Hyperbolic 3-manifolds and Cluster Algebras
aa r X i v : . [ m a t h . G T ] S e p SUBMIT
Hyperbolic -manifolds and Cluster Algebras KENTARO NAGAO, YUJI TERASHIMA andMASAHITO YAMAZAKI
Abstract.
We advocate the use of cluster algebras and their y -variables in thestudy of hyperbolic 3-manifolds. We study hyperbolic structures on the map-ping tori of pseudo-Anosov mapping classes of punctured surfaces, and showthat cluster y -variables naturally give the solutions of the edge-gluing condi-tions of ideal tetrahedra. We also comment on the completeness of hyperbolicstructures. §
0. Introduction0.1. Cluster Algebras
Cluster algebras were introduced by Fomin and Zelevinsky ([FZ02])around 2000. Since then, many authors have uncovered beautiful connec-tions between the theory of cluster algebras and a wide range of mathematicssuch as • dual canonical bases and their relations with preprojective algebrasand quiver varieties ([BFZ05], [Lec10], [Nak11a], [Kim12]) • total positivity ([Fom10]) • (higher) Teichm¨uller theory and its quantization ([FG06, FG07, FG09],[Tes07, Tes11]) • • cluster categories ([Kel10], [Ami09], [Pla11]) • discrete integrable systems ([Ked08], [KNS11]) • Donaldson-Thomas theory ([KS], [Nag13]) • supersymmetric gauge theories ([GMN13], [CNV], [EF12]) KENTARO NAGAO, YUJI TERASHIMA AND MASAHITO YAMAZAKI
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The goal of this paper is to add yet another item to this list: the theoryof hyperbolic -manifolds . This paper is a companion to [TY14], whichdiscusses the application of cluster algebras to the physics of 3d N = 2supersymmetric gauge theories. -manifolds A hyperbolic 3-manifold (with cusps) has a decomposition into idealtetrahedra. This makes it possible for us to compute invariants of the 3-manifold, such as the hyperbolic volume and the Chern-Simons invariant[NZ85, Neu92].An ideal tetrahedron is parametrized by a complex number called a shape parameter . Given a topological decomposition of the 3-manifold intoideal tetrahedra, we need to find shape parameters which satisfy edge-gluingequations ( § cusp equations ( § M ϕ of mapping classes ϕ of asurface Σ with punctures. We mainly discuss the case that the mappingtorus admits a hyperbolic structure.The main results of this paper are summarized as follows: • Solving the periodicity equation in Theorem 4.4 for cluster transfor-mations, we get a solution of the edge-gluing equations of the mappingtorus M ϕ with an ideal triangulation induced by the cluster transfor-mations. • Shape parameters of tetrahedra are given by the cluster y -variables,where the initial values of the y -variables are taken to be the solutionof the periodicity equation. • The cusp condition is written as a simple condition on a product ofthe initial values of the y -variables. Remark
Remark yperbolic 3-manifolds and Cluster Algebras SUBMIT
In [TY11], the authors conjectured an equivalence of the partition func-tion of a 3 d N = 2 gauge theory on a duality wall and that of the SL(2 , R )Chern-Simons theory on a mapping torus. This is a 3 d/ d counterpart ofthe 4 d/ d correspondence, known as the AGT relation ([AGT10]).In [TY13], the authors demonstrated that a limit of the 3d N = 2 par-tition function reproduces the hyperbolic volume of the mapping torus inthe case of the once-punctured torus by using quantum cluster transfor-mations. The key observation in [TY13] was that the shape parameterssatisfying edge-gluing equations (as previously analyzed in [Gu´e06]) appearat the saddle point.In [KN11], it was shown that classical dilogarithm identities ([Nak11c])naturally emerge from quantum dilogarithm identities ([Kel11], [Nag11]) bythe saddle-point method.It will be interesting to learn from physics about the “quantum” aspectsof hyperbolic geometry of 3-manifolds. Remark • We have periodicity conditions on the cluster y -variables both in (a)and in (b). However, in (a), periodicity is imposed as identities ofrational functions on y i ’s, whereas in (b) we solve the periodicityequations to determine values of y i , which in turn determines thehyperbolic structure of the mapping tori. • In (a), the product of the quantum dilogarithms associated to thesequence of mutations is equal to 1 (quantum dilogarithm identity[Kel11]). In (b), the product gives a non-trivial action of the mappingclass in the quantum Teichm¨uller theory. • In terms of surface triangulations and flips, after a sequence of flips,in (a) we get the original triangulation (up to a permutation of ver-
KENTARO NAGAO, YUJI TERASHIMA AND MASAHITO YAMAZAKI
SUBMIT tices), while in (b) we get the original triangulation pulled back bythe mapping class. • A mutation provides a derived equivalence of 3-dimensional Calabi-Yau categories associated to quivers with potential ([KY11]). In (a),the composition of the derived equivalences is an identity functor,while in (b) it gives the action of the mapping class on the derivedcategory (see [Nag]). • The derived equivalence induced by mutation corresponds to a wall-crossing in the space of stability conditions, and a sequence of muta-tions gives a new chamber. In (a), the new chamber coincides with theoriginal one, while in (b) the chamber is obtained from the original oneby the action of the mapping class on the space of stability conditions.In other words, the former is the wall crossing associated with a con-tractible cycle, whereas the latter corresponds to a non-contractiblecycle with non-trivial monodromies ( cf. [ADJM12]).
Acknowledgments
K. N. is supported by the Grant-in-Aid for Research Activity Start-up(No. 22840023) and for Scientific Research (S) (No. 22224001). Y. T. issupported in part by the Grants-in-Aid for Scientific Research, JSPS (No22740036). M. Y. would like to thank PCTS and its anonymous donor forgenerous support. We would like to thank H. Fuji, A. Kato, S. Kojima,H. Masai, T. Nakanishi, S. Terashima, T. Yoshida and D. Xie for helpfulconversation. §
1. Cluster Algebras1.1. Quiver Mutation
In this paper, we always assume that a quiver has • the vertex set I = { , . . . , n } , and • no loops and oriented 2-cycles (see Figure 1).For vertices i and j ∈ I , we define Q ( i, j ) = ♯ { arrows from i to j } , Q ( i, j ) = Q ( i, j ) − Q ( j, i ) . yperbolic 3-manifolds and Cluster Algebras SUBMIT
Figure 1: A loop (left) and an oriented 2-cycle (right) of a quiver.Note that the quiver Q is uniquely determined by the skew-symmetric ma-trix Q ( i, j ) (or equivalently Q ( i, j )) under the assumption above. For the vertex k , we define a new quiver µ k Q (mutation of Q at vertex k ) by an anti-symmetric matrix µ k Q ( i, j ) = ( − Q ( i, j ) i = k or j = k,Q ( i, j ) + Q ( i, k ) Q ( k, j ) − Q ( j, k ) Q ( k, i ) i, j = k. Given a sequence k = ( k , . . . , k l ) of vertices and “time” parameters t = 0 , . . . , l , we define Q := Q, Q t := µ k t − · · · µ k Q ( t > . For initial values x i (0) = x i and y i (0) = y i , we define the cluster x -variables x i ( t ) and the cluster y -variables (coefficients) y i ( t ) ( i ∈ I ) by x i ( t + 1) = Q j x j ( t ) Q t ( i,j ) + Q j x j ( t ) Q t ( j,i ) x i ( t ) , (1)and y i ( t + 1) = ( y k ( t ) − i = k,y i ( t ) y k ( t ) Q t ( k,i ) (cid:0) y k ( t ) (cid:1) Q t ( i,k ) i = k. (2) §
2. Triangulated Surfaces and Quivers
Let Σ be a closed connected oriented surface and M be a finite set ofpoints on Σ, called punctures . We assume that M is non-empty and (Σ , M )is not a sphere with less than four punctures.We choose an ideal triangulation τ of Σ, i.e. , we decompose Σ intotriangles whose vertices are located at the punctures. We will not allowself-folded arcs (see Figure 2) in this paper. In this paper we restrict ourselves to cluster algebras associated with skew-symmetricmatrices.
KENTARO NAGAO, YUJI TERASHIMA AND MASAHITO YAMAZAKI
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Figure 2: We do not allow self-folded triangles as in this Figure.
For a triangulation τ without self-folded arcs we will define a quiver Q τ whose vertex set I is the set of arcs in τ .For a triangle ∆ and arcs i and j , we define a skew-symmetric integermatrix Q ∆ by Q ∆ ( i, j ) := i and j , with i following j in the clockwise order, − Q τ := X ∆ ∈ τ Q ∆ , where the sum is taken over all triangles in τ . Let Q τ denote the quiverassociated to the matrix Q τ .For an arc i in the triangulation τ , we can flip the edge i to get a newtriangulation f i ( τ ). This operation is compatible with a mutation at vertex i : Q f i ( τ ) = µ i ( Q τ ) . For a triangulation τ , we define T = T ( τ ) := C ( y e ) e ∈ τ , T ∨ = T ∨ ( τ ) := C ( x e ) e ∈ τ . For a puncture m ∈ M , take a sufficiently small circle around m and let e , . . . , e n be the sequence of arcs which intersect with the circle, where e , . . . , e n may have multiplicity. We define y m := n Y i =1 y e i (3) yperbolic 3-manifolds and Cluster Algebras SUBMIT and T = T ( τ ) := C [ y e , y − e ] e ∈ τ (cid:14) ( y m ) m ∈ M . Let us fix a mapping class ϕ . Then the two triangulations τ and ϕ ( τ ) arerelated by a sequence of flips, together with appropriate changes of labels.More formally, there exists a sequence k = ( k , . . . , k l ) ∈ ( τ ) l such that the two triangulations τ and ϕ ( τ ) are related by the sequence offlips associated to k (see [FST08, Proposition 3.8]). Note that a flip providesa canonical bijection of the edges of the triangulations. We can representthe composition of the bijections by a permutation σ ∈ S I . We define theautomorphismsCT ϕ : T ( τ ) = T ( ϕ ( τ )) ∼ −→ T ( τ ) , CT ∨ ϕ : T ∨ ( τ ) = T ∨ ( ϕ ( τ )) ∼ −→ T ∨ ( τ )by CT ϕ ( y e ) = y σ ( e ) ( l ) , CT ∨ ϕ ( x e ) = x σ ( e ) ( l )Thanks to the result [FST08, Theorem 3.10] and the pentagon relation ofcluster transformations, CT ϕ and CT ∨ ϕ are independent of the choices of thesequences of flips and provides a well-defined action of the mapping classgroup on T ( τ ). §
3. Pseudo-Anosov Mapping Tori
Let τ , ϕ , k and σ be as in § h = h ( t ) be the edge flipped at t and h ′ be the edge after the flip.Let a , b , c and d be the edges of the quadrilateral in the triangulations whosediagonals are h and h ′ . We associate a topological tetrahedron ∆ = ∆( t )whose edges are labeled by a , b , c , d , h and h ′ (see Figure 3).For any pseudo-Anosov mapping class ϕ , this provides a topologicaltetrahedron decompositions of the mapping torus ([Ago11]). A mappingclass ϕ is pseudo-Anosov if and only if the mapping torus has a hyperbolicstructure. KENTARO NAGAO, YUJI TERASHIMA AND MASAHITO YAMAZAKI
SUBMIT h h ′ hh ′ acb ddab c c dab flip tetrahedron Figure 3: A flip in a 2d triangulation can be traded for a 3d tetrahedron. §
4. Equations for Hyperbolic Structure4.1. Shape Parameters
For an ideal tetrahedron in H with vertices 0, 1, z and ∞ (Figure 4), weassociate the shape parameter z with the edge connecting 0 and ∞ . For anideal tetrahedron, a pair of mutually non-intersecting edges has a commonshape parameter, and the shape parameters for the three pairs of mutuallynon-intersecting edges are given by (Figure 5) z, − z − , − z . (4)We take a sequence of flips and associated topological decompositionof the mapping torus as in Section 3. For t ∈ Z , let ∆( t ) denote the t -thtetrahedron, where ∆( t ) and ∆( t + l ) are identified for any t . Let Z ( t )denote the shape parameter of ∆( t ) at the edge h ( t ), the edge flipped attime t . Note that the sequence ( Z ( t )) satisfies shape parameter periodicity Z ( t + l ) = Z ( t ) . (5)For a tetrahedron ∆, let ∆ be the set of edges of ∆. We define E := a t ∈ Z ∆( t ) . yperbolic 3-manifolds and Cluster Algebras SUBMIT z ∞ Figure 4: An ideal tetrahedron with shape parameter z . z / (1 − z ) 01 1 1 − z − z / (1 − z ) 1 − z − Figure 5: The three shape parameters of an ideal tetrahedron. KENTARO NAGAO, YUJI TERASHIMA AND MASAHITO YAMAZAKI
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Let e E denote the set of all edges in the tetrahedron decomposition of Σ × R and π : E → e E be the canonical surjection.Given parameters ( Z ( t )) t ∈ Z , we can define associated parameter Z e = Z e ( t ) for any t ∈ Z and e ∈ ∆( t ) as the shape parameter of ∆( t ) on theedge e , which is determined as in (4). Suppose that the shape parameters ( Z ( t )) t ∈ Z gives an ideal tetrahedrondecomposition . This holds if and only if the following three conditions aresatisfied.First, we need the shape parameter periodicity condition as alreadydiscussed in (5). Second, we needIm Z ( t ) > t (positivity condition) , so that the tetrahedron is positively oriented. Third, for each edge g ∈ e E ,the product of all the shape parameters associated to the elements in π − ( g )must be 1 ([Thu79], see Figure 6): Y ¯ g ∈ π − ( g ) Z ¯ g = 1 (edge-gluing equation) .z z z z z z z z z z Figure 6: The edge-gluing equation around an edge. Here we do not require completeness. See § yperbolic 3-manifolds and Cluster Algebras SUBMIT y -variables and Gluing Conditions Proposition e ( t ) ∈ τ ( t ) be the edge which we flip at t and e ′ ( t +1) ∈ τ ( t +1) be the edge which appears after the flip. The edge-gluingequation is satisfied for the shape parameters Z ( t ) := − y e ( t ) ( t ) (cid:0) = − y e ′ ( t +1) ( t + 1) − (cid:1) . (6) Proof..
Let g ∈ e E be an edge which appears at the t -th flip at ¯ g ′ anddisappear at the t -th flip ¯ g ′′ (Figure 7). Let ¯ g (resp. ¯ g ) be the uniqueFigure 7: An edge g in a tetrahedron decomposition appears at time t anddisappears at time t .element in ∆( t ) ∩ π − ( g ) (resp. in ∆( t ) ∩ π − ( g )). The gluing equationassociated with g is1 = t Y t = t Y ¯ g ∈ ∆( t ) ∩ π − ( g ) Z ¯ g = Z ( t ) × t − Y t = t +1 Y ¯ g ∈ ∆( t ) ∩ π − ( g ) Z ¯ g × Z ( t ) . For this equation, we will show Z ( t ) × T Y t = t +1 Y ¯ g ∈ ∆( t ) ∩ π − ( g ) Z ¯ g = − y g ( T + 1) − , (7) KENTARO NAGAO, YUJI TERASHIMA AND MASAHITO YAMAZAKI
SUBMIT where g is the edge corresponding to g which τ ( t ) ( t = t + 1 , . . . , t ) havein common. We show the equation above by induction with respect to T .The claim for T = t trivially follows from the definition (6). Let us assumethe above statement for T → T −
1. To show the statement for T , we needto show Y ¯ g ∈ ∆( t ) ∩ π − ( g ) Z ¯ g = y g ( t ) /y g ( t + 1) . We will show this by classifying the positional relation of g and e ( t ). • g and e ( t ) have no triangle in common: both side of the equationabove is 1. • g and e ( t ) have a single triangle in common : – Q t ( e ( t ) , g ) = 1 (see Figure 8) : e ( t ) g A ( t )1 − A ( t ) − g Figure 8: The case with Q t ( e ( t ) , g ) = 1.(LHS) equation(4) = 1 − Z ( t ) − = (RHS) , – Q t ( e ( t ) , g ) = − equation(4) = (1 − Z ( t )) − = (RHS) , • g and e ( t ) have two triangles in common: – Q t ( e ( t ) , g ) = ± equation(4) = (1 − Z ( t ) ∓ ) ± = (RHS) , – Q t ( e ( t ) , g ) = 0 : this can not happen because we prohibit self-folded edges in this paper (see Figure 10). yperbolic 3-manifolds and Cluster Algebras SUBMIT e ( t ) g A ( t )1 − A ( t ) − g g e ( t ) = g g − A ( t ) − Figure 9: The case with Q t ( e ( t ) , g ) = 2. e ( t ) g g g g = g Figure 10: The case with Q t ( e ( t ) , g ) = 0. KENTARO NAGAO, YUJI TERASHIMA AND MASAHITO YAMAZAKI
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Patching ideal tetrahedra with corners removed, we get a hyperbolic3-manifold with boundaries, each of which is isomorphic to a torus. Notethat such a boundary torus has two directions: the direction of “time”parameter t ( time direction ) and the direction of the original surface ( surfacedirection ) .The intersection of a removed corner and a boundary torus gives atriangle on the torus with a shape parameter for each angle. Example A , B , C , D , O the punctures and σ (resp. σ or σ ) be the Dehn half-twist along a circlecontaining A and B (resp. B and C , or C and D ) in the anti-clock direction(see Figure 11). Note that σ , σ and σ generate the braid group B . WeFigure 11: A Dehn half-twist σ along a circle containing A and B .take • σ σ σ − as a mapping class. • the triangulation as in Figure 13, •
8, 9, 5, 7, 1, 8 as a sequence of edges which we flip .The mapping torus is the complement of the two-component link in S × S (Figure 12), and hence we have two boundary components. We show thetriangulation of the universal cover of one of the components in Figure 14.Fix a puncture m ∈ M of the surface and a time parameter t . Let F i ( i ∈ Z /n Z ) be the triangle in τ ( t ) which is adjacent to e i − , m and e i ,where ( e , . . . , e n ) is the sequence of arcs around m as before. We avoid to use the terms “longitude” and “meridian” to avoid a confusion. Flipping at 8, 6 (resp. 6, 9, 5, 7 or 1, 8) corresponds to the half-twist σ (resp. σ or σ − ). Canceling the doubled 6, we get the sequence above. yperbolic 3-manifolds and Cluster Algebras SUBMIT
Figure 12: The link corresponding to σ σ σ − .1 2 3 45 6 78 9 OA B C D
Figure 13: A triangulation of the five-punctured sphere. KENTARO NAGAO, YUJI TERASHIMA AND MASAHITO YAMAZAKI
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Figure 14: The triangulation of the boundary torus. A triangle with number t represents the t -th tetrahedron ∆( t ), whose modulus Z ( t ) corresponds toa dihedral angle represented by a black dot. yperbolic 3-manifolds and Cluster Algebras SUBMIT
On the boundary torus, e i represents a vertex and F i represents anedge connecting e i − and e i . The union of F i ’s provides a (piecewise linear)closed curve on the boundary torus . We call this a vertical line (see Figure15). Figure 15: Vertical lines, drawn in the triangulation of Figure 14.The holonomy of along a cycle in the surface direction is given as follows.A vertical line divides the boundary torus into two parts. We fix one of them.For a vertex e i on the vertical line, we take all angles in the universal coverwhich have e i as the vertex and which are on the given side of the verticalline. We denote by H i the product of the shape parameters associated tothese angles. Then we have(the holonomy in the surface direction) = Y i ( − H i ) . (8)A hyperbolic structure given by a sequence of shape parameters is completeif and only if the following condition holds:the holonomy along the surface direction of each boundary istrivial ( cusp condition , [NZ85]). As a cycle, this represents the homology generator in the surface direction. KENTARO NAGAO, YUJI TERASHIMA AND MASAHITO YAMAZAKI
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In Figure 7, we study the set of all tetrahedra which are adjacent to anedge. In this setting, the vertical line divides the set of these tetrahedra intotwo groups: tetrahedra which appear before/after t = t . Hence we have H i = Z ( t ) × t − Y t = t +1 Y ¯ e i ∈ ∆( t ) ∩ π − ( e i ) Z ¯ e i . By (7), the right hand side equals to − y e i ( t ). Therefore the holonomy (8) isequivalent with y m ( t ) = Q ni =1 y e i ( t ) (recall (3)). We can show this productis independent on the choice of t , either by induction or by using the edge-gluing conditions (vertical lines at different choices of t are homologous inthe triangulation of the boundary torus).In summary, we get the following description of the holonomy: Proposition m in thesurface direction is equal to y m . Let us summarize our results in the form of a theorem:
Theorem y e ) e ∈ τ be non-zero complex numbers such that y m = 1 for any puncture m ∈ M . Assume that y h ( t ) (cid:12)(cid:12)(cid:12) y e (0)= y e is well-definedfor any h and t and that the periodicity equation is satisfied y σ ( h ) = y h ( l ) (cid:12)(cid:12)(cid:12) y e (0)= y e . Let us define the shape parameters Z ( t ) by Z ( t ) := − y e ( t ) ( t ) (cid:12)(cid:12)(cid:12) y e (0)= y e , where e ( t ) is the edge flipped at time t , and suppose that Z ( t ) = 0 , t . Then ( Z ( t )) satisfies the edge-gluing equations in § § Z ( t )) >
0, see theexamples in the next section. yperbolic 3-manifolds and Cluster Algebras SUBMIT §
5. Examples
In the last section, we demonstrate Theorem 4.4 in the case of a once-punctured torus and of a five-punctured sphere. The examples are chosenfor the sake of simplicity, and the same methods apply to more generalmapping classes of more general punctured surfaces (recall Remark 0.1).We also discuss an example of the six-punctured disc, to show that ourformulation covers the non-hyperbolic cases not covered in Theorem 4.4. LR Let us start with a once-punctured torus. We take a sequence of twoflips as in Figure 16. This is the mapping class studied in [TY13, § y = y − (cid:0) y − (1 + y − ) (cid:1) − ,y = y (1 + y ) (cid:0) y (1 + y − ) − (cid:1) ,y = y − (1 + y − ) , and the cusp condition is y y y = 1 . Solving these equations, we get a solution y = 1 , y = − − √− , y = − √− . By Theorem 4.4, shape parameters Z (0) = − y (0) = − y = 1 + √− ,Z (1) = − y (1) = − y (1 + y − ) − = 1 + √− , satisfy edge-gluing conditions. Moreover the imaginary parts of Z (0) and Z (1) are positive, and we obtain a complete hyperbolic structure on themapping torus. The parameters coincide with the ones in [TY13, § KENTARO NAGAO, YUJI TERASHIMA AND MASAHITO YAMAZAKI
SUBMIT y (0) = y , y (0) = y , y (0) = y . y (1) = y (1 + y − ) − , y (1) = y − , y (1) = y (1 + y ) . y (2) = y − (1 + y − ) , y (2) = y − (cid:0) y − (1 + y − ) (cid:1) − , y (2) = y (1 + y ) (cid:0) y (1 + y − ) − (cid:1) .flip at the edge 1flip at the edge 2Figure 16: Example: once punctured torus and LR . yperbolic 3-manifolds and Cluster Algebras SUBMIT σ σ σ − Let us take the example of a five-punctured sphere in Example 4.2.The cusp conditions are y y y y y y = y y = y y y y = y y y y = y y = 1 , and the shape parameter periodicity conditions are y = 1 y y y y (cid:16) (1 + y + y y )((1 + y + y y )(1 + y + y y )+ y (1 + (1 + y )(1 + y ) y )(1 + (1 + y + y y ) y )) (cid:17) ,y = y (1 + y + y y ) ,y = y y y (1 + y + y y + y (1 + y )(1 + (1 + y + y y ) y ))(1 + y + y y )(1 + y + y y ) + y (1 + (1 + y )(1 + y ) y )(1 + (1 + y + y y ) y ) ,y = y (1 + (1 + y )(1 + y ) y )(1 + y + y y + y (1 + y )(1 + (1 + y + y y ) y ))(1 + y + y y )(1 + y + y y ) ,y = y y y y + y y + y (1 + y )(1 + (1 + y + y y ) y ) ,y = y y y y + y y ,y = (1 + y + y y )(1 + y + y y ) + y (1 + (1 + y )(1 + y ) y )(1 + (1 + y + y y ) y ) y y y ,y = y (1 + y + y y )(1 + (1 + y )(1 + y ) y )(1 + y + y y + y (1 + y )(1 + (1 + y + y y ) y )) ,y = y y y y y (1 + y + y y )(1 + y + y y ) + y (1 + (1 + y )(1 + y ) y )(1 + (1 + y + y y ) y ) . Solving the shape parameter periodicity conditions with cusp conditions,we get 14 solutions. We take one of the solutions y = 1 . − . × √− ,y = 1 ,y = − . . × √− ,y = 0 . . × √− ,y = 0 . . × √− ,y = 1 . . × √− ,y = 0 . − . × √− ,y = 0 . − . × √− ,y = − . − . × √− . KENTARO NAGAO, YUJI TERASHIMA AND MASAHITO YAMAZAKI
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Following the algorithm in Theorem 4.4, we get the following six parameters0 . × √− − . , . × √− . , . × √− − . , . × √− . , . × √− . , . × √− . , whose imaginary parts are positive, which provide a complete hyperbolicstructure. The volume of the mapping torus computed from the parametersabove is . . This coincides with the value computed by SnapPea/SnapPy [CDW].
Our formalism discussed in this paper applies to in general non-hyperbolic3-manifolds which are themselves not covered in Theorem 4.4. To illustratethis point, let us consider a disk with six points, and we consider the 1 / y = y (3) , y = y (3) , y = y (3) ,y = y (3) , y = y (3) , y = y (3) , y = y (3) , y = y (3) , y = y (3) . Note that indices of edges in the boundary are rotated. The y -variables are The hyperbolic volume of an ideal tetrahedron with modulus z is given by the Bloch-Wigner function D ( z ) = Im(Li ( z )) + arg(1 − z ) log | z | , (9)where Li ( z ) = − R z − t ) t dt is the Euler classical dilogarithm function. When a 3-manifold is triangulated by ideal tetrahedra, the hyperbolic volume of the 3-manifold isthe sum of the hyperbolic volumes of the tetrahedra. yperbolic 3-manifolds and Cluster Algebras SUBMIT ,
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Figure 17: A non-hyperbolic example, associated with the 1 / y (3) = y − (1 + y ) − ,y (3) = y − (1 + y (1 + y ))(1 + y (1 + y )) ,y (3) = y − (1 + y ) − ,y (3) = y (1 + y − (1 + y ) − ) − ,y (3) = y (1 + y (1 + y )) ,y (3) = y (1 + y − ) − (1 + y − (1 + y ) − ) − ,y (3) = y (1 + y − (1 + y ) − ) − ,y (3) = y (1 + y (1 + y )) ,y (3) = y (1 + y − ) − (1 + y − (1 + y ) − ) − . A solution of the periodicity conditions is y = 12 , y = 3 , y = 12 , y = √ , y = 1 √ ,y = 2 √ , y = √ , y = 1 √ , y = 2 √ . Shape parameters of three tetrahedra evaluated at the solution above are Z (0) = − y (0) = − y = − ,Z (1) = − y (1) = − y (1 + y ) = − ,Z (2) = − y (2) = − y (1 + y ) = − L ( x ), we have(with Z ( i ) ′′ = (1 − Z ( i )) − ) L ( Z (0) ′′ ) + L ( Z (1) ′′ ) + L ( Z (2) ′′ ) = π , KENTARO NAGAO, YUJI TERASHIMA AND MASAHITO YAMAZAKI
SUBMIT which is the complexified volume of the 3-manifold. This is identified withthe central charge of ˆsl(2) WZW model at the level 4 (see Remark 0.3).
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