Teichm\"uller spaces as degenerated symplectic leaves in Dubrovin--Ugaglia Poisson manifolds

In this paper we study the Goldman bracket between geodesic length functions both on a Riemann surface $\Sigma_{g,s,0}$ of genus $g$ with $s=1,2$ holes and on a Riemann sphere $\Sigma_{0,1,n}$ with one hole and $n$ orbifold points of order two. We show that the corresponding Teichm\"uller spaces $\mathcal T_{g,s,0}$ and $\mathcal T_{0,1,n}$ are realised as real slices of degenerated symplectic leaves in the Dubrovin--Ugaglia Poisson algebra of upper--triangular matrices $S$ with 1 on the diagonal.


Introduction
In this paper we study some special symplectic leaves in the Poisson algebra (1) of upper-triangular matrices S with 1 on the diagonal. This algebra appears as the semi-classical limit of the famous Nelson-Regge algebra in 2 + 1-dimensional quantum gravity [27,28], and in Chern-Simons theory as Fock-Rosly bracket [13]. At classical level, this algebra was discovered in the context of Frobenius manifold theory by Dubrovin and Ugaglia [8,32] and in the study of non-symmetric bilinear forms by Bondal [1].
In this paper we adopt the isomonodromic deformations perspective. According to Dubrovin's isomonodromicity theorem part III [8], the metric, the flat coordinates, the pre-potential and the structure constants of a n dimensional semi-simple Frobenius manifold are given by the space of parameters u = (u 1 , . . . , u n ) together with an n×n skew-symmetric matrix function V (u) such that the linear differential operator is given by complicated transcendent functions, the Poisson bracket on the space of Stokes matrices is given by very simple quadratic formulae: {s ik , s jl } = 0, for i < k < j < l, {s ik , s jl } = 0, for i < j < l < k, {s ik , s jl } = iπ (s ij s kl − s il s kj ) , for i < j < k < l, (1) {s ik , s kl } = iπ 2 (s ik s kl − 2s il ) , for i < k < l, {s ik , s il } = − iπ 2 (s ik s il − 2s kl ) , for i < k < l.
This bracket was obtained in [8] in the case n = 3, then for any n > 3 in [32], and for this reason it is called the Dubrovin-Ugaglia bracket.
The same bracket appeared in Teichmüller theory as the Goldman bracket [16] between geodesic length functions both on a Riemann surface Σ g,s,0 of genus g with s = 1, 2 holes and on a Riemann sphere Σ 0,1,n with one hole and n orbifold points of order two. Let us denote the two Teichmüller spaces by T g,s,0 , s = 1, 2 and T 0,1,n respectively. These are real symplectic manifolds of dimension respectively dim R (T g,s,0 ) = 3n − 7 for n odd 3n − 8 for n even, with g = n − 1 2 , s = 1 for n odd, 2 for n even, and dim R (T 0,1,n ) = 2(n − 2), while the generic symplectic leaves L generic in the Dubrovin-Ugaglia bracket have dimension dim C (L generic ) = n(n − 1) 2 − n 2 .
It is natural to ask whether the Teichmüller spaces arise as real slices of some subvarieties of a generic leaf or of a degenerated leaf. In this paper, we prove that in the both cases the Teichmüller spaces correspond to degenerated symplectic leaves whose complex dimension is equal to the real dimension of the Teichmüller space itself (see Theorem 4.1). As a consequence, we give flat coordinates on such degenerated symplectic leaves by introducing a suitable complexification of the shear coordinates. The paper is organised as follows. In Section 2 we recall some facts about the isomonodromic deformations of the operator Λ(z) and about the Dubrovin-Ugaglia bracket. This part is mainly a review, apart perhaps the minor Remark 2.2. In Section 3, we review some basics on Teichmüller theory recalling the characterization of the Stokes matrices whose entries arise as geodesic length functions on a Riemann surface Σ g,s,0 of genus g with s = 1, 2 holes and on a Riemann sphere Σ 0,1,n with one hole and n orbifold points of order two. Section 4 is original and contains the characterization of the symplectic leaves arising in Teichmüller theory, including the proof of Theorem 4.1 stating that in the both cases (i.e., for the Riemann surface Σ g,s,0 of genus g with s = 1, 2 holes and Riemann sphere Σ 0,1,n with one hole and n orbifold points of order two) the Teichmüller spaces T g,s,0 and T 0,1,n are the real slices of degenerated symplectic leaves complex dimension equal to the real dimension of the Teichmüller space itself. In subsection 4.1 we discuss an interesting interpretation in terms of a n particle model in Minkowski space and in subsection 4.4 we discuss the complexification of the shear coordinates. Section 5 contains a heuristic discussion about some minor progress towards the characterisation of the Frobenius manifold structure on the Teichmüller spaces T g,s,0 and T 0,1,n .
Acknowledgements. The authors are grateful to Boris Dubrovin, who put them in contact and gave them many helpful suggestions. We would like to thank also Jörgen Andersen, Alexei Bondal

Dubrovin-Ugaglia bracket
In this Section, we recall some facts about the monodromy data of the operator Λ(z), its monodromy preserving deformations, and the construction of the Dubrovin-Ugaglia bracket by the so-called duality [9] which allows to map Λ(z) to a Fuchsian differential operator.
The Dubrovin-Ugaglia bracket is a Poisson bracket on the group of uppertriangular matrices S with 1 on the diagonal. These matrices S arise as monodromy data of the following system of n first order ODEs where U = diagonal(u 1 , . . . , u n ), (u 1 , . . . , u n ) ∈ X n , X n = {(u 1 , . . . , u n ) ∈ C n |u i = u j for i = j} and V = −V T is a skew symmetric n × n matrix with eigenvalues µ 1 , . . . , µ n .
2.1. Monodromy data. A general description of monodromy data of linear systems of ODE can be found in [21,22,20]. Here we use the same notations as in [8], where most results of this sub-section are proved. We fix a real number ϕ ∈ [0, 2π[ and consider the open subset U ∈ X n such that the rays L 1 , ..., L n defined by do not intersect. We assume that the points (u 1 , . . . , u n ) ∈ U are ordered in such a way that the rays L 1 , . . . , L n exit from infinity in counter-clockwise order.
In [8] it was proved that for a fixed line l there exists ε > 0 small enough, Z ∈ R large enough, two sectors Π L and Π R defined as (4) Π R = {z : arg(l) − π − ε < arg(z) < arg(l) + ε, |z| > |Z|} Π L = {z : arg(l) − ε < arg(z) < arg(l) + π + ε, |z| > |Z|} and two unique fundamental solutions Y L (z) in Π L and Y R (z) in Π R such that In the narrow sectors obtained by the intersection of Π L and Π R , we have two fundamental matrices with the same asymptotic behaviour (5). They are related by multiplication by a constant invertible matrix The matrices S + and S − are called Stokes matrices. Due to the skew symmetry of V , they satisfy the following relation Thanks to the choice of the order of u 1 , . . . , u n , S is upper triangular with 1 on the diagonal.
Near the regular singular point 0, there exists a fundamental matrix of the system (2) of the form where a branch cut between zero and infinity has been fixed along the negative part l − of l, the matrix Γ is the eigenvector matrix of V , V Γ = Γµ and R is a nilpotent matrix satisfying the following relation: The monodromy M 0 of the system (2) with respect to the normalized fundamental matrix (6) generated by a simple closed loop around the origin is The central connection matrix C between 0 and ∞ is defined by The monodromy data of the system (2) consist of (µ, R, C, S) and are related by 2.2. Dual Fuchsian system and its monodromy data. Following [9], we consider a n × n Fuchsian system of the form and ν is an arbitrary parameter. This system is dubbed dual to the system (2). Let us remind how the monodromy data of this system (9) are related to the monodromy data of system (2): Theorem 2.1. [9] Let q = e 2πiν and assume that q is not a root of the characteristic equation Then there exist n linearly independent solutions φ (1) , . . . , φ (n) of the system (9) analytic in λ ∈ C \ ∪ j L j such that the monodromy transformations M 1 , . . . , M n along the small loops encircling counter-clockwise the points u 1 , . . . , u n are given by The monodromy around infinity is given by Monodromy preserving deformations. The monodromy preserving deformations equations for the system (2) are the following non-linear differential equations (13) ∂V where E i is the matrix with entries E i kl = δ ik δ il . For any solutions V (u) of equation (13), the monodromy data (µ, R, C, S) of the system are constant in a disk in X n . These same equations describe the isomonodromic deformations of (9), namely the monodromy data M 1 , . . . , M n of the system are constant in a disk in X n . Indeed equations (13) are equivalent to the Schlesinger equations [30] for A 1 , . . . , A n . In [19] it was proved that the spectral curve of these two systems is the same. The set of equations (13) can be written as a n-times Hamiltonian system on the space of skew-symmetric matrices V equipped with the standard linear Poisson bracket for so(n) ∋ V : Indeed equation (13) can be rewritten as where the Hamiltonian functions H i depend on the times u 1 , . . . , u n (17) Equivalently the isomonodromic deformations equations for A 1 , . . . , A n can be written as where {·, ·} are the standard linear Poisson bracket for gl(n) ∋ A k and the Hamiltonian functions H i are given by: and coincide with the previous ones thanks to the fact that This bracket does not satisfy the Jacobi identity, but it reduces to a Poisson bracket on the adjoint invariant objects, i.e., on the traces of the matrices M 1 , . . . , M n and their products. If q is chosen in such a way that condition (11) is satisfied, the monodromy matrices of the dual Fuchsian system have the form (12) so that (21) Tr where S ij is the ij entry in the Stokes matrix S. As a consequence the entries of the Stokes matrix S are adjoint invariant, and the Korotkin-Samtleben bracket reduces to a Poisson bracket on them. This was precisely the main idea by Ugaglia, she assumed q = 1 and proved that for det(S + S T ) = 0, the restriction of the Korotkin-Samtleben bracket to the entries of the Stokes matrix leads to a closed Poisson algebra given by the formulae (1). The Casimirs of this Poisson bracket are the eigenvalues of the matrix S −T S so that the generic Poisson leaves L generic have dimension Remark 2.2. Note that actually it is not necessary to choose q = 1. In fact, given any q such that condition (11) is satisfied, it is always true that where ǫ(k) is the sign of k. By brute force computation, using the specific form of the matrices M k , one obtains always the same Poisson bracket (1). This observation is quite important when we want to study the case when the rank of the matrix S + S T is very low. In this case we can pick q = 1 and prove that the Poisson algebra is (1) anyway. We will discuss this case further in Section 4.

Poisson algebras of geodesic length functions
In this Section we discuss the Dubrovin-Ugaglia bracket in the context of Teichmüller theory. We first recall some key facts which will be needed below.
Due to E. Verlinde and H. Verlinde [33] the configuration space of Einstein gravity in 2 + 1 dimensions is a Riemann surface with boundary components (or holes) and orbifold points times an interval representing the time variable. The algebra of observables is identified with the collection of geodesic length functions of geodesic representatives of homotopy classes of closed curves together with its natural mapping class group action.
The Poisson structure on geodesic length functions is provided by the Goldman brackets [16] and coincides with the Poisson brackets that follow from the Chern-Simons theory [13].
The Poisson algebra of geodesic functions is always closed (and linear) on the subset of geodesic functions corresponding to multi-curves, which are sets of curves without intersections and self-intersections. However, these sets are always infinite whereas the Teichmüller spaces T g,s,n are spaces of (real) dimension 6g − 6 + 2s + 2n, where g is the genus of the Riemann surface, s is the number of boundary components (or holes) and n is the number of orbifold points.
Multi-curve geodesic functions are therefore algebraically dependent, and one encounters the problem of constructing an algebraically independent (or, at least, finite) basis of observables such that the Poisson brackets become closed on this set. In the general case this problem is still open.
In the special case of Riemann surfaces with one or two holes [5], [6], and in the case of a Riemann sphere with one hole and n orbifold points of order 2 [3], the Poisson algebra generated by the Goldman bracket on geodesic length functions closes and coincides with the Dubrovin-Ugaglia bracket.
Here we recall the basics of this construction, which is based on the graph description of the Teichmüller space. Denote by Σ g,s,n a Riemann surface of genus g with s holes and n orbifold points of order two. We assume the hyperbolicity condition 2g − 2 + s > 0, so that by the Poincaré uniformization theorem, we have Σ g,s,n ∼ H + 2 /∆ g,s,n , where H + 2 is the upper half plane and ∆ g,s,n is a Fuchsian group, the fundamental group of the surface Σ g,s,n : In particular, for orbifold Riemann surfaces, the Fuchsian group ∆ g,s,n is such that all its elements are either hyperbolic or have trace equal to zero. We recall the Thurston shear-coordinate description [29], [12] of the Teichmüller spaces of Riemann surfaces with holes and, possibly, orbifold points (see [3]). The main idea is to decompose each hyperbolic matrix γ ∈ ∆ g,s,n as a product of the form where I is a set of integer indices and the matrices R, L and X Zi are defined as follows: and to decompose each traceless element as where γ is decomposed as in (22) and The main point of this construction is that one can obtain the decompositions (22) and (23) by looking at closed loops on the fat-graph. The fat-graph, or spine, Γ g,s,n is a connected graph that can be drawn without self-intersections on Σ g,s,n that has all vertices of valence three except exactly n one-valent vertices situated at the orbifold points, has a prescribed cyclic ordering of labeled edges entering each vertex, and it is a maximal graph in the sense that its complement on the Riemann surface is a set of disjoint polygons (faces), each polygon containing exactly one hole (and becoming simply connected after gluing this hole). Since a graph must have at least one face, only Riemann surfaces with holes, s > 0, can be described in this way. These fat graphs (or spines) constructed originally in [12] [14] in the case of surfaces without orbifold points are dual to ideal triangle decompositions of Penner [29]. We obtain the decomposition of an element of the Fuchsian group ∆ g,s,n using the one-to-one correspondence between closed paths in the fat graph (spine) Γ g,s,n and conjugacy classes of the Fuchsian group ∆ g,s,n . The decomposition (22) can be obtained by establishing a one-to-one correspondence between elements of the Fuchsian group itself and closed paths in the spine starting and terminating at the same directed edge. Each time the path A corresponding to the element γ A (or, equivalently, to its invariant closed geodesic) passes through the αth edge, an edgematrix X Zα with the real coordinate Z α (related to the length of that edge) appears in the decomposition of γ. At the end of the edge, the path can either turn right or left, and a matrix R or L respectively appears in the decomposition [12]. To obtain decomposition (23), we observe that when a path reaches a one-valent vertex (a pending vertex), it undergoes an inversion [2], which corresponds to inserting the matrix F into the corresponding string of 2 × 2-matrices. The edge terminating at a pending vertex is called a pending edge.
The algebras of geodesic length functions were constructed in [2] by postulating the Poisson relations on the level of the shear coordinates Z α of the Teichmüller space: where the sum ranges all the three-valent vertices of a graph and α i are the labels of the cyclically (counterclockwise) ordered (α i+3 ≡ α i ) edges incident to the vertex with the label α. This bracket gives rise to the Goldman bracket on the space of geodesic length functions [16].
In terms of geodesic length functions the bracket (24) corresponds to So we see that every time we consider the bracket between the geodesics lengths of two loops A and B, we produce the geodesics lengths of two new loops A B and A B −1 . To close the Poisson algebra one must use the skein relation valid for two arbitrary matrices in P SL (2): . We can use this relation for resolving the crossing between the two geodesics A and B as in Fig. 1. The skein relation is often not enough to close the Poisson algebra on a finite set of generators. In this paper, we shall consider two special cases in which we indeed can close the algebra just by means of skein relation: the case of Riemann surfaces of genus g and one or two holes (which we dub CF P due to the fact that it was mostly developed in [6,4]) in subsection 3.1 and the case of a Riemann sphere with one hole and with n ≥ 3 orbifold points of order two (dubbed A n case due to its close ties to cluster algebra theory [15]) in subsection 3.2.
3.1. The CF P case. This is the case of a Riemann surface of genus g with one or two holes, the fat-graph on which graph-simple geodesics constitute a convenient algebraic basis is shown in Fig. 2. The genus g = n−1 2 , where n is the number of vertical edges, and the number s of holes is s = 1 for n odd, 2 for n even.
Graph-simple closed geodesics in this picture are those and only those that pass through exactly two different vertical edges; we can then enumerate them by ordered pairs of edge indices denoting by G ij (i < j) the corresponding geodesic functions. Denoting by Z 1 , . . . , Z n the coordinates on the vertical edges and by Y 1 , . . . , Y 2n−6 those on the horizontal edges, we obtain so, for example, The Poisson algebra for the functions G ij is described by 3.2. The A n case. The simplest case of orbifold Riemann surface is a Riemann sphere Σ 0,1,n with one hole and n ≥ 3 orbifold points of order two. In this case, the fat-graph Γ 0,1,n is a tree-like graph with n pending vertices depicted in Fig. 3 for n = 3, 4. We enumerate the n pending vertices counterclockwise, i, j = 1, . . . , n, and consider the algebra of all geodesic functions.
We consider a basis γ 1 , . . . , γ n in the Fuchsian group ∆ 0,1,n such that (The sign convention is such that when we interpret G i,j as being the geodesic functions related to lengths ℓ i,j of closed geodesics, we have G i,j = 2 cosh(ℓ i,j /2) ≥ 2.) In this case, for convenience we let Z i denote the coordinates of pending edges  and Y j all other coordinates. This basis in the Fuchsian group ∆ 0,1,n is given by the following: . . .
Observe that Trγ i = 0, i = 1, . . . , n. It is not hard to check that the matrix has eigenvalues (−1) n−1 e ±P/2 , where P is the length of the perimeter around the hole: Let G i,j = −Tr(γ i γ j ) with i < j denote the geodesic function corresponding to the geodesic line that encircles exactly two pending vertices with the indices i and j. Examples for n = 3 and n = 4 are in figure 3. It turns out that these geodesic functions suffice for closing the Poisson algebra: Note that this is again a simple rescaling of the Dubrovin-Ugaglia bracket.
Remark 3.1. The formulae for G ij in terms of the shear coordinates Z 1 , . . . , Z n , Y 1 , . . . , Y n−3 in the A n case coincide with a specialization of the formulae of the geodesics G ij given by (27) in which we assume Y n−3+i = Y i for i = 1, . . . n − 3 and we take the double lengths 2Z 1 , . . . , 2Z n In other words:

Symplectic leaves corresponding to the Teichmüller space
As mentioned in the introduction, since the Poisson bracket for geodesic length functions both in the CF P case (Riemann surface Σ g,s,0 of genus g with s = 1, 2 holes) and in the A n case (Riemann sphere Σ 0,1,n with one hole and n orbifold points of order two) coincides with the Dubrovin-Ugalgia bracket, it makes sense to characterize the symplectic leaves to which the two Teichmüller spaces T g,s,0 , s = 1, 2, and T 0,1,n belong.
In particular we recall that dim R (T g,s,0 ) = 3n − 7 for n odd 3n − 8 for n even, where g = n − 1 2 , s = 1 for n odd, 2 for n even, and dim R (T 0,1,n ) = 2(n − 2), while the generic symplectic leaves L generic in the Dubrovin-Ugaglia bracket have dimension It is natural to ask whether the Teichmüller spaces arise as real slices of some subvarieties of a generic leaf or of a degenerated leaf. In this section we prove that in both cases the Teichmüller spaces correspond to degenerated symplectic leaves complex dimension equal to the real dimension of the Teichmüller space itself: Theorem 4.1. Denote by L An and by L CF P the symplectic leaves to which the Stokes matrices of with entries s ij = G ij where G ij are given respectively by (29,30) or by (27) belong. Then dim C (L CF P ) = 3n − 7 for n odd 3n − 8 for n even and dim C (L An ) = 2(n − 2), Proof. In order to compute the dimension of the symplectic leaf to which a particular Stokes matrix belongs we use a formula by Bondal [1] which is based on the block diagonal form of the Jordan normal form J 0 of S −T S: Given an arbitrary upper triangular matrix S with 1 on the diagonal, the Jordan normal form J 0 of S −T S decomposes as follows where J λ,k denotes the k × k Jordan block with eigenvalue λ, i.e. , and n λ,k and m (−1) k+1 ,k are the multiplicities of the blocks J λ,k ⊕J 1 λ ,k and J (−1) k+1 ,k respectively.
The dimension of the symplectic leaf L S to which S belongs is min(k, l)n λ,k n λ,l + 2 λ=±1 min(k, l)n λ,k n λ,l + +2 λ=±1 min(k, l)n λ,k m λ,l + 1 2 λ=±1 min(k, l)m λ,k m λ,l − (34) Proof. The proof of the first statement is a trivial consequence of Section 5.5 in Bondal's paper. The formula (34) is (5.10) in [1] (with two small corrections: a factor 2 in the first term of the second row and the last term in the last row were missing).
In order to use this result to compute the dimension of our symplectic leaves we need to describe the Jordan normal form J 0 of S −T S for a Stokes matrix S with entries s ij = G ij where G ij are given either by (27) or by (29,30). This is achieved in the next two theorems which will be proved in subsections 4.2 and 4.3 respectively. with γ 1 , . . . , γ n given in terms of shear coordinates by formula (30). Then for n even, the matrix S −T S, has the following Jordan form: with γ 1 , . . . , γ n given in terms of shear coordinates by formula (39). Then the matrix S −T S, has the following Jordan form for n even: , and for n odd: where I n−4 and I n−5 are respectively the (n − 4) × (n − 4) and (n − 5) × (n − 5) identity matrices and j=1 Y j are the perimeters of the 2 holes in the case of n even, and P = n i=1 Z i + 2n−6 j=1 Y j is the perimeter of the one hole for n odd.
Remark 4.5. Very similar Jordan normal forms appear for the matrix S −T S where S is the Stokes matrix associated to the Frobenius manifold structure on Hurwitz space (see Theorem 4 in [31]). However, in that case the central elements P or P 1 , P 2 are rational multiples of 2πi rather than real numbers.

A first step in the direction of proving Theorems 4.3 and 4.4 is carried out in the next Lemma:
Lemma 4.6. The matrix of the symmetric form G ij = (S T + S) ij has at most rank four in the case of a Riemann surface Σ g,s,0 of genus g with s = 1, 2 and at most rank three in the A n case.
Proof. We prove this lemma in the next subsection where an interesting interpretation in terms of n particle model in Minkowski space is studied.

4.1.
Minkowski space model. In both CF P and A n cases, each element G ij can be presented as G ij = − Tr γ i γ j , where γ k , k = 1, . . . , n, are given by (30) for the A n case and, thanks to Remark 3.1, by the following matrices in CF P case: . . . Expand γ 1 , . . . , γ n as where σ 1 , . . . , σ 4 are the real Pauli matrices in the CF P case. In the latter case we have: is the metric tensor of the Minkowski 3 + 1-dimensional space-time.
In the A n case, because each γ i is a conjugate of F , Tr γ i = 0, no fourth component occurs. We then have and η αβ = diag (+, −, −) is here the metric tensor of the Minkowski 2+1-dimensional space-time. This concludes the proof of Lemma 4.6.
Remark 4.7. It is interesting to notice that in the both cases, we can therefore associate v (i) α with the components of 4-or 3-dimensional vector v (i) in the corresponding Minkowski space. Due to the fact that Trγ 2 i = 2 we obtain the restriction , v (i) ) = 2 ∀i, where we use the standard repeated indices summation. This implies that all the vectors v (i) , i = 1, . . . , n lie in the upper sheet of the hyperboloid of two sheets (they are time-like vectors in the physical terminology). In this case G ij is the scalar product of the corresponding vectors, and since the difference of two different time-like vectors lying on the same sheet is a space-like vector with negative norm, v (i) − v (j) 2 = 4 − 2G ij < 0, and all G ij are greater than two, as expected.  Proof. This is a simple consequence of the fact that the eigenvalues of the matrix S −T S are part of the monodromy data of the system (2) and therefore they must be central elements in the Dubrovin-Ugaglia bracket and therefore of the Goldman bracket (24). As a consequence the determinant of any linear combination λ −1 S T + λS is a modular-invariant function. On the other hand, this determinant is a Laurent polynomial of order not higher than n in each of e Zα/2 , which inevitably means that this determinant is a Laurent polynomial of order not higher than n of e P/2 alone.
The idea of the proof of Theorem 4.3 is to use the modular invariance to choose in a special way the parameters Z i , i = 2, . . . , n − 1 and Y j , j = 2, . . . , n − 1, leaving Z 1 and Z n arbitrary. In fact, since the eigenvalues are modular invariants, if we change some of Z i and Y j by preserving their total sum, the eigenvalues must remain the same.
Because the determinant of λ −1 S T + λS is a rational function in e Zα/2 and e Y β /2 , it has a unique analytic continuation in the domain of complex values of Z 1 , . . . , Z n , Y 1 , . . . , Y n−3 . The value of the determinant must then be conserved provided the exponential e P , P = n α=1 Z α + n−3 β=1 Y β , remains invariant. We now present a convenient choice of these, complex, parameters. We take the representation graph (the spine) of the form depicted in Fig. 4, in which we specially indicated geodesic functions that will play an important role in the proof. We choose all the Y j to be −iπ, then X Yj = 0 i i 0 for any j = 1, . . . , n − 3.
This special matrix is characterised by that LX Y L = RX Y R = X Y . We also use extensively that R = −L 2 and L = R 2 . We next take Z 2 = −Z 3 = Z 4 = · · · = (−1) n−1 Z n−1 and leave Z 1 and Z n arbitrary. Under this choice of the parameters, the entriesG ij := S ij + S ji simplify considerably. Namely, we obtaiñ G n−1,n = −G n−2,n =G n−3,n = · · · = (−1) n−1G 2,n = Tr LX 2Zn RX 2Zn−1 G 1,n = e Zn+Z1 + e −Zn−Z1 , even n Tr LX 2Z1 RX 2Zn odd n (45) All the entries of the matrix S are either ±2, or ±G 1,2 , or ±G n−1,n orG 1,n . We are now going to show that we can re-arrange the rows and columns of the matrix λS + λ −1 S T in order to obtain the form: For such matrix form (46) we can easily compute the determinant: where I k is (λ − λ −1 ) k times the determinant of the skew-symmetric matrix with all the entries above the diagonal equal to the unity; this determinant is zero for odd k and 1 for even k. So, we obtain (48) Let us prove formula (46) and deduce the values of the eigenvalues of J 0 in the even and in the odd dimensional cases separately.
For even n, we have that det(λS + λ −1 S T ) is given by and multiplying the odd columns and rows by −1 and cyclically permuting rows and columns {1, 2, . . . , n − 1, n} → {n, 1, 2, . . . , n − 1}, we obtain the matrix of the form (46) with a = −G n−1,n , b = G 1,2 , and c = G 1,n . Neither a nor b however contribute to the determinant (48) for even n, whereas, from (45), G 1,n = e P + e −P (because the contribution from other Z i vanish for even n, Z 2 + · · · + Z n−1 = 0). For even n we therefore have For odd n we have that det(λS + λ −1 S T ) is given by and multiplying the odd columns and rows by −1 and cyclically permuting rows and columns {1, 2, . . . , n − 1, n} → {n, 1, 2, . . . , n − 1}, we obtain the matrix of the form (46) with a =G n−1,n , b = G 1,2 , and c = G 1,n . Now, the elements G 1,2 ,G n−1,n , and G 1,n (see their explicit expressions in (45)) constitute the Markov triple, that is, abc − a 2 − b 2 − c 2 = (e P − e −P ) 2 , where P = Z 1 + Z 2 + Z n is the perimeter of the hole (the remaining Z i are mutually canceled). For odd n, we therefore have and the roots of the characteristic equation det(S −T S − η) = 0 (η = −λ 2 ) are now η = {e P , e −P , 1, −1, . . . , −1}. Since the rank of S −T S + 1 is less or equal three, all these numbers are eigenvalues (for P = 0) and the Jordan form is diagonal. This concludes the proof of (36) for n odd.

4.2.1.
Symplectic leaves corresponding to A n . We are now ready to prove that the dimension of the symplectic leaves L An corresponding to A n is which is the double the real dimension of the Teichmüller space.
Proof. Thanks to Theorem 4.3, for n even, while for n odd, Using (34) we get precisely This concludes the proof of Theorem 4.1 in the A n case.

4.3.
Proof of Theorems 4.4 and 4.1 in the CF P case. The idea of the proof is the same as for Theorem 4.3. We already proved in Lemma 4.6 that rk(S −T S +I) = 4, so we only need to compute the remaining 4 eigenvalues. Lemma 4.8 is still valid and we will now show how to pick the parameters Z i ,and Y j , in a way to simplify computations.
Even n. In this case, we have two modular-invariant parameters P 1 and P 2 such that n α=1 Z α + Y 2n−6 β=1 = P 1 +P 2 . We cannot now set all the variables Y j to be iπ in the graph in Fig. 2 because those are the variables that distinguish between the perimeters of these two holes. We can set however Z 1 = −Z 2 = Z 3 = · · · = −Z n , take two of the variables Y , say, the variables Y 1 and Y n−2 of the two edges (above and below) that separate Z 2 and Z 3 to be arbitrary and set all the remaining Y j to be iπ (and X Y = 0 i i 0 ). We then have six basic matrix elements, a = G 1,n , b = G 2,n , c = G 3,n , d = G 1,2 , e = G 1,3 , and f = G 2,3 , and the matrix λ −1 S T + λS reduces by the same operations of row/column multiplication by −1 and cyclic permutations of row/columns to the form We express the remaining determinant through two invariant determinants: For the determinant in question, we have +4 cosh 2 (P 1 /2) + 4 cosh 2 (P 2 /2) (51) The roots of (51) for ϕ = −λ 2 are n − 4-fold root ϕ = −1 and four simple roots ϕ = −e (P1+P2)/2 , ϕ = −e −(P1+P2)/2 , ϕ = −e (P1−P2)/2 , and ϕ = −e −(P1−P2)/2 . When all these roots are distinct, the Jordan form is diagonal and all the roots correspond to eigenvectors. These completes the analysis of the Jordan forms for the CF P case.

4.3.1.
Symplectic leaves corresponding to the CF P case. We can now prove that the dimension of the symplectic leaves L CF P corresponding to the CF P case is dim C (L An ) = 3n − 7 for n odd 3n − 8 for n even which is the double the real dimension of the Teichmüller space.
Proof. Thanks to Theorem 4.4, we have that in the case of n even the Jordan normal form decomposes as: and for n odd By using (34) we conclude the proof of Theorem 4.1 in the CF P case.

4.4.
Complexification. In this section we observe that the Stokes matrices belonging to the Teichmüller symplectic leaves L An and L CF P can be parameterized in terms of complex coordinates Z 1 , . . . , Z n , Y 1 , . . . , Y k where k = n − 3 in the A n case and k = 2n − 6 in the CF P case by the same formulae where γ i , γ j are now matrices in SL 2 (C) still given by formulae (30) and (39) with complex Z 1 , . . . , Z n , Y 1 , . . . , Y k .
When the coordinates Z i become complex, we can still use the same parameterization of elements of the discretely acting group, which becomes now a finitely generated subgroup of P SL(2, C), not P SL(2, R), i.e., a Kleinian group. This Kleinian group ∆ g ′ ⊂ P SL(2, C) describes now a handlebody, that is, the quotient of the upper half-space H + 3 := C × R + by the action of ∆ g ′ . The handlebody is geometrically a filled Riemann surface whose boundary is a closed Riemann surface of genus g ′ = 2g + s − 1 obtained from the action of this group on the boundary of H + 3 , i.e., on the complex plane C, and admits a Schottky uniformisation. Note that in this approach we do not present the three-dimensional manifold as a direct product of a Riemann surface (with holes) and a time interval; instead we have an actual handlebody endowed with the set of closed geodesics inside it; each closed geodesic corresponds, as before, to a conjugacy class of the Kleinian group.
Note that, in this case, we loose the distinction between holes and handles of the original Riemann surface Σ g,s : if we consider two Riemann surfaces Σ g1,s1 and Σ g2,s2 such that they are described by the same number of shear coordinates, or in other words such that dim (T g1,s1 × R s1 ) = dim (T g2,s2 × R s2 ), they can be considered as different parameterisations of the same handle-body, as we demonstrate on the example below.
Example 4.9. Complexification of the Teichmüller space T 1,1 of a torus with one hole and of the Teichmüller space T 0,3 of a sphere with three holes.
In Fig. 5 the original (two-dimensional) Riemann surface is obtained under the action of a Kleinian group in H + 3 restricted to the real vertical slice H + 2 . Of course, this is possible only when the real slice of the Kleinian group is simultaneously a Fuchsian group itself, i.e., a discrete subgroup of P SL(2, R). However, we can continuously vary the parameters X i in the complex domain to ensure a smooth transition between two patterns, as shown in Fig. 5.
Note that on the intermediate stages of the transition process in Fig. 5 we have no embedded two-dimensional (geodesically closed) Riemann surface inside the handlebody; it is reconstructed only when the group again becomes Fuchsian.
Although the two Riemann surfaces in Fig. 5, Σ 1,1 and Σ 0,3 , have different topologies, their sets of geodesic lengths are the same, so we say they are isospectral.
Note that equations (52) not always admit real solutions in terms of Z i for a given real X i : the obstruction is provided by the Markov element, In the case of the torus Σ 1,1 with real Z i , we have the inequality M ≥ 0, whereas in the case of the sphere Σ 0,3 with real X i , we have the inequality M ≥ −4, so Eqs. 52 admit real solutions both in Z i and in X i iff M ≥ 0.
In Fig. 6 we depict the explicit relation between the geodesic functions and indicate the image of the boundary curve. In Fig. 7 the same correspondence is presented for the spines Γ 1,1 and Γ 0,3 . Note that neither the intersection indices between the curves nor the Poisson brackets are preserved under this identification. Figure 6. The transformation between geodesics on Σ1,1 and Σ0,3.

Conclusion
Theorems 4.3 and 4.4 characterise the Stokes matrices arising in the Teichmüller theory of a Riemann sphere with one hole and n orbifold points and of a Riemann surface of genus g with one or two holes respectively. Figure 7. The transformation between geodesics in Fig. 6 depicted for the spines Γ1,1 and Γ0,3.
In section 4, we have seen that all Strokes matrices belonging to the degenerated symplectic leaves L An and L CF P can be parameterised in terms of complex coordinates Z 1 , . . . , Z n , Y 1 , . . . , Y k , where k = n − 3 in the A n case and k = 2n − 6 in the CF P case.
In order to characterise the Frobenius Manifold structure corresponding to these degenerated symplectic leaves one possible strategy is to determine the solution V (u 1 , . . . , u n ) of the isomonodromic deformation equation (13) and then to use Dubrovin's isomonodromicity theorem part III in [8] to reconstruct the metric, the flat coordinates, the pre-potential and the structure constants of the Frobenius manifold. Unfortunately at the moment this strategy fails at the very first step, i.e. we are unable to determine V (u 1 , . . . , u n ), even in the simplest case, i.e. for n = 3. We are going to explain what happens in this case in the next subsection and then in subsection 5.2 we will say a few words about the case of n > 3. 5.1. Case n = 3. For n = 3 we deal only with L A3 (the CFP case for n = 3 is completely equivalent to this one up to doubling of the shear coordinates). The geodesics G ij are given by the following formula in which we use cyclic notation: which for Z 1 , Z 2 , Z 3 ∈ R are strictly bigger than 2. In this case all Stokes matrices in the generic symplectic leaves can be parameterised in terms of the complexified shear coordinates Z 1 , Z 2 , Z 3 , simply imposing where now G i,j are given by (54) with complex Z 1 , Z 2 , Z 3 .
For generic values of the central element p = Z 1 + Z 2 + Z 3 , the Jordan normal form J 0 of the monodromy around 0 of system (2) is actually diagonal, so that the matrix V is non resonant and the Stokes matrix S determines uniquely the local solutions V (u 1 , u 2 , u 3 ) of the isomonodromic deformation equations (13).
The generic solutions of the sixth Painlevé equation are irreducible transcendental functions, i.e. they cannot be expressed via elementary or classical transcendental functions by simple operations. Of course some special solutions may be reducible: indeed all algebraic solutions of (55) were classified in [10] and [26], and the so called classical solutions, solutions that can be expressed in terms of hypergeometric functions were classified in [34]. However, in the geometric case, i.e. for Z 1 , Z 2 , Z 3 ∈ R, the solutions are certainly irreducible: in [10] and [26] it was proved that in order to have algebraic solutions, a necessary condition is that |S i,j | < 2, which is clearly violated in the geometric case. Moreover, using the results of [25], it is rather straightforward to prove that these solutions are never of hypergeometric type.
Another nasty surprise is given by looking at the asymptotic behaviour of the geometric solutions near the critical points. Indeed most PVI solutions have asymptotic behaviour of algebraic type, namely given σ i , i = 1, 2, 3 complex numbers such that 2 sin πσ i 2 = S jk , i = j, k, and ℜ(σ i ) ∈]0, 1[, the corresponding PVI solution has the following asymptotic behaviours of algebraic type [23]: However, for S i,j = G i,j > 2, we have σ i = 1 + iν i , ν i ∈ R for all i = 1, 2, 3. In this case the asymptotics are no longer of algebraic type, but become very complicated [18]. For example near 0 we have: where φ is a phase parameter and F (x), F 1 (x) are the two Jacobi elliptic integrals. This makes all asymptotic computations of V and of the metric, the flat coordinates, the pre-potential and the structure constants of the Frobenius manifold extremely involved if not impossible.

5.2.
Higher n. First observe that the discrepancy d between the dimension of the generic symplectic leaves and the dimension of the leaves L An and L CF P is given by: we see that for n = 3, 4 the leaves L An are generic, while for n = 4, 5 the leaves L CF P are generic. As we have observed above, this fact is at the root of why we can't actually solve the isomonodromic deformation equations (13): for small n we deal with "generic solutions" which, as we have seen above, are irreducible transcendental functions. For n > 5 the discrepancy d between the dimension of the generic symplectic leaves and the dimension of the leaves L An and L CF P is non zero. In terms of solutions V (u 1 , . . . , u n ) of the isomonodromic deformation equation (13), this means that the matrix function V (u 1 , . . . , u n ) satisfies extra d independent equations. These are algebraic equations that can be obtained by observing that as soon as n is large enough, J 0 has a block diagonal form in which one block is the minus identity. This means that V is resonant, and in principle we should have where R is a nilpotent matrix satisfying (7) which can be recursively determined in terms of the entries of V . When a minus identity diagonal block appears, all off diagonal entries corresponding to that diagonal block must be zero, leading to extra equations for V . For example in the A n case, for n even we have n − 2 eigenvalues equal to −1, so we should expect R to have (n−2)(n−3) 2 off diagonal entries. Since on our degenerated symplectic leaf only one of those in non zero, we expect (n−4)(n−1) 2 equations of which only d = (n−4) 2 2 are independent. Following the same train of thoughts as in [11], this implies that the solution V (u 1 , . . . , u n ) of the isomonodromic deformation equation (13) corresponding to the degenerated symplectic leaves can be in fact reduced to the Garnier system both in the A n and in the CF P case. Work on this reduction is still in progress.