne the map I q D. To do ths, let l be a pont of D PGL 2,S(Z t ), represented by a collecton of curves on S and S. Wrte C for the collecton of curves on S and C for the collecton of curves on S. There s an deal trangulaton T C of S such that each curve n C concdes wth an edge of T C, and there s an deal trangulaton T C of S such that each curve n C concdes wth an edge of T C. Fx an deal trangulaton T of S. Then each c C determnes an ntegral g-vector g c = g T c,t C. The trangulaton T determnes a correspondng deal trangulaton of S, and so we can lkewse assocate to each c C an ntegral vector g c = g T c,t C. Lemma 6.4. The number where g C = c C s ndependent of the trangulaton T. N l := Λ T (g C,g C ), g c, g C = c C g c, Proof. Let T and T be two dfferent trangulatons. Let gc,t T C and gc T,T C be the g-vectors assocated to the arcs c C and c C, respectvely, defned usng the trangulaton T. Smlarly, let gc,t T C and gc T,T be the vectors defned usng the trangulaton C T. Number the edges of T so that each edge s dentfed wth some number 1,...,m. Usng the recurrence relatons dscussed n Secton 3 of [21], we can wrte g T c,t C = P T c,t C (e 1,...,e m ), g T c,t C = PT c,t C (e 1,...,e m ) for lnear polynomals Pc,T T C and Pc T,T C, and we also have gc,t T C = Pc,T T C (g1,t T,...,gT m,t ), gt c,t = C PT c,t C (gt 1,T,...,gT m,t ) 28