Algebras Leonid Chekhov

Marta Mazzocco

Vladimir Rubtsov

PAINLEV É MONODROMY MANIFOLDS, DECORATED CHARACTER VARIETIES AND CLUSTER

In this paper we introduce the concept of decorated character variety for the Riemann surfaces arising in the theory of the Painlevé differential equations. Since all Painlevé differential equations (apart from the sixth one) exhibit Stokes phenomenon, it is natural to consider Riemann spheres with holes and bordered cusps on such holes. The decorated character is defined as complexification of the bordered cusped Teichmüller space introduced in [8]. We show that the decorated character variety of a Riemann sphere with s holes and n > 1 cusps is a Poisson manifold of dimension 3s + 2n -6 and we explicitly compute the Poisson brackets which are naturally of cluster type. We also show how to obtain the confluence procedure of the Painlevé differential equations in geometric terms.

Introduction

It is well known that the sixth Painlevé monodromy manifold is the SL 2 (C) character variety of a 4 holed Riemann sphere. The real slice of this character variety is the decorated Teichmüller space of a 4 holed Riemann sphere, and can be combinatorially described by a fat-graph and shear coordinates. By complexifying the shear coordinates, flat coordinates for the character variety of a 4 holed Riemann sphere were found in [START_REF] Chekhov | Shear coordinates on the versal unfolding of the D 4 singularity[END_REF].

For the other Painlevé equations, the interpretation of their monodromy manifolds as "character varieties" of a Riemann sphere with boundary is still an extremely difficult problem due to the fact that the linear problems associated to the other Painlevé equations exhibit Stokes phenomenon. This implies that some of the boundaries have bordered cusps on them [START_REF] Chekhov | Colliding holes in Riemann surfaces and quantum cluster algebras[END_REF]. Being on the boundary, these bordered cusps escape the usual notion of character variety leading to the necessity of introducing a decoration.

In this paper we present a decoration which truly encodes the geometry of each cusped boundary. On the real slice of our decorated character variety, this decoration corresponds to choosing some horocycles to associate a λ-length to each bordered cusp. 1 This geometric description allows us to introduce flat coordinates in the corresponding bordered cusped Teichmüller space (see [START_REF] Chekhov | Colliding holes in Riemann surfaces and quantum cluster algebras[END_REF] for the definition of this notion) and by complexification on the decorated character variety.

This leads us to define explicitly a set of coordinates on the decorated character variety of the Riemann spheres with bordered cusps which arise in the theory of the Painlevé differential equations and to compute the Poisson brackets in these coordinates. Such Poisson brackets coincide with the cluster algebra Poisson structure as predicted in [START_REF] Chekhov | Colliding holes in Riemann surfaces and quantum cluster algebras[END_REF].

We note that another approach to this problem was developed in [START_REF] Boalch | Geometry and braiding of Stokes data; fission and wild character varieties[END_REF] where the definition of wild character variety was proposed following a construction by Gaiotto, Moore and Neitzke [START_REF] Gaiotto | Wall-crossing, Hitchin systems, and the WKB approximation[END_REF] which consisted in introducing spurious punctures at the points of intersection between the Stokes lines and some fixed circles around each irregular singularity. This description does not seem compatible with the confluence procedure of the Painlevé equations, which is one of our motivations to propose a new approach.

We show that, if we exclude P V I, we have nine possible Riemann surfaces with bordered cusps, for which we define the decorated character variety. We show that in each case there is a singled Poisson sub-algebra which is the coordinate ring of an affine variety (the monodromy manifold of one of the Painlevé differential equations) We shall explain it in details in the Section 5. Indeed, all the Painlevé differential equations arise as monodromy preserving deformations of an auxiliary linear system of two first order ODEs. The monodromy data of such auxiliary linear system are encoded in their monodromy manifolds which can all be described by affine cubic surfaces in C 3 defined by the zero locus of the corresponding polynomials in C[x 1 , x 2 , x 3 ] given in Table 1, where ω 1 , . . . , ω 4 are some constants related to the parameters appearing in the corresponding Painlevé equation as described in Section 2.

P-eqs

Polynomials P V I x 1 x 2 x 3 + x 2 1 + x 2 2 + x 2 3 + ω 1 x 1 + ω 2 x 2 + ω 3 x 3 + ω 4 P V

x 1 x 2 x 3 + x 2 1 + x 2 2 + ω 1 x 1 + ω 2 x 2 + ω 3 x 3 + 1 + ω 2 3 -ω3(ω2+ω1ω3)(ω1+ω2ω3)

(ω 2 3 -1) 2 P V deg x 1 x 2 x 3 + x 2 1 + x 2 2 + ω 1 x 1 + ω 2 x 2 + ω 1 -1 P IV x 1 x 2 x 3 + x 2 1 + ω 1 x 1 + ω 2 (x 2 + x 3 ) + ω 2 (1 + ω 1 -ω 2 ) P III x 1 x 2 x 3 + x 2 1 + x 2 2 + ω 1 x 1 + ω 2 x 2 + ω 1 -1 P III D7 x 1 x 2 x 3 + x 2 1 + x 2 2 + ω 1 x 1 -x 2 P III D8 x 1 x 2 x 3 + x 2 1 + x 2 2 -x 2 P II JM x 1 x 2 x 3 -x 1 + ω 2 x 2 -x 3 -ω 2 + 1 P II F N x 1 x 2 x 3 + x 2 1 + ω 1 x 1 -x 2 -1 P I x 1 x 2 x 3 -x 1 -x 2 + 1
Table 1.

Note that in Table 1, we distinguish ten different monodromy manifolds, the P III P III , P III P III D 7 and P III P III D 8 correspond to the three different cases of the third Painlevé equation according to Sakai's classification [START_REF] Sakai | Rational Surfaces Associated with Affine Root Systems and Geometry of the Painlevé Equations[END_REF], and the two monodromy manifolds P II F N and P II JM associated to the same second Painlevé equation correspond to the two different isomonodromy problems found by Flaschka-Newell [START_REF] Flaschka | Monodromy and spectrum preserving deformations I[END_REF] and Jimbo-Miwa [START_REF] Miwa | Monodromy preserving deformations of linear ordinary differential equations with rational coefficients II[END_REF] respectively.

Our methodology consists in reproducing the famous confluence scheme for the Painlevé equations:

P D6 III " " ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ / / P D7 III " " ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ / / P D8 III P V I / / P V / / = = ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ! ! ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ P deg V " " ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ < < ③ ③ ③ ③ ③ ③ ③ ③ P JM II / / P I P IV < < ② ② ② ② ② ② ② ② ② / / P F N II < < ③ ③ ③ ③ ③ ③ ③ ③ ③
in terms of the following two basic operations on the underlying Riemann sphere:

• Chewing-gum: hook two holes together and stretch to infinity by keeping the area between them finite (see Fig. 1). • Cusps removal: pull two cusps on the same hole away by tearing off an ideal triangle (see Fig. 2).

Figure 1. The process of confluence of two holes on the Riemann sphere with four holes: as a result we obtain a Riemann sphere with one less hole, but with two new cusps on the boundary of this hole. The red geodesic line which was initially closed becomes infinite, therefore two horocycles (the green dashed circles) must be introduced in order to measure its length. As shown by the first two authors in [START_REF] Chekhov | Colliding holes in Riemann surfaces and quantum cluster algebras[END_REF], these two operations correspond to certain asymptotics in the shear coordinates and perimeters. We will deal with such asymptotics in Section 5. The confluence process on the underlying Riemann spheres with cusped boundaries is described in Fig. 3. Note that these results agree with the work by T. Sutherland [28] who used the auxiliary linear problem to produce a quadratic differential on the same underlying Riemann spheres with cusped boundaries. In his work, Sutherland associated a quiver to each of the above Painlevé cusped Riemann spheres and explicitly exhibit the canonical connected component of the space of numerical stability conditions of the Painlevé quivers.

In our work, cluster algebras appear naturally when describing the bordered cusped Teichmüller space of each Riemann sphere with bordered cusps. Indeed, as shown in [START_REF] Chekhov | Colliding holes in Riemann surfaces and quantum cluster algebras[END_REF], when bordered cusps arise, it is possible to introduce a generalised lamination on the Riemann surface consisting only of geodesics which start and terminate at the cusps. The geodesic length functions (well defined by fixing horocycles at each cusp) in this lamination are the coordinates in the bordered cusped Teichmüller space, while the decoration itself is given by the choice of horocycles. In the Poisson structure given by the Goldman bracket, these coordinates satisfy the cluster algebra Poisson bracket. This is due to the fact that the geodesics in the lamination do not intersect in the interior of the Riemann sphere, but come together asymptotically in the bordered cusps. We also study the corresponding cluster mutations and show that in the case of a Riemann sphere with four holes they correspond to the procedure of analytic continuation for solutions to the sixth Painlevé equation, thus showing that this procedure of analytic continuation satisfies the Laurent phenomenon. For the other Painlevé equations, the cluster algebra mutations correspond to the action of the Mapping Class Group on the cusped lamination.

We define the decorated character variety as the complexification of the bordered cusped Teichmüller space so that by complexifying the coordinates given by the generalised laminations we obtain coordinates on the decorated character varieties. We show that in the case of the Painlevé differential equations, the decorated character variety is a Poisson manifold of dimension 3s + 2n -6, where s is the number of holes and n > 1 is the number of cusps. We show that in each case the decorated character variety admits a special Poisson sub-manifold defined by the set of functions which Poisson commute with the frozen cluster variables. This submanifold is defined as a cubic surface M φ := Spec(C[x 1 , x 2 , x 3 ]/ φ = 0 ), where φ is one of the polynomials in Table 1, with the natural Poisson bracket defined by:

(1.1) {x 1 , x 2 } = ∂φ ∂x 3 , {x 2 , x 3 } = ∂φ ∂x 1 , {x 3 , x 1 } = ∂φ ∂x 2 .
This paper is organised as follows: in Section 2, we recall the link between the parameters ω 1 , . . . , ω 4 and the Painlevé parameters α, β, γ and δ in each Painlevé equation and discuss the natural Poisson bracket (1.1) on each cubic. In Section 3, we remind some important notions on the combinatorial description on the bordered cusped Teichmüller space. In Section 4, we introduce the notion of decorated character variety. In Section 5, we present the flat coordinates for each cubic and describe the laminations and the corresponding cluster algebra structure. In Section 6, we explain the generalised cluster algebra structure appearing in the case of P V I, P V , P III and P IV . In the Appendix we discuss the singularity theory.

Acknowledgements. The authors are grateful to P. Clarkson and T. Sutherland for helpful discussions. We are thankful to B.V. Dang for his help with SINGULAR package. The work of L.O.Ch. was partially supported by the center of excellence grant "Centre for Quantum Geometry of Moduli Spaces" from the Danish National Research Foundation (DNRF95) and by the Russian Foundation for Basic Research (Grant Nos. 14-01-00860-a and 13-01-12405-ofi-m2). This research was supported by the EPSRC Research Grant EP/J007234/1, by the Hausdorff Institute, by ANR "DIADEMS", by RFBR-12-01-00525-a, MPIM (Bonn) and SISSA (Trieste).

The Painlevé the monodromy manifolds and their Poisson structure

According to [START_REF] Saito | Moduli spaces for linear differential equations and the Painlevé equations[END_REF], the monodromy manifolds M (d) have all the form

(2.2) x 1 x 2 x 3 + ǫ (d) 1 x 2 1 + ǫ (d) 2 x 2 2 + ǫ (d) 3 x 2 3 + ω (d) 1 x 1 + ω (d) 2 x 2 + ω (d) 3 x 3 + ω (d)
4 = 0, where d is an index running on the list of the Painlevé cubics P V I, P V, P V deg , P IV , P III D6 , P III D7 , P III D8 , P II JM , P II F N , P I and the parameters ǫ

(d) i , ω (d) i , i = 1,
ω (d) 1 = -G (d) 1 G (d) ∞ -ǫ (d) 1 G (d) 2 G (d) 3 , ω (d) 2 = -G (d) 2 G (d) ∞ -ǫ (d) 2 G (d) 1 G (d) 3 , ω (d) 3 = -G (d) 3 G (d) ∞ -ǫ (d) 3 G (d) 1 G (d) 2 , (2.3) ω (d) 4 = ǫ (d) 2 ǫ (d) 3 G (d) 1 2 + ǫ (d) 1 ǫ (d) 3 G (d) 2 2 + ǫ (d) 1 ǫ (d) 2 G (d) 3 2 + G (d) ∞ 2 + +G (d) 1 G (d) 2 G (d) 3 G (d) ∞ -4ǫ (d) 1 ǫ (d) 2 ǫ (d) 3 ,
where

G (d) 1 , G (d) 2 , G (d) 3 , G (d)
∞ are some constants related to the parameters appearing in the Painlevé equations as follows: where the parameters θ 0 , θ 1 , θ t , θ ∞ are related to the Painlevé equations parameters in the usual way [START_REF] Miwa | Monodromy preserving deformations of linear ordinary differential equations with rational coefficients II[END_REF]. Note that for P III D7 the parameters G 1 and G 2 tend to infinity -we take this limiti in such a way that

G (d) 1 =        2 cos πθ 0 d = P V I, P V, P III, P V deg , P IV, P II F N 1 d = P III D8 , P II JM , P I ∞ d = P III D7 , 0 d = P III D8 , G (d) 2 =            2 cos πθ 1 d = P V I, P V, 2 cos πθ ∞ d = P III, P V deg , P IV, e iπθ0 d = P II JM 1 d = P III D8 , P II F N P I ∞ d = P III D7 , P III D8 , G (d) 3
=        2 cos πθ t d = P V I, 2 cos πθ ∞ d = P V, P IV 1 d = P II JM , 0 d = P III, P V deg , P III D7 , P III D8 , P II F N , P I (2.4)
G (d) ∞ =        2
ω 1 = -G 1 G ∞ and ω 2 = -G 2 G ∞
are not zero, while ω 4 = 0. Similarly for P III D8 .

Remark 2.1. Observe that in the original article [START_REF] Saito | Moduli spaces for linear differential equations and the Painlevé equations[END_REF] the cubic corresponding to the Flaschka-Newell isomonodromic problem [START_REF] Flaschka | Monodromy and spectrum preserving deformations I[END_REF] is in the form x 1 x 2 x 3 + x 1 -x 2 + x 3 + 2 cos πθ 0 = 0. This can be obtained from our cubic P II :

E * 7 x 1 x 2 x 3 + x 2 1 + ω 1 x 1 -x 2 + 1
by the following diffeomorphism (away from x 2 x 3 = 0):

x 1 → -sx 1 , x 2 → 1 s x 2 , x 3 → s 2 x 2 1 -1+x1x2 s x 3 x 1 x 2 ,
where s = 2 cos πθ 0 . The reason to choose the cubic in the form P II : E * 7 will be clear in Section 5.

Remark 2.2. Note that the P III D7 , P III D8 and P I cubics have different signs in [START_REF] Saito | Moduli spaces for linear differential equations and the Painlevé equations[END_REF], which can both be obtained by a trivial rescaling of the variables x 1 , x 2 , x 3 .

2.1. Natural Poisson bracket on the monodromy manifold. We would like to address here some natural facts that arise when comparing the various descriptions of family of affine cubics surfaces with 3 lines at infinity (2.2).

First of all, the projective completion of the family of cubics 2.2 with ǫ (d) i = 0 for all i = 1, 2, 3 has singular points only in the finite part of the surface and if any of ǫ (d) i , i = 1, 2, 3 vanish, then M (d) is singular at infinity with singular points in homogeneous coordinates X i = 1 and X j = 0, j = i ( [START_REF] Oblomkov | Double Affine Hecke Algebras of Rank 1 and Affine Cubic Surfaces[END_REF]). Here x i = Xi X0 . One can consider this family of cubics as a variety S = {(x, ω) ∈ C 3 × Ω) : S(x, ω) = 0} where x = (x 1 , x 2 , x 3 ), ω = (ω 1 , ω 2 , ω 3 , ω 4 ) and the "x-forgetful" projection π : S → Ω : π(x, ω) = ω. This projection defines a family of affine cubics with generically non-singular fibres π -1 (ω) (we will discuss the nature of these singularities in Subsection 6.3).

The cubic surface S ω has a volume form ϑ ω given by the Poincaré residue formulae:

(2.5)

ϑ ω = dx 1 ∧ dx 2 (∂S ω )/(∂x 3 ) = dx 2 ∧ dx 3 (∂S ω)/(∂x 1 ) = dx 3 ∧ dx 1 (∂S ω )/(∂x 2 )
.

The volume form is a holomorphic 2-form on the non-singular part of S ω and it has singularities along the singular locus. This form defines the Poisson brackets on the surface in the usual way as

(2.6) {x 1 , x 2 } ω =
∂S ω ∂x 3 and the other brackets are defined by circular transposition of x 1 , x 2 , x 3 . It is a straightforward computation to show that for (i, j, k) = (1, 2, 3):

(2.7) {x i , x j } ω = ∂S ω ∂x k = x i x j + 2ǫ d i x k + ω d i
and the volume form (2.5) reads as (2.8)

ϑ ω = dx i ∧ dx j (∂S ω )/(∂x k ) = dx i ∧ dx j (x i x j + 2ǫ d i x k + ω d i )
.

In a special case of P V I , i.e. the D 4 cubic with parameters ω i = 0 for i = 1, 2, 3 and ω 4 = -4, there is an isomorphism π : C * × C * /η → S ω : [START_REF] Cantat | Dynamics on character varieties and Malgrange irreducibility of Painlevé VI equation[END_REF] (2.9)

π(u, v) → (x 1 , x 2 , x 3 ) = (-u -1/u, -v -1/v, -uv -1/uv),
where η is the involution of C * ×C * given by u → 1/u, v → 1/v. The log-canonical 2-form θ = du∧dv uv defines a symplectic structure on C * × C * which is invariant with respect the involution η and therefore defines a symplectic structure on the nonsingular part of the cubic surface S ω for ω i = 0 for i = 1, 2, 3 and ω 4 = -4.

The relation between the log-canonical 2-form θ = du∧dv uv and the Poisson brackets on the surface S ω can be extended to all values of the parameters ω and for all the Painlevé cubics as we shall show in this paper. In fact the flat coordinates that we will introduce in Section 5 are such that their exponentials satisfy the log-canonical Poisson bracket.

Remark 2.3. The cubic M (P V I) appears in many different contexts outside of the Painlevé theory. For example, it was studied in Oblomkov' s work (see [START_REF] Oblomkov | Double Affine Hecke Algebras of Rank 1 and Affine Cubic Surfaces[END_REF]) in relation to Cherednik algebras and M. Gross, P. Hacking and S.Keel (see Example 5.12 of [START_REF] Gross | Mirror symmetry for log Calabi-Yau surfaces I[END_REF]) claim that the family 2.2 can be "uniformized" by some analogues of theta-functions related to toric mirror data on log-Calabi-Yau surfaces. More precisely, the projectivisation Y of 2.2 with the cubic divisor ∆ : X 1 X 2 X 3 = 0 is an example of so called Looeijnga pair and Y \ ∆ is a log-symplectic Calabi-Yau variety with the holomorphic 2-form 2.5. We shall quantize this log-Calabi-Yau variety in the subsequent paper [START_REF] Chekhov | Quantised Painleve monodromy manifolds and Calabi-Yau algebras[END_REF].

Combinatorial description of the bordered cusped Teichmüller space

Let us start with the standard case, i.e. when no cusps are present. According to Fock [START_REF] Fock | Combinatorial description of the moduli space of projective structures[END_REF] [START_REF] Fock | Dual Teichmüller spaces[END_REF], the fat graph associated to a Riemann surface Σ g,n of genus g and with n holes is a connected three-valent graph drawn without self-intersections on Σ g,n with a prescribed cyclic ordering of labelled edges entering each vertex; it must be 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). In the case of a Riemann sphere Σ 0,4 with 4 holes, the fat-graph is represented in Fig. 4.

The geodesic length functions, which are traces of hyperbolic elements in the Fuchsian group ∆ g,s such that Σ g,s ∼ H/∆ g,s are obtained by decomposing each hyperbolic matrix γ ∈ ∆ g,s into a product of the so-called right, left and edge matrices:

R := 1 1 -1 0 , L := 0 1 -1 1 , X si := 0 -exp si 2 exp -si 2 0 ,
where s i is the shear coordinate associated to the i-th edge in the fat graph.

In [START_REF] Chekhov | Colliding holes in Riemann surfaces and quantum cluster algebras[END_REF] the notion of fat-graph was extended to allow cusps. Here we present this definition adapted to the special cases dealt in the current paper: Definition 3.1. We call cusped fat graph (a graph with the prescribed cyclic ordering of edges entering each vertex) G g,s,n a spine of the Riemann surface Σ g,s,n with g handles, s and n > 0 decorated bordered cusps if (a) this graph can be embedded without self-intersections in Σ g,s,n ; (b) all vertices of G g,s,n are three-valent except exactly n one-valent vertices (endpoints of the open edges), which are placed at the corresponding bordered cusps; (c) upon cutting along all nonopen edges of G g,s,n the Riemann surface Σ g,s,n splits into s polygons each containing exactly one hole and being simply connected upon contracting this hole.

Definition 3.2. We call geometric cusped geodesic lamination (CGL) on a bordered cusped Riemann surface a set of nondirected curves up to a homotopic equivalence such that (a) these curves are either closed curves (γ) or arcs (a) that start and terminate at bordered cusps (which can be the same cusp); (b) these curves have no (self)intersections inside the Riemann surface (but can be incident to the same bordered cusp); (c) these curves are not empty loops or empty loops starting and terminating at the same cusp.

In the case of arcs, the geodesic length functions are now replaced by the signed geodesic lengths of the parts of arcs contained between two horocycles decorating the corresponding bordered cusps; the sign is negative when these horocycles intersect.

Combinatorially speaking this corresponds to calculating the lengths of such arcs by associating to each arc a matrix in SL 2 (R) in the same way as before, i.e. by taking products of left, right and edge matrices, but then by taking the trace of such product of matrices multiplied by the cusp matrix: K = 0 0 -1 0 at the right hand side of the whole expression. For example the arc b in Fig. 6 has length

l b such that exp l b 2 = Tr (X(k 1 )RX(s 3 )RX(s 2 )RX(p 2 )RX(s 2 )LX(s 3 )LX(k 1 )K) .
Note that in all fat graphs in this paper we distinguish the shear coordinates s 1 , s 2 , s 3 which correspond to the edges in the central T shaped part of the graph and the shear coordinates k 1 , . . . , k 6 which arise when breaking holes.

In [START_REF] Chekhov | Colliding holes in Riemann surfaces and quantum cluster algebras[END_REF] it is proved that for every cusped fat-graph with the additional property that the polygons containing holes with no cusps are monogons, there exists a complete cusped geodesic lamination which consists only of arcs and simple loops around the un-cusped holes. Loosely speaking, this means that all lengths of any closed geodesic or of any arc in the Riemann surface is a Laurent polynomial of the lengths of the elements in the lamination.

The Poisson brackets between lengths of arcs and closed geodesics can be computed by using the Weil-Petersson bracket , which is shear coordinates becomes [START_REF] Chekhov | A quantum Techmüller space[END_REF][START_REF] Chekhov | Quantum mapping class group, pentagon relation, and geodesics[END_REF] (3.10)

f (Z), g(Z) = 4g+2s+|δ|-4 3-valent vertices α = 1 3 mod 3 i=1 ∂f ∂Z αi ∂g ∂Z αi+1 - ∂g ∂Z αi ∂f ∂Z αi+1
,

where Z α are the shear coordinates on each edge and the sum ranges all threevalent vertices of a graph and α i are the labels of the cyclically (clockwise) ordered (α 4 ≡ α 1 ) edges incident to the vertex with the label α. This bracket gives rise to the Goldman bracket on the space of geodesic length functions [START_REF] Goldman | Invariant functions on Lie groups and Hamiltonian f lows of surface group representations[END_REF] and in [START_REF] Chekhov | Colliding holes in Riemann surfaces and quantum cluster algebras[END_REF] it is proved that on the lengths of the elements of a complete cusped geodesic lamination which consists only of arcs and simple loops this Poisson bracket gives rise to the cluster algebra Poisson structure.

In order to describe this Poisson structure more explicitly, notice that for cusped fat-graph with the additional property that the polygons containing holes with no cusps are monogons, every hole with no associated bordered cusps is contained inside a closed loop, which is an edge starting and terminating at the same threevalent vertex. Vice versa, every such closed loop corresponds to a hole with no associated bordered cusps. Therefore every open edge corresponding to a bordered cusp "protrudes" towards the interior of some face of the graph, and we have exactly one hole contained inside this face.

As a consequence of these facts, we can fix an orientation of the fat graph and of each open edge which allows us to determine a natural partition of the set of bordered cusps into nonintersecting (maybe empty) subsets δ k , k = 1, . . . , s of cusps incident to the corresponding holes, and to set a cyclic ordering in every such subset. This means that all arcs in the lamination are uniquely determined by 4 indices, telling us in which cusp they originate and terminate and in what order they enter or exit the cusp. For example, if we orient the fat graph in Fig. 6 counterclockwise so that the arc d originates in cusp 2 and is the first arc in that cusp, then it terminates in cusp 1 and it is the eight arc in that cusp (we count arcs starting from the side of open edge that goes into the fat-graph), we can denote d = g 21, [START_REF] Gross | Mirror symmetry for log Calabi-Yau surfaces I[END_REF] . Analogously b = g 13,14 and so on. The formula for the Poisson brackets is then completely combinatorial: (3.11) {g si,tj , g pr ,q l } = g si,tj g pr ,q l ǫ i-r δ s,p + ǫ j-r δ t,p + ǫ i-l δ s,q + ǫ j-l δ t,q 4 , where ǫ k := sign(k).

In [START_REF] Chekhov | Colliding holes in Riemann surfaces and quantum cluster algebras[END_REF] it is proved that the abstract bracket defined by (3.11) is indeed a Poisson bracket.

Decorated character variety

The classical character varieties are moduli spaces of monodromy data of regular or singular connections, which can be considered like representation spaces of the fundamental group of a Riemann surface. N. Hitchin proved that they are endowed with a holomorphic symplectic structure [START_REF] Hitchin | Frobenius manifolds (with notes by David Calderbank)[END_REF].

It is well-known that so-called Stokes data should be added to the classical monodromy in the case of non-fuchsian irregular singularities. That is why we want to generalise the previous representation space description to define an appropriate generalisation of the classical (or "tame") character variety.

Various descriptions of generalised character varieties as spaces of representations of a "wild fundamental groupoid" [START_REF] Paul | Dynamics on Wild Character Varieties[END_REF], "Stokes groupoid" [START_REF] Gualtieri | The Stokes groupoids[END_REF] or as "fissions" varieties of Stokes representations associated with a complex reductive linear algebraic group G [START_REF] Boalch | Geometry and braiding of Stokes data; fission and wild character varieties[END_REF].

In this paper we propose a different notion of decorated character variety which is based on the combinatorial description of Teichmüller space explained in the previous section. Our construction is based on the fact that topologically speaking a Riemann surface Σ g,s,n with n holes, with s bordered cusps is equivalent to a Riemann surface Σg,s,n of genus g, with s holes and n marked points m 1 , . . . , m n on the boundaries.

We introduce the fundamental groupoid of arcs A as the set of all directed paths γ ij : [0, 1] → Σg,s,n such that γ ij (0) = m i and γ ij (1) = m j modulo homotopy. The groupoid structure is dictated by the usual path-composition rules.

In this groupoid we have

n subgroups Π j = {γ jj |γ jj : [0, 1] → Σg,s,n , γ jj (0) = m j , γ jj (1) = m j }.
Each of these subgroups is isomorphic to the usual fundamental group and Π j = γ -1 ij Π i γ ij for any arc γ ij ∈ A. Now, we use the geometry: using the decoration at each cusp, we associate to each arc γ ij a matrix M ij ∈ SL 2 (R) as explained in the previous section, for example

M ij = X(k j )LX(z n )R • • • LX(z 1 )RX(k i ).
In order to associate a matrix in SL 2 (C), we complexify the coordinates Z i ∈ C in all formulas. We define two different characters:

Tr K : SL 2 (C) → C M → Tr(M K
) and the usual character (i.e. trace) which is only defined for the images of Π i , i = 1, . . . n. Finally, we define the decorated character variety as

s 3 p 3 s 1 p 1 s 2 p 2
Hom (A, { Tr K (M ij )} ∪ { Tr (M ii )})
To equip the decorated character variety with a Poisson bracket, we extend Poisson brackets (3.10) to the complexified shear coordinates. For Z i , Z j ∈ C we postulate

{Z i , Z j } = {Z i , Z j } := {Z i , Z j } R and {Z i , Z j } ≡ 0 or, explicitly, {ℜZ i , ℜZ j } = -{ℑZ i , ℑZ j } = 1
2 {Z i , Z j } R , {ℜZ i , ℑZ j } ≡ 0 where we let {Z i , Z j } R denote the (constant) Poisson bracket (3.10). All formulas for Poisson brackets between characters then remain valid irrespectively on whether we consider real or complexified generalised shear coordinates Z i .

Decorated character varieties and Painlevé monodromy manifolds

In the case of a Riemann sphere with 4 holes and no cusps, the fat graph is given in Fig. 4 and the three geodesics lengths x 1 , x 2 , x 3 of the thee geodesics which go around two holes without self-intersections are enough to close the Poisson algebra.

By following the rules explained in Section 3, the following parameterization of x 1 , x 2 , x 3 in shear coordinates on the fat-graph of a 4-holed sphere was found in [START_REF] Chekhov | Shear coordinates on the versal unfolding of the D 4 singularity[END_REF]:

x 1 = -e s2+s3+ p 2 2 + p 3 2 -e -s2-s3-p 2 2 - p 3 2 -e s2-s3+ p 2 2 - p 3 2 -G 2 e -s3-p 3 2 -G 3 e s2+ p 2 , x 2 = -e s3+s1+ p 3 2 + p 1 2 -e -s3-s1-p 3 2 - p 1 2 -e s3-s1+ p 3 2 - p 1 2 -G 3 e -s1-p 1 2 -G 1 e s3+ p 3 , x 3 = -e s1+s2+ p 1 2 + p 2 2 -e -s1-s2-p 1 2 - p 2 2 -e s1-s2+ p 1 2 - p 2 2 -G 1 e -s2-p 2 2 -G 2 e s1+ p 1 , (5.12)
where

G i = e p i 2 + e -p i 2 , i = 1, 2, 3,
and

G ∞ = e s1+s2+s3+ p 1 2 + p 2 2 + p 3 2 + e -s1-s2-s3-p 1 2 - p 3 2 - p 3 2 .
Note that by complexifying s 1 , s 2 , s 3 , p 1 , p 2 , p 3 , we obtain a parameterisation of the P V I cubic, i.e. of the character variety of SL 2 (C) character variety of a Riemann sphere with 4 holes.

We are now going to produce a similar coordinate description of each of the other Painlevé cubics. We will provide a geometric description of the corresponding Riemann surface and its fat-graph and discuss the corresponding decorated character variety.

5.1. Shear coordinates for P V . The confluence from the cubic associated to P V I to the one associated to P V is realised by

p 3 → p 3 -2 log[ǫ],
in the limit ǫ → 0. We obtain the following shear coordinate description for the P V cubic:

x 1 = -e s2+s3+ p 2 2 + p 3 2 -G 3 e s2+ p 2 2 , x 2 = -e s3+s1+ p 3 2 + p 1 2 -e s3-s1+ p 3 2 - p 1 2 -G 3 e -s1-p 1 2 -G 1 e s3+ p 3 2 , x 3 = -e s1+s2+ p 1 2 + p 2 2 -e -s1-s2-p 1 2 - p 2 2 -e s1-s2+ p 1 2 - p 2 2 -G 1 e -s2-p 2 2 -G 2 e s1+ p 1 2 , (5.13)
where

G i = e p i 2 + e -p i 2 , i = 1, 2, G 3 = e p 3 2 , G ∞ = e s1+s2+s3+ p 1 2 + p 2 2 + p 3 2 .
These coordinates satisfy the following cubic relation:

x 1 x 2 x 3 + x 2 1 + x 2 2 -(G 1 G ∞ + G 2 G 3 )x 1 -(G 2 G ∞ + G 1 G 3 )x 2 - -G 3 G ∞ x 3 + G 2 ∞ + G 2 3 + G 1 G 2 G 3 G ∞ = 0. (5.14)
Note that the parameter p 3 is now redundant, we can eliminate it by rescaling. To obtain the correct P V cubic, we need to pick

p 3 = -p 1 -p 2 -2s 1 -2s 2 -2s 3 so that G ∞ = 1.
Geometrically speaking, sending the perimeter p 3 to infinity means that we are performing a chewing-gum move: two holes, one of perimeter p 3 and the other of perimeter s 1 + s 2 + s 3 + p1 2 + p2 2 + p3 2 , become infinite, but the area between them remains finite, thus leading to a Riemann sphere with three holes and two cusps on one of them. In terms of the fat-graph, this is represented by Fig. 5.

The geodesic x 3 corresponds to the closed loop obtained going around p 1 and p 2 (green and red loops), while x 1 and x 2 are arcs starting at one cusp, going around p 1 and p 2 respectively, and coming back to the other cusp.

As explained in Section 3, according to [START_REF] Chekhov | Colliding holes in Riemann surfaces and quantum cluster algebras[END_REF], the Poisson algebra related to the character variety of a Riemann sphere with three holes and two cusps on one boundary is 7-dimensional. The fat-graph admits a complete cusped lamination as displayed in Fig. 6 so that a full set of coordinates on the character variety is given by the complexification of the five elements in the lamination and of the two parameters G 1 and G 2 which determine the perimeter of the two non-cusped holes.

Notice that there are two shear coordinates associated to the two cusps, they are denoted by k 1 and k 2 , their sum corresponds to what we call p 3 in (5.13). These shear coordinates do not commute with the other ones, they in fact satisfy the following relations: As a consequence, the elements G 3 and G ∞ are not Casimirs in this Poisson algebra, despite being frozen variables in the cluster algebra setting (see Section 6) In terms of shear coordinates, the elements in the lamination correspond to two loops (whose hyperbolic cosin length is denoted by G 1 and G 2 respectively) and five arcs whose lengths are expressed as follows: {b, c} = 0, {b, d} = -1 2 bd, {b, e} = 1 2 be, (5.17)

{s 3 , k 1 } = {k 1 , k 2 } = {k 2 , s 3 } = 1. s 3 k 1 k 2 s 1 p 1 s 2 p 2
a = e k1+s1+2s2+s3+ p 1 2 +p2 , b = e k1+s2+s3+ p 2 2 , e = e k 1 2 + k 2 2 , c = e k1+s1+s2+s3+ p 1 2 + p 2 2 , d = e k 1 2 + k 2 2 +s1+s2+s3+ p 1 2 + p 2
{c, d} = - 1 2 cd, {c, e} = 1 2 ce, {d, e} = 0, {G 1 , •} = {G 2 , •} = 0, (5.18) so that the elements G 1 , G 2 and G 3 G ∞ = de are central.
The generic family of symplectic leaves are determined by the common level set of the three Casimirs G 1 , G 2 and G 3 G ∞ = de and are 4-dimensional (rather than 2-dimensional like in the P V I case).

On each symplectic leaf, the P V monodromy manifold (5.14) is the subspace defined by those functions of a, b, c, d (and of the Casimir values G 1 , G 2 , G 3 G ∞ = de) which commute with the frozen variables, i.e. with G 3 = e (and therefore with d as well, since de is a Casimir). To see this, we can use relations (5.15) to determine the exponentiated shear coordinates in terms of a, b, c, d, and then deduce the expressions of x 1 , x 2 , x 3 in terms of the lamination. We obtain the following expressions:

x 1 = -e a c -d b c , x 2 = -e b c -G 1 d b a -d b 2 ac -d c a , (5.19) x 3 = -G 2 c b -G 1 c a - b a - c 2 ab - a b , (5.20)
which automatically satisfy (5.14).

Due to the Poisson relations (5.16) the functions that commute with e are exactly the functions of a b , b c , c a , d. Such functions may depend on the Casimir values G 1 , G 2 and G 3 G ∞ and e itself, so that d = G ∞ becomes a parameter now. With this in mind, it is easy to prove that x 1 , x 2 , x 3 are algebraically independent functions of a b , b c , c a , d so that x 1 , x 2 , x 3 form a basis in the space of functions which commute with e. 5.2. Shear coordinates for P V deg . The confluence from P V to P V deg is realised by the substitution s 3 → s 3 -log[ǫ], in formulae (5.13). In the limit ǫ → 0 we obtain:

x 1 = -e s2+s3+ p 2 2 + p 3 2 , x 2 = -e s3+s1+ p 3 2 + p 1 2 -e s3-s1+ p 3 2 - p 1 2 -G 1 e s3+ p 3 2 , x 3 = -e s1+s2+ p 1 2 + p 2 2 -e -s1-s2-p 1 2 - p 2 2 -e s1-s2+ p 1 2 - p 2 2 -G 1 e -s2-p 2 2 -G 2 e s1+ p 1 2 , (5.21)
where

G i = e p i 2 + e -p i 2 , i = 1, 2, G ∞ = e s1+s2+s3+ p 1 2 + p 2 2 + p 3
2 . These coordinates satisfy the following cubic relation:

(5.22) x 1 x 2 x 3 + x 2 1 + x 2 2 -G 1 G ∞ x 1 -G 2 G ∞ x 2 + G 2 ∞ = 0.
Note that the parameter p 3 is now redundant, we can eliminate it by rescaling. To obtain the correct P V deg cubic, we need to pick

p 3 = -p 1 -p 2 -2s 1 -2s 2 -2s 3 .
Geometrically speaking, sending the shear coordinate s 3 to infinity means that we are performing a cusp-removing move. In terms of the fat-graph, this is represented by Fig. 7.

The character variety of a Riemann sphere with three holes and one cusp on one boundary is 5-dimensional. The fat-graph admits a complete cusped lamination so that a full set of coordinates on the character variety is given by the complexification of the geodesic length functions of the elements in the lamination. Now we have only one shear coordinate associated to the cusp, denoted by s 3 , which does not commute with the other shear coordinates.

s 3 s 1 p 1 s 2 p 2
We omit the picture of the P V deg lamination as it is very similar to Fig. 6, in which the edges labelled by k 1 and k 2 are removed and the geodesics d and e are lost.

In terms of shear coordinates, the elements in the lamination are two loops corresponding to the parameters G 1 and G 2 and three arcs for which the lengths are expressed as follows:

(5.23)

a = e s1+2s2+s3+ p 1 2 +p2 , b = e s2+s3+ p 2 2 , c = e s1+s2+s3+ p 1 2 + p 2 2 ,
They satisfy the following Poisson relations, which can be deduced by formula (3.11):

(5.24) {a, b} = ab, {a, c} = 0, {b, c} = 0, so that the element c is a Casimir as well as the parameters G 1 , G 2 . Each symplectic leaf is two-dimensional and corresponds to the P V deg monodromy manifold (5.22). Indeed, we can use relations (5.23) to determine the exponentiated shear coordinates in terms of a, b, c, and then deduce he expressions of x 1 , x 2 , x 3 in terms of the lamination. We obtain the following expressions:

x 1 = -b, x 2 = -G 1 bc a - b 2 a - c 2 a , (5.25) x 3 = -G 2 c b -G 1 c a - b a - c 2 ab - a b , (5.26)
which automatically satisfy (5.22).

In terms of lamination, the confluence from P V to P V deg is given by the following rules:

a → a, b → b, c → c, d → c, e → 0.
p 2 → p 2 -2 log[ǫ],
in formulae (5.13). In the limit ǫ → 0 we obtain:

x 1 = -e s2+s3+ p 2 2 + p 3 2 -G 3 e s2+ p 2 2 , x 2 = -e s3+s1+ p 3 2 + p 1 2 -e s3-s1+ p 3 2 - p 1 2 -G 3 e -s1-p 1 2 -G 1 e s3+ p 3 2 , x 3 = -e s1+s2+ p 1 2 + p 2 2 -G 2 e s1+ p 1
2 , (5. [START_REF] Sakai | Rational Surfaces Associated with Affine Root Systems and Geometry of the Painlevé Equations[END_REF] where

G 1 = e p 1 2 + e -p 1 2 , G 2 = e + p 2 2 , G 3 = e + p 3 2 , G ∞ = e s1+s2+s3+ p 1 2 + p 2 2 + p 3
2 . These coordinates satisfy the following cubic relation:

x 1 x 2 x 3 + x 2 1 -(G 1 G ∞ + G 2 G 3 )x 1 -G 2 G ∞ x 2 - -G 3 G ∞ x 3 + G 2 ∞ + G 1 G 2 G 3 G ∞ = 0. (5.28)
Note that the parameters p 3 , p 2 are now redundant, we can eliminate it by rescaling. To obtain the correct P IV cubic, we need to pick

p 2 = p 3 = -p 1 -2s 1 -2s 2 -2s 3 so that G 2 = G 3 = G ∞ .
Similarly to the previous case, this means that we send the perimeter p 2 to infinity, which is a chewing-gum move leading to a Riemann sphere with two holes, one of which has 4 cusps on it. The corresponding fat-graph is given in Fig. 8, where we see 4 new shear coordinates, one for each cusp, so that in formulae (5.27)

p 2 = k 3 + k 4 and p 3 = k 1 + k 2 .
The character variety is now 8 dimensional and the complete cusped lamination is given in Fig. 9.

In terms of shear coordinates, the elements in the lamination are expressed as follows:

a = e s1+s2+k3+ p 1 2 , b = e s 2 2 + s 3 2 + k 1 2 + k 4 2 , c = e s1+ s 2 2 + s 3 2 + p 1 2 + k 1 2 + k 3 2 , (5.29) d c a f h b e k 1 k 2 k 3 k 4 s 3 s 1 s 2 p 1
d = e s1+ s 2 2 + s 3 2 + p 1 2 + k 2 2 + k 3 2 , e = e k 1 2 + k 2 2 , f = e s 2 2 + s 3 2 + k 1 2 + k 3 2 , g = e k 3 2 + k 4 2 .
We omit the formulae for the Poisson brackets as these can be easily extracted from (3.11). Let us notice that the element bdeh is a Casimir as well as the perimeter G 1 . Each symplectic leaf is six-dimensional and the PIV monodromy manifold (5.28) is the subspace of those functions of a, b, . . . , g which commute with e and g. The proof of this statement is quite similar to the previously considered analogous assertion for the PV-case and we omit it. 5.4. Shear coordinates for P III. The confluence from P V to P III is obtained by the following substitution:

s 1 → s 1 + 2 log[ǫ], p 2 → p 2 -2 log[ǫ], p 1 → p 1 -2 log[ǫ].
In the limit as ǫ → 0 we obtain:

x 1 = -e s2+s3+ p 2 2 + p 3 2 -G 3 e s2+ p 2 2 , x 2 = -e s3-s1+ p 3 2 - p 1 2 -G 3 e -s1-p 1 2 -G 1 e s3+ p 3 2 , x 3 = -e s1+s2+ p 1 2 + p 2 2 -e -s1-s2-p 1 2 - p 2 2 -G 1 e -s2-p 2 2 -G 2 e s1+ p 1 2 , (5.30)
where

G i = e p i 2 , i = 1, 2, 3 G ∞ = e s1+s2+s3+ p 1 2 + p 2 2 + p 3
2 . These coordinates satisfy the following cubic relation:

(5.31) x 1 x 2 x 3 +x 2 1 +x 2 2 -( G 1 G ∞ + G 2 G 3 )x 1 -( G 2 G ∞ + G 1 G 3 )x 2 + G 1 G 2 G 3 G ∞ = 0.
We can pick p 2 = p 3 = 0 in order to obtain the correct P III cubic. Note that there is a slight discrepancy between the G i s in the cubic (5.31) and the G i s dictated by our formulae (2.4). This is easily solved by a simple transformation

G ∞ = G 1 G ∞ , G 1 = G 1 G ∞ + 1 G 1 G ∞ , G 2 = G ∞ G 1 + G 1 G ∞ .
To understand the geometry of this confluence, we first need to flip the P V fat-graph to the equivalent graph given in Fig. 10. The new shear coordinates are expressed in terms of the old ones as follows:

ŝ1 = -s 1 -p 1 -log(1 + e s2 ), ŝ2 = -s 2 + log (1 + e p1+s1 + e p1+s1+s2 )(1 + e s1 + e s1+s2 ) , ŝ3 = s 3 -log(1 + e -s2
), p1 = p 1 , (5.32) p2 = p 2 + s 2 + p 1 + 2s 1 + 2 log(1 + e s2 ) + + log (1 + e p1+s1 + e p1+s1+s2 )(1 + e s1 + e s1+s2 ) .

Remark 5.1. Note that this flip is obtained by composing two mapping class group transformations described in Figures 3 and4 of [START_REF] Chekhov | Colliding holes in Riemann surfaces and quantum cluster algebras[END_REF]. This means that the fat-graph of P V is mapped to an intermediate fat-graph which does not satisfy the property that the polygons containing holes with no cusps are monogons. This is not a problem as in fact we can map the lamination in Fig. 6 to this new fat-graph and then to Fig. 10.

In the new shear coordinates the substitution (5.30) becomes simply:

p1 → p1 -2 log[ǫ],
which geometrically speaking corresponds to the fat-graph in Fig. 11, where we see 4 new shear coordinates, one for each cusp, so that p2 = k3 + k4 and p1 = k5 + k6 . This is the fat-graph of a Riemann sphere with two holes each of them with two cusps. Note that the coordinates in Fig. 11 are the true shear coordinates, namely they satisfy the Poisson brackets:

{ k2 , ŝ3 } = {ŝ 3 , k1 } = { k1 , k2 } = {ŝ 3 , ŝ2 } = {p 2 , ŝ3 } = 1, {ŝ 2 , p2 } = 2, {ŝ 1 , ŝ2 } = {p 2 , ŝ1 } = {ŝ 1 , k5 } = { k6 , ŝ1 } = { k5 , k6 } = 1, (5.
k1 = k 1 , k2 = k 2 , k5 = k 5 , k6 = k 6 , p2 = p 2 + s 2 + k 5 + k 6 + 2s 1 + 2 log(1 + e s2 ) + (5.34) + log (1 + e k5+k6+s1 + e k5+k6+s1+s2 )(1 + e s1 + e s1+s2 .
to go back to s 1 , s 2 , s 3 , p 2 , k 1 , k 2 , k 5 , k 6 , we see that k 5 , k 6 have non standard Poisson brackets with s 1 , s 2 , s 3 . This is due to the fact that this limiting transformation (5.34) destroys the geometry, as it essentially maps from a Riemann sphere with two holes each of them with two cusps to a Riemann sphere with two holes one of which has 4 cusps, and the other has no cusps (the P IV case). This implies that the correct coordinates to describe the character variety of a Riemann sphere with two holes each of them with two cusps are the complexified ŝ1 , ŝ2 , ŝ3 , p2 , k1 , k2 , k5 , k6 .

This character variety is 8-dimensional. The fat-graph admits a complete cusped lamination as displayed in Fig. 12 so that a full set of coordinates on the character variety is given by the eight complexified elements in the lamination.

In terms of the shear coordinates ŝ1 , ŝ2 , ŝ3 , p2 , k1 , k2 , k5 , k6 the elements in the P III lamination are expressed as follows: .

a = e
The Poisson brackets admit two Casimirs, de and hg so that the symplectic leaves are 6-dimensional. Within each symplectic leaf, the P III monodromy manifold is given by the set of functions which commute with d, e, g, h. To see this, we proceed in the same way as in the case of PV. On the monodromy manifold we set e = G3 , d = G∞ , h = G2 and g = G1 .

Note that, unlike the cases of PV (Fig. 6), PIV (Fig. 9), and PII (Fig. 19 below), the expressions for the λ-lengths of arcs in (5.35) are not monomials in the exponentiated shear coordinates. This is because, unlike the other named cases, the fat graph in Fig. 12 is not dual to the maximum cusped lamination {a, b

We obtain the dual graph out of the one in Fig. 12 if we first flip the edge ŝ1 subsequently flipping the edge ŝ2 . We depict the resulting fat graph and lamination in III system depicted on the fat graph dual to this variety.

It is interesting to construct the lamination and pattern in Fig. 12 as a limit of the corresponding system of P V . When we flip the fat-graph for P V , the lamination is flipped too as in Fig. 14.

When we open the inside hole to obtain the fat-graph for PIII, the P V arcs a and b break giving rise to the new arcs a, b and f and G 1 breaks up giving rise to g so that 5.5. Shear coordinates for P III D7 . The confluence from the generic P III cubic (5.31) to the P III D7 one is realised by the substitution

a → af, b → bf, c → c, d → d, e → e, G 2 → h, G 1 → g.
s 3 → s 3 -log[ǫ],
in formulae (5.30). in the limit ǫ → 0 we obtain:

x 1 = -e s2+s3+ p 2 2 + p 3 2 , x 2 = -e s3-s1+ p 3 2 - p 1 2 -G 1 e s3+ p 3 2 , x 3 = -e s1+s2+ p 1 2 + p 2 2 -e -s1-s2-p 1 2 - p 2 2 -G 1 e -s2-p 2 2 -G 2 e s1+ p 1 2 , (5.37)
where

G i = e p i 2 , i = 1, 2, G 3 = 0, G ∞ = e s1+s2+s3+ p 1 2 + p 2 2 + p 3
2 . These same expressions can equivalently obtained by the substitution:

s 1 → s 1 + 2 log(ǫ), p 1 → p 1 -2 log(ǫ), p 2 → p 2 -2 log(ǫ),
in formulae (5.21) and by taking the limit as ǫ → 0. These coordinates satisfy the following cubic relation:

(5.38)

x 1 x 2 x 3 + x 2 1 + x 2 2 -G 1 G ∞ x 1 -G 2 G ∞ x 2 = 0. We can pick p 2 = p 3 = 0 and s 3 = -s 1 -s 2 -p1
2 in order to obtain the correct P III D7 cubic.

In terms of fat graph, we need to work with the coordinates (5.32), for which the confluence gives ŝ3 → ŝ3 -log[ǫ], which corresponds to the fat-graph in Fig. 15. This corresponds to a Riemann sphere with two holes, one with one cusp and one with two cusps. The character variety is 6 dimensional and we omit the picture of the P III D7 lamination as it is very similar to Fig. 12, in which the edges labelled by k 1 and k 2 are removed and the arcs d and e are lost. Again the Poisson brackets can be calculated using (3.11) and it is easy to check that there are two Casimirs so that the symplectic leaves are 4 dimensional. The P III D7 monodromy manifold is given by those functions of a, b, c, d, f, h that commute with h. 5.6. Shear coordinates for P III D8 . The confluence from the generic P III D7 cubic (5.38) to the P III D8 one is realised by the substitution

s 1 → s 1 + log[ǫ], p 2 → p 2 -2 log[ǫ]
in formulae (5.37). In the limit ǫ → 0 we obtain:

x 1 = -e s2+s3+ p 2 2 + p 3 2 , x 2 = -e s3-s1+ p 3 2 - p 1 2 , x 3 = -e s1+s2+ p 1 2 + p 2 2 -e -s1-s2-p 1 2 - p 2 2 -G 2 e s1+ p 1 2 , (5.39)
where

G 1 = 0, G 2 = e p 2 2 , i = 1, 2, G 3 = 0, G ∞ = e s1+s2+s3+ p 1 2 + p 2 2 + p 3 2 .
These coordinates satisfy the following cubic relation:

(5.40) x 1 x 2 x 3 + x 2 1 + x 2 2 -G 2 G ∞ x 2 = 0. We can pick p 2 = p 3 = 0 and s 3 = -s 1 -s 2 -p1
2 in order to obtain the correct P III D8 cubic.

In terms of fat graph, we need to work with the coordinates (5.32), for which the confluence gives ŝ1 → ŝ1 -log[ǫ], which corresponds to the fat-graph in Fig. 16. This corresponds to a Riemann sphere with two holes, each with one cusp on it. The character variety is 4 dimensional and the Poisson brackets can be calculated using (3.11). It is easy to check that there are two Casimirs so that the symplectic leaves are 2 dimensional and coincide with the P III D8 monodromy manifold. 5.7. Shear coordinates for P II JM . The confluence from the generic P IV cubic (5.28) to the P II JM cubic is realised by the substitution

p 1 → p 1 -2 log[ǫ],
in formulae (5.27). In the limit ǫ → 0 we obtain:

x 1 = -e s2+s3+ p 2 2 + p 3 2 -G 3 e s2+ p 2 2 , x 2 = -e s3+s1+ p 3 2 + p 1 2 -G 1 e s3+ p 3 2 , x 3 = -e s1+s2+ p 1 2 + p 2 2 -G 2 e s1+ p 1 2 , (5.41)
where

G i = e p i 2 , i = 1, 2, 3, G ∞ = e s1+s2+s3+ p 1 2 + p 2 2 + p 3
2 . These coordinates satisfy the following cubic relation:

x 1 x 2 x 3 -G 1 G ∞ x 1 -G 2 G ∞ x 2 - -G 3 G ∞ x 3 + G 2 ∞ + G 1 G 2 G 3 G ∞ = 0. (5.42)
Note that the parameters p 3 , p 2 , p 1 are now redundant, we can eliminate it by rescaling. To obtain the correct P II JM cubic, we need to pick p 2 = p 3 = 0 and p 1 = -2s 1 -2s 2 -2s 3 .

Again, this means that we send the perimeter p 1 to infinity, which is a chewinggum move leading to a Riemann sphere with one hole with 6 cusps on it. The corresponding fat-graph is given in Fig. 17 where we have 6 cusp shear coordinates k 1 , . . . , k 6 such that in formulae (5.41) we must set p 1 = k 5 + k 6 , p 2 = k 3 + k 4 and p 3 = k 1 + k 2 . The character variety is now 9-dimensional, with only one Casimir and the P II JM monodromy manifold is given by the set of functions that Poisson commute with p 1 = k 5 + k 6 , p 2 = k 3 + k 4 and p 3 = k 1 + k 2 . We omit all details in this case. 5.8. Shear coordinates for P II F N . The confluence from the generic P IV cubic (5.28) to the P II F N cubic is realised by the substitution

s 3 → s 3 -log[ǫ],
in formulae (5.27). In the limit ǫ → 0 we obtain:

x 1 = -e s2+s3+ p 2 2 + p 3 2 , x 2 = -e s3+s1+ p 3 2 + p 1 2 -e s3-s1+ p 3 2 - p 1 2 -G 1 e s3+ p 3 2 , x 3 = -e s1+s2+ p 1 2 + p 2 2 -G 2 e s1+ p 1 2 ,
where

G 1 = e p 1 2 + e -p 1 2 , G 2 = e p 2 2 , G 3 = 0, G ∞ = e s1+s2+s3+ p 1 2 + p 2 2 + p 3
2 .. They satisfy the following cubic relation (5.43) x

1 x 2 x 3 + x 2 1 -G 1 G ∞ x 1 -G 2 G ∞ x 2 + G 2 ∞ = 0.
Observe that we can obtain exactly the same formulate by starting from the generic P V deg cubic (5.22) by the substitution

p 2 → p 2 -2 log[ǫ],
and by taking the limit as ǫ → 0.

To obtain the P II F N cubic we pick p 2 = p 3 = 0 and p 1 = -2s 1 -2s 2 -2s 3 . Geometrically speaking, sending the shear coordinate s 3 to infinity means that we are performing a cusp-removing move. This gives a Riemann sphere with two holes, one of them having three cusps on its boundary. In terms of the fat-graph, this is represented by Fig. 18, where we wee three cusps of coordinates k 3 , k 4 and s 3 , so that in formulae (5.43) we must set p 2 = k 3 + k 4 . . The decorated character variety in this case is 6 dimensional. The lamination is given by the loop around the un-cusped-hole and the five arcs in Fig. 19.

d a f h b k 3 k 4 s 3 s 1 s 2 p 1
d = e s1+ s 2 2 + s 3 2 + p 1 2 + k 3 2 , f = e s 2 2 + s 3 2 + k 3 2 , g = e k 3 2 + k 4 2 .
To show that our decorated character variety is not the same as the wild character variety (see [START_REF] Boalch | Wild Character Varieties, points on the Riemann sphere and Calabi's examples[END_REF] for the P II F N case), we deal with this case in all details. The Poisson brackets among the complexified lamination arcs lengths are given by

{a, b} = {d, f } = 0, {a, d} = - ad 2 , {a, f } = af 2 , {a, h} = af 2 , {b, d} = bd 4 , {b, f } = bf 4 , {b, h} = - bh 4 , {d, h} = dh 4 , {f, h} = f h 4 .
It is easy to check that there are two Casimirs, G 1 and bdh, so that the symplectic leaves are 4-dimensional and the P II F N monodromy manifold (5.43) is the subspace of those functions of a, b, d, f, h which commute with h. To check that x 1 , x 2 and x 3 commute with h it is enough to express them in terms of the lamination:

x 1 = -bf, x 2 = -G 1 df a - d 2 a - f 2 a , x 3 = - dh f - ab f .
Vice versa all functions commuting with h must have the form a α b β d δ f φ where α, β, γ, δ, φ are some numbers satisfying 2α -β + δ + φ = 0. Using this fact, it is easy to prove that on each symplectic leaf the set of functions which commute with h is 2-dimensional.

5.9. Shear coordinates for P I. The confluence from the generic P II JM cubic to the P I one is realised by

s 3 → s 3 -log[ǫ],
in formulae (5.41). In the limit ǫ → 0 we obtain:

x 1 = -e s2+s3+ p 2 2 + p 3 2 , x 2 = -e s3+s1+ p 3 2 + p 1 2 -G 1 e s3+ p 3 2 , x 3 = -e s1+s2+ p 1 2 + p 2 2 -G 2 e s1+ p 1
2 , (5.45) where

G i = e p i 2 , i = 1, 2, G 3 = 0, G ∞ = e s1+s2+s3+ p 1 2 + p 2 2 + p 3 2 .
These coordinates satisfy the following cubic relation:

(5.46) x 1 x 2 x 3 -G 1 G ∞ x 1 -G 2 G ∞ x 2 + G 2 ∞ = 0.
Observe that we can obtain exactly the same formulate by starting from the generic P II F N cubic (5.43) in Flashcka-Newell form by the substitution

p 1 → p 1 -2 log[ǫ],
and by taking the limit as ǫ → 0. Note that the parameters p 3 , p 2 , p 1 and s 3 are now redundant, we can eliminate them by rescaling. We pick p 1 = p 2 = p 3 = 0 and s 3 = -s 1 -s 2 , we obtain the correct P I by changing the sign of x 1 and x 2 .

Geometrically speaking, sending the shear coordinate s 3 to infinity means that we are performing a cusp-removing move. In terms of the fat-graph, this is represented by Fig. 20.

The character variety is now 7-dimensional, with only one Casimir and the P I monodromy manifold is given by the set of functions that Poisson commute with p 1 = k 5 + k 6 , p 2 = k 3 + k 4 . We omit all details in this case.

s 1 → s 1 -log[ǫ],
in formulae (5.45). In the limit ǫ → 0 we obtain:

x 1 = -e s2+s3+ p 2 2 + p 3 2 , x 2 = -e s3+s1+ p 3 2 + p 1 2 , x 3 = -e s1+s2+ p 1 2 + p 2 2 -G 2 e s1+ p 1 2 , (5.47) where G 2 = e p 2 2 , G 1 = G 3 = 0, G ∞ = e s1+s2+s3+ p 1 2 + p 2 2 + p 3
2 . These coordinates satisfy the following cubic relation:

(5.48) x 1 x 2 x 3 -G 2 G ∞ x 2 + G 2 ∞ = 0.
Note that the parameters p 3 , p 2 , p 1 and s 3 are now redundant, we can choose p 1 = p 2 = p 3 = 0 and s 3 = -s 1 -s 2 so that G 2 = G ∞ = 1. This gives us the following cubic (5.49)

x 1 x 2 x 3 -x 2 + 1 = 0.

In order to relate this cubic to the Weiestrass elliptic curve, we need to projectivise it first:

(5.50) x 1 x 2 x 3 -x 2 x 2 0 + x 3 0 = 0. This cubic is now invariant under the following transformation

x 0 → αx 0 , x 1 → βx 1 , x 2 → αx 2 , x 3 → α 2 β x 3 ,
so that we can rescale x 2 → 1 and x 3 → x 1 , leading to the Weiestrass elliptic curve:

x 2 1 -x 2 0 + x 3 0 = 0. 6. Painlevé cluster algebras: braid-group and affine MCG actions 6.1. Painlevé VI: analytic continuation and cluster mutations. In [START_REF] Dubrovin | Monodromy of certain Painlevé-VI transcendents and reflection group[END_REF][START_REF] Mazzocco | Rational solutions of the Painleve' VI equation, Kowalevski Workshop on Mathematical Methods of Regular Dynamics[END_REF] it was proved that the procedure of analytic continuation of a local solution to the sixth Painlevé equation corresponds to the following action of the braid group on the monodromy manifold: (6.51) β 1 :

x 1 → -x 1 -x 2 x 3 + ω 1 , x 2 → x 3 , x 3 → x 2 , (6.52) β 2 : x 1 → x 3 , x 2 → -x 2 -x 1 x 2 + ω 2 , x 3 → x 1 , (6
.53) β 3 :

x 1 → x 2 , x 2 → x 1 , x 3 → -x 3 -x 1 x 2 + ω 3 .
In [START_REF] Chekhov | Shear coordinates on the versal unfolding of the D 4 singularity[END_REF] it was shown that flips on the shear coordinates correspond to the action of the braid group on the cubic. The flips f 1 , f 2 , f 3 of the shear coordinates which give rise to the braid transformations β 1 β 2 and β 3 respectively have the following form (6.54) f 1 :

s 1 → -p 1 -s 1 , p 2 → p 3 , p 3 → p 2 , s 2 → s 3 + log [1 + e s1 ] + log [1 + e s1+p1 ] , s 3 → s 2 -log [1 + e -s1 ] + log [1 + e -s1-p1 ] , (6.55) f 2 : s 1 → s 3 -log [1 + e -s2 ] -log [1 + e -s2-p2 ] , s 2 → -p 2 -s 2 , p 1 → p 3 , p 3 → p 1 , s 3 → s 1 + log [1 + e s2 ] + log [1 + e s2+p2 ] ,
(6.56) f 3 :

s 1 → s 2 + log [1 + e s3 ] + log [1 + e s3+p3 ] , s 2 → s 1 -log [1 + e -s3 ] -log [1 + e -s3-p3 ] , s 3 → -p 3 -s 3 p 1 → p 2 , p 2 → p 1 .
Remark 6.1. Observe that in [START_REF] Chekhov | Teichmüller spaces of Riemann surfaces with orbifold points of arbitrary order and cluster variables[END_REF] it was proved that shear coordinate flips (6.54), (6.55), (6.56) are indeed dual to the generalised cluster mutations (6.59) for the corresponding λ-lengths.

We are now going to show that when G ∞ = 2 (geometrically this means that we have a puncture at infinity), the action of the braid group coincides with a generalised cluster algebra structure [START_REF] Chekhov | Teichmüller spaces of Riemann surfaces with orbifold points of arbitrary order and cluster variables[END_REF].

In order to see this let us compose each braid with a Okamoto symmetry in order to obtain the following (6.57)

β i : x i → -x i -x j x k + ω i , j, k = i, x j → x j , for j = i
By using (2.2) this transformation acquires a cluster flavour: (6.58)

β i : x i x ′ i = x 2 j + x 2 k + ω j x j + ω k x k + ω 4 j, k = i.

Indeed let us introduce the shifted variables:

y i := x i -G i , i = 1, 2, 3,
they satisfy the generalised cluster algebra relation: (6.59) µ i : y i y ′ i = y 2 j + y 2 k + G i y j y k j, k = i. Note that generalised cluster algebras satisfy the Laurent phenomenon. In particular this result implies that procedure of analytic continuation of the solutions to the sixth Painlevé equation satisfies the Laurent phenomenon: if we start from a local solution corresponding to the point (y 0 1 , y 0 2 , y 0 3 ) on the shifted Painlevé cubic y 1 y 2 y 3 + y 2 1 + y 2 2 + y 2 3 + G 1 y 2 y 3 + G 2 y 1 y 3 + G 3 y 1 y 2 = 0 any other branch of that solution will corresponds to points (y 1 , y 2 , y 3 ) on the same cubic such that each y i is a Laurent polynomial of the initial (y 0 1 , y 0 2 , y 0 3 ).

6.2. Generalised cluster algebra structure for P V and P V deg . In this case, we have a Riemann surface Σ 0,3,2 with two bordered cusps on one hole. The only nontrivial Dehn twist is around the closed geodesic γ encircling these two holes (this geodesic is unique). We now consider the effect of this MCG transformation on the system of arcs in Fig. 6.

e d c b a γ ω 2 ω 1 M a e d c b a ′ ω 2 ω 1 M b e d b ′ a ′ c ω 2 ω 1
Here the generalized mutations M a and M b are given by the formulas

a ′ a = b 2 + c 2 + ω 1 bc; b ′ b = (a ′ ) 2 + c 2 + ω 2 a ′ c, or, explicitly, (6.60) a b →    b 2 + c 2 + ω 1 bc a (b 2 + c 2 + ω 1 bc) 2 a 2 b + ω 2 c b 2 + c 2 + ω 1 bc ba + c 2 b    .
The geodesic function of γ is (6.61)

G γ = ω 2 c b + ω 1 c a + a b + b a + c 2
ab and this function is the so-called Hamiltonian MCG invariant: it generates the corresponding Dehn twist (see [START_REF] Kashaev | On the spectrum of Dehn twists in quantum Teichmüller theory[END_REF]), has nontrivial Poisson brackets with a and b, and is preserved by the MCG action (6.60).

In the case of P V deg , all the above formulas remain valid provided we replace c by the λ-length d of the boundary arc. 6.3. Generalised cluster algebra structure for P III D6 , P III D7 , and P III D8 . In all cases of P III, we have a Riemann surface Σ 0,2,n with n 1 > 0 and n 2 > 0, n 1 + n 2 = n, bordered cusps on the respective holes. For any n 1 and n 2 , the only nontrivial Dehn twist is around the closed geodesic γ separating the holes (this geodesic is unique). Its geodesic function G γ is the Hamiltonian MCG invariant. Besides this invariant, we have (non-Hamiltonian) invariants, which are λ-lengths of all arcs starting and terminating at the same boundary component.

We begin with the case of P III D6 and consider the MCG action on the system of arcs in Fig. 12: (6.64)

The case of P III D7 coincides with that of P III D6 , the geodesic λ 1,1 now becomes the boundary geodesic after erasing the bordered cusp 2.

In the case of P III D8 we erase bordered cusps 2 and 3; the only MCG transformation is For example P IV is the equation corresponding to E 6 and it has a two singular fibres with singularity of type A 2 and at three singular fibres with singularity of type A 1 .

The scope of this section is to show that the non singular fibres of each family of affine cubics are locally diffeomorphic to the versal unfolding [START_REF] Arnol | Critical points of smooth functions and their normal forms[END_REF] of the singularity of the type given in the second column of the table .   A.3.1. D 4 . This case corresponds to the sixth Painlevé equation. The cubic in this case is (we drop the indices ( D 4 ) for convenience):

(A.1)

x 1 x 2 x 3 + x 2 1 + x 2 2 + x 2 3 + ω 1 x 1 + ω 2 x 2 + ω 3 x 3 + ω 4 = 0. To show that this is diffeomorphic to the versal unfolding of D 4 we need to map this cubic to Arnol'd form. To this aim we first shift all variables by x i → x i + 2, i = 1, 2, 3 to obtain (A.2) x 2 1 +x 2 2 +x 2 3 +2x 1 x 2 +2x 2 x 3 +2x 1 x 3 +x 1 x 2 x 3 + ω 1 x 1 + ω 2 x 2 + ω 3 x 3 + ω 4 = 0, where ω i = ω i + 8, for i = 1, 2, 3, ω 4 = ω 4 + 2(ω 1 + ω 2 + ω 3 ) + 20.

As a second step we use the following diffeomorphism around the origin:

x → x - 1 2 y, y → x + 1 2 x, z → z + y 2 8 -2x - x 2 2
ω 3 2 so that the new cubic (up to a Morse singularity that we throw away and after a shift x → x-ω3 4 ) becomes indeed the versal unfolding of a D 4 singularity in Arnol'd form:

-2x 3 1 +

x 1 x 2 2 2 + ω 1 x 1 + ω 2 x 2 + ω 3 x 2 1 + ω 4 , where

ω 1 = ω 1 + ω 2 -8 -4ω 3 - ω 2 3 8 , ω 2 = ω 2 -ω 1 2 ,

Figure 2 .

 2 Figure 2. The process of breaking up a Riemann surface with boundary cusps: by grabbing together two cusps and pulling we tear apart an ideal triangle.

Figure 3 .

 3 Figure 3. The table of confluences of Riemann surfaces from the Painlevé perspective. The red arrows correspond to chewing-gum moves, the green ones to cusp removal.

  d = P V I, P V, P III, P V deg , P III D7 , P III D8 , P IV, P II F N , 0 for d = P II JM , P I, ǫ d = P V I, P V, P III, P V deg , P III D7 , P III D8 0 for d = P IV, P II F N , P II JM , P I, ǫ (d) 3 = 1 for d = P V I, 0 for d = P V, P III, P V deg , P III D7 , P III D8 , P IV, P II F N , P II JM , P I.

  while

  cos πθ ∞ d = P V I, P IV 1 d = P V, P V deg , D 8 , P II JM , P II F N , P I e iπθ0 d = P III 0 d = P III D7 , P III D8 .

Figure 4 .

 4 Figure 4. The fat graph of the 4 holed Riemann sphere. The red dashed geodesic is x 1 .

Figure 5 . 1 Figure 6 .

 516 Figure 5. The fat graph corresponding to PV. The geodesic x 3 remains closed, while x 1 (see red arc) and x 2 become arcs.

  following Poisson relations, which can be deduced by formula (3.11): {a, b} = ab, {a, c} = 0, {a, d} = -1 2 ad, {a, e} = 1 2 ae, (5.16)

Figure 7 .

 7 Figure 7. The fat graph corresponding to P V deg . The red arc corresponds to x 1 .

2 Figure 8 . 5 . 3 .

 2853 Figure 8. The fat graph corresponding to PIV. The variable x 1 now corresponds to the product of the two red arcs.

Figure 9 .

 9 Figure 9. The system of arcs for PIV.

Figure 10 .

 10 Figure 10. The flipped fat graph corresponding to PV. The red arc corresponds to x 1 .

Figure 11 .

 11 Figure 11. The fat graph corresponding to PIII.

Figure 12 .

 12 Figure 12. The character variety for the P D 6 III system.

Fig. 13 ..Figure 13 .

 1313 Figure 13. The character variety for the P D 6

1 Figure 14 .

 114 Figure 14. Transformation of the P V system of arcs from Fig. 6 under a sequence of two flips.

Figure 15 .

 15 Figure 15. The fat graph corresponding to P III D7 .

Figure 16 .

 16 Figure 16. The fat graph corresponding to P III D8 .

3 Figure 17 .

 317 Figure 17. The fat graph corresponding to P II JM .

3 Figure 18 .

 318 Figure 18. The fat graph corresponding to P II F N .

Figure 19 . 2 ,

 192 Figure 19. The system of arcs for PII FN .

6 Figure 20 .

 620 Figure 20. The fat graph corresponding to PI.

  These transformations are governed by the standard mutation rules,bb ′ = hc + f a, f f ′ = gc + ab ′ ;in order to describe them in a more regular way, let us introduce the notation: we let λ(i)α,β denote the λ-length of the arc that goes between bordered cusps α and β (belonging to different boundary components) winding i times around the lower hole. For example, , b ′ = λ

  , c = λ 1,1 etc. Note that λ α,β with the labels α and β pertaining to the same boundary component are unique and invariant under the MCG action.The net result of the Dehn twist on the triple {λ

.

  This action admits two invariants: G γ and λ 4,4 (the latter is obtained by the mutation of the element f , or λ

Table 2 .

 2 ) 1,4 ) 2 + λ 1,1 λ 4,4 λThe cases of P IV , P II, and P I correspond to finite cluster algebras admitting no nontrivial modular transformations. singular fibre with singularity of type specified by any Dynkin sub-diagram of the Dynkin diagram given in the second column of the table.

	
	  .
	(i-1)
	1,4

We use the term bordered cusp meaning a vertex of an ideal triangle in the Poincaré metric in order to distinguish it from standard cusps (without borders) associated to punctures on a Riemann surface. 1

Appendix A Singularity theory approach to the Painlevé cubics As mentioned above, for special values of ω the fibre may have a singularity. Such singularities were classified in [START_REF] Inaba | Dynamics of the sixth Painlevé equation, in Théories asymptotiques et équations de Painlevé[END_REF] for P V I and in [START_REF] Saito | Moduli spaces for linear differential equations and the Painlevé equations[END_REF] for all other Painlevé equations. These results can be summarised in the following table :  The meaning of the table is the following: for each Painlevé equation of type specified by the first column in the table, there is at least one singular fibre with singularity of the type given in the second column of the table, and at least one

The above formulae show that the versal unfolding parameters ω 1 , . . . , ω 4 are independent as long as ω 1 , . . . , ω 4 are.

A.4. D 5 . This case corresponds to the fifth Painlevé equation. The cubic in this case is (we drop the indices ( D 5 ) for convenience):

, where only three parameters are free:

Again we want to show that this is diffeomorphic to the versal unfolding of A 3 . To this aim we impose the following change of variables: (A.4)

where u(x 2 ) is a function to be determined. This maps the P V cubic to:

It is easy to prove that any solution u(x 2 ) of the equation

will define a diffeomorphism by (A.4) mapping (A.3) to the versal unfolding of A 3 .

A.5. D 6 . This case corresponds to the third Painlevé equation. The cubic in this case is (we drop the indices ( D 3 ) for convenience):

, where only two parameters are free:

The most singular fibre is given by G ∞ = 1 and G 2 = 2 and has two singular points at (1, 0, 2) and (0, 1, 2) respectively. We can define two local diffeomorphisms, one around (1, 0, 2), the other around (0, 1, 2), which map our cubic to the versal unfolding of a A 1 singularity.

The first diffeomorphism is given by:

The second diffeomorphism is:

A.5.1. E 6 . This case corresponds to the fourth Painlevé equation. The cubic in this case is (we drop the indices ( E 6 ) for convenience):

, where only two parameters are free:

Again we want to show that this is diffeomorphic to the versal unfolding of A 2 . To this aim we impose the following change of variables:

where u is function of x 3 satisfying the following

It is easy to prove that this transformation is a local diffeomorphism mapping our cubic to

∞ , the versal unfolding of the A 2 singularity. A.5.2. E 7 . This case corresponds to the second Painlevé equation. Since the treatment of the two cubics E * 7 and E * * 7 is completely equivalent, we choose to work with the former:

where:

The following change of variables:

where u is a function of x 2 satisfying

, is a local diffeomorphism mapping our cubic to the versal unfolding of the A 1 singularity:

x 2 1 -x 2 3 + x 2 2 + ω 4 .


Orale Fostered

Celhto

Talking about Someone's Objects of Belief Dialogical Language Games, Epistemic Acquisition and Intentional Identity Contribution to the meeting fourth meeting of the Réseau LACTO (Langage, Argumentation et Cognition dans les Traditions Orales) JEU ET ORALITE DANS LES SOCIETES A TRADITION

According to the main stream approaches to epistemic notions, knowledge and belief are understood as propositional operators. Thus,  Gildas believes that there is a witch in his village is understood as expressing a proposition. Moreover,  Gildas knows that Gödel proved the completeness of first-order logic, expresses a proposition too. The difference between both is that a belief-proposition can be true if it is "factually" false: in our case the proposition can be true in relation to Gildas's belief, that is, despite the fact that there is no witch in his village. Furthermore, assuming that there is not witch in his village, the following proposition is false  Gildas knows that there is a witch in his village.

If the latter is false, from the point of view of the mainstream, there was not knowledge at all, rather it was from the start a belief that has not been verified. Now, against this recent way of analyzing such kind of phenomena there is an older tradition from Plato and Aristotle up to Kant, that sees knowledge and belief as the product of certain acts. Indeed, from the point of view of this tradition knowledge and belief are the result of acts of judgement, made public by suitable assertions. Thus, a proposition that expresses the knowledge/belief of an agent is rooted in acts of judgement. More precisely, knowledge and belief are dispositions or states, whereas judgements are acts. Assertions are acts by which the acquisition of knowledge is made public. What we know/believe is what we have judged. However, our judgements may turn out to be wrong, they might, for instance be incompatible with other of our judgements and we might then withdraw them. Thus, the dynamics involved when passing from belief to knowledge or vice versa is not really about changing the epistemic operator, but rather about changing our assertions because of changes on the grounds brought forward to support them. This suggests that our story is still incomplete: acts of judgements, are not performed in isolation, acts of judgment are the product of social interaction. When I make my judgement public by means of an assertion I engage myself in a game, to make use of Brandom-Searle's terminology, of giving and asking for reasons. Interaction is the basis on which knowledge and belief are rooted. Knowledge and belief are, as stressed by Hegel, product of social interaction. In fact, the interaction in question is an interaction by means of which I am said to have knowledge if I am able, not only to understand what counts as supporting my assertion (the reason that grounds it); but also be able to produce this reason when asked for it and to defend in the context of further assertions to which my first assertion commits me and in the context of other previous assertions.

As stressed in the text that motivates the present LACTO meeting, oral traditions offer a striking case for the study of this view on epistemic acquisition. Indeed, as pointed out by N. Agblemagnon (1969), and M. [START_REF] Diagne | Critique de la raison orale (Les pratiques discursives en Afrique noire)[END_REF], within the context of the oral traditions of Africa, the initiation to epistemic acquisition is deployed by games of make-belief such as tales telling, legends, epopees, proverbs and riddles. In this context it is also worth mentioning G. Nzokou's work (2013) that provides the first logical analysis of the dynamics of epistemic acquisition and justification involved in debates within the frame of traditional law based on proverbs. Now let us delve deeper into the matter: An epistemic state by the means of which we are able to bring forward that could count as a ground for asserting a proposition, say A, does not constitute the knowledge sufficient to assert that proposition A is true. In order to be able to assert that A is true, I must be able to produce the ground that justifies A.

If Gildas believes, that there is a witch in his village, it is sufficient for the truth-of the belief-assertion, him being able to bring forward (when asked) that what could count as a witch. Gildas has knowledge only in the case, that there is some ground, say, a, that justifies A, and that he can identify his objects of belief with a. In other words, as pointed out by A. B. [START_REF] Dango | Des dialogues aux tableaux dans le contexte de révision de croyances : De l'oralité à l'écriture[END_REF]2015) based on the work of [START_REF] Ranta | Type-Theoretical Grammar[END_REF][START_REF] Ranta | Constructing possible worlds[END_REF] and [START_REF] Primiero | Constructive modalities for information[END_REF]2012) while an object of belief involves some kind of variable such as x , knowledge involves being able to produce the identity between x and a. This can be expressed by the difference between hypothetical and categorical assertions. So, whereas

 c(x) : C (x : A)
expresses that C is justified provided the hypothesis A. Or better that the justification of A is dependent upon the hypothesis A (the function c(x) indicates such a dependence),  c(a) : C, given that a : A, expresses that C is justified, given that we know that A is justified.

In the first case we have belief: C is true under the hypothesis of A. We do not know if either A or C are justified, we make the hypothesis that they are. In the second case, we know that A is justified, and a is the ground on which we build our justification, and so we know C is justified. Now the passage from belief to knowledge involves recognizing the object of belief x (the hypothetical object x) as being indeed nothing else that the ground a. In other words the passage from belief to knowledge involves being able to bring forward the following identity  c(a) : C, given that x = a : A, Now in the contribution to the meeting of Dakar (2015, Rahman showed that such kind of identity statements are based on dialogical interaction. Hence, those expressions of identity that signalise the passage from belief to knowledge are also based on games of asking and giving reasons.

The main aims of our present contribution is to provide a dialogical study of the development of identity assertions in the context of epistemic attributions (including the case of anaphora as initiated by [START_REF] Dango | Approche dialogique de la révision des croyances dans le contexte de la théorie constructive des types[END_REF]). For short we would like to deal with the distinction between epistemic attributions de re, de dicto and de objecto for assertions attributing belief involving anaphora such as  Gildas believes that there is a witch in the village, but there is no witch in that village  Gildas believe that there is a witch in the village and Sebastian believes that she (Gilda's witch) is beautiful

In order to develop our study we will make use of examples extracted of existing oral practices in the African tradition. The point is to stress that not only it is the case that epistemic acquisition is result of a specific form language games involving social interaction. We claim that also identity of the objects of believe are the product of dialogical interaction. This relates to a more general tenet of us: knowledge and belief are the result of the interface between action and propositions. Oral traditions deployed this interface since the times of yore.

Some Bibliographic References

  Shahid Rahman* and Adjoua Bernadette Dango (*Univ. Lille, CNRS, UMR 8163 -STL -Savoirs Textes Langage, F-59000 MESH, Lille, France, ** Université de Bouaké) 1

The present paper is part of an ongoing project in the context of the research-program Argumentation, Decision, Action (ADA) and the project Logique, Argumentation et Cognition dans les Traditions Orales (LACTO) both supported by the Maison Européenne des Sciences de l' Homme et de la Société -USR 318 and by the laboratory UMR 8163: STL and it also is a contribution to the ANR Project SÊMAINÔ.