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Factorization rules in quantum Teichmuller theory

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Factorization rules in quantum Teichmuller theory

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Content FACTORIZATION RULES IN QUANTUM TEICHM ¨ ULLER THEORY by Julien Roger A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (MATHEMATICS) August 2010 Copyright 2010 Julien Roger Dedication To Marie and Rapha¨ el ii Table of Contents Dedication ii List of Figures v Abstract vi Chapter 1: Introduction 1 Chapter 2: Motivations 6 2.1 Modular functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 The volume conjecture . . . . . . . . . . . . . . . . . . . . . . . . . 9 Chapter 3: Geometric background 10 3.1 Teichm¨ uller spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 The augmented Teichm¨ uller space . . . . . . . . . . . . . . . . . . . 11 3.3 Exponential shear coordinates . . . . . . . . . . . . . . . . . . . . . 12 3.4 The Weil-Petersson Poisson structure . . . . . . . . . . . . . . . . . 13 Chapter 4: Pinching along curves: geometric properties 15 4.1 Induced ideal triangulations . . . . . . . . . . . . . . . . . . . . . . . 15 4.2 Extension of the shear coordinates . . . . . . . . . . . . . . . . . . . 17 4.3 Extension of the Weil-Petersson Poisson structure . . . . . . . . . . . 20 Chapter 5: Pinching along curves: quantum aspect 25 5.1 The Chekhov-Fock algebra . . . . . . . . . . . . . . . . . . . . . . . 25 5.2 A homomorphism between Chekhov-Fock algebras . . . . . . . . . . 27 5.3 Application to the representation theory of Chekhov-Fock algebras . . 29 5.3.1 Representations of the Chekhov-Fock algebra . . . . . . . . . 29 5.3.2 Convergence of representations in the augmented Teichm¨ uller space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 iii Chapter 6: Behavior under changes of coordinates: the quantum Teichm¨ uller space 36 6.1 The quantum Teichm¨ uller space . . . . . . . . . . . . . . . . . . . . 36 6.2 Representations of the quantum Teichm¨ uller space . . . . . . . . . . 37 6.3 Main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 References 52 iv List of Figures 2.1 Degeneration of Riemann surfaces . . . . . . . . . . . . . . . . . . . 8 4.1 Elementary pieces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.2 Disallowed cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.3 Extension of shear parameters . . . . . . . . . . . . . . . . . . . . . 19 6.1 Diagonal exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 6.2 A diagonal exchange in a non-crossed square . . . . . . . . . . . . . 41 6.3 A diagonal exchange in a crossed square . . . . . . . . . . . . . . . . 42 6.4 Two non-adjacent triangles . . . . . . . . . . . . . . . . . . . . . . . 50 6.5 Final step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 v Abstract For a punctured surfaceS, a point of its Teichm¨ uller spaceT(S) determines an irre- ducible representation of its quantization T q (S). We analyze the behavior of these representations as one goes to infinity inT(S), or in the moduli spaceM(S) of the surface. Our main result states that an irreducible representation ofT q (S) tend in an appropriate sense to a direct sum of representations ofT q (S ), whereS is obtained from S by pinching a multicurve to a set of nodes. The result is analogous to the factorization rule found in conformal field theory. vi Chapter 1: Introduction LetS be a surface of genusg obtained from a closed compact surfaceS by remov- ings puncturesv 1 , . . . ,v s . The Teichm¨ uller spaceT(S) of a surfaceS is the space of isotopy classes of complete hyperbolic metrics onS with finite area. It comes equipped with a natural K¨ ahler metric, called the Weil-Petersson metric which is invariant under the action of the mapping class group MCG(S) onto T(S). A quantization of the Teichm¨ uller space was successfully described by L. Chekhov and V . Fock in [11], and, in a slightly different setting, by R. Kashaev in [17]. In the work of Chekhov and Fock, the main geometric ingredient is the notion of shear coordinates on the enhanced Teichm¨ uller space e T(S) which were introduced by W. Thurston [27]. On the algebraic side, they make use of the quantum dilogarithm as described by L. Faddeev and Kashaev [13]. In the physics literature, the interest for the quantization of Teichm¨ uller theory can be traced back to the work of E. Verlinde and H. Verlinde [29, 30] among others. In particular, H. Verlinde conjectured in [30] that quantum Teichm¨ uller theory should give rise to a family of representations of the mapping class groups which could be identified with a modular functor obtained from Liouville conformal field theory. The existence of such a modular functor associated to the quantum Teichm¨ uller space was conjectured also by Fock [14] and was studied further by J. Teschner [26]. The goal of this work is to investigate a similar question in the context of the exponential quantum Teichm¨ uller space studied by H. Bai, F. Bonahon and X. Liu 1 [19, 3, 10]. Given a parameter q 2 C , The quantum Teichm¨ uller space T q (S) is a non-commutative algebra, deformation of the algebra of functions on T(S). For q a root of unity, Bonahon and Liu [10] describe a complete classification of the finite dimensional irreducible representations : T q (S) ! End(V ). In partic- ular, if we choose q to be a primitive N-th root of unity with N odd and we fix weights p 1 ;:::;p s 2 f0;:::;N 1g labeling the punctures v 1 , . . . , v s of S, one can associate to every hyperbolic metricm2T(S) a unique irreducible representation m :T q (S)! End(V ). Using a construction similar to the one described in [4], one can then construct a projective vector bundle K q = K q (p 1 ;:::;p s ) over T(S) with fiberPV , where the fiber atm2 T(S) is endowed with the action of the irreducible representation m ofT q (S). This construction behaves well under the action of the mapping class groupMCG(S) and we obtain a projective vector bundle e K q over the moduli spaceM(S) =T(S)=MCG(S). In the spirit of conformal field theory one can then ask how this bundle extends to the Deligne-Mumford compactificationM(S) of the moduli space. To study this question we analyze how the representation m ofT q (S) breaks down when the metric m approaches a point ofM(S)rM(S), that is when the lengths of a finite number of geodesics ofS tend to 0 for this metric. The result we obtain can then be interpreted as a factorization rule for this theory. More precisely, let be an ideal triangulation ofS, that is a triangulation ofS with vertices at the punctures. Following the construction of Chekhov and Fock in [11], Bonahon and Liu [19, 10] associate toS, the triangulation and a parameterq2C , an algebraT q (S) called the Chekhov-Fock algebra. It is the skew-commutative alge- bra overC with generatorsX 1 1 ;:::;X 1 n associated to the edges of and relations 2 X i X j = q 2 ij X j X i , where q 2 C and the ij 2 f2;1; 0; 1; 2g are the coef- ficients of the Weil-Petersson Poisson structure on the enhanced Teichm¨ uller space e T(S), parametrized using Thurston shear coordinates. Approaching a point in the boundary of M(S) corresponds to shrinking a finite number of non intersecting geodesics of S to a point. Hence we give ourselves a finite union of non-intersecting, non homotopic, essential simple closed curves = [ i i S and we consider the surfaceS = Sr . One should think ofS as being obtained fromS by pinching the multicurve tok nodes and removing them. It is a possibly disconnected surface with two new punctures for each curve removed. The ideal triangulation on S induces an ideal triangulation on S whose edges are given by taking homotopy classes of edges of\S . An essential step is to relate the quantum Teichm¨ uller spaces ofS andS . Proposition 1 For every ideal triangulation and every multicurve on S, there exists an algebra homomorphism q ; :T q (S )!T q (S) described explicitly by sending each generator of T q (S ) to certain monomials in T q (S). The existence of this homomorphism is the algebraic translation of the fact that the Weil-Petersson Poisson structure extends naturally to the completionT(S) of the Teichm¨ uller space for the Weil-Petersson metric (H. Masur [20]). This completion is called the augmented Teichm¨ uller space and was introduced by W. Abikoff [1] and L. Bers [8]. As a set it is the union ofT(S) and of theT(S ) for every multicurve . The action ofMCG(S) onT(S) extends toT(S) and the quotientT(S)=MCG(S) can be identified topologically withM(S). 3 Whenq is anN-th root of unity withN odd, Bonahon and Liu associate an irre- ducible representation m :T q (S)! End(V ) of the Chekhov-Fock algebra to a met- ricm2T(S) and weightsp 1 ;:::;p s 2f0;:::;N 1g labeling the puncturesv 1 , . . . , v s ofS. m is defined uniquely up to isomorphism and varies continuously with the metricm. One can then ask what happens whenm approaches a lower-dimensional stratumT(S ) inT(S). For simplicity, let us restrict attention, in the introduction, to the case where consists of a single curve. Theorem 2 Letm t 2 T(S),t2 (0; 1], be a continuous family of hyperbolic metrics such that, ast! 0,m t converges tom 2T(S ) inT(S). Let t :T q (S)! End(V ), t2 (0; 1] be a continuous family of irreducible representations classified bym t and weightsp 1 ;:::;p s 2f0;:::;N 1g labeling the puncturesv 1 , . . . ,v s ofS. Then, ast! 0, the representation t q ; :T q (S )! End(V ) approaches N1 M i=0 i :T q (S )! End( N1 M i=0 V i ) where, for eachi, i :T q (S )! End(V i ) is the irreducible representation classified bym , the weightsp 1 ;:::;p s labeling the old punctures and the weighti labeling the two new puncturesv 0 andv 00 ofS . The next step is to show that this decomposition is well-behaved under changes of triangulations. More precisely, for any two triangulations and 0 ofS, Chekhov and Fock introduce coordinate change isomorphisms q 0 : b T q 0 (S)! b T q (S) between the rings of fractions associated to the Chekhov-Fock algebrasT q (S) andT q 0 (S). The 4 quantum Teichm¨ uller spaceT q (S) ofS is then defined to be the union of theT q (S) for every triangulation, modulo the relation which identifiesX 0 2T q 0 (S) to q 0 (X 0 )2 T q (S). The coordinate change isomorphisms are only defined between fraction algebras. For certain representations ofT q (S) the composition with q ; 0 still makes sense and we obtain a representation q ; 0 : T q 0 (S)! End(V ). A representation of T q (S) is then a family of representationsf g of T q (S) such that q 0 = 0 for every triangulations and 0 . In particular, given m 2 T(S) and weights p 1 ,. . . ,p s labeling the punctures ofS, one obtain a representation m =f m; g where m; :T q (S)! End(V ) is the irreducible representation ofT q (S) classified by these data. Theorem 3 Let t = f t; g , t 2 (0; 1], be a continuous family of irreducible representations of T q (S) classified by weights p 1 , . . . , p s and a continuous family m t 2 T(S), t2 (0; 1] such that m t approaches m 2 T(S ) as t! 0. For each triangulation ofS, we let the limit lim t!0 t; q ; = M i i ; be given as in Theorem 2. Then for any triangulations and 0 and weighti, we have i ; 0 = i ; q 0 : Hence the family of representationsf i ; g determines an irreducible representa- tion i :T q (S)! End(V ), which is classified bym, the weightsp 1 , . . . ,p s labeling the old punctures, and the weighti labeling the new puncturesv 0 andv 00 ofS . 5 Chapter 2: Motivations As a motivation for this work we briefly describe two conjectures which are related to the study of the quantization of Teichm¨ uller space. 2.1 Modular functors Modular functor comes from the study of two dimensional conformal field theory (see D. Friedan and S. Shenker [15], G. Moore and N. Seiberg [21], G. Segal [24] for exam- ple). There are several descriptions of what a modular functor should be. Some of them are purely topological (see for example V . Turaev [28]). We give here a more geomet- ric definition, following A. Beauville [7] closely. One can find in [5] a discussion on the equivalence of these definitions. As an initial data we will need a finite set L called the label set of the theory. In general L is endowed with an involution l ! l , although in our situation this involution is probably the identity. A marked Riemann surface (C;p;l) is a non- necessarily connected compact Riemann surface with a finite number of distinguished pointsp = (p 1 ;:::;p s ) and a labeling of the pointsl = (l 1 ;:::;l s )2L s by elements ofL. A modular functor (C;p;l)!V (C;p;l) 6 associates to any marked Riemann surface a finite dimensional complex vector space and satisfy, among others, the following axioms: M1 (Disjoint union) Let (C;p;l) be the disjoint union of two marked Riemann sur- faces (C 0 ;p 0 ;l 0 ) and (C 00 ;p 00 ;l 00 ). Then V (C;p;l) =V (C 0 ;p 0 ;l 0 ) V (C 00 ;p 00 ;l 00 ) M2 (Naturality) Let (C t ) t2D be a holomorphic family of compact Riemann surfaces parametrized by the unit disk D C with marked points p 1 (t), . . . , p s (t) depending holomorphically on t. Then, for any t 2 D there is a canonical isomorphism V (C t ;p(t);l) !V (C 0 ;p(0);l) For the next axiom we need to introduce the notion of nodal surface. A Riemann surfaceC with nodes is such that every point2C has a neighborhood biholomorphic to either a disc or the conef(z;w)2C 2 =zw = 0 andj(z;w)j< 1g, where is sent to 0. In the second case we call a node ofC. We then denote by e C the compact Riemann surface obtained fromC by removing and adding two new marked points 0 and 00 (See Figure 2.1). In a similar situation as in the previous axiom, if the family C t is parametrized over Dr 0 it may happen that the surface develops a node as t! 0. That is, the underlying topological surface S gets pinched along an essential simple closed curve to obtain a surfaceS 0 and the complex structure ofC t approaches, in a suitable sense, a complex structure onS 0 with a node at the pinching. We will describe this phenomenon more prcisely in the next section in terms of hyperbolic geometry, and refer to Figure 2.1 for now. 7 Figure 2.1: Degeneration of Riemann surfaces M3 (Factorization rule) In the situation of Figure 2.1, For anyt2Drf0g, there is an isomorphism V (C t ;p(t);l) ! M 2L V (C 0 ; (p 1 (0);:::;p s (0); 0 ; 00 ); (l 1 ;:::;l s ;; )) A well known example of a modular functor is the so-called Wess-Zumino-Witten model based on the representation theory of the affine Lie algebrab g associated to a simple Lie algebra g. The label setL parametrizes a certain set of irreducible repre- sentations ofb g. The vector spaces associated to marked Riemann surfaces (C;p;l) form a vector bundle over the moduli space of Riemann surfaces or type (g;n) called the conformal block bundle or the bundle of vacua. Finally axioms M2 and M3 are satisfied because of the existence of a projectively flat connection on this bundle, called the Knizhnik-Zamolodchikov connection. We refer to T. Kohno [18]) for an introduc- tion to the subject with applications to low dimensional topology. In [30], H. Verlinde conjecture that the Hilbert space associated to the quantization of the Teichm¨ uller space could be identified with the space of conformal blocks for the representation theory of the Virasoro algebra. A similar conjecture was made by Fock 8 in [14]. In both cases the label set is infinite and is identified with the set of possible lengths of a geodesic representing the curve being pinched, in the hyperbolic metric associated to the complex structure ofC. More recently this conjecture was studied by Teschner in [26]. 2.2 The volume conjecture We letK be a knot inS 3 and we suppose that the knot is hyperbolic, that is that the complementS 3 rK admits a complete hyperbolic metric, which is necessarily finite and unique up to isometry by Mostow rigidity. Then the volume ofS 3 rK for this metric, denoted byVol(S 3 rK), is a topological invariant ofK. Another family of invariants of knots are given by the colored Jones polynomials J N (K;t)2Z [t;t 1 ], indexed by an integerN > 1. The volume conjecture stated by Kashaev and refined by H. Murakami and J. Murakami relates these two invariants Conjecture 4 (Kashaev-Murakami-Murakami) For any hyperbolic knot K in S 3 , we have 2 lim N!1 log J N (K;e 2i=N ) N =Vol(S 3 rK) This conjecture generalizes to non-hyperbolic knots by specifying the right-hand side to be equal to 0. We refer to [22] and reference therein for details. In the original conjecture of Kashaev [16], the Jones polynomial was replaced by another invarianthKi N (the Kashaev invariant) which was later shown in [23] to be equal to the Jones polynomial. On the other hand, the original descrition ofhKi N made use of the quantum dilogarithm [17], which is at the center of the quantization of the Teichm¨ uller space. A study of the quantum Teichm¨ uller space and its relationship with quantum hyperbolic geometry as constructed by S. Baseilhac and R. Benedetti (see for example [6]) should provide some insights for the study of this conjecture. 9 Chapter 3: Geometric background From now on, S will be a surface of genus g obtained from a closed surface S without boundary by removings puncturesv 1 ;:::;v s . We will assume thatS has at least one puncture and has Euler characteristic(S) = 2 2gs< 0. The simplest such surfaces are the spheres with 3 or 4 punctures and the once-punctured torus. 3.1 Teichm¨ uller spaces WFor the next sections, we will need two variants of Teichm¨ uller space. The first and most classical one, denoted simply asT(S), will be the set of isotopy classes of complete hyperbolic metrics onS with finite area. We will call this space simply the Teichm¨ uller space of S. however we will need to drop the finite area condition to describe the exponential shear coordinates, which are essential to the definition of the quantum Teichm¨ uller space. LetConv(S;m) denote the convex core ofS, that is the smallest non-empty closed convex subset of (S;m). Conv(S;m) is a surface with cusps and geodesic boundaries and is homeomorphic to S. If (S;m) has finite area then its convex core consists of the whole surface. Otherwise some of the punctures of (S;m) will have a neighborhood isometric to an infinite area funnel bounded by one of the geodesic boundaries ofConv(S;m). Then, we let e T(S) be the space of isotopy classes of complete hyperbolic metrics onS, possibly with infinite volume, together with an orientation of all the boundary components ofConv(S;m). e T(S) will be called the enhanced Teichm¨ uller space of S. In particular, since for a complete hyperbolic 10 metricConv(S;m) has no boundary component, there is a natural embedding ofT(S) into e T(S). 3.2 The augmented Teichm¨ uller space We will also need to consider the augmented Teichm¨ uller spaceT(S) which was intro- duced by Abikoff [1, 2] and Bers [8], and was further studied by Masur [20] (See more recently S. Wolpert [32] and references therein). We will briefly recall its construction and some of its properties. Denote by C(S) the complex of curves on S and let =f 1 ;:::; k g be a k- simplex in C(S), that is a set of k curves 1 , . . . , k in S which are disjoint, non- homotopic and essential. We will denote byS the surface obtained fromS by remov- ing the curves 1 , . . . , l . It is a possibly disconnected surface with two new punctures for each curve removed. As a set we define T(S) =T(S)[ [ 2C(F ) T(S ); whereT(S ) is the product of Teichm¨ uller spaces associated to the connected compo- nents ofS . TheT(S ) are called the strata ofT(S). A topology onT(S) can be defined as follows: a sequence of metrics (m n ) n in T(S) converges tom 2 T(S ) if, asn!1,l mn ( i ) tends to 0 for everyi andm n converges uniformly tom on every compact ofS . The action of the mapping class group MCG(S) of S on Teichm¨ uller space extends to T(S) and, as a topological space, the quotient T(S)=MCG(S) can be identified with the Deligne-Mumford compactification M(S) of the moduli space M(S) =T(S)=MCG(S) [1]. 11 3.3 Exponential shear coordinates One of the main ingredients for the quantization of Teichm¨ uller space as first described in [11] is the notion of shear coordinates introduced by W. Thurston [27]. We will describe here their exponential version as described for example in [9] (see also [19]). Since (S) < 0 and S has at least one puncture, it admits an ideal triangula- tion =f 1 ;:::; n g, that is a triangulation of S with verticesv 1 ;:::;v s and edges 1 ;:::; n , where the edges are considered up to isotopy. The number of edges in an ideal triangulation only depends on the Euler characteristic of S and is given by n =3(S) = 6g + 3s 6. If we endowS with a hyperbolic metricm, each edge i is isotopic to a unique geodesicg i for this metric. One can then associate to each edge i of a numberx i 2R + , called the exponential shear parameter ofm along i , obtained as follows: let ~ g i be a lift ofg i to the universal cover of (S;m), which we identify with the upper half-spaceH 2 . ~ g i separates two triangles ~ T 1 i and ~ T 2 i , bounded by lifts of edges of so that the union ~ Q i = ~ g i [ ~ T 1 i [ ~ T 2 i forms a square inH 2 with vertices on the real line boundingH 2 . For a given orientation of ~ g i , we name the ver- tices of ~ Q i z ,z + ,z r ,z l where ~ g i goes fromz toz + , andz r andz l are respectively to the right and to the left of ~ g i . The exponential shear parameter ofm along i is then x i =cross-ratio(z r ;z l ;z ;z + ) = (z r z )(z l z + ) (z r z + )(z l z ) 2R + : One can conversely construct a hyperbolic metricm from any choice of parameters x 1 ;:::;x n 2 R + associated to the edges of , by gluing hyperbolic triangles into squares whose vertices have the prescribed cross-ratio. This defines a homeomorphism ' : e T(S)!R n + for every ideal triangulation ofS. Givenx 1 ;:::;x n the shear parameters ofm2 e T(S) associated to a triangulation , one can read the geometry of (S;m) around each puncturev j ofS as follows: let 12 p j = x k 1j 1 x k nj n where k ij 2 f0; 1; 2g is the number of ends of the edge i that converge to v j . Then if p j = 1, v j is a cusp. Otherwisejlogp j j is the length of the boundary component ofConv(S;m) facingv j . In particular we obtain a natural embedding ofT(S) intoR n + , by setting all thep j equal to 1. The shear parameters are defined along each edge of or equivalently for pairs of adjacent triangles. More generally let ~ be the lift of to the universal cover e S of (S;m). LetP andQ be two ideal triangles in e S delimited by ~ . Let ~ i 1 ;:::; ~ i l , lifts of i 1 ;:::; i l respectively, be the set of edges of ~ separatingP andQ. We include in this set the edges ofP andQ which are closest to each other. The shearing cocycle ofm2T(S) associated to is defined for such triangles by (P;Q) = l X j=1 logx i j wherex i j is the shearing parameter ofr along the edge i j . We refer to [9] for more details. 3.4 The Weil-Petersson Poisson structure The enhanced Teichm¨ uller space can be endowed with the Weil-Petersson Poisson structure which admits a simple expression in the logarithmic shear coordinates asso- ciated to a triangulation ofS (see for example [14]).Sr has 2n spikes converging toward the punctures, each of them being delimited by edges i , j non necessarily distinct. Fori, j2f1;:::;ng, leta ij 2f0; 1; 2g be the number of spikes ofSr which are delimited on the left by i and on the right by j , when looking toward the 13 end of the spikes. The Weil-Petersson Poisson structure is then given in coordinates by the following bi-vector WP = X i;j ij @ @ logx i ^ @ @ logx j where ij =a ij a ji 2f2;1; 0; 1; 2g: It is closely related to the Weil-Petersson symplectic form onT(S) which in turn is associated to a K¨ ahler metric for the natural complex structure on the Teichm¨ uller space. 14 Chapter 4: Pinching along curves: geometric properties We suppose once again that S is a surface of genus g with s 1 punctures v 1 , . . . , v s and such that (S) < 0. Let =f 1 ;:::; k g be a k-simplex in C(S) and S =Sr[ i i =Sr where, by abuse of notation, we identify with the multicurve [ i i . It is homeomorphic to a surface withs + 2k punctures: the “old” onesv 1 , . . . ,v s and two new puncturesv 0 i andv 00 i corresponding to the removal of i fori = 1;:::k. Alternatively one can think ofS as being obtained fromS by pinching the multicurve tok nodes and removing them. Note thatS may be disconnected. The goal of this section is to describe explicitly the behavior of the shear coordi- nates and of the Weil-Petersson Poisson structure when approaching the stratumT(S ) coming fromT(S), in the topology of the augmented Teichm¨ uller spaceT(S). 4.1 Induced ideal triangulations Given an ideal triangulation =f 1 ;:::; n g of S, we want to define an induced ideal triangulation ofS . We choose =[ i i so that the i never cross the same edge twice in a row. Letr denote the family of arcs obtained from the edges of intersected withS . We regroup these arcs into distinct homotopy classes 1 , . . . , l inS , each consisting of a certain number of segments of 1 , . . . , n .f 1 ;:::; l g 15 is then a family of non-intersecting and non-homotopic (homotopy classes of) arcs in S . Lemma 5 l =n and =f 1 ;:::; n g is an ideal triangulation ofS . Proof. Since the i do not backtrack, r decomposes S into pieces of the form given in Figure 4.1, where the dashed lines represent and the first one can have 0, 1, 2 or 3 such sides. The first piece is an ideal triangle in S and the other two are Figure 4.1: Elementary pieces bigons. One can collapse these bigons successively to arcs with one or two vertices beingv 0 i orv 00 j for somei, j. This can be done without changing the homotopy type of S unless we are in one of the situations described in Figure 4.2. This cannot happen since the i are essential and non-homotopic to each other. The remaining arcs Figure 4.2: Disallowed cases correspond to the homotopy classes 1 ;:::; l and decomposeS into ideal triangles. 16 Since(S ) = (S) and the number of edges of an ideal triangulation only depends on the Euler characteristic, the ideal triangulation =f 1 ;:::; l g of S has the same number of edges as. We call the induced triangulation ofS by. Remark 6 In practice each edge i is obtained by considering a maximal sequence of adjacent bigons in the decomposition ofS byr and collapsing it to an edge. Via this process, ideal triangles for onS are identified naturally with ideal triangles for onS. 4.2 Extension of the shear coordinates Let be an ideal triangulation ofS and =f 1 ;:::; n g be the induced triangulation of S . We suppose that, for i = 1;:::;n, the edge i corresponds to the homotopy class ofk ij segments from j forj = 1;:::;n. Proposition 7 Letm t 2T(S),t2 (0; 1] be a continuous family of hyperbolic metrics onS with (x 1 (t);:::;x n (t))2R n + their shear parameters for, andm 2T(S ) with (y 1 ;:::;y n )2R n + its shear parameters for . Then m t ! t!0 m inT(S)) lim t!0 x k i1 1 (t)x k in n (t) =y i fori = 1;:::;n: Proof. We suppose thatm t ! m ast! 0. Let i be an edge of inS , ~ i one of its lifts to the universal cover of (S ;m ). Note that the universal cover of (S ;m ) is isometric to several copies ofH 2 one for each connected component ofS . We denote by f S the one containing ~ i . Then ~ i is the diagonal of a square consisting of two ideal trianglesP i 0 andQ i 0 in f S . 17 By Remark 6, i corresponds to a succession of bigons in the decompositionr ofS ending at the sides of two (non-necessarily distinct) triangles. We consider a lift of this bigon to the universal cover e S t of (S;m t ). It ends at the sides of two triangles P i t andQ i t which are separated byk ij lifts of the edge j forj = 1;:::;n. Hence the shearing cocycle t associated tom t satisfies t (P i t ;Q i t ) = X j k ij logx j (t): In addition, these lifts can be chosen so that, with the right identification of f S with e S t ,P i t andQ i t approachP i 0 andQ i 0 respectively ast! 0. We can then use the following estimate, obtained from Lemma 8 in [9]: letg t and h t be the edges ofP i t andQ i t which are closest to each other. Then, ifa t (resp. b t ) is the projection of the third vertex of P i t (resp. Q i t ) onto g t (resp. h t ), and if b 0 t is the geodesic projection ofb t ontog t , we have t (P i t ;Q i t )d(a t ;b 0 t ) l mt ( ): We refer to Figure 4.3 for an example with notations. SinceP i t andQ i t approachP i 0 andQ i 0 respectively, we havea t !a 0 andb t ;b 0 t !b 0 wherea 0 andb 0 are the projections of the third vertex ofP i 0 andQ i 0 respectively onto g 0 =h 0 = ~ i . Then, sincel mt ( )! 0 ast! 0, lim t!0 X j k ij logx j (t) = lim t!0 t (P i t ;Q i t ) = lim t!0 d(a t ;b 0 t ) =d(a 0 ;b 0 ) = 0 (P i 0 ;Q i 0 ) = logy i : 18 Figure 4.3: Extension of shear parameters We denote byc i the number of times intersects the edge i . In the case where consists of a single curve, andm t 2T(S),t2 (0; 1], is a family of hyperbolic metrics with shear parametersx 1 (t), . . . ,x n (t), we consider the monomial x (t) =x c 1 1 (t)x cn n (t) associated to . Lemma 8 If, ast! 0,l mt ( ) approaches 0, thenx (t) approaches 1. Proof. This is a consequence of the formula for the length of in shear coordinates as described for example in [11]. In particular we have 2 cosh(l mt ( )=2) =x 1=2 (t) +x 1=2 (t) +::: (4.1) where all the terms in the sum are positive. Since the left hand side approaches 2, we necessarily havex (t) approaches 1 as t! 0. 19 It also follows from this that the other terms on the right hand side of 4.1 approach 0 ast! 0. This implies thatx 1=2 +x 1=2 is the leading term in the expression of the cosh of the length functionl : ( ) when approaching the stratumT(S ) inT(S). This explains the essential rˆ ole the quantum analogueX ofx will play later on. 4.3 Extension of the Weil-Petersson Poisson structure Masur [20] (see also Wolpert [31]) proved that the Weil-Petersson metric on T(S) extends in an appropriate sense toT(S) and can be identified to the Weil-Petersson metric on the lower dimensional strataT(S ). We would like to know how this fact together with Proposition 7 translate in terms of the expression of the Weil-Petersson Poisson structure on T(S) and T(S ) in the shear coordinates associated to and respectively. To do so we use a homological interpretation of the Weil-Petersson structure as described for example in [9]. Proposition 10 below can then be interpreted as a topological analogue of the result of Masur. Let = ( ij ) ij be the matrix of coefficients of the Weil-Petersson Poisson structure onT(S) in the coordinates (logx 1 ;:::; logx n ) associated to an ideal triangulation = f 1 ;:::; n g, as was described in Section 3.4. Setting( i ; j ) = ij , extends to an antisymmetric form onH(;Z) =Z n , the free abelian group generated over the set of edges of. Following for example in [9], we can give a homological interpretation of . See also [10] Section 2 for details. LetG be the dual graph of and b G the oriented graph obtained fromG by keeping the same vertex set and replacing each edge ofG by two oriented edges which have the same endpoints as the original edge but with opposite orientations. There is a unique way to thicken b G into a surface b S such that: 1. b S deformation retracts to b G; 20 2. as one goes around a vertexb v of b G in b S, the orientations of the edges of b G adjacent tob v alternatively point toward and away fromb v; 3. the natural projectionp: b G!G extends to a 2-fold branched covering b S!S, branched along the vertex set of b G. Let : b S! b S be the covering involution of the branched coveringp: b S!S. The following was proven in [10] (Lemma 6 and 7). Lemma 9 The groupH(;Z) can be identified with the subgroup ofH 1 ( b S) consisting of thoseb such that (b ) =b . Then if,2H(;Z) correspond tob , b 2H 1 ( b S), we have(;) =b b , their algebraic intersection number. The identification of Lemma 9 is given as follows: ife i 2H(;Z) is the element associating weight 1 to the edge i and weight 0 to the other edges, p 1 (e i ) = b e i is the lift in b G of the edge ofG dual to i . The closed curveb e i comes with a natural orientation given by the one on b G, and we identifyb e i with its homology class inH 1 ( b S). More generally to = P i e i , we can then associate the homology classb = P i b e i . Suppose now that = ( ij ) ij is the matrix of coefficients of the Weil-Petersson Poisson structure onT(S ) for the induced ideal triangulation =f 1 ;:::; n g. We suppose that, fori = 1;:::;n, i corresponds to the homotopy class ofk ij segments from j for j = 1;:::;n. We let k i = (k i1 ;:::;k in )2 Z n and identify with a bilinear form onZ n =H(;Z). The following proposition relates the entries of and . 21 Proposition 10 With the notations above, the coefficients of the Weil-Petersson Pois- son structure onT(S) andT(S ) are related by the following formula: ij =(k i ;k j ) = X u;v k iu k jv uv ; fori;j = 1;:::;n: Proof. LetG be the graph dual to inS,G be the graph dual to inS andG 0 be the graph dual tor inS . We recall thatr denotes the family of arcs obtained from the edges of intersected withS . It decomposesS into triangles and bigons and hence the vertices ofG 0 are either bivalent or trivalent. We will call a maximal chain inG 0 any chain of edges connected via bivalent vertices and with endpoints at trivalent vertices. In particular edges connecting trivalent vertices directly are maximal chains. Following Remark 6, the maximal chains ofG 0 are in on-to-one correspondence with the edges ofG , and this defines a natural homeomorphismG 0 =G . In addition the identification of ideal triangles for and gives an identification ofG andG 0 in a neighborhood of each of their trivalent vertices. We also consider the oriented graphs b G and b G defined as before as well as the oriented graph b G 0 obtained fromG 0 by keeping the same set of trivalent vertices and replacing each maximal chain by two such chains connected to the same (trivalent) endpoints, endowed with opposite orientations. This graph is naturally homeomor- phic to b G . As described above b G thickens into b S, and both b G and b G 0 thicken into b S and we have covering maps p: b S ! S and p : b S ! S which restrict to the corresponding graphs. Recall that, by definition, we haveS = Sr S. Similarly we claim that we can identify b S with b Srb whereb = p 1 ( ). Indeed, letU S S be a union of small discs around the trivalent vertices ofG 0 whereG 0 is identified withG. By 22 construction, we havep 1 (U) =p 1 (U) b S. Outside ofU both coverings are trivial, so we also have a natural identificationp 1 (S rU) =p 1 (S rU) b S. Hence we obtain that b S =p 1 (S ) b S. Note in addition that can be chosen so that it doesn’t pass through any of the ramification points of p (that is the vertices of G). Hence p 1 (S ) = b Srb wherep 1 ( ) =b consists of two non intersecting multicurves and we can identify b S with b Srb sitting in b S. Accordingly, the mapp is identified with the restriction ofp toSr . At the level of homology, the inclusionb : b S ,! b S induces a mapb : H 1 ( b S )! H 1 ( b S) such that, if : b S !S is the covering involution, thenb = b . On the other hand there is a natural map : G 0 !G defined by sending each edge of G 0 onto the edge of G dual to the same edge of the triangulation . It lifts to a mapb : b G 0 ! b G which is also such thatb = b . One can then consider retractionsb r : b S! b G andb r : b S ! b G 0 such thatb b r is equal tob rb , and hence we obtain the following commutative diagram: H 1 ( b S ) b // b r o H 1 ( b S) b r o H 1 ( b G 0 ) b // H 1 ( b G): (4.2) The mapsb r andb r can be naturally chosen so that b r =b r and b r = b r so that, ifb is an element ofH( ;Z) via the identification of Lemma 9, then b b r (b ) =b r b (b ) corresponds to an element ofH(;Z). Lete i be the generator ofH(;Z) assigning weight 1 to i and 0 to the other edges of andf i be the generator ofH( ;Z) assigning weight 1 to i and 0 to the other edges of . As described before, we associate toe i the homology classb e i 2 H 1 (S) corresponding to the lift in b G of the edge ofG dual to i . Similarly, tof i we associate b f i 2H 1 (S ) which corresponds to the lift in b G 0 of the maximal chain ofG 0 dual to i . 23 By construction, this chain consists ofk iu edges dual to u foru = 1;:::;n. Hence, as curves,b ( b f i ) coversk iu timesb e i , positively with the given orientations. Homologically we obtain b ( b f i ) = n X u=1 k iu b e u fori = 1;:::;n: (4.3) Combining the commutative diagram (4.2) and equation (4.3) we obtain the fol- lowing equalities: ij = b f i b f j =b ( b f i )b ( b f i ) =b ( b f i )b ( b f j ) = X u k iu b e u X v k jv b e v = X u;v k iu k jv b e u b e v = X u;v k iu k jv uv : 24 Chapter 5: Pinching along curves: quantum aspect 5.1 The Chekhov-Fock algebra Let = f 1 ;:::; n g be an ideal triangulation of S, and fix a non-zero complex numberq2C . The Chekhov-Fock algebraT q (S) ofS associated to is the algebra generated by the elements X 1 i associated to the edges i of and subject to the relations X i X j =q 2 ij X j X i for every i, j, where the ij 2 f2;1; 0; 1; 2g are the coefficients of the Weil-Petersson Poisson structure on e T(S) in the shear coordinates associated to , as described in Section 3.4. We will sometimes use the notation T q (S) = C [X 1 ;:::;X n ] q to specify the generators of the algebra. If A and B are two monomials in the variables X 1 , . . . , X n , then they satisfy a relation of the typeAB = q 2 BA, for some integer, and we will use the notation (A;B) =. This coefficient is independent of the order of the generators inside each monomial. 25 For k = (k 1 ;:::;k n ) 2 Z n and if A is a monomial consisting of k i times the generator X i for i=1;:::;n, in any given order, we define the following element in T q (S): [A] =X k =q P i<j k i k j ij X k 1 1 :::X kn n : This is known as the Weyl quantum ordering. These monomials satisfy the following relations: X k X l =q (k;l) X k+l =q 2(k;l) X l X k ; where we once again identify with a bilinear form onZ n =H(;Z). The different notations for coincide in the sense that(X k ;X l ) =(k;l). If is a path inG connecting two vertices, and we suppose that does not back- track, we associate to it the element a = (a 1 ;:::;a n ) of H(;Z), where a i is the number of times the path passes through the edge ofG dual to i . Then, to we can associate the elementX =X a defined as above. If is another path inG we have X X =q 2b b X X by Lemma 9, whereb and b are the associated elements inH 1 ( b S). Note that in this caseb = p 1 () is endowed with the orientation induced by b G, and similarly for b . Of particular interest will be the elementX associated to a multicurve inS, which we identify with its retraction to a cycle inG. If is another surface with ideal triangulation, a homomorphism betweenT q (S) andT q () doesn’t in general preserve the quantum orderings. However we have the following elementary lemma which will be useful later on. 26 Lemma 11 Let A 1 , . . . , A r be monomials in T q (S), B 1 , . . . , B r be monomials in T q (). If :T q (S)!T q () is an algebra homomorphism such that ([A i ]) = [B i ] for alli, then ([A 1 A r ]) = [B 1 B r ]. 5.2 A homomorphism between Chekhov-Fock alge- bras Following Propositions 7 and 10 we want to construct a homomorphism between the Chekhov-Fock algebras associated toS andS . We recall that, by Lemma 5, induces an ideal triangulation =f 1 ;:::; n g onS , where i is the homotopy class inS ofk ij segments from j , forj = 1;:::;n. We letk i = (k i1 ;:::;k in ) fori = 1;:::;n. Proposition 12 The map q ; :T q (S ) =C [Y 1 ;:::;Y n ] q !C [X 1 ;:::;X n ] q =T q (S) defined on the generators by q ; (Y i ) =X k i extends to an algebra homomorphism. Proof. We check it on the generators ofT q (S ): q ; (Y i Y j ) = q ; (Y i ) q ; (Y j ) =X k i X k j =q 2(k i ;k j ) X k j X k i =q 2 ij q ; (Y j Y i ): 27 where the last equality is given by Proposition 10. The following lemma states that if we pinch the curves constituting in different orders, the resulting homomorphisms given by Proposition 12 are the same. Lemma 13 Consider any sequence of integersi 1 , . . . ,i k such that fi 1 ;:::;i k g =f1;:::;kg and let l =[ l j=1 i j for anylk. Then q ; = q i k ; k1 q i k1 ; k2 q i 1 ; : Proof. The definition of the edges i of as homotopy classes of segments from 1 , . . . , n does not depend on the order in which the curves 1 , . . . , k are pinched. Hence the generatorsY i ofT q (S ) are sent to the same monomials by the two maps, up to ordering. Since all the maps considered send generators to quantum ordered monomials, Lemma 11 implies that the quantum orders are respected on each side and hence the maps coincide. For future reference we want to interpret q ; in terms of dual graphs. As in the proof of Proposition 10 we letG be the graph dual to inS andG 0 the graph dual to r inS . There is a natural map : G 0 !G defined by sending each edge ofG 0 to the edge ofG dual to the same edge of. If is any path inG = G 0 , we denote byY the quantum ordered product of generators associated to the edges crossed by, and we define similarlyX for any path inG. Then, by definition of q ; , we have q ; (Y ) =X () : 28 5.3 Application to the representation theory of Chekhov-Fock algebras 5.3.1 Representations of the Chekhov-Fock algebra The irreducible finite dimensional representations ofT q (S) forq a root of unity have been studied in details in [10]. An important step is to describe the center of these algebras. Consider the elements P i =X p i = [X p i1 1 X p i2 2 X p in n ] ofT q (S) associated to the puncturev i ofS, wherep ij 2f0; 1; 2g is the number of ends of the edge j that converge tov i andp i = (p i1 ;:::;p in ). NamelyP i is the product of generators associated to the edges ending atv i , with the quantum ordering. If i is a small loop aroundv i and we identify it with its retraction to a cycle in the dual graph G, we also haveP i =X i with the notations introduced previously. In addition, leth = (1;:::; 1) and H =X h = [X 1 X n ]: The following proposition describes the center of the Chekhov-Fock algebras for cer- tain values ofq. Proposition 14 (Bonahon-Liu) If q 2 is a primitive N-th root of unity with N odd, the centerZ q ofT q (S) is generated by the elementsX N i fori = 1;:::;n, theP j for j = 1;:::;s andH. 29 A complete classification of the finite dimensional irreducible representations of T q (S), where q 2 is a root of unity was obtained in [10]. A particular case of this classifications is given by the following theorem. Theorem 15 (Bonahon-Liu) Suppose thatq is a primitiveN-th root of unity withN odd. Every irreducible finite dimensional representation : T q (S) ! End(V ) has dimension N 3g+p3 and is determined completely by its restriction to the centerZ q ofT q (S). In particular, givenp j 2f0;:::;N 1g integers labeling each of the punctures v j of S and m2 T(S) an hyperbolic metric with finite area and shear parameters x 1 ;:::;x n associated to , there is a finite dimensional irreducible representation :T q (S)! End(V ) such that: (X N i ) =x i Id V fori = 1;:::;n; (P j ) =q p j Id V forj = 1;:::;s. These conditions determine uniquely up to isomorphism. Proof. The first part is given by Theorem 20 in [10]. The second part is a specialization of Theorem 21 to the case where thex i are positive real numbers andx p j1 x p jn = 1 for j = 1;:::;s, corresponding to the shear parameters of a complete hyperbolic metric onS. In this case(H) is completely determined by the fact that(H N ) = Id V andH 2 =P 1 P s . We call p = (p 1 ;:::;p s ) the labeling of the puncturesv 1 ,. . . ,v s associated to the representation. 5.3.2 Convergence of representations in the augmented Teichm¨ uller space From now on we suppose thatq is a primitiveN-th root of unity and thatN is odd. 30 Theorem 15 implies that to p = (p 1 ;:::;p s )2f0;:::;N 1g s and a continu- ous family of hyperbolic metricsfm t g t T(S) one can associate a continuous fam- ily of irreducible representations p t : T q (S) ! End(V ) as follows: let x 1 (t), . . . , x n (t) be the shearing parameters associated to m t for the triangulation and let 1 be an irreducible representation classified by m 1 and weights p 1 , . . . , p n . Then, by Theorem 15, there are elements A 1 , . . . , A n of End(V ) such that A N i = Id V and p 1 (X i ) = N p x i (1)A i . One can then define a family of representations p t defined on the generators by p t (X i ) = N p x i (t)A i . In this way we obtain a family of rep- resentationsf p t g t classified byfm t g t and labeling (p 1 ;:::;p s ). It is continuous in the sense that for any element X 2 T q (S),f p t (X)g t is a continuous family in End(V ) =C N (3g+p3) 2 . By composing with the homomorphism q ; from Proposition 12, any represen- tation ofT q (S) gives a representation q ; ofT q (S ). If p is an irreducible representation classified by m2 T(S) and labeling p, then p q ; is a reducible representation. We would like to know how this representation decomposes into irre- ducible subrepresentations whenm approachesm 2T(S ). We recall that the punctures of S are v 1 , . . . , v s corresponding to the same punctures in S together with the new punctures v 0 1 , v 00 1 , . . . , v 0 k , v 00 k corresponding to the removal of the curves 1 , . . . , k . We say that p 2 f0;:::;N 1g s+2k is a compatible labeling of (v 1 ;:::;v s ;v 0 1 ;v 00 1 ;:::;v 0 k ;v 00 k ) if it is of the form p = (p 1 ;:::;p s ;i 1 ;i 1 ;:::;i k ;i k ). Theorem 16 Letm t 2T(S),t2 (0; 1], be a continuous family of hyperbolic metrics such thatm t converges tom 2 T(S ) inT(S) ast approaches 0. Let t :T q (S)! End(V ),t2 (0; 1] be a continuous family of irreducible representations classified by m t and a labeling (p 1 ;:::;p s )2f0;:::;N 1g s of the puncturesv 1 , . . . ,v s ofS. 31 Then, ast approaches 0, the representation t q ; :T q (S )! End(V ) approaches M p p :T q (S )! End( M p V p ) where the direct sum is over all possible compatible labelings p on S and p : T q (S )! End(V p ) is an irreducible representation classified by the metric m and the labelingp . Proof. We first suppose thatk = 1, that is consists of a single curve. Let P 1 , . . . , P s be the central elements ofT q (S) associated to the punctures v 1 , . . . ,v s ofS andP 1 , . . . ,P s be the central elements ofT q (S ) associated to the same punctures inS . We also have the two central elementsP 0 andP 00 associated to the two puncturesv 0 andv 00 . Let also =X 2T q (S) be the monomial associated to the retraction of to a cycle inG. In practice if crossesc i times the edge i of for i = 1;:::;n we have = [X c 1 1 X cn n ]: Lemma 17 The elements defined above satisfy: 1. q ; (P i ) =P i fori = 1;:::;s; 2. q ; (P 0 ) = q ; (P 00 ) = . Proof. Note first that q ; respects the quantum ordering. 32 For (1) we let i be a small curve going aroundv i once inS S. If we identify i with its retraction to a cycle inG 0 we have thatP i =Y i . Since( i ) corresponds to the retraction of i ontoG we see that q ; (P i ) =X ( i ) =P i : For (2), let 0 be a curve parallel to such that 0 is homotopic tov 0 inS . Then 0 retracts to a cycle inG andP 0 = Y . In addition( 0 ) corresponds to the retraction of 0 , and hence of , ontoG, so q ; (P 0 ) =X ( 0 ) =X = : The same argument holds forP 00 if one consider a curve 00 parallel to and homotopic tov 00 inS . Lemma 18 There exists 2T q (S) such that =q 4 . Proof. We recall that, by Lemma 9,H(;Z) is identified with the subgroupf[b ]2 H 1 ( b S)j ([b ]) =[b ]g ofH 1 ( b S). To prove the lemma it suffices to find a path inG such thatb b =2 (sinceX X =q 2b b X X ) and take =X 1 . If is non separating, and since is an essential simple closed curve, there exists another simple closed curve inS which interects exactly once. Then b =p 1 () is such thatb b = 2[ ] [] =2. If is separating then it divides the set of vertices V of G into two non-empty subsets. Let be an arc with endpoints in V on each side of and intersecting exactly once. Then b = p 1 () is a simple closed curve in b S and by construction b b =2. 33 Lemma 18 implies that under the action of t (), V decomposes into eigenspaces V i of dimension N 3g+p4 with associated eigenvalues c(t)q i , where i2 f0;:::;N 1g andc(t)2C . In addition t ( N ) = t ((X N 1 ) c 1 (X N n ) cn ) =x c 1 1 (t)x cn n (t)Id V =x (t)Id V wherex (t) is the combinatorial length of . Hence, after a shift by some N-th root of 1, we can consider thatc(t) = N p x (t)2R + . P 0 and P 00 are central in T q (S ), so the eigenspaces of t q ; (P 0 ) = t q ; (P 00 ) = t () are invariant under the action of t q ; . In other words t q ; = M i i t; ; where i t; :T q (S )! End(V i ) is such that i t; (P 0 ) = i t; (P 00 ) =c(t)q i Id V i : For dimensional reasons these representations are irreducible by Theorem 15. By Lemma 17, we have that t q ; (P j ) = t (P j ) =q p j Id V , hence i t; (P j ) =q p j Id V i ; and by definition of q ; , t q ; (Y N j ) = t ((X N 1 ) k 1 (X N n ) kn ) =x k j1 1 (t)x k jn n (t)Id V ; hence i t; (Y N j ) =x k j1 1 (t)x k jn n (t)Id V i 8i: 34 Then by Proposition 7, ast! 0,x k j1 1 (t)x k jn n (t)!y j , the shear parameters of m , and by Lemma 8,we havec (t)! 1. This implies that, ast! 0, i t; approaches the irreducible representation i ofT q (S ) which is classified by the weightsp 1 , . . . , p s associated to the puncturesv 1 , . . . ,v s , the weighti associated tov 0 andv 00 , and the hyperbolic metricm 2T(S ). Using Lemma 13, the case of a multicurve =f 1 ;:::; n g follows by induction onk. 35 Chapter 6: Behavior under changes of coordinates: the quantum Teichm¨ uller space 6.1 The quantum Teichm¨ uller space We want to apply the results of the preceding section to the representations of the quan- tum Teichm¨ uller spaceT q (S). Let us first recall its construction as given in [19]. If is an ideal triangulation ofS, we denote by b T q (S) the fraction division algebra of the Chekhov-Fock algebraT q (S). Chekhov and Fock constructed in [11] a family of iso- morphisms q 0 : b T q 0 (S)! b T q (S), called coordinate change isomorphisms, defined for any two triangulations, 0 ofS. In particular, if 00 is another triangulation, they satisfy the composition relation q 00 = q 0 q 0 00 . The main example is given by the case where and 0 differ by a diagonal exchange in an embedded square inS as in Figure 6.1. Then q 0 (X 0 n ) =X n ifn6= 1;:::; 5 and q 0 (X 0 0 ) =X 1 0 ; q 0 (X 0 1 ) = (1 +qX 0 )X 1 ; q 0 (X 0 2 ) = (1 +qX 1 0 ) 1 X 2 ; q 0 (X 0 3 ) = (1 +qX 0 )X 3 ; q 0 (X 0 4 ) = (1 +qX 1 0 ) 1 X 4 : 36 Figure 6.1: Diagonal exchange We refer to [19] or [10] for similar formulas when some of the edges of the square are identified. Using the fact that one can get from any triangulation to another triangulation 0 by a succession of diagonal exchanges and reindexings (see [25]), one can then construct q 0 , for any triangulations, 0 by composition. Following [19], we define the quantum Teichm¨ uller spaceT q (S) as the quotient T q (S) = G b T q (S)= where the disjoint union is over all ideal triangulations of S, and the equivalence relation identifiesX 0 2 b T q 0 (S) to q 0 (X 0 )2 b T q (S). 6.2 Representations of the quantum Teichm¨ uller space A first attempt at defining a representation ofT q (S) would be to consider a family of representations of b T q (S) for every triangulation ofS such that q 0 = 0 for every , 0 . However one can easily check that such representations cannot be finite dimensional. On the other hand the Chekhov-Fock algebrasT q (S) admit many 37 finite dimensional representations. Hence, for our purpose, a representation of the quantum Teichm¨ uller spaceT q (S) will be a family of representations ofT q (S), for every triangulation of S satisfying certain compatibility relations when changing triangulations. More precisely (see [10] for details), if :T q (S)! End(V ) is an algebra homo- morphism, we say that the algebra homomorphism q 0 :T q 0 (S)! End(V ) makes sense if, for every Laurent polynomial X 0 2 T q 0 (S), the rational fraction q 0 (X 0 )2T q 0 (S) can be written as quotients q 0 (X 0 ) =P 1 Q 1 1 =Q 1 2 P 2 of Laurent polynomials P 1 , Q 1 , P 2 , Q 2 2 T q (S) such that (Q 1 ) and (Q 2 ) are invertible in End(V ). We then define q 0 (X 0 ) = (P 1 ) (Q 1 ) 1 = (Q 2 ) 1 (P 2 )2 End(V ): One can check that this definition doesn’t depend on the decomposition of q 0 (X 0 ) as a quotient of polynomials, and that this defines an algebra homomorphism q 0 : T q 0 (S)! End(V ). Definition 19 A representation =f g of the quantum Teichm¨ uller spaceT q (S) over the vector space V consists of the data of an algebra homomorphism : T q (S)! End(V ) for every triangulation in such a way that, for every , 0 , the representation q 0 :T q 0 (S)! End(V ) makes sense and is equal to 0. 38 We will sometimes use the notation : T q (S) ! End(V ) for such a repre- sentation, keeping in mind that consists in fact of a family of homomorphisms f :T q (S)! End(V )g . Such representations were called representations of the polynomial core ofT q (S) in [10]. To prove that a family of representationsf g of the Chekhov-Fock algebras is in fact a representation ofT q (S), one can use the following lemma (Lemma 25 in [10]). Lemma 20 Let an algebra homomorphism :T q (S)! End(V ) be given for every ideal triangulation. Suppose that q 0 :T q 0 (S)! End(V ) makes sense and is equal to 0 whenever and 0 differ by a diagonal exchange or a re-indexing. Then =f g is a representation ofT q (S). Given an irreducible representation :T q (S)! End(V ) for some ideal triangu- lation, It is shown in [10] that the composition q 0 :T q (S)! End(V ) makes sense for any other ideal triangulation 0 and defines an irreducible representation 0 ofT q 0 (S). We will say that the family =f g is an irreducible representation of T q (S). In addition, if is classified by the labeling (p 1 ;:::;p n ) and the metricm2T(S) expressed in the shear coordinates for, then the representation 0 is also classified by the same labeling and the metricm expressed in the shear coordinates for 0 . In this case, we say that =f g is the irreducible representation ofT q (S) classified bym and (p 1 ;:::;p n ). note that this definition depends on the choice of the original trian- gulation. However another choice of starting triangulation would give an isomorphic representation. If m t 2 T(S) is a continuous family of hyperbolic metrics for t and t; is a continuous family of representations ofT q (S) classified bym t and weightsp 1 , . . . ,p n , then we will say that t =f t; g is a continuous family of representations ofT q (S). 39 6.3 Main theorem The next step is to study how the decomposition obtained in Theorem 16 is affected by changing the triangulation. Given two triangulations , 0 , we can consider the following diagram: b T q 0 (S) q 0 //b T q (S) b T q 0 (S ) q ; 0 OO q 0 //b T q (S ): q ; OO (6.1) This diagram is in general non-commutative. We focus on the case where and 0 differ by a diagonal exchange in a squareQ as in Figure 6.1. If the curve never crossesQ vertically or horizontally, that is never crosses suc- cessively 1 , 0 , 3 or 2 , 0 , 4 , the triangulations and 0 also differ by a diagonal exchange and can be identified outside of a squareQ (cf Figure 6.2, the doted curves represent ) If does crossQ vertically or horizontally at leas once, the triangulations and 0 can be identified (cf Figure 6.3). In this case we introduce two maps q 0 ;v and q 0 ;h , defined for each case of iden- tifications on the boundary ofQ as follows: in each case q 0 ;v (X i ) = q 0 ;h (X i ) = X i fori 5, q 0 ;v (X 0 ) = q 0 ;h (X 0 ) =X 1 0 and ifQ is embedded, q 0 ;v (X 0 1 ) =qX 0 X 1 ; q 0 ;v (X 0 2 ) =X 2 q 0 ;v (X 0 3 ) =qX 0 X 3 ; q 0 ;v (X 0 4 ) =X 4 40 Figure 6.2: A diagonal exchange in a non-crossed square and q 0 ;h (X 0 1 ) =X 1 ; q 0 ;h (X 0 2 ) =q 1 X 0 X 2 q 0 ;h (X 0 3 ) =X 3 ; q 0 ;h (X 0 4 ) =q 1 X 0 X 4 ; if 1 = 3 and 2 6= 4 , q 0 ;v (X 0 1 ) =q 4 X 2 0 X 1 ; q 0 ;v (X 0 2 ) =X 2 ; q 0 ;v (X 0 4 ) =X 4 41 Figure 6.3: A diagonal exchange in a crossed square and q 0 ;h (X 0 1 ) =X 1 ; q 0 ;h (X 0 2 ) =q 1 X 0 X 2 ; q 0 ;h (X 0 4 ) =q 1 X 0 X 4 ; if 1 = 2 and 3 6= 4 , q 0 ;v (X 0 1 ) =X 0 X 1 ; q 0 ;v (X 0 3 ) =qX 0 X 3 ; q 0 ;v (X 0 4 ) =X 4 42 and q 0 ;h (X 0 1 ) =X 1 ; q 0 ;h (X 0 3 ) =X 3 ; q 0 ;h (X 0 4 ) =q 1 X 0 X 4 ; if 1 = 3 and 2 = 4 , that isS is a once punctured torus, q 0 ;v (X 0 1 ) =q 4 X 2 0 X 1 ; q 0 ;v (X 0 2 ) =X 2 and q 0 ;h (X 0 1 ) =X 1 ; q 0 ;h (X 0 2 ) =q 4 X 2 0 X 2 : The other cases are inverses of the ones above. Note that we exclude the case when 1 = 2 and 3 = 4 corresponding to a sphere with three holes since there are no essential simple closed curves onS in this case. One can easily check that q 0 ;v and q 0 ;h are algebra homomorphisms. Proposition 21 If and 0 differ by a diagonal exchange in a squareQ as in figure 6.1 then either one of the following situations occurs 1. the multicurve doesn’t crossQ horizontally or vertically and q 0 q ; 0 = q ; q 0 ; 43 2. crossesQ vertically at least once and q 0 ;v q ; 0 = q ; ; 3. crossesQ horizontally at least once and q 0 ;h q ; 0 = q ; : Proof. We use the notations of Figures 6.2 and 6.3. The following lemma is a simple computation. Lemma 22 Let A i , i = 1; 2; 3; 4 monomials inT q (S) be the products of generators associated to the edges of converging to the cornerc i ofQ, A 5 be the products of generators when one crosses Q vertically, A 6 the product of generators when one crosses Q horizontally. Define similarly B 1 , . . . , B 6 monomials in T q 0 (S) for the triangulation 0 . Then fori;j = 1;:::; 4 q 0 ([B i ]) = q 0 ;v ([B i ]) = q 0 ;h ([B i ]) = [A i ] and q 0 ;v ([B 5 ]) = [A 5 ] and q 0 ;h ([B 6 ]) = [A 6 ]: Lemma 22 says that q 0 , q 0 ;v and q 0 ;h send the products of generators around a corner of Q for 0 to the respective products for , respecting the quantum order- ings. In addition q 0 ;v and q 0 ;h respect the quantum ordered products of generators crossingQ vertically and horizontally respectively. We first consider the case where bothQ andQ are embedded inS andS respec- tively and we look at the generatorsY 1 andY 0 1 associated to the edges 1 and 0 1 . 44 For (1), we notice using Remark 6 that 1 (resp 0 1 ) corresponds to a rectangle for r which starts along 1 (resp 0 1 ), and then crossesl i timesQ around the corner c i for i = 1; 2; 3; 4. It also crosses k i times i (resp 0 i ) for i = 5;:::;n. We let Z =X k 5 5 X kn n andZ 0 =X 0 k 5 5 X 0 kn n . Then q ; (Y 1 ) = X 1 A l 1 1 A l 2 2 A l 3 3 A l 4 4 Z =q X 1 A l 1 1 A l 2 2 A l 3 3 A l 4 4 Z where is some integer. We note that(X 1 ;A i ) =(X 0 1 ;B i ) fori = 1; 2; 3; 4 hence q ; (Y 0 1 ) = X 0 1 B l 1 1 B l 2 2 B l 3 3 B l 4 4 Z 0 =q X 0 1 B l 1 1 B l 2 2 B l 3 3 B l 4 4 Z 0 with the same coefficientq in front. We also note that 0 corresponds to the edge 0 and hence q ; (Y 0 ) =X 0 . Using Lemma 22 together with Lemma 11 we have q 0 q ; 0 (Y 0 1 ) = q 0 (q X 0 1 B l 1 1 B l 2 2 B l 3 3 B l 4 4 Z 0 ) =q (1 +qX 0 )X 1 A l 1 1 A l 2 2 A l 3 3 A l 4 4 Z = q ; ((1 +qY 0 )Y 1 ) = q ; q 0 (Y 0 1 ): A similar computation works for Y 0 2 , Y 0 3 and Y 0 4 . For i > 4, i corresponds to a rectangle with neither side ending alongQ but which may still cross it at the corners, and one shows in the same way that the equality holds forY 0 5 , . . . , Y 0 n . Hence (1) is true in the case of embedded squares. For (2) 1 corresponds to a rectangle which starts along 1 then crosses l i times the square Q around the corner c i for i = 1; 2; 3; 4, and may also cross Q vertically l 5 times. 0 1 corresponds to a rectangle which starts along 0 0 , crosses 1 , then crosses 45 Q in the same way as for 1 . Each also crosses k i times i or 0 i respectively, for i = 5;:::;n. Then q ; (Y 1 ) = X 1 A l 1 1 A l 2 2 A l 3 3 A l 4 4 A l 5 5 Z =q X 1 A l 1 1 A l 2 2 A l 3 3 A l 4 4 A l 5 5 Z where is some integer. We note that(X 1 ;A i ) = (X 0 0 X 0 1 ;B i ) fori = 1; 2; 3; 4; 5, hence q ; (Y 0 1 ) = X 0 0 X 0 1 B l 1 1 B l 2 2 B l 3 3 B l 4 4 B l 5 5 Z 0 =q [X 0 0 X 0 1 ] B l 1 1 B l 2 2 B l 3 3 B l 4 4 B l 5 5 Z 0 =q 1 X 0 0 X 0 1 B l 1 1 B l 2 2 B l 3 3 B l 4 4 B l 5 5 Z 0 : Using the definition of q 0 ;v in case 1 together with Lemma 22 and Lemma 11 we obtain q 0 ;v q ; 0 (Y 0 1 ) = q 0 ;v (q 1 X 0 0 X 0 1 B l 1 1 B l 2 2 B l 3 3 B l 4 4 B l 5 5 Z 0 ) =q 1 X 1 0 qX 0 X 1 A l 1 1 A l 2 2 A l 3 3 A l 4 4 A l 5 5 Z =q X 1 A l 1 1 A l 2 2 A l 3 3 A l 4 4 A l 5 5 Z = q ; (Y 1 ): A similar argument works for the other generatorsY 0 2 , . . . ,Y 0 n . The case (3) is similar to (2), where now 1 corresponds to a rectangle which doesn’t crossQ vertically but crosses it horizontallyl 6 times. The arguments are the same replacingB 5 andl 5 withB 6 andl 6 . If some of the edges of Q or Q are identified, the same arguments work using the corresponding formulas for q 0 , q 0 ;v and q 0 ;h . One also needs to take into account that in this case an edge i of may correspond to a rectangle in Sr 46 with both ends along edges inQ. The formulae for q ; (Y i ) and q ; (Y 0 i ) have to be changed accordingly. We would like to interpret the maps q 0 ;v and q 0 ;h as the limit of the coordinate change q 0 when the length of approaches 0, depending on whether crosses Q vertically or horizontally. Lemma 23 Let t; ,t2 (0; 1], be a continuous family of irreducible representations of T q (S) classified by a continuous familym t 2T(S) such that, ast! 0,m t approaches m 2T(S ) inT(S). Then, if 0 differs from by a diagonal exchange as in Figure 6.1 and crossesQ vertically (i=v) or horizontally (i=h), we have t; q 0 t; q 0 ;i ast! 0: Proof. We suppose that (x 1 (t);:::;x n (t)) are the shear parameters associated tom t , t2 (0; 1] for the triangulation. Then, as in section 5.3.2, there are matricesA 1 , . . . , A n such that t; (X i ) = N p x i (t)A i for everyt. For simplicity we consider the case whenQ is embedded inS. The computations are similar in the non-embedded cases. If crossesQ vertically thenx 0 (t)!1 as t! 0 and, considering for example the generatorX 1 , we have t; q 0 (X 1 ) = (1 +q N p x 0 (t)A 0 ) N p x 1 (t)A 1 q N p x 0 (t) N p x 1 (t)A 0 A 1 = t; q 0 ;v (X 1 ); and similarly for the other generators. 47 If crossesQ horizontally thenx 0 (t)! 0, and considering for example the gen- eratorX 2 , we have t; q 0 (X 1 ) = (1 +q( N p x 0 (t)A 0 ) 1 ) 1 N p x 1 (t)A 1 q 1 N p x 0 (t) N p x 1 (t)A 0 A 1 = t; q 0 ;h (X 1 ); and similarly for the other generators. Proposition 24 Suppose that t; : T q (S)! End(V ) and t; 0 : T q 0 (S)! End(V ) are two continuous families of irreducible representations classified by the same label- ing and by a continuous family m t 2 T(S), t 2 (0; 1) such that, as t ! 0, m t approachesm 2T(S ) inT(S). Using the notations of Theorem 16 we have lim t!0 t; q ; = M p p ; and lim t!0 t; 0 q ; 0 = M p p ; 0 : Suppose in addition that for allt we have t; q 0 = t; 0. Then p ; 0 = p ; q 0 for all compatiblep : Proof. We show the result for the case when = 1 is a simple closed curve. The general case follows by induction onk using Lemma 13. By Lemma 20, it suffices to show the result for a diagonal exchange in a squareQ as in Figure 6.1. We letP be the quantum ordered product of generators associated to the edges of ending at one of the new punctures ofS . If 0 is different from , we define similarly P 0 . Following the proof of Theorem 16 we have that t; q ; = L i i t; where i t; are representations ofT q (S ) onto V i the eigenspace of t; q ; (P ) with eigenvalueq i . Then, by definition, i ; is the limit ast approaches 0 of i t; . 48 We have t; q ; q 0 ! ( M i i ; ) q 0 and, by hypothesis, t; q 0 q ; 0 = t; 0 q ; 0 ! M i i ; 0: If doesn’t cross Q vertically or horizontally then, by Proposition 21, q 0 q ; 0 = q ; q 0 . Composing on both sides by t; on the left and taking the limit ast approaches 0 we get L i i ; 0 = ( L i i ; ) q 0 . Note that q 0 (P 0 ) =P so q 0 sends the eigenspaces ofP onto those ofP with the same eigenvalues. Hence i ; 0 = i ; q 0 for everyi. If crossesQ say vertically, then q 0 is the identity. By Proposition 21 q 0 ;1 q ; 0 = q ; and, by Lemma 23, t; q 0 t; q 0 ;1 . Composing on both sides by q ; 0 on the right and taking the limit as t approaches 0, we get that L i i ; 0 = L i i ; . The decompositionV = L V i is the same on each side, hence i ; 0 = i ; for alli in this case. Proposition 24 implies that p ; differs from p ; 0 only if is different from 0 . Hence we can rename these representations p ; . Lemma 25 Any ideal triangulation ofS is induced by some triangulation ofS. Proof. Let be a given triangulation ofS and be the induced triangulation ofS . It suffices to show that for any triangulation obtained from by an exchange of diagonal, there exists 0 a triangulation ofS such that = 0 . Suppose i is the diagonal of some square for onS . Then i corresponds to the homotopy of segments of i 1 , . . . , im , in order when going from one triangle forming the square to the other (see Figure 6.4). 49 Figure 6.4: Two non-adjacent triangles We claim that ifm 2, a diagonal exchange along im doesn’t change the induced triangulation. Indeed im is then necessarily the diagonal of a square which is crossed vertically or horizontally at least once by . We denote by (m) the triangulation obtained from by the diagonal exchange along im . Then, in addition, the edge i of (m) = corresponds to the homotopy class of segments of (m) i 1 , . . . , (m) i m1 (See Figure 6.5 for a typical case). Figure 6.5: Final step By induction, we obtain a triangulation (1) such that (1) = and 1 corresponds to a segment from exactly one edge (1) i 1 of (1) . This edge is then contained in a square which is not crossed by except possibly at the corner, and doing a diagonal exchange along (1) i 1 induces the desired diagonal exchange along. Combining Proposition 24 and Lemma 25 we obtain the following theorem. 50 Theorem 26 Let t =f t; g be a continuous family of irreducible representations of T q (S) classified by the labeling (p 1 ;:::;p s ) and a continuous family of metrics m t 2T(S) such thatm t approachesm 2T(S ) ast! 0. For each triangulation, we let the limit lim t!0 t; q ; = M p p ; be given as in Theorem 16. Then, for every compatible labelings p on S , the family of representations f p ; g 2(S) defines an irreducible representation p ofT q (S ) classified by the met- ricm and the labelingp . 51 References [1] William Abikoff, Degenerating families of Riemann surfaces, Ann. of Math. (2) 105 (1977), no. 1, 29–44. [2] William Abikoff, Augmented Teichm¨ uller spaces, Bull. Amer. Math. Soc. 82 (1976), no. 2, 333–334. 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Abstract (if available)

Abstract For a punctured surface, a point of its Teichmuller space determines a representation of its quantized Teichmuller space. We analyze the behavior of these representations as one goes to infinity in the Teichmuller space, or in the moduli space of the surface. The result we obtain is analogous to the factorization rule found in conformal field theory.

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Creator Roger, Julien (author)

Core Title Factorization rules in quantum Teichmuller theory

Contributor Electronically uploaded by the author (provenance)

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Mathematics

Publication Date 08/05/2010

Defense Date 05/07/2010

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag hyperbolic geometry,OAI-PMH Harvest,quantum topology

Language English

Advisor Bonahon, Francis (committee chair), Honda, Ko (committee member), Pilch, Krzysztof (committee member)

Creator Email jroger@usc.edu,jujuroger@yahoo.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m3284

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