Moduli spaces of local systems and higher Teichmüller theory

V.V. Fock, A. B. Goncharov

Contents

1 Introduction 1 2 The moduli spaces A_{G,Ŝ} and X_{G,Ŝ} 12 3 Surfaces, graphs, mapping class groups and groupoids 19 4 Elements of positive algebraic geometry 23 5 Positive configurations of flags 27 6 A positive structure on the moduli space X_{G,Ŝ} 38 7 A positive structure on the moduli space A_{G,Ŝ} 51 8 Special coordinates on the A and X spaces when G is of type A_{m} 60 9 The pair (X_{P GL_{m},Ŝ}, A_{SL_{m},Ŝ}) corresponds to an orbi-cluster ensemble 76 10 The classical Teichmüller spaces 84 11 Laminations and canonical pairings 86 12 Positive coordinate systems for X -laminations 100 13 A Weil–Petersson form on A_{G,Ŝ} and its motivic avatar 103

1 Introduction

1 An algebraic-geometric approach to higher Teichmüller theory. Let S be a surface with n holes. The Teichmüller space T_{S} is the moduli space of complex structures on S modulo diffeomorphisms isotopic to the identity. We will usually assume that S is hyperbolic, i.e. χ(S) < 0. Then the Poincaré uniformisation theorem identifies it with the space of all faithful representations π_{1}(S) → P SL_{2}(ℝ) with discrete image, modulo conjugations by P SL_{2}(ℝ). Traditionally the Teichmüller theory is considered as a part of analysis and geometry.

The Teichmüller space is by no means a set of real points of an algebraic variety. Indeed, discreteness is not a condition of an algebraic nature. In this paper we show that, nevertheless, there is an algebraic-geometric approach to the Teichmüller theory. Moreover we show that it admits a generalization where P SL_{2} is replaced by a split semi-simple algebraic group G with trivial center. Here are the main features of our story.

Let G be a split reductive algebraic group over ℚ. Let Ŝ be a pair consisting of a compact oriented surface S with n holes and a finite (possibly empty) collection of marked points on the boundary considered modulo isotopy. We define two new moduli spaces, denoted X_{G,Ŝ} and A_{G,Ŝ}, closely related to the classical moduli space L_{G,S} of representations π_{1}(S) → G modulo conjugations. We will always assume that G has trivial center (resp. simply-connected) for the X - (resp. A-) moduli space. In particular when Ŝ is a disk with marked points on the boundary we get moduli spaces of cyclically ordered configurations of points in the flag variety B := G/B and twisted cyclic configurations of points of the principal affine variety A := G/U.

We show that if S does have holes, i.e. n > 0, then these moduli spaces carry interesting additional structures. In particular for each of them we define a distinguished collection of coordinate systems, equivariant under the action of the mapping class group of S. We prove that their transition functions are subtraction free, and thus provide a positive atlas on the corresponding moduli space.

We say that X is a positive variety if it admits a positive atlas, i.e. a distinguished collection of coordinate systems as above. If X is a positive variety, we can take points of X with values in any semifield, i.e. a set equipped with operations of addition, multiplication and division, e.g. ℝ_{>0}. We define higher Teichmüller spaces X_{G,Ŝ}^{+} and A_{G,Ŝ}^{+} as the sets of ℝ_{>0}-points of the corresponding positive spaces X_{G,Ŝ} and A_{G,Ŝ}. Precisely, they consist of the real points of the corresponding moduli spaces whose coordinates in one, and hence in any, of the constructed coordinate systems are positive. Our approach for general G uses G. Lusztig’s theory of positivity in semisimple Lie groups [L1]-[L2]. For G = SL_{m} we have an elementary and self-contained approach developed in Sections 8-9.

We prove that the representations π_{1}(S) → G(ℝ) underlying the points of X_{G,Ŝ}^{+} are discrete and hyperbolic. For G = P SL_{2}(ℝ) we recover the Fuchsian representations. If S is compact, a component in the moduli space L_{G,S} was defined and studied by N. Hitchin [H1] by a completely different method. A partial generalization to punctured S see in [BAG].

Here is how the classical Teichmüller spaces appear. If n > 0, the space T_{S} has a boundary. Let us consider its 2^{n}-sheeted cover T_{S}^{+} → T_{S}, obtained by choosing orientations of each of the boundary components. It is ramified exactly at the boundary of T_{S}. This space was studied in [F]. We show that in the absence of marked points, i.e. when S = Ŝ, the space X_{P GL_{2},S}^{+} is identified with the Teichmüller space T_{S}^{+}, and A_{SL_{2},S}^{+} is identified with the decorated Teichmüller space T_{S}^{d} defined by R. Penner [P1]. The restrictions of our canonical coordiantes to the corresponding Teichmüller spaces are the coordinates defined in loc. cit. Our two moduli spaces, their positivity, and especially the discussed below motivic data seem to be new even in the classical setting.

It makes sense to consider points of our moduli spaces with values in an arbitrary semifield. An interesting situation appears when we consider exotic tropical semifields 𝔸^{t}. Let us recall their definition. Let 𝔸 be one of the three sets ℤ, ℚ, ℝ. Replacing the operations of multiplication, division and addition in 𝔸 by the operations of addition, subtraction and taking the maximum we obtain the tropical semifield 𝔸^{t}. We show that the set of points of the moduli space A_{SL_{2},S} with values in the tropical semifield ℝ^{t} is identified with Thurston’s transversal measured laminations on S [Th]. The ℚ^{t}- and ℤ^{t}-points of A_{SL_{2},S} give us rational and integral laminations. Now taking the points of the other moduli spaces with values in tropical semifields we get generalizations of the lamination spaces. We show that for any positive variety X the projectivisation of the space X(ℝ^{t}) serves as the Thurston type boundary for the space X^{+} := X(ℝ_{>0}).

We suggest that there exists a remarkable duality between the A-moduli space for the group G and the X -moduli space for the Langlands dual group ^{L}G. Here is one of (conjectural) manifestations of this duality. Let X be a positive variety. A rational function F on X is called a good positive Laurent polynomial if it is a Laurent polynomial with positive integral coefficients in every coordinate system of the positive atlas on X. We conjecture that the set of ℤ^{t}-points the A/X -moduli space for G should parametrize vectors of a canonical basis in the space of all good positive Laurent polynomials on the X /A-moduli space for the Langlands dual group ^{L}G, for the positive atlases defined in our paper. For G = SL_{2} or P GL_{2} we elaborate this theme in Section 11.

If S has no holes, the moduli space L_{G,S} has a natural symplectic structure. If n > 0, the moduli space X_{G,Ŝ} has a natural Poisson structure (it is provided by the construction in [FR]). We introduce a degenerate symplectic structure on the moduli space A_{G,Ŝ}, an analog of the Weil-Petersson form. To define the Weil-Petersson form we construct its K_{2}-avatar. We show that it admits an even finer motivic avatar related to the motivic dilogarithm and the second motivic Chern class c_{2}^{M} of the universal G-bundle on BG. In fact the explicit construction of the class c_{2}^{M} for G = SL_{m} given in [G3] delivers the canonical coordinates on the A_{SL_{m},Ŝ}-space. It was our starting point.

The Poisson structure and the Weil-Petersson form are compatible in a natural way: the quotient of the A-space along the null foliation of the Weil-Petersson form is embedded into the X -space as a symplectic leaf of the Poisson structure. The representations of π_{1}(S) corresponding to its points are the ones with unipotent monodromies around each hole.

Recall that there is an embedding SL_{2} ↪ G, as a principal SL_{2}–subgroup, well defined up to a conjugation. It leads to natural embeddings (G′ denotes the adjoint group)

A_{SL_{2},S} ↪ A_{G,S}, X_{P GL_{2},S} ↪ X_{G′,S}

and their counterparts for the Teichmüller and lamination spaces.

Summarizing, we suggest that

The pair of moduli spaces (A_{G,Ŝ}, X_{G′,Ŝ}) is the algebraic-geometric avatar of higher Teichmüller theory.

After the key properties of the (X , A)-moduli spaces were discovered and worked out, we started to look for an adequate framework describing their structures. It seems that such a framework is provided by the object which we call (orbi)-cluster ensemble and develop in [FG2]. It generalizes the cluster algebras, discovered and studied in a remarkable series of papers by S. Fomin and A. Zelevinsky [FZI], [FZII], [FZ3]. A cluster ensemble is a pair of positive spaces X_{E} and A_{E} defined for any integral valued skew-symmetrizable function E on the square of a finite set. The cluster algebra is closely related to the positive space A_{E}. The orbi-cluster ensemble is even a more general structure.

For any G there exists a orbi-cluster ensemble (A_{E(G,Ŝ)}, X_{E(G,Ŝ)}) closely related to the pair of moduli spaces (A_{G,Ŝ}, X_{G,Ŝ}). For G = SL_{2} the corresponding pairs of positive spaces coincide. For other G’s the precise relationship between them is more complicated. We want to stress that usually, e.g. when Ŝ = S and S has at least two punctures, the pair of positive spaces (A_{G,Ŝ}, X_{G,Ŝ}) corresponds not to a cluster, but rather to an orbi-cluster ensemble. In Section 9 we explain this for G = SL_{m}. A more technical general case is treated in the sequel to this paper. The fact that Penner’s decorated Teichmuller space is related closely to a cluster algebra was independently observed in [GSV2].

We show in [FG2] that cluster ensembles have essentially all properties of the two moduli spaces described above, including the relation with the motivic dilogarithm. Our point is that

A big part of the higher Teichmüller theory can be generalized in the framework of (orbi)-cluster ensembles, providing a guide for investigation of the latter.

The relationship between the higher Teichmüller theory and cluster ensembles is very fruitful in both directions. Indeed, the higher Teichmüller theory suggests what kind of structures and results need to be developed in the theory of cluster ensembles. For example, we suggest in Section 4 loc. cit. that there exists a remarkable duality between the X_{E} and A_{E} spaces, which in the special case of our pair of moduli spaces includes the one discussed above. This duality should include the canonical bases of Lusztig and the Laurent phenomenon of Fomin and Zelevinsky [FZ3] as special cases.

We show in [FG2] that any cluster ensemble can be quantized. In the sequel to this paper we apply this to quantize the moduli space X_{G,Ŝ}. The quantization is given by means of the quantum dilogarithm. Remarkably the quantization is governed by the motivic avatar of the Weil-Petersson form on the A_{G,Ŝ} moduli space.

The rest of the Introduction contains a more detailed discussion of main definitions and results.

2. The two moduli spaces. Let

S = \overline{S} − D_{1} ∪ ... ∪ D_{n}

where \overline{S} is a compact oriented two dimensional manifold without boundary of genus g, and D_{1}, ..., D_{n} are non intersecting open disks. The boundary ∂S of S is a disjoint union of oriented circles C_{1}, ..., C_{n}.

Definition 1.1 A marked surface Ŝ is a pair (S, {x_{1}, ..., x_{k}}), where S is a compact oriented surface and {x_{1}, ..., x_{k}} is a finite set of distinct boundary points, considered up to an isotopy. The set of boundary points can be empty.

We have Ŝ = S if there are no boundary points. Let us define the punctured boundary ∂Ŝ of Ŝ by

∂Ŝ := ∂S − {x_{1}, ..., x_{k}}

It is a union of circles and (open) intervals. Let N be the number of all connected component of ∂Ŝ. We say that a marked surface Ŝ is hyperbolic if g > 1, or g = 1, N > 0, or g = 0, N ≥ 3. We will usually assume that Ŝ is hyperbolic, and N > 0.

Recall that an H–local system is a principle H-bundle with a flat connection. There is a well known bijection between the homomorphisms of the fundamental group π_{1}(X, x) to a group H modulo conjugation, and the isomorphism classes of H–local systems on a nice topological space X. It associates to a local system its monodromy representation.

Let G be a split reductive algebraic group over ℚ. Then L_{G,S} is the moduli space of G-local systems on S. It is an algebraic stack over ℚ, and its generic part is an algebraic variety over ℚ. We introduce two other moduli spaces, denoted A_{G,Ŝ} and X_{G,Ŝ}, closely related to L_{G,S}. They are also algebraic stacks over ℚ.

The flag variety B parametrizes Borel subgroups in G. Let B be a Borel subgroup. Then B = G/B. Further, U := [B, B] is a maximal unipotent subgroup in G.

Let L be a G-local system on S. We assume that G acts on L from the right. We define the associated flag bundle L_{B} and principal affine bundle L_{A} by setting

L_{B} := L ×_{G} B = L/B;  L_{A} := L/U  (1)

Definition 1.2 Let G be a split reductive group. A framed G-local system on Ŝ is a pair (L, β), where L is a G-local system on S, and β a flat section of the restriction of L_{B} to the punctured boundary ∂Ŝ. The space X_{G,Ŝ} is the moduli space of framed G–local systems on Ŝ.

Let us assume now that G is simply-connected. The maximal length element w_{0} of the Weyl group of G has a natural lift to G, denoted \overline{w}_{0}. Set s_{G} := \overline{w}_{0}^{2}. Then one shows that s_{G} is in the center of G and s_{G}^{2} = e. In particular s_{G} = e if the order of the center of G is odd, i.e. G is of type A_{2k}, E_{6}, E_{8}, F_{4}, G_{2}. If s_{G} ≠ e, e.g. G = SL_{2k}, the definition of the moduli space A_{G,Ŝ} is more subtle. So in the Introduction we do not consider this case.

Definition 1.3 Let G be is a simply-connected split semisimple algebraic group, and s_{G} = e. A decorated G-local system on Ŝ is a pair (L, α), where L is G-local system on S, and α a flat section of the restriction of L_{A} to ∂Ŝ. The space A_{G,Ŝ} is the moduli space of decorated G-local systems on Ŝ.

Let X be a G–set. The elements of G\X^{n} are called configurations of n elements in X. The coinvariants of the cyclic translation are called cyclic configurations of n elements in X.

Example. Let Ŝ be a disk with k ≥ 3 boundary points, so L is a trivial G-local system. Then X_{G,Ŝ} is identified with the cyclic configuration space of k flags in G, that is coinvariants of the cyclic shift on G\B^{k}. Indeed, choosing a trivialization of L, we identify its fibers with G. Then the section β over the punctured boundary of the disk determines a cyclically ordered collection of flags. A change of the trivialization amounts to a left diagonal action of G. So a cyclic configuration of k flags is well defined. It determines the isomorphism class of the framed G-local system on the marked disk. Similarly, assuming s_{G} = e, the space A_{G,Ŝ} is identified with the cyclic configuration space of k points in the principal affine space A := G/U.

Forgetting β we get canonical projection

π : X_{G,Ŝ} ⟶ L_{G,S}

If Ŝ = S, it is a finite map at the generic point of degree |W |^{n}, where W is the Weyl group of G.

We show that the moduli spaces X_{G,Ŝ} and A_{G,Ŝ} have some interesting additional structures. To explain this we need a few simple definitions.

3. Positive structure of the moduli spaces. Let H be a split algebraic torus over ℚ, so H =^{∼} 𝔾_{m}^{N}. Following [BeK], a rational function f on H is called positive if it can be written as f = f_{1}/f_{2} where f_{1}, f_{2} are linear combinations of characters with positive integral coefficients. A positive rational map between two split algebraic tori H_{1}, H_{2} is a rational map f : H_{1} → H_{2} such that for any character χ of H_{2} the composition χ ∘ f is a positive rational function on H_{1}. A composition of positive rational functions is positive .

Example. The map x = a + b, y = b is positive, but its inverse b = y, a = x − y is not.

We define a positive divisor in H as the divisor of a positive rational function on H. Here is our working definition of a positive atlas.

Definition 1.4 A positive atlas on an irreducible scheme/stack X over ℚ is a family of birational isomorphisms

ψ_{α} : H_{α} ⟶ X, α ∈ C_{X},  (2)

called positive coordinate systems on X, parametrised by a non empty set C_{X}, such that: i) each ψ_{α} is regular at the complement to a positive divisor in H_{α}; ii) for any α, β ∈ C_{X} the transition map ψ_{β}^{−1} ∘ ψ_{α} : H_{α} ⟶ H_{β} is a positive rational map.

A positive scheme is a scheme equipped with a positive atlas. In particular a positive variety is rational. Given a positive scheme X, we associate to it an open domain X(ℝ_{>0}) ↪ X(ℝ) called the positive part of X. Namely, a point p ∈ X(ℝ) lies in X(ℝ_{>0}) if and only if its coordinates in one, and hence any positive coordinate system on X are positive. Composing ψ_{α}^{−1} with an isomorphism H_{α} →^{∼} 𝔾_{m}^{N} we get coordinates {X_{i}^{α}} on X, providing an isomorphism of manifolds

φ_{α} : X(ℝ_{>0}) ⟶^{∼} ℝ^{dimX}, p ∈ X(ℝ_{>0}) ⟼ {log X_{i}^{α}(p)} ∈ ℝ^{dimX}  (3)

Let Γ be a group of automorphisms of X. We say that a positive atlas (2) on X is Γ-equivariant, if Γ acts on the set C_{X}, and for every γ ∈ Γ there is an isomorphisms of algebraic tori i_{γ} : H_{α} ⟶^{∼} H_{γ(α)} making the following diagram commutative:

\begin{matrix}H_{α} & ⟶^{ψ_{α}} & X \\ ↓ i_{γ} & & ↓ γ_{∗} \\ H_{γ(α)} & ⟶^{ψ_{γ(α)}} & X \end{matrix}  (4)

Here γ_{∗} is the automorphism of X provided by γ. If a positive atlas on X is Γ-equivariant, then the group Γ acts on X(ℝ_{>0}).

Recall that the mapping class group Γ_{S} of S is the group of all outer automorphisms of π_{1}(S) preserving the conjugacy classes provided by simple loops around each of the punctures. It acts naturally on the moduli spaces X_{G,Ŝ}, A_{G,Ŝ}, and L_{G,S}.

Theorem 1.5 Let G be a split semisimple simply-connected algebraic group, and G′ the corresponding adjoint group. Assume that n > 0. Then each of the moduli spaces X_{G′,Ŝ} and A_{G,Ŝ} has a positive Γ_{S}–equivariant atlas.

Here is an outline of our construction of positive atlas on X_{G′,Ŝ}. Let us picture the boundary components of S without marked points as punctures. In the presence of marked points let us choose a single distinguished point on each connected component of the punctured boundary ∂Ŝ. We define an ideal triangulation T of Ŝ as a triangulation of S with all vertices at the punctures and distinguished points. Each triangle of this triangulation can be considered as a disc with three marked points. These points are located on the edges of the triangle, one per each edge. The restriction of an element of X_{G′,Ŝ} to a triangle t of an ideal triangulation provides a framed G-local system on the described above marked triangle t̂. It is an element of X_{G′,t̂}. So we get a projection p_{t} : X_{G′,Ŝ} → X_{G′,t̂}. An edge of the triangulation T is called internal if it is not on the boundary of S. Given an oriented internal edge e of T we define a rational projection p_{e} : X_{G′,Ŝ} → H. Let us orient all internal edges of an ideal triangulation. Then the collection of projections p_{t} and p_{e} provide us a rational map

π_{T} : X_{G′,Ŝ} ⟶ ∏ ; triangles t of T X_{G′,t̂}× H{internal edges of T }_{G′}  (5)

Theorem 1.6 (Decomposition Theorem). For any ideal triangulation T of Ŝ, the map (5) is a birational isomorphism.

Several different proves of this theorem are given in Section 6. A similar Decomposition Theorem for the space A_{G,Ŝ} is proved in Section 7, see (104).

We define a positive atlas on X_{G′,t̂}. Then the right hand side of (5) provides a positive atlas given by the product of the ones on X_{G′,t̂} and H. We prove that the transition functions between the atlases provided by different triangulations are subtraction free. Therefore the collection of birational isomorphisms {π_{T} } for all ideal triangulations of Ŝ provides a (bigger) positive atlas on X_{G′,Ŝ}. It is Γ_{S}-equivariant by the construction.

For G = SL_{m} we have a stronger result. A positive atlas is called regular if the maps ψ_{α} are regular open embeddings.

Theorem 1.7 The moduli spaces X_{P GL_{m},Ŝ} and A_{SL_{m},Ŝ} have regular positive Γ_{S}–equivariant atlases.

Theorem 1.5 is proved in Sections 6-7 using the results of Section 5. Theorem 1.7 is proved in Sections 8-9 using simpler constructions of positive atlases on the SL_{m}-moduli spaces given in Section 8. These constructions can be obtained as special cases of the ones given in Sections 5-7, but they have some extra nice properties special for SL_{m}, and do not rely on Sections 5-7. In particular our X- (resp. A-) coordinates on the configuration space of triples of flags (resp. affine flags) in an m-dimensional vector space are manifestly invariant under the cyclic (resp. twisted cyclic) shift. One can not expect this property for a general group G, and this creates some technical issues, making formulations and proofs heavier. In addition to that, the construction of positive coordinate systems given in Section 8 is simpler and can be formulated using the classical language of projective geometry. We continue discussion of the SL_{m} case in the Section 1.8.

4. Applications of positivity: higher Teichmüller spaces and laminations.

Definition 1.8 The higher Teichmüller spaces X_{G′,Ŝ}^{+} and A_{G′,Ŝ}^{+} are the real positive parts of the moduli spaces X_{G,Ŝ} and A_{G,Ŝ}:

X_{G′,Ŝ}^{+} := X_{G′,Ŝ}(ℝ_{>0}) ↪ X_{G,Ŝ}(ℝ), A_{G,Ŝ}^{+} := A_{G′,Ŝ}(ℝ_{>0}) ↪ A_{G,Ŝ}(ℝ)

The mapping class group Γ_{S} acts on X_{G′,Ŝ}^{+} and A_{G,Ŝ}^{+}. Let

L_{G′,S}^{+} := π(X_{G′,S}^{+}) ⊂ L_{G′,S}(ℝ)

We call the points of L_{G′,S}^{+} positive local systems on S, and their monodromy representations positive representations of π_{1}(S). There is a natural embedding

L_{G′,S}^{+} ↪ X_{G′,S}^{+}

splitting the canonical projection X_{G′,S}^{+} → L_{G′,S}^{+}. The following result is proved in Section 6.8.

Theorem 1.9 Let G be a split real reductive algebraic group with trivial center. Then the image of a positive representation of π_{1}(S) is a discrete subgroup in G(ℝ).

Since the isomorphisms ψ_{α} describing the positive atlas on X_{G,Ŝ} are defined explicitly, we obtain a parametrisation of a class of discrete subgroups in G(ℝ) which generalizes the Fuchsian subgroups of P SL_{2}(ℝ).

In Section 10 we show that our construction specialized to the case G = SL_{2} for the A-space and G = P GL_{2} for the X -space, gives the classical Teichmüller spaces T^{d}(S) and T^{h}(S):

Theorem 1.10 Let S be a hyperbolic surface with n > 0 holes. Then

a) The Teichmüller space T_{S}^{+} is isomorphic to X_{P GL_{2},S}^{+}. Further, L_{P GL_{2},S}^{+} is the classical Teichmüller space T_{S}.

b) The decorated Teichmüller space T_{S}^{d} is isomorphic to A_{SL_{2},S}^{+}.

Combining with isomorphisms X_{P GL_{2},S}^{+} =^{∼} ℝ^{6g−6+3n} and A_{SL_{2},S}^{+} =^{∼} ℝ^{6g−6+3n} we arrive at isomorphisms of manifolds

T_{S}^{+} =^{∼} ℝ^{6g−6+3n} and T_{S}^{d} =^{∼} ℝ^{6g−6+3n}

The first gives an elementary proof of the classical Teichmüller theorem for surfaces with holes.

When S is compact, a Teichmüller component of the space of representations π_{1}(S) → G(ℝ) modulo conjugation was defined by N. Hitchin [H1]. His approach is analytic. It is based on the deep analytic results of N. Hitchin [H2], K. Corlette [C], C. Simpson [S] and S. Donaldson [D], and makes an essential use of a complex structure on S. It is very natural but rather non explicit. Our approach is different: it is combinatorial, explicit, and does not use any complex structure on S. The two approaches are completely independent. The precise relationship between the two versions of Teichmüller spaces is not clarified at the moment.

We introduce integral, rational and real G–lamination spaces as the tropical limits of the positive moduli spaces X_{G′,Ŝ} and A_{G,Ŝ}, i.e. the sets of points of these spaces in the semifields ℤ^{t}, ℚ^{t}, ℝ^{t}. The projectivisations of the real lamination spaces are the Thurston type boundaries of the corresponding Teichmüller spaces.

5. Positivity and hyperbolicity of the monodromy representations. Consider the universal G-local system on the moduli space S ×X_{G,S}. Its fiber over S ×p is the local system corresponding to the point p of the moduli space X_{G,S}. Let 𝔽_{G,S} be the field of rational functions on the moduli space X_{G,S}. The monodromy of the universal local system around a loop on S is a conjugacy class in G(𝔽_{G,S}).

Observe that given a positive variety X, there is a subset ℚ_{+}(X) of the field of rational functions of X consisting of the functions which are positive in one, and hence in any of the positive coordinate systems on X. Clearly ℚ_{+}(X) is a semifield, positive semifield of the positive variety X.

Theorem 1.5 delivers a positive atlas on the moduli space X_{G,S}. Therefore we have the corresponding positive semifield 𝔽_{G,S}^{+} in the field 𝔽_{G,S}.

According to Lusztig [L1] the group G has a natural positive atlas. It is defined using a pair (B^{+}, B^{−}) of opposite Borel subgroups in G. So we have a well defined subset G(𝔽_{G,S}^{+}).

Theorem 1.11 The monodromy of the universal G-local system on S × X_{G,S} around any homotopy non-trivial non-boundary loop on S is conjugated in G(𝔽_{G,S}) to an element of G(𝔽_{G,S}^{+}).

The monodromy around a boundary component is conjugated to an element of B^{±}(𝔽_{G,S}^{+}), where B^{±} means one of the Borel subgroups B^{+} or B^{−}.

Observe that although this theorem is about monodromies of local systems on S, it can not be even stated without introducing the moduli space X_{G,S} and a positive atlas on it. Here is a corollary formulated entirely in classical terms.

An element of G(ℝ) is called regular hyperbolic if it is contained in a unique ℝ-split maximal torus of G(ℝ). The corresponding conjugacy class bears the same name. In particular it is regular and semisimple. Regular hyperbolic elements of GL_{n}(ℝ) are the elements with real distinct eigenvalues.

Theorem 1.12 The monodromy of a positive local system on S is faithful. Moreover its monodromy around any homotopy nontrivial non-boundary loop is regular hyperbolic.

It follows immediately from Theorem 1.11. Indeed, Definition 1.8 and Theorem 1.11 imply that the monodromy of a positive representation around a homotopy nontrivial non-boundary loop is conjugated to an element of G(ℝ_{>0}). Further, by the Gantmacher-Krein theorem [GKr] for G = GL_{n}, and Theorem 5.6 in [L1] in general, any element of G(ℝ_{>0}) is regular hyperbolic. To prove faithfulness observe that the identity element does not belong neither to G(ℝ_{>0}) nor to B^{±}(ℝ_{>0}).

6. The universal higher Teichmüller spaces and positive G(ℝ)-opers. In the special case when Ŝ is a disc with k boundary points, the higher Teichmüller space X_{G,Ŝ}^{+} parametrises positive configurations of k flags for G(ℝ). One of their key properties is invariance under the cyclic shift. Using this, we introduce universal higher Teichmüller spaces X_{G}^{+}:

Definition 1.13 The universal higher Teichmüller space X_{G}^{+} consists of all maps

β : ℙ^{1}(ℚ) ⟶ B(ℝ)  (6)

considered modulo the action of G(ℝ), which are positive in the following sense: for any finite subset x_{1}, ..., x_{k} of ℙ^{1}(ℚ), whose cyclic order is induced by a natural one on ℙ^{1}(ℚ), the configuration of flags (β(x_{1}), ..., β(x_{n})) is positive.

For G = P GL_{2} we recover the universal Teichmüller space considered by Bers and Penner [P3].

The higher Teichmüller spaces X_{G,S}^{+} can be embedded in the universal one as follows. Let Δ be a torsion free subgroup of P SL_{2}(ℤ) and S_{Δ} := H/Δ, where H denotes the hyperbolic plane. Then S_{Δ} is equipped with a distinguished ideal triangulation, given by the image of the Farey triangulation of the hyperbolic plane. Conversely, the subgroup Δ ⊂ P SL_{2}(ℤ) is determined uniquely up to conjugation by an ideal triangulation of S. A framed G(ℝ)-local system on S_{Δ} provides a representation ρ : Δ → G(ℝ), and naturally gives rise to a ρ-equivariant map (6). This property characterizes the higher Teichmüller space for S_{Δ}:

Theorem 1.14 There is a natural bijection between the points of the space X_{G,S_{Δ}}^{+} and pairs (a representations ρ : Δ → G(ℝ), a ρ-equivariant map (6)), considered modulo G(ℝ)-conjugation.

The universal higher Teichmüller space X_{G}^{+} is equipped with a natural positive coordinate atlas. Namely, recall the Farey triangulation of the hyperbolic plane, see Picture (7). The set of vertices of the Farey triangulation is identified with ℙ^{1}(ℚ). Using the Farey triangulation just the same way as ideal triangulations of S, we obtain the following result, the Decomposition Theorem for the universal higher Teichmüller space X_{G}^{+}, providing coordinate systems on X_{G}^{+}.

Theorem 1.15 There exists a canonical isomorphism

φ_{G} : X_{G}^{+} ⟶ X_{G,3}^{+}{Farey triangles} × H(ℝ_{>0})^{{Farey diagonals}}  (8)

In Section 7.5 we define universal higher Teichmüller spaces A_{G}^{+} and prove similar results for them.

Positive continuous maps

ℙ^{1}(ℝ) ⟶ B(ℝ)  (9)

are objects of independent interest. We prove in Section 6.9 that positive smooth maps (9) are integral curves of a canonical non-integrable distribution on the flag variety, provided by the simple positive roots, whose dimension equals to the rank of G. So they can be viewed as the G(ℝ)-opers in the sense of Beilinson and Drinfeld on the circle. We call them positive G(ℝ)-opers.

It is well known that P GL_{m}(ℝ)-opers are nothing else but smooth curves in ℝℙ^{m−1}: a smooth projective curve gives rise to its osculating curve in the flag variety. Recall (cf. [Sch]) that a curve C ⊂ ℝℙ^{m−1} is convex if any hyperplane intersects it in no more then m−1 points. We show in Section 8.12 that positive P GL_{m}(ℝ)-opers correspond to smooth convex curves in ℝℙ^{m−1}. More generally, positive continuous maps (9) give rise to C^{1}-smooth convex curves. Thus positive continuous maps (9) generalize projective convex curves to the case of an arbitrary reductive group G. It is interesting to note that convex curves appeared in some classical problems of analysis ([GKr]).

7. The Weil-Petersson form on the moduli space A_{G,Ŝ} and its K_{2} and motivic avatars. We define the Weil-Petersson form Ω_{G,Ŝ} on the moduli space A_{G,Ŝ} by constructing a K_{2}-class on A_{G,Ŝ}. We suggest that many interesting symplectic structures can be upgraded to their K_{2}-avatars. Let us explain how a K_{2}-class provides a symplectic form.

Recall that thanks to the Matsumoto theorem [Mi] the group K_{2}(F) of a field F is the quotient of the abelian group Λ^{2}F^{∗} by the subgroup generated by the elements (1 − x) ∧ x, x ∈ F^{∗} − {1}, called the Steinberg relations. If F = ℚ(Y) is the field of rational functions on a variety Y there is a homomorphism

d log : K_{2}(ℚ(Y)) ⟶ Ω_{log}^{2}(Spec(ℚ(Y))); f_{1} ∧ f_{2} ⟼ d log f_{1} ∧ d log f_{2}

Applying it to a class w ∈ K_{2}(ℚ(Y)) we get a rational 2-form d log(w) with logarithmic singularities. Let us formulate a condition on w which guarantees that the 2-form d log(w) is non singular, and hence defines a degenerate symplectic structure of Y. Let Y_{1} be the set of all irreducible divisors in a variety Y. There is a tame symbol homomorphism:

Res : K_{2}(ℚ(Y)) ⟶ ⊕_{D∈Y_{1}}ℚ(D)^{∗};  {f, g} ⟼ Rest_{D}(f^{v_{D}(g)}/g^{v_{D}(f)})  (10)

where Rest_{D} is restriction to the generic point of D, and v_{D}(g) is the order of zero of g at the generic point of the divisor D. Set

H^{2}(Y, ℤ_{M}(2)) := Ker(Res) ↪ K_{2}(ℚ(Y)),  H^{2}(Y, ℚ_{M}(2)) := H^{2}(Y, ℤ_{M}(2)) ⊗ ℚ (11)

The map d log transforms the tame symbol to the residue map. So its restriction to (11) provides a homomorphism

d log : H^{2}(Y, ℤ_{M}(2)) ⟶ Ω^{2}(Y_{0})

where Y_{0} is the nonsingular part of Y. We define a class

W_{G,Ŝ} ∈ H^{2}(A_{G,Ŝ}, ℚ_{M}(2))^{Γ_{S}}  (12)

and set

Ω_{G,Ŝ} := d log(W_{G,Ŝ}) ∈ Ω^{2}(A_{G,Ŝ})^{Γ_{S}}

We also define the class W_{G,S} for a compact oriented surface S. One can show that for G = SL_{2} the form Ω_{SL_{2},S} coincides with W. Goldman’s symplectic structure [Gol1] on the space L_{SL_{2},S}. Its restriction to the Teichmüller component is the Weil-Petersson symplectic structure. The K_{2}-avatar of the Weil-Petersson form is a new object even in the classical case G = SL_{2}.

Motivic avatar of the Weil-Petersson form. We show that the K_{2}-class W_{G,Ŝ} is a manifestation of a richer algebraic structure described by using the weight two motivic cohomology understood via the Bloch-Suslin complex, and the motivic dilogarithm. Namely, we construct a motivic avatar

𝕎_{G,Ŝ} ∈ H_{Γ_{S}}^{2}(A_{G,Ŝ}, ℚ_{M}(2))

of the form Ω_{G,Ŝ}. It lives in the Γ_{S}–equivariant weight two motivic cohomology of A_{G,Ŝ}.

The 2-form Ω_{G,Ŝ} as well as its avatars are lifted from similar objects on the moduli space U_{G,Ŝ} of the unipotent framed G-local systems on Ŝ. The corresponding 2-form on the moduli space U_{G,Ŝ} is non degenerate, and thus provides a symplectic structure on this space.

Our construction uses as a building block the second motivic Chern class c_{2}^{M} of the universal G–bundle over the classifying space BG_{•}. When G = SL_{m}, there is an explicit cocycle for the class c_{2}^{M} constructed in [G3]. Applying to it our general procedure we arrive at an explicit cocycle for the class 𝕎_{SL_{m},Ŝ}. Magically this construction delivers canonical coordinates on the space A_{SL_{m},Ŝ}. Their properties are outlined in the next subsection.

8. The (orbi)-cluster ensemble structure in the P GL_{m}/SL_{m} case. Let T be an ideal triangulation of Ŝ. Take the triangle

x + y + z = m, x, y, z ≥ 0

and consider its triangulation given by the lines x = p, y = p, z = p where 0 ≤ p ≤ m is an integer. The m–triangulation of a triangle is a triangulation isotopic to this one. Let us m–triangulate each triangle of the triangulation T, so each side gets m + 1 vertices. We get a subtriangulation, called the m–triangulation of T. An edge of the m–triangulation of T is called an internal edge if it does not lie on a side of the original triangulation T. The orientation of S provides orientations of the internal edges. Indeed, take a triangle t of the triangulation T. The orientation of S provides an orientation of its boundary. An internal edge e sitting inside of t is parallel to a certain side of t, so the orientation of this side induces an orientation of e. This is illustrated on Figure 1. Here on the right there are two adjacent triangles of the triangulation T. On the left we show the 4–triangulation of these triangles. Their internal edges are oriented accordantly to the cyclic structure coming from the clockwise orientation of the plain.

Given an ideal triangulation T of Ŝ, we define a canonical coordinate system {Δ_{i}} on the space A_{SL_{m},Ŝ}, parametrized by the set

I_{m}^{T} := {vertices of the m–triangulation of T } - {vertices at the punctures of S}  (13)

Further, we define a canonical coordinate system {X_{j}} on X_{P GL_{m},Ŝ} parametrised by the set

J_{m}^{T} := I_{m}^{T} − { the vertices at the boundary of S }

Figure 1: The 4-triangulation arising from a triangulation of a surface

Figure 2: The set I_{2}^{T} is naturally identified with the set of all internal edges of T.

Definition 1.16 A skew symmetric ℤ–valued function ε_{pq} on the set of vertices of the m-triangulation of T is given by

ε_{pq} := #{oriented edges from p to q} − #{oriented edges from q to p}  (14)

The Weil-Petersson form and the Poisson structure in these coordinates are given by

Ω_{SL_{m},Ŝ} = ∑_{i_{1},i_{2}} ε_{i_{1}i_{2}}d log Δ_{i_{1}} ∧ d log Δ_{i_{2}},  {X_{j_{1}}, X_{j_{2}}}_{P GL_{m},Ŝ} = ∑_{j_{1}j_{2}} ε_{j_{1},j_{2}}X_{j_{1}}X_{j_{2}}  (15)

The K_{2}-avatar of the Weil-Petersson form is given by W_{SL_{m},Ŝ} = ∑_{i_{1},i_{2}} ε_{i_{1}i_{2}}{Δ_{i_{1}}, Δ_{i_{2}}}

We call an ideal triangulation T of Ŝ is special if it contains, as a part, one of the graphs shown on Figure. A marked surface Ŝ is special if it admits a special triangulation, and regular otherwise.

Figure 3: Special graphs, called a virus and an eye.

Lemma 1.17 A marked hyperbolic surfaces Ŝ is special if and only if it contains at least two holes, and in addition one of the holes has no marked points on the boundary.

The sets I_{m}^{T} and J_{m}^{T} can be defined for any, not necessarily finite, triangulation T. An especially interesting example is provided by the Farey triangulation. It is obviously regular. It has no boundary, so the I and J sets coincide, and denoted by I_{m}. Since the function ε_{pq} for the set I_{m} has a finite support for every given p, all structures of the cluster ensemble but the W -class make sense.

Theorem 1.18 a) The universal pair of positive spaces (X_{P GL_{m}}, A_{SL_{m}}) is a part of the cluster ensemble for the function ε_{pq} on the set I_{m}.

b) If Ŝ is regular, the pair of positive spaces (X_{P GL_{m},Ŝ}, A_{SL_{m},Ŝ}) is a part of the cluster ensemble for the function ε_{pq} from Definition 1.16.

We assign a coordinate system on the X - and A-space to every triangulation T of Ŝ. For every T the function ε_{pq} from Definition 1.16 describes the Poisson structure on X_{P GL_{m},Ŝ} and the W - class on X_{SL_{m},Ŝ} in the corresponding coordinate system via formulas (15). However the transition functions between the coordinate systems assigned to special triangulations are different then the ones prescribed by the cluster ensemble prescriptions. Therefore, although Definition 1.16 assigns a cluster ensemble structure for every Ŝ, if Ŝ is special, the pair of positive spaces (X_{P GL_{m},Ŝ}, A_{SL_{m},Ŝ}) is no longer a part of this, or in fact any other cluster ensemble structure. We show that it is related to a more general orbi-cluster ensemble structure.

Theorem 1.19 If Ŝ is special, the pair of positive spaces (X_{P GL_{m},Ŝ}, A_{SL_{m},Ŝ}) is a part of the orbicluster ensemble for the function ε_{pq} from Definition 1.16.

The precise meaning of this theorem is discussed in Section 9. In the sequel to this paper we will prove a similar result for an arbitrary G, quantize the higher Teichmüller spaces and as a result construct an infinite dimensional unitary projective representation of the group Γ_{S}.

9. Coda. We suggest that investigation of the pair of moduli spaces (A_{G,Ŝ}, X_{G′,Ŝ}) can be viewed as a marriage of representation theory and the theory of surfaces. Indeed, when the surface Ŝ is simple, i.e. is a disc with marked points on the boundary or an annulus, we recover many aspects of the representation theory for the group G, quite often from a new point of view: positivity in G(ℝ), invariants in the tensor products of finite-dimensional representations, W -algebras, quantum groups, canonical bases, Springer variety etc. On the other hand, when G = SL_{2} but Ŝ is general we get the theory of Riemann surfaces.

Figure 4: Leitfaden

Acknowledgments. N. Reshetikhin informed us about his work in progress with R. Kashaev, aimed at quantization of the moduli spaces of flat G-connections on S, containing a result similar to our decomposition (104). After the first version of this work was posted in the ArXive, we learned about the very interesting preprint of F. Labourie [Lab] who, in particular, proved discreteness and hyperbolicity of representations of π_{1}(S) → P SL_{m}(ℝ), when S has no holes, lying in the Hitchin component, and proved that their limit sets are described by convex projective curves.

This work was developed when V.F. visited Brown University. He would like to thank Brown University for hospitality and support. He wish to thank grants RFBR 01-01-00549, 0302-17554 and NSh 1999.2003.2. A.G. was supported by the NSF grant DMS-0099390. The paper was finished when A.G. enjoyed the hospitality of IHES (Bures sur Yvette) and MPI (Bonn) during the Summer and Fall of 2003. He is grateful to NSF, IHES and MPI for the support. We are grateful to J. Bernstein, M. Gromov, D. Kazhdan and M. Kontsevich for useful discussions.

2 The moduli spaces A_{G,Ŝ} and X_{G,Ŝ}

Let us replace the holes on S by punctures, getting a surface S′. The local systems on S and S′ are the same. Let us equip the surface S′ with a complex structure. Then we get a complex algebraic curve 𝕊. The moduli space of local systems on 𝕊 has two different algebraic structures: the monodromic and the De Rham structures. Nevertheless the corresponding complex analytic spaces are isomorphic. The monodromic structure depends only on the fundamental group π_{1}(𝕊(ℂ)). Therefore it is determined by the topological surface S′ underlying the curve 𝕊. The corresponding moduli stack is denoted L_{G,S}. To define the De Rham structure recall the Riemann-Hilbert correspondence ([D1]) between the isomorphism classes of G–local systems on S′ = 𝕊(ℂ) and algebraic G–connections with regular singularities on the algebraic curve 𝕊. The moduli space L_{G,𝕊} of regular G-connections on 𝕊 is an algebraic stack over ℚ. Its algebraic structure depends on the algebraic structure of the curve 𝕊. It is the De Rham algebraic structure. The Riemann-Hilbert correspondence provides an isomorphism of the complex analytic spaces L_{G,S}(ℂ) and L_{G,𝕊}(ℂ).

1. The moduli spaces X_{G,Ŝ}. The flag variety B parametrises all Borel subgroups in G. Choosing a Borel subgroup B we identify it with G/B. Let L be a G-local system on S. Recall the associated flag bundle L_{B} := L ×_{G} B. We call the elements of a fiber of L_{B} over x by flags over x in L. A reduction of a G–local system L on a circle to B is just the same as a choice of a flag in a fiber of L invariant under the monodromy.

Examples. a) A flag in a vector space V is given by a filtration

0 = V_{0} ⊂ V_{1} ⊂ V_{2} ⊂ ... ⊂ V_{m} = V, dimV_{i} = i  (16)

The flag variety for G = GL_{m} parametrizes all flags in V.

b) Let G = GL_{m}. Recall the bijective correspondence between the GL_{m}–local systems and local systems of m-dimensional vector spaces. Namely, let V be a standard m-dimensional representation of GL_{m}. A GL_{m}–local system L provides a local system of vector spaces L′ := L×_{GL_{m}} V. Conversely, given an m-dimensional local system L′ we recover the corresponding GL_{m}–local system by taking all bases in fibers of L. The flag bundle L_{B} consists of all flags in the fibers of the local system L′. If G = SL_{m}, L′ is a local system with an invariant volume form.

Definition 2.1 Let G be an arbitrary split reductive group. An enhanced G-local system on Ŝ is a pair (L, β) where L is a G-local system on S and β is a flat section of the restriction of L_{B} to the punctured boundary ∂Ŝ. X_{G,Ŝ} is the moduli space of enhanced G–local systems on Ŝ.

One can rephrase this definition as follows:

i) If C_{i} is a boundary component without marked points, we choose a flag F_{x} over a certain point x ∈ C_{i} invariant under the monodromy around C_{i}. Equivalently, we choose a reduction to the subgroup B of the restriction of the G-local system L to the boundary component C_{i}. The parallel transport of the flag F_{x} provides the restriction of the section β to C_{i}, and vice versa.

ii) If there is a non empty set {x_{1}, ..., x_{p}} of the boundary points on the component C_{i}, the complement C_{i} − {x_{1}, ..., x_{p}} is a union of p arcs. Let us pick a point y_{j} inside of each of these arcs, and choose a flag F_{j} over y_{j}. These flags may not be invariant under the monodromy. The flags F_{j} determine the restriction of the flat section β to the corresponding arc, and vice versa.

The Cartan group of G. A split reductive algebraic group G determines the Cartan group H. Namely, let B be a Borel subgroup in G. Then H := B/U where U is the unipotent radical of B. If B′ is another Borel subgroup then there is a unique isomorphism B/U ⟶ B′/U′. So the quotient B/U is canonically isomorphic to the Cartan subgroup H in G. Let

i_{B} : H → B/U  (17)

be the canonical isomorphism. The Cartan group H is not considered as a subgroup of G. The canonical projection B → H = B/U provides a natural projection

κ : X_{G,S} ⟶ H^{{punctures of S}}  (18)

The moduli space X_{G,Ŝ} as a quotient stack. Let F be an arbitrary field. Choose a point x ∈ S. Let g be the genus of the surface \overline{S}. Let us choose generators α_{1}, ..., α_{g}, β_{1}, ..., β_{g}, γ_{1}, ..., γ_{n} ∈ π_{1}(S, x) such that

∏_{j=1}^{g}[α_{j}, β_{j}] ∏_{i=1}^{n} γ_{i} = 1  (19)

and where γ_{i} belongs to the conjugacy class in π_{1}(S, x) determined by the i-th hole in S. It is well known that such a system of generators exists, and that (19) is the only relation between them in π_{1}(S, x).

Definition 2.2 An enhanced representation π_{1}(S, x) → G(F) is the following data:

(X_{α_{1}}, ..., X_{α_{g}}, X_{β_{1}}, ..., X_{β_{g}}, X_{γ_{1}}, ..., X_{γ_{n}}, {B_{j}}),  X_{?} ∈ G(F), j ∈ π_{0}(∂Ŝ), B_{j} ∈ B(F) (20)

where if j corresponds to a connected component C_{j} of the boundary of S without marked points on it, then X_{γ_{j}} ∈ B_{j}(F).

Obviously the elements of the set (20) are the F -points of an affine algebraic variety denoted by X̃_{G,Ŝ}. The algebraic group G acts on this variety, and, by definition, the stack X_{G,Ŝ} is the quotient stack X̃_{G,Ŝ}/G.

Examples. 1. Let D̂_{n} be a disc with n marked points on the boundary. Then X̃_{G,D̂_{n}} = B^{n}, and X_{G,D̂_{n}} = G\B^{n} is the space of configurations of n flags.

2. Let S be an annulus. Then X̃_{G,S} = {(g, B_{1}, B_{2})} where B_{i} are Borel subgroups in G and g ∈ B_{1}, g ∈ B_{2}. The quotient X_{G,S} := X̃_{G,S}/G is the Steinberg variety, see [CG].

2. Pinnings. Let us introduce some notations following [L1]. Let G be a split reductive simplyconnected connected algebraic group over ℚ. Let H be a split maximal torus of G and B^{+}, B^{−} a pair of opposed Borel subgroups containing H, with unipotent radicals U^{+}, U^{−}. Let U_{i}^{+} (i ∈ I) be the simple root subgroups of U^{+} and let U_{i}^{−} be the corresponding root subgroups of U^{−}. Here I is a finite set indexing the simple roots. Denote by χ′_{i} : H ⟶ 𝔾_{m} the simple root corresponding to U_{i}^{+}. Let χ_{i} : 𝔾_{m} ⟶ H be the simple coroot corresponding to χ′_{i}. We assume that for each i ∈ I we are given isomorphisms x_{i} : K ⟶ U_{i}^{+} and y_{i} : K ⟶ U_{i}^{−} such that the maps

( \begin{matrix}1 & a \\ 0 & 1 \end{matrix} ) ⟼ x_{i}(a), ( \begin{matrix}1 & 0 \\ b & 1 \end{matrix} ) ⟼ y_{i}(b), ( \begin{matrix}t & 0 \\ 0 & t^{−1} \end{matrix} ) ⟼ χ_{i}(t)

provide a homomorphism φ_{i} : SL_{2}(K) ⟶ G. The datum (H, B^{+}, B^{−}, x_{i}, y_{i}; i ∈ I) is called a pinning for G. The group G acts by conjugation on pinnings, making it into a principal homogeneous space for the group G/Center(G). In particular any two pinnings for G are conjugated in G.

There is unique involutive antiautomorphism Ψ : G ⟶ G such that for all i ∈ I and t ∈ H one has Ψ(x_{i}(a)) = y_{i}(a), Ψ(y_{i}(a)) = x_{i}(a) and Ψ(t) = t.

Example. If G = SL_{m} then B^{+}, B^{−} are the subgroups of upper and lower triangular matrices, so H is the subgroup of diagonal matrices. Ψ is the transposition. If G = P SL(V) where V is a vector space then a choice of pinning is the same thing as a choice of a projective basis in V_{m}.

Let X^{∗}(T) and X_{∗}(T) be the groups of characters T → 𝔾_{m} and cocharacters 𝔾_{m} → T for a split tori T. There is a pairing

< ·, · >: X^{∗}(T) × X_{∗}(T) ⟶ ℤ, χ′ ∘ χ : 𝔾_{m} ⟶ 𝔾_{m}, t ⟼ t^{<χ′,χ>}

Let A = (a_{ij}) be the Cartan matrix given by a_{ij} :=< χ′_{j}, χ_{i} >. Recall the weight lattice P ⊂ X^{∗}(H). It is the subgroup of all γ ∈ X^{∗}(H) such that < γ, χ >∈ ℤ for all i ∈ [1, ..., r]. Let {ω_{1}, ..., ω_{r}} be the ℤ–basis of fundamental weights given by < ω_{j}, χ_{i} >= δ_{ij}.

3. The element s_{G}. A choice of pinning for G provides an inclusion of the Cartan group H into G. Abusing notation we will denote the obtained subgroup by H and call it the Cartan subgroup corresponding to the pair (B^{−}, B^{+}) of opposite Borel subgroups. Then the element

\overline{s}_{i} := y_{i}(1)x_{i}(−1)y_{i}(1) = φ_{i} ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} )

lifts to Norm_{G}(H) the generator s_{i} ∈ W. It is well known that the elements \overline{s}_{i}, i ∈ I, satisfy the braid relations. Therefore we can associate to each w ∈ W its standard representative \overline{w} ∈ Norm_{G}(H) in such a way that for any reduced decomposition w = s_{i_{1}}...s_{i_{k}} one has \overline{w} = \overline{s}_{i_{1}}...\overline{s}_{i_{k}}. Let w_{0} be the maximal length element of W. We set s_{G} := \overline{w}_{0}^{2}. Then evidently s_{G}^{2} = e.

Example. If G = SL_{m} then s_{SL_{m}} = (−1)^{m−1}e.

Let R be the system of roots for the group G, and R = R_{+} ∪ R_{−} its decomposition into a union of the sets of positive and negative roots. Then for any element w ∈ W its length l(w) equals to |R_{−} ∩ w(R_{+})|. For any root α = ∑ n_{i}α_{i} we set χ_{α}(t) := ∏ χ_{i}(t)^{n_{i}}.

Lemma 2.3 Let w ∈ W. Then

\overline{w}\overline{w^{−1}} = ∏_{α∈R_{−}∩w(R_{+})} χ_{α}(−1)

Proof. By the induction on the length l(w). If l(w) = 1 this is clear. Assume w = w_{i}w′ where l(w′) = l(w) − 1. Then by the induction assumption

\overline{w}\overline{w^{−1}} = \overline{w}_{i}\overline{w′}\overline{w^{−1}}\overline{w}_{i} = w_{i}( ∏_{α∈R_{−}∩w′(R_{+})} χ_{α}(−1))w_{i}^{−1}\overline{w}_{i}\overline{w_{i}^{−1}} = ∏_{α∈R_{−}∩w′(R_{+})} χ_{w_{i}(α)}(−1)χ_{α_{i}}(−1)

Since w_{i}(R_{−} ∩ w′(R_{+})) ∪ −α_{i} = R_{−} ∩ w(R_{+}), the lemma is proved.

Corollary 2.4 s_{G} is a central element of G of order 2.

Proof. Let 2ρ be the sum of all positive roots. Then (see Proposition 29, Chapter VI, paragraph 1 in [Bo]) we have 2ρ = ω_{1} + ... + ω_{r}. Denote by χ′_{2ρ} the character of H corresponding to 2ρ. Then for any coroot χ one has < χ′_{2ρ}, χ >∈ 2ℤ. Since w_{0}^{−1} = w_{0}, this combined with Lemma 2.3 implies that s_{G} commutes with the elements x_{i}(t), and thus is a central element. The corollary is proved.

Recall the well defined up to a conjugation embedding i : SL_{2} ↪ G called the principal SL_{2} subgroup of G.

Lemma 2.5 s_{G} = i(s_{SL_{2}}) for the principal SL_{2} subgroup of G.

Proof. An easy exercise.

4. The moduli spaces A_{G,Ŝ}. Let us choose a maximal unipotent subgroup U in G. The normalizer of U in G is a Borel subgroup B. Conversely, U is the commutant of B. The Cartan group H acts from the right on the principal affine variety A := G/U: if A = gU ∈ A we set

A · h = gU · h := gi_{B}(h)U  (21)

Since H is commutative it, of course, can be considered as a left action. This action is free. The quotient A/H is identified with the flag variety B = G/B. The canonical projection A → B is a principal H-bundle. More generally, let X be a right principal homogeneous space for G. Then X/U is an AutX–homogeneous space. Recall the principal affine bundle L_{A} := L/U associated to a G–local system L on S.

Let T′_{S} the punctured tangent bundle to the surface S, i.e. the tangent bundle with the zero section removed. Its fundamental group π_{1}(T′S, x) is a central extension of π_{1}(S, y) by ℤ, where y ∈ T_{x}S:

0 ⟶ ℤ ⟶ π_{1}(T′S, x) ⟶ π_{1}(S, y) ⟶ 0  (22)

Let T′_{y}S be the punctured tangent space at a point y. The inclusion T′_{y}S ↪ T′S induces an isomorphism of π_{1}(T′_{y}S) =^{∼} ℤ with the central subgroup ℤ in (22). We denote by σ_{S} a generator of this central subgroup. It is well defined up to a sign.

Definition 2.6 A twisted G-local system on S is a local system on T′S with the monodromy s_{G} around σ_{S}.

Observe that since s_{G} is of order two, this does not depend on the choice of the generator σ_{S}. Let \overline{π}_{1}(S, y) be the quotient of π_{1}(S, y) by the central subgroup 2ℤ ⊂ ℤ, so it is a central extension

0 ⟶ ℤ/2ℤ ⟶ \overline{π}_{1}(T′S, x) ⟶ π_{1}(S, y) ⟶ 0

We denote by \overline{σ}_{S} the order two central element in the subgroup ℤ/2ℤ. The twisted local systems on S are in bijective correspondence with the representations ρ : \overline{π}_{1}(T′S, x) → G, considered modulo conjugation, such that ρ(\overline{σ}_{S}) = s_{G}.

Remark. Let us choose a complex structure on S. Then the uniformisation provides an embedding i : π_{1}(S, y) ↪ P SL_{2}(ℝ). It is easy to see that the group \overline{π}_{1}(T′S, x) is isomorphic to the preimage of the subgroup i(π_{1}(S, y)) in SL_{2}(ℝ). In particular \overline{σ}_{S} corresponds to the element −e ∈ SL_{2}(ℝ).

The group \overline{π}_{1}(T′S, x) is isomorphic to the direct product ℤ/2ℤ × π_{1}(S, y), although this isomorphism is by no means canonical. The set of all isomorphisms

\overline{π}_{1}(T′S, x) ⟶ ℤ/2ℤ × π_{1}(S, y)

which send the central subgroup ℤ/2ℤ on the left to the left factor on the right, and reduce to the canonical isomorphism modulo the ℤ/2ℤ-subgroups, is a principal homogeneous space over Hom(π_{1}(S, y), ℤ/2ℤ). Therefore choosing such an isomorphism we can identify the space of twisted G-local systems on S with the space of G-local systems on S. Observe that a choice of such an isomorphism is the same thing as a choice of a spin structure on S.

Let C_{i} be a little annulus containing a boundary component C_{i} of S as a component of its boundary. Observe that π_{1}(T′C_{i}) is canonically identified with ℤ × ℤ: the first factor is π_{1}(T′_{x}C_{i}), and the second is generated by the tangent vectors to C_{i}. Let L be the isomorphism class of the G-local system on C_{i} whose monodromy along the element (a, b) ∈ ℤ × ℤ = π_{1}(T′C_{i}) is s_{G}^{a}.

Let x_{1}, ..., x_{p} be the marked points on the component C_{i}. Let us present the annulus C_{i} as a product C_{i} × [0, 1], and let C′_{i} := (C_{i} − {x_{1}, ..., x_{p}}) × [0, 1]. Then C′_{i} has k connected components.

Definition 2.7 Let G be a split simply-connected reductive group. Let L be a G-local system on T′S representing a twisted local system on S. A decoration on L is a choice of locally constant section α of the restriction of the principal affine bundle L_{A} to ∪_{i}C′_{i}.

A decorated twisted G–local system on a marked surface Ŝ is a pair (L, α), where L is a twisted G-local system on S, and α is a decoration on L. The space A_{G,Ŝ} is the moduli space of decorated twisted G–local systems on Ŝ.

When s_{G} = e this definition reduces to the one given in the Definition 1.3. Just like in the X -case, the moduli space A_{G,Ŝ} can be understood as the quotient moduli stack. We leave the obvious details to the reader.

Let Conf_{n}^{∗}(A) := G\A^{n} be the configuration space of n affine flags in G.

Definition 2.8 Twisted cyclic shift map S : Conf_{n}^{∗}(A) ⟼ Conf_{n}^{∗}(A) is given by

S : (A_{1}, A_{2}, ..., A_{n}) ⟼ (A_{2}, ..., A_{n}, s_{G}A_{1})

Since s_{G} is a central element one has A · s_{G} = s_{G}A. Therefore S^{n} = Id because

S^{n}(A_{1}, A_{2}, ..., A_{n}) = (s_{G}A_{1}, s_{G}A_{2}, ..., s_{G}A_{n}) ∼ (A_{1}, A_{2}, ..., A_{n})

The coinvariants of twisted cyclic map acting on the space Conf_{n}^{∗}(A) are called twisted cyclic configurations of n affine flags in G. Let { \atop Conf}_{n}^{∗}(A) be the moduli space of of twisted cyclic configurations of n affine flags in G.

Lemma 2.9 Let D̂_{k} be a disc D with k marked points at the boundary. Then A_{G,D̂_{k}} is identified

with the twisted cyclic configuration space { \atop Conf}_{k}^{∗}(A).

Proof. The punctured tangent bundle to a disc is homotopy equivalent to a circle. So there is unique up to an isomorphism twisted G-local system on D̂_{k}. Consider a path in T′D obtained by choosing a non zero tangent vector field on a simple loop near the boundary of D. The monodromy around this path is s_{G}. The marked points x_{1}, ..., x_{k} on the boundary C = ∂D divide C into a union of the arcs C_{1}^{∗}, ..., C_{k}^{∗}. We assume they follow the boundary clockwise. Let us choose a vector v_{i} tangent to D at a certain point of the arc C_{i}^{∗}. We assume that the vectors v_{i} look inside of the disc D. For each vector v_{i} there is canonical homotopy class of path from v_{i} to v_{1} in T′D obtained by counterclockwise transport of v_{i} along C towards v_{1}. Restricting the decoration section α to v_{1}, ..., v_{k} and then translating the obtained affine flags in the fibers over v_{2}, ..., v_{k} to the fiber over v_{1} we get a configuration of affine flags (A_{1}, ..., A_{k}) over v_{1}. If we choose v_{2} instead of v_{1} as the initial vector, this configuration will be replaced by its twisted cyclic shift (A_{2}, ..., A_{n}, s_{G}A_{1}). So we identified A_{G,D̂_{k}}

with { \atop Conf}_{k}^{∗}(A). The lemma is proved.

Recall the adjoint group G′ corresponding to the group G. Since the canonical map G → G′ kills the center, and s_{G} is in the center, a twisted G-local system L on T′S provides a G′-local system L′ on S. The canonical projection p : L_{A} ⟶ L′_{B} provides a map

p : A_{G,Ŝ} ⟶ X_{G′,Ŝ},  (L, α) ⟼ (L′, β), β := p(α)  (23)

Lemma 2.10 Let k be the number of boundary points. Then dimX_{G′,Ŝ} + k · dimH = dimA_{G,Ŝ}. If k = 0, one has dimX_{G′,S} = dimA_{G,S} = dimL_{G,S} = (2g + n − 2)dimG.

Proof. The map (18) is surjective over the generic point. Consider a boundary component C_{i} without boundary points. The condition that a G–local system has a unipotent monodromy around C_{i} decreases the dimension by rank(G), while adding a decoration at s we increase the dimension by dimH = rank(G). On the other hand over an arc the space of α’s is G/U, while the space of β’s is G/B. The lemma is proved.

5. Remarks, complements and examples. 1. If Ŝ = S, i.e. there is no boundary points, it may be convenient to assume that S = \overline{S}−\{s_{1}, ..., s_{n}\} is a surface with n punctures. For each puncture there is a well defined isotopy class of a little oriented circle making a simple counterclockwise loop around s. Let L_{s} be the isomorphism class of the restriction of a G–local system L to it. An enhanced G–local system on S is then a G–local system L equipped with the following additional structure: for every puncture s we choose a reduction of the local system L_{s} to the Borel subgroup B.

One can always avoid talking about holes D_{i} in \overline{S}. Indeed, let S be a surface with n punctures. Denote by Sp_{s_{i}}L the specialization of the local system L to the punctured tangent space T_{s_{i}} \overline{S} − \{0\} at s_{i}. It is a local system on the punctured tangent space. The marked points are replaced by a collection of distinct rays in the punctured tangent bundle T_{s_{i}} \overline{S} − \{0\}. An enhanced structure on L is given by the following data at the punctures s_{i}: restrict the local system Sp_{s_{i}}L to the complement of the chosen rays, and take its flat section β. This language is convenient when we work with an algebraic structure on S.

2. Let us look more closely at the projection (23) in the case when Ŝ = S. The monodromy of a G–local system on a circle is described by a conjugacy class in G. The monodromy provides a bijective correspondence between the conjugacy classes in G and the isomorphism classes of G–local system on a circle. A G–local system L on S is unipotent if for every boundary component C_{i} the monodromy of the restriction L_{|C_{i}} is unipotent.

Definition 2.11 U_{G,S} is the moduli space of the enhanced unipotent G–local systems on S.

Observe that a B-local system L_{s} on a circle has a unipotent monodromy if and only if it can be reduced to a U-local system. A decoration on L_{s} is the same as a reduction of the structure group to the subgroup U. So a decoration on a B–local system L_{s} on a circle exists if and only if L_{s} is unipotent. Thus a decorated G–local system on S is unipotent. The right action of H on L_{s} provides an action of H on the set of all decorations, making it into a principal homogeneous space over H.

Therefore forgetting the decoration we get a canonical projection

p′ : A_{G,S} ⟶ U_{G,S}  (24)

It provides A_{G,S} with a structure of a principal H{punctures}_{G}-bundle over U_{G,S}, where H_{G} is the

Cartan group of G. Observe that the map κ is the projection onto the group H{punctures}_{G′}. There is a commutative diagram, where i is the natural closed embedding:

A_{G,S}
p′ ↓  ↘ p
U_{G′,S}  ⟶^{i} X_{G′,S} ⟶^{κ} H{punctures}_{G′}

(25)

One has U_{G′,S} = p(A_{G,S}) = κ^{−1}(e).

3. Affine flags. The points of L_{B} parametrize the flags in the fibers of L. In many cases, e.g. when the center of G is trivial, there is a similar interpretation of the principal affine bundle based on the notion of an affine flag.

Let A be a direct sum of one dimensional H-modules corresponding to all simple roots. Let U be the Lie algebra of U. The H-module U/[U, U] is isomorphic to A.

Definition 2.12 Suppose that the center of G is trivial. An affine flag is a pair

{B, i : A ⟶^{∼} U/[U, U]}  (26)

where B is the Borel subgroup containing U, and i is an isomorphism of H-modules.

The group G acts by conjugation on affine flags. This action is transitive. Since the center of G is trivial, the stabilizer of the affine flag (26) is U. So the principal affine bundle parametrises the affine flags in the fibers of L. The same conclusion can be maid for the classical groups with non trivial center. Here are some examples.

The classical affine flags. i). G = GL_{m}. Recall that an affine flag in a vector space V is given by a flag (16) in V plus a choice of non zero vectors v_{i} at each quotient V_{i}/V_{i−1} for all i = 1, ..., m. Equivalently, an affine flag in V can be defined by a sequence of non zero volume elements η_{i} ∈ detV_{i}:

η_{1} := v_{1}, η_{2} := v_{1} ∧ v_{2}, ..., η_{m} := v_{1} ∧ ... ∧ v_{m};

Alternatively, consider the dual volume forms ω_{i} ∈ detV_{i}^{∗}, < η_{i}, ω_{i} >= 1. We may parametrise the affine flags by the sequence of the volume forms (ω_{1}, ..., ω_{m}). The variety of all affine flags is identified with GL_{m}/U where U is the subgroup of upper triangular matrices in GL_{m} with units at the diagonal.

ii). The SL_{m} case. An affine flag in V determines a volume form in V. By an affine flag in a vector space with a given volume form ω we understand an affine flag whose volume form is the form ω. The variety all affine flags with a given volume form is identified with SL_{m}/U.

Example. An affine flag in a two dimensional vector space V is a pair {a non zero vector v_{1} and a non zero 2–form ω in V }. The variety of all affine flags in a two dimensional symplectic vector space is identified with 𝔸^{2} − {0}.

3 Surfaces, graphs, mapping class groups and groupoids

This Section contains some background material, suitably adopted or generalized. We start from the ribbon graphs and their generalizations needed to describe two dimensional surfaces with marked points on the boundary. We use it throughout the paper. Then we introduce a convenient combinatorial model of the classifying orbi-space for mapping class groups which will be used in Section 12. In the end we recall some basic facts about the classical Teichmüller spaces which will be used in Section 10 and some other parts of the paper.

1. Ribbon graphs and hyperbolic surfaces. Recall that a flag of a graph Γ is a pair (v, E) where v is a vertex of Γ, E is an edge of Γ, and v is an an endpoint of E. A ribbon graph is a finite graph such that for each vertex a cyclic order of the set of all flags sharing this vertex is chosen. An example of a ribbon graph is provided by a graph drawn on an oriented surface S. The cyclic order of the flags sharing the same vertex is given by the counterclockwise ordering of the flags.

We say that an oriented path on a ribbon graph turns left at a vertex if we come to the vertex along the edge which is next (with respect to the cyclic order) to the edge used to get out of the vertex. A face path on a ribbon graph is a closed oriented path turning left at each vertex, which ends as soon as the path starts repeat itself. To visualize a face path imagine a ribbon graph glued of actual thin ribbons. Then a face path is a boundary component of such a ribbon graph. For example the punctured torus has one face path. A flag determines a face path. This face path can be visualized by a strip bounded by the face path. Conversely a face path plus a vertex (resp. edge) on this path determine uniquely the flag containing this vertex (resp. edge) situated on the face path.

Given a ribbon graph Γ we can get an oriented compact surface S_{Γ,c} with a ribbon graph isomorphic to Γ lying on the surface. Namely, we take a disc for each of the face paths and glue its boundary to the graph along the face path. If we use punctured discs in the above construction we get an oriented surface S_{Γ} which can be obviously retracted onto Γ. Clearly S_{Γ} is obtained by making n punctures on S_{Γ,c}, where n = F (Γ). Any hyperbolic surface can be obtained as S_{Γ} for certain ribbon graph Γ. Indeed, consider a triangulation of the surface with the vertices at the punctures, and take the dual graph.

2. Ideal triangulations of marked hyperbolic surfaces and marked trivalent graphs. Let Ŝ be a marked hyperbolic surface. If its boundary component C_{i} has no boundary points on it, let us picture the corresponding hole on S as a puncture.

A marked trivalent graphs on S of type Ŝ is a graph Γ on S with the following properties:

i) The vertices of Γ are of valence three or one.

ii) The set of univalent vertices coincides with the set of marked points on the boundary ∂S. iii) The surface S can be shrinked onto Γ. We define an ideal triangulation of Ŝ as a triangulation of S with the vertices either at the

punctures or at the boundary arcs, such that each arc carries exactly one vertex of the triangulation, and each puncture serves as a vertex. The edges of an ideal triangulation either lie inside of S, or belong to the boundary of S.

The duality between triangulations and trivalent graphs provides a bijection between the isotopy classes of ideal triangulations of Ŝ and isotopy classes of marked trivalent graphs on Ŝ. The ends of the dual graph are dual to the boundary edges of the ideal triangulation.

A marked ribbon graph is a graph with trivalent and univalent vertices equipped with a cyclic order of edges sharing each trivalent vertex. Given a marked ribbon graph one can construct the corresponding marked surface.

Consider a marked graph (\overline{Γ}, v) which has just one 4-valent vertex in addition to trivalent and univalent vertices. Then there are two marked trivalent graphs Γ and Γ′ related to (\overline{Γ}), as shown on the picture. We say that Γ′ is obtained from Γ by a flip at the edge E shrinked to v. A similar operation on triangulations consists of changing an edge in a 4-gon of the triangulation. It is well known that any two isotopy classes of marked trivalent graphs on a surface are related by a sequence of flips.

Figure 5: Flip at the edge E

3. The mapping class group of S. The mapping class group Γ_{S} is the quotient of the group of all orientation preserving diffeomorphisms of S modulo its connected component consisting of the diffeomorphisms isotopic to the identity. It has the following well known group theoretic description. Every hole D_{i} on S provides a conjugacy class in the fundamental group π_{1}(S, x). The mapping class group Γ_{S} consists of all outer automorphisms of π_{1}(S, x) preserving each of these conjugacy classes. The group Γ_{S} acts by automorphisms of the A and X moduli spaces. We will use a combinatorial description of the mapping class group as the automorphism group of objects of the modular groupoid of S defined below.

4. The graph complex G_{•}(S) of a hyperbolic surface S. It is a version of the complex used by Boardman and Kontsevich. To define it, recall the ℤ/2ℤ–torsor Or_{Γ} of orientations of a graph Γ, given by invertible elements in the determinant of free abelian group generated by the edges of Γ. Let G_{k}(S) be the abelian group of ℤ–valued functions φ(Γ, ε_{Γ}) with finite support on the set of pairs (Γ, ε_{Γ}), where Γ is an isotopy class of an embedded graph on S such that S can be retracted to it, ε_{Γ} ∈ Or_{Γ}, one has φ(Γ, −ε_{Γ}) = −φ(Γ, ε_{Γ}), and k is given by

k − 3 = ∑_{v∈V (Γ)}(val(v) − 3)  (27)

Here val(v) is the valency of a vertex v of Γ. Observe that if all except one vertices of Γ are of valency three, then k is the valency of this one vertex. The differential ∂ : G_{•}(S) ⟶ G_{•−1}(S) is provided by contraction of the edges of Γ:

∂φ(Γ, ε_{Γ}) := ∑_{E′,Γ′} φ(Γ′, ε_{Γ′})  (28)

Here the sum is over all pairs (E′, Γ′), where E′ is an edge of Γ′, such that the ribbon graph Γ′/E′ obtained by shrinking of E′ is isomorphic Γ, and ε_{Γ′} = E′ ∧ ε_{Γ}. Then ∂^{2} = 0. We place the trivalent graphs in the degree 2, and consider G_{•}(S) as a homological complex.

The graph complex of S is the chain complex of modular complex a finite dimensional contractible polyhedral complex 𝔾(S), called the modular complex of S, constructed in the next subsection.

5. The modular complex of S. Recall the n–dimensional Stasheff polytop K_{n}. It is a convex polytop whose combinatorial structure is described as follows. Consider plane trees with n + 3 ends lying on an oriented circle. The ends are marked by the set {1, ..., n + 3}, so that its natural order is compatible with the circle orientation. The codimension p faces of K_{n} are parametrized by the isotopy classes of such plane trees with p + 1 internal vertices. Let F_{T} be the face corresponding to a tree T. Then F_{T} belongs to the boundary of F\overline{T} if and only if \overline{T} is obtained from T by shrinking some internal edges. The Stasheff polytop K_{n} can be realized as a cell in the configuration space of n + 3 cyclically ordered points on a circle, see [GM] for a survey.

Example. K_{0} is a point, K_{1} is a line segment, and K_{2} is a pentagon.

Let us define a finite dimensional contractible polyhedral complex 𝔾(S), called the modular complex of S. The faces of the modular complex 𝔾(S) are parametrized by the isotopy classes of graphs Γ on S, homotopy equivalent to S. The polyhedron P_{Γ} corresponding to Γ is a product of the Stasheff polytops:

P_{Γ} := ∏_{v∈V (Γ)} K_{val(v)−3}  (29)

Here V (Γ) is the set of vertices of Γ, and val(v) is the valency of a vertex v. The face P_{Γ} is at the boundary of the face P\overline{Γ} if and only if \overline{Γ} is obtained from Γ by shrinking of some edges. Precisely, the inclusion P_{Γ} ↪ P\overline{Γ} is provided by the similar data for the Stasheff polytops appearing in (29). The mapping class group D(S) acts on 𝔾(S) preserving the polyhedral structure.

Examples. 1. The vertices of 𝔾(S) correspond to the isotopy classes of the trivalent graphs on S. The edges – to the graphs with one 4–valent vertex. The two dimensional faces are either pentagons corresponding to graphs with unique 5–valent vertex, or squares corresponding to graphs with two 4–valent vertices.

2. Let S be a torus punctured at a single point. The mapping class group of S is isomorphic to SL_{2}(ℤ). Then 𝔾(S) is the tree dual to the classical modular triangulation of the hyperbolic plane:

Figure 6: The tree dual to the modular triangulation

Proposition 3.1 The graph complex G_{•}(S) is the chain complex of the modular complex 𝔾(S).

Proof. The faces of the modular complex are glued according to the boundary map (28). An orientation ε_{Γ} of Γ provides an orientation of the face P (Γ). The lemma now follows from the very definition.

Proposition 3.2 The polyhedral complex 𝔾(S) is contractible. The stabilisers of the mapping class group acting on 𝔾(S) are finite.

The proof is deduced from Strebel’s theory of quadratic differentials on Riemann surfaces [S]. One can show that there exists a Γ_{S}-equivariant contraction of the Teichmüller space to the modular complex of S. So the modular complex is a ”combinatorial version” of the Teichmüller space.

6. Modular groupoid and combinatorial description of the mapping class group. The quotient 𝔾(S)/Γ_{S} is a polyhedral orbifold. By proposition 3.2 its orbifold fundamental group is isomorphic to Γ_{S}. The modular groupoid of S is a combinatorial version of the Poincare groupoid of this orbifold. Precisely, the objects of the modular groupoid of S are Γ_{S}–orbits of the vertices of modular complex 𝔾(S), i.e. isomorphism classes of of the trivalent ribbon graphs of type S. The morphisms are represented by paths on the 1-skeleton composed with the graph automorphisms, i.e. the elements of Γ_{S} stabilizing the vertices. The two paths represent the same morphism if an only if they are homotopic in the 2–skeleton 𝔾_{≤2}(S)/Γ_{S}.

Here is a description of the modular groupoid by generators and relations. Recall the flip determined by an oriented edge of the modular complex, see the picture (5). Every morphism in the modular groupoid is a composition of the elementary ones, given by flips and graph symmetries. There are two types of relations between the elementary morphisms provided by the two types of the two dimensional faces of 𝔾(S):

i) (Square). The flips performed at two disjoint edges commute.

ii) (Pentagon). The composition of the five flips corresponding to the (oriented) boundary of a pentagonal face of 𝔾(S) is zero.

They together with the graph symmetries generate all the relations between the morphisms in the modular groupoid.

7. Background on the Teichmüller spaces. The classical Teichmüller space T_{S} for a hyperbolic surface S can be defined as the moduli space of any of the following four different objects related to S:

1) Complex structures on S.

2) Faithful representations π_{1}(S) → P SL_{2}(ℝ) with discrete image, modulo P SL_{2}(ℝ)-conjugation.

3) Complete Riemannian metrics on S of curvature −1, called complete hyperbolic metrics.

4) Hyperbolic metrics on S with geodesic boundary ∂S.

Indeed, the uniformisation theorem implies 1) <=> 2). Taking the quotient of the hyperbolic plane H by the action of the image of π_{1}(S) in P SL_{2}(ℝ) we go from 2) to 3). A Riemannian metric on S provides a complex structure on S, and hence the arrow 3) => 1). Taking the universal cover of S we see the equivalence 2) <=> 3). For each hole on a surface S with complete hyperbolic structure there is a unique geodesic, called boundary geodesic, homotopic to the boundary of the hole, except the case when the hole degenerates to a puncture, i.e. the corresponding end of S is a cusp. Cutting out the ends of the surface S along the boundary geodesics we get a surface with geodesic boundary. This leads to the equivalence 3) <=> 4).

The universal cover of a surface S equipped with a hyperbolic metric with geodesic boundary can be realized as the hyperbolic plane minus an infinite collection of disjoint discs bounding geodesics. These geodesics project onto the boundary geodesics on S. Precisely, let S′ be a surface with complete hyperbolic metric corresponding to S, i.e. S = S′ − T_{1} ∪ ... ∪ T_{n} where T_{i} is the tube cutted out from S′ by the boundary geodesic corresponding to the i-th hole. The preimage of each boundary geodesic to the hyperbolic plane is a disjoint collection of geodesics on H, with the monodromy group Γ acting transitively on it. The tube T_{i} lifts to a union of discs bounding these geodesics. Cutting out these discs we get a universal cover H_{S} of S sitting inside of H. The cusps on the boundary of H_{S} form a Cantor set, which is the limit set of Γ.

If the number n of holes is positive, the Teichmüller space T_{S} is a manifold with corners. Choosing orientations of all boundary geodesics we get a version of the Teichmüller space denoted T_{S}^{+}. Forgetting orientations of the geodesics we get a 2^{n} : 1 covering T_{S}^{+} → T_{S}. On the other hand the orientation of S provides canonical orientations of the geodesics, and hence an embedding T_{S} ↪ T_{S}^{+}.

8. Special graphs and flips. Recall that a trivalent graph on S is special if it contains, as a part, one of the two graphs shown on Figure 7 by punctured lines. We call them an eye and a virus. The triangulations dual to the eye and virus are shown on Figure 7 by solid lines. The sides shown by arrows are identified. The triangulations dual to the graphs containing a virus are characterised by the property that they contain a triangle with two sides identified. To obtain a diagram of the triangulation corresponding to an eye we cut one of the edges of the triangulation, the top one on the middle picture, getting the right diagram on Figure 7.

Figures 8 and 9 show that the special graphs can be obtained by flips from the usual graphs. Namely, on Figure 8 we obtain an eye by making a flip at the bottom horizontal edge of the graph

Figure 7: A virus and an eye, with the corresponding triangulations.

on the left. The bottom of Figure 9 shows that a flip at the leg of a virus produces an eye, and vice versa. A flip at the loop of a virus is not defined.

Figure 8: An eye is obtained by a flip at the bottom edge on the left.

Figure 9 shows that a flip from an eye to a virus, lifted to a 2 : 1 cover of the surface ramified at the puncture at the center of the eye, turns out to be a composition of flips at the edges E_{1} and E_{2} of a regular graph. The edges E_{1} and E_{2} are the preimage of the edge E. The non-trivial automorphism of the cover is given by the central symmetry. Below we always treat a special triangulation of S by going to a cover of S where the triangulation becomes regular.

Proof of Lemma 1.17. Let S be a surface with a trivalent graph Γ on S, which is homotopy equivalent to S. Let us make a puncture of S, getting a new surface S′. Then adding a virus to Γ as shown on the picture we get a trivalent graph Γ′ homotopy equivalent to S′. So S′ is special. Since any marked surface with at least two holes, such that the boundary of one of the holes is without marked points, is special. Let us prove the converse claim. A virus separates the surface into two domains, each of them containing a whole. Moreover the one of them surrounded by the head of the virus has no marked points on its boundary. The lemma is proved.

4 Elements of positive algebraic geometry

1. Semifields and positive varieties. Following [BFZ96], we define a semifield as a set P endowed with the operations of addition and multiplication, which have the following properties:

i) Addition in P is commutative and associative.

ii) Multiplication makes P an abelian group.

Figure 9: A flip from an eye to a virus is a folding of a composition of two flips on regular graphs

Figure 10: Adding a virus to a graph

iii) distributivity: (a + b)c = ac + bc for a, b, c ∈ P. Here are important examples of semifields:

a) P = ℚ_{>0} = {x ∈ ℚ|x > 0},

b) P = ℝ_{>0} = {x ∈ ℝ|x > 0},

c) Let K be a field and P ⊂ K a semifield inside of K, with the addition and multiplication in P induced from K, like in the examples a) and b). Then there is a new semifield

K((ε))_{P} := {f = a_{s}ε^{s} + a_{s+1}ε^{s+1} + ...|a_{s} ∈ P } ⊂ K((ε))

consisting of all Laurent power series with the leading coefficient from P. For example starting from ℚ_{>0} we get a semifield denoted ℚ((ε))_{>0}. Similarly one defines ℝ((ε))_{>0} starting from ℝ_{>0}.

d) The tropical semifields: P = ℤ, P = ℚ or P = ℝ with the multiplication ⊗ and addition ⊕ given by

a ⊗ b := a + b, a ⊕ b := max(a, b)  (30)

Observe that in the examples a), b) and c) the semifield P is naturally embedded to a field, while in the example d) this is impossible since a ⊕ a = a.

Lemma 4.1 Assigning to f ∈ K((ε))_{P} the minus degree of its leading coefficient we get a homomorphism of semifields

−deg : K((ε))_{P} ⟶ ℤ^{t}

Proof. Clear. There is a unique way to assign to a split tori H an abelian group H(P) so that 𝔾_{m}(P) = P and

H → H(P) is a functor from the category of split algebraic tori to the one of abelian groups. One has H(ℤ^{t}) = X_{∗}(H) and H(P) = X_{∗}(H) ⊗_{ℤ} P, where P is considered here as an abelian group.

Let Pos be a category whose objects are split algebraic tori and morphisms are positive rational maps. A positive rational map φ : H_{1} → H_{2} gives rise to a map of sets φ_{∗} : H_{1}(P) → H_{2}(P), providing us a functor Pos ⟶ Sets. In particular if φ is invertible in the category Pos then φ_{∗} is an isomorphism. Therefore given a positive variety (X, {ψ_{α}}) we get a collection of abelian groups H_{α}(P) and set isomorphisms ψ_{α,β} : H_{α}(P) → H_{β}(P) between them.

Recall the definition of a positive structure on a scheme/stack given in Section 1.3.

Definition 4.2 Let P be a semifield and (X, {ψ_{α}}) a positive scheme. Then its P –part X(P) is a set whose elements are collections {p_{α} ∈ H_{α}(P)} such that ψ_{α,β}(p_{α}) = p_{β} for any α, β ∈ C_{X}.

Decomposing H_{α} into a product of 𝔾_{m}’s we get an isomorphism φ̃_{α} : P^{dimH_{α}} ⟶^{∼} X(P).

The P -part X(P) of X depends only on the positive birational class of X, and compatible positive structures give rise to the same P -part.

If P = K_{>0} is a semifield in a field K we can realize the K_{>0}–part of X inside of the set of Kpoints of X. Indeed, thanks to the condition i) of definition 1.4 the map ψ_{α} is regular at H_{α}(K_{>0}). Therefore there is a subset

X_{K_{>0}}^{+} := ψ_{α}(H_{α}(K_{>0})) ↪ X(K)  (31)

Thanks to the condition ii) of definition 1.4 it does not depend on α.

We say that a positive atlas ({ψ′_{α}}, C′) on X dominates another one ({ψ_{α}}, C) if there is an inclusion i : C ↪ C′ such that for any α ∈ C the coordinate systems ψ_{α} : H_{α} → X and ψ′_{i(α)} : H′_{i(α)} → X are identified. So the positive atlases on X are partially ordered. If two positive atlases C′ and C′′ dominate the same one C, then there is a new positive atlas C′ ∪C′′ dominating all of them. Therefore for a given positive atlas on X there is the maximal positive atlas on X dominating it.

Let (X, {ψ_{α}}, C) be a positive scheme. Then ψ_{α}(H_{α}) is a Zariski open subset of X. Their union over all α ∈ C is a regular Zariski open subset of X. Applying this construction to the maximal positive atlas on X dominating the original one we get a regular Zariski open subset X^{reg} = X_{C}^{reg} ↪ X called the regular part of a positive variety X.

The P -part of X can be defined in a more general set up of positive spaces.

2. A category of positive spaces. Recall that a groupoid is a category where all morphisms are isomorphisms. Below we will assume that the set of isomorphisms between every two objects of a groupoid is non-empty, i.e. all groupoids are connected. Let G be a connected groupoid and α its object. The fundamental group Γ_{G,α} of the groupoid G based at α is the group of automorphisms of the object α. The fundamental groups based at different objects are isomorphic, but the isomorphism is well defined only up to an inner automorphism.

Definition 4.3 A positive space X is a pair consisting of a groupoid G_{X} and a functor

ψ_{X} : G_{X} ⟶ Pos  (32)

The groupoid G_{X} is called the coordinate groupoid of a positive space.

Thus for every object α of the coordinate groupoid G_{X} there is an algebraic tori H_{α}, and for every morphism f : α ⟶ β in the groupoid there is a positive birational isomorphism ψ_{f} : H_{α} ⟶ H_{β}.

Let X and Y be two positive spaces defined via the coordinate groupoids G_{X} and G_{Y} and functors ψ_{X} and ψ_{Y}. A morphism X → Y is given by a functor μ_{X ,Y} : G_{X} → G_{Y} and a natural transformation of functors m : ψ_{X} → ψ_{Y} ∘ μ_{X ,Y}. A morphism is called a strict morphism if for every object α ∈ G_{X} the map m_{α} : ψ_{X} (α) → ψ_{Y} ∘ μ_{X ,Y}(α) is a homomorphism of algebraic tori.

Given a semifield P and a positive space X defined via (32) we get a functor

ψ_{X}^{P} : G_{X} ⟶ Sets, α ⟼ H_{α}(P), H_{α} := ψ_{X} (α)

It describes the set X (P) of P -points of X. The fundamental group of the groupoid G_{X} acts on X (P).

Example 1. A positive variety X provides a functor (32) as follows. The fundamental group of the coordinate groupoid G_{X} is trivial, so it is just a set. Namely, it is the set parametrising positive coordinate charts of the atlas defining the positive structure of X. The functor ψ_{X} is given by ψ_{X} (α) := H_{α} and ψ_{X} (α → β) := ψ_{α,β}.

Example 2. Let X be a Γ-equivariant positive scheme. It provides a positive space X given by a functor (32) such that the fundamental group of the groupoid of G_{X} is isomorphic to Γ.

3. The semiring of good positive Laurent polynomials for a positive space X. Let T_{i,α} be the coordinates on the tori H_{α}. Then ℤ[H_{α}] is the ring of Laurent polynomials in T_{i,α}. Let us say that a Laurent polynomial F in the variables T_{i,α} is good it for every morphism f : β → α in G_{X} the rational function f^{∗}F is a Laurent polynomial in the coordinates T_{i,β}. Denote by 𝕃(X) the ring of all good Laurent polynomials for X. We say that F is a good positive Laurent polynomials if for every morphism f : β → α in G_{X} the function f^{∗}F is a Laurent polynomial in the coordinates T_{i,β} with positive integer coefficients. Denote by 𝕃_{+}(X) the semiring of good positive Laurent polynomials for X.

Let ℚ_{+}(X) be the semifield of all positive rational functions for X. Set 𝕃̃_{+}(X) = ℚ_{+}(X) ∩ 𝕃(X). It contains 𝕃_{+}(X), but may be bigger, as the following example shows: x^{2} − x + 1 = (x^{3} + 1)/(x + 1).

The semiring 𝕃_{+}(X) determines the set of extremal elements E(X), that is the elements which can not be decomposed into a sum of two non zero good positive Laurent polynomials. Any element of 𝕃_{+}(X) is a linear combination of the extremal elements with non negative integer coefficients.

Remark. If we replace a positive atlas on a scheme X by a one dominating it, the set of points of X with values in a semifield P does not change, while the (semi)ring of good (positive) Laurent polynomials can get smaller.

4. The schemes X^{∗∗} and X^{∗}. Let X = (ψ_{X} , G_{X}) be a positive space. Let Γ := Γ_{X ,α} be the fundamental group of the groupoid G_{X}. Below we define two Γ-equivariant positive schemes, X^{∗} ↪ X^{∗∗}, corresponding to X in the sense that X is the positive space assigned to the positive schemes X^{∗} and X^{∗∗}. We start with the scheme X^{∗∗}.

Lemma 4.4 Let X be a positive space. Then there exists unique positive regular Γ-equivariant scheme X^{∗∗} corresponding to X which is ”minimal” in the following sense. For any positive regular Γ-equivariant scheme Y corresponding to X there exists unique morphism X^{∗∗} ↪ Y.

Proof. Let us assume first that the fundamental group of the coordinate groupoid of X is trivial. Then the scheme X^{∗∗} is obtained by gluing the coordinate tori H_{α} according to the birational isomorphisms ψ_{α,β}. For any field K the set of its K-points is given by collections of elements p_{α} ∈ H_{α}(K) such that ψ_{α,β}p_{α} = p_{β}. The general case is obtained by considering the universal covering groupoid for G_{X}. The lemma is proved.

The ring 𝕃[X] is identified with the ring of global regular functions on X^{∗∗}.

Let {ψ_{α}^{∗∗}} be the positive atlas of the scheme X^{∗∗}. The scheme X^{∗} is defined as the subscheme of X^{∗∗} given by the intersection of all coordinate charts of this atlas:

X^{∗} := ∩_{α∈C_{X}} Imψ_{α}^{∗∗}(H_{α}) ⊂ X^{∗∗}

It is a Γ-equivariant positive scheme: the positive atlas for X^{∗∗} induces the one for X^{∗}. For any α ∈ C_{X} the scheme X^{∗} is the complement to a possibly infinite collection of positive divisors in the torus H_{α}.

If the fundamental group of the groupoid G_{X} is trivial, X^{∗} is characterised by the following universality property. It is the maximal scheme such that for every α there exists a regular open embedding i_{α} : X^{∗} ↪ H_{α} and ψ_{α,β}i_{α} = i_{β}. Reversing the arrows we arrive at the universality property for the scheme X^{∗∗}.

Let (X, {ψ_{α}}, C) be a positive regular scheme/stack. Then X^{∗∗} = ∪_{α∈C}Imψ_{α}(H_{α}) and one has X^{∗} ⊂ X^{∗∗} ⊂ X.

Lemma 4.5 Let (X, {ψ_{α}}, C) be a positive regular scheme/stack, and X is noetherian. Then X^{∗∗} can be covered by a finite collection of the open subsets ψ_{α}(H_{α}).

Proof. Since X is noetherian, any open subset of X is quasi-compact, i.e. any cover by open subsets admits a finite subcover. The subset X^{∗∗} is open, and covered by the open subsets ψ_{α}(H_{α}). Therefore there exists a finite subcover. The lemma is proved.

Therefore for a positive regular noetherian scheme X the ring 𝕃(X) as well as the semiring 𝕃_{+}(X) are determined by a finite number of conditions.

5. ℤ–, ℚ– and ℝ–tropicalizations and the Thurston boundary of a positive space. Let 𝔸 be either ℤ, or ℚ, or ℝ. Denote by 𝔸^{t} the corresponding tropical semifield. Then a positive rational map φ : H_{1} → H_{2} gives rise to a piecewise-linear map φ_{∗} : H_{1}(𝔸^{t}) → H_{2}(𝔸^{t}).

Let X be a positive space. Then we have the manifold X (ℝ_{>0}), and the 𝔸-tropicalization X (𝔸^{t}) of X. They are related as follows. First of all, one has X (ℤ^{t}) ↪ X (ℚ^{t}) ↪ X (ℝ^{t}). The multiplicative group 𝔸_{>0}^{∗} of positive elements in 𝔸 acts by automorphisms of the tropical semifield 𝔸^{t}. Let us denote by {0} the zero element in the abelian group H(𝔸^{t}). Let 𝔸 be either ℚ or ℝ. Set

ℙH(𝔸^{t}) := (H(𝔸^{t}) − {0})/𝔸_{>0}^{∗}

Let φ : H → H′ be a positive rational map. Then the piecewise-linear map φ_{∗} : H(𝔸^{t}) → H′(𝔸^{t}) commutes with the multiplication by 𝔸_{>0}^{∗}, and φ_{∗}{0} = {0}. Thus we get a morphism

\overline{φ}_{∗} : ℙH(𝔸^{t}) ⟶ ℙH′(𝔸^{t})

Definition 4.6 Let X be a positive space. Let 𝔸 be either ℚ or ℝ. The projectivisation ℙX (𝔸^{t}) of the 𝔸-tropicalisation of X is

ℙX (𝔸^{t}) := \{\{p_{α} ∈ ℙH_{α}(𝔸^{t})\} | \overline{ψ}_{α,β}(p_{α}) = p_{β} for any α, β ∈ G_{X}\}

Observe that ℙH_{α}(ℝ^{t}) is a sphere, and the maps ψ_{α,β} are piecewise-linear maps of spheres. The set ℙX (ℚ^{t}) is an everywhere dense subset of ℙX (ℝ^{t}).

The degree map from Lemma 4.1 provides a map

−deg : X (ℝ((ε))_{>0}) ⟶ X (ℤ^{t})

An ℝ((ε))_{>0}-point x(ε) of X, considered modulo action of the group of positive reparametrizations of the formal ε-line, i.e. modulo substitutions ε = ϕ(ε), where ϕ(ε) ∈ ℝ((ε))_{>0}, gives rise to a well defined element of ℙX (ℚ^{t}), provided that degx(ε) ≠ {0}. Thus ℙX (ℝ^{t}) serves as the Thurston type boundary of X (ℝ_{>0}).

5 Positive configurations of flags

The first two subsections contain background material about positivity in algebraic semisimple groups, due to Lusztig, and Fomin and Zelevinsky. The rest of the Section is devoted to definition and investigation of positive structures on configurations of three and four generic flags in G.

1. A positive regular structure on U and G. We recall some basic results of G. Lusztig [L1], [L2]. The only difference is that we present them as results about existence and properties of regular positive structures on U and G. The proofs in loc.cit. give exactly what we need. However, as explained in Section 5.8, one needs to be careful when relating this point of view with the one used by Lusztig.

Let W be the Weyl group of G. Let s_{i} ∈ W be the simple reflection corresponding to x_{i}, y_{i}. Let w_{0} be the maximal length element in the Weyl group W, and w_{0} = s_{i_{1}}...s_{i_{n}} its reduced expression into a product of simple reflections. It is encoded by the sequence i := {i_{1}, ..., i_{n}} of simple reflections in the decomposition of w_{0}.

Proposition 5.1 a)Each reduced expression of w_{0}, encoded by i, provide an open regular embedding

ψ_{i} : 𝔾_{m}^{n} ↪ U,  (a_{1}, ..., a_{n}) ⟼ x_{i_{1}}(a_{1})...x_{i_{n}}(a_{n})  (33)

b) The collection of embeddings ψ_{i}, when i run through the set of all reduced expression of w_{0}, provides a positive structure on U.

c) Replacing x_{i_{p}} by y_{i_{p}} in (33) we get an open regular embedding ψ\overline{i} : 𝔾_{m}^{n} ↪ U^{−}, providing a positive regular structure on U^{−}. The map Ψ : U^{−} → U^{opp} is a positive regular isomorphism.

Proof. Proposition 2.7 in [L1] and the proof of its part a) prove the proposition.

2. Positive regular structures on double Bruhat cells [FZ99]. The double Bruhat cell corresponding to an element (u, v) ∈ W × W is defined by

G^{u,v} := BuB ∩ B^{−}vB^{−}  (34)

A double reduced word for the element (u, v) ∈ W × W is a reduced word for an element (u, v) of the Coxeter group W × W. We will use the indices \overline{1}, \overline{2}, ..., \overline{r} for the simple reflections in the first copy of W, and 1, 2, ..., r for the second copy. Then a double reduced word j for (u, v) is simply a shuffle of a reduced word for u written in the alphabet [\overline{1}, \overline{2}, ..., \overline{r}], and a reduced word for v written in the alphabet [1, 2, ..., r]. Recall that l(u) is the length of an element u ∈ W. We set x\overline{i}(t) := y_{i}(t). The following result is Theorem 1.2 in [FZ99].

Proposition 5.2 a) Given a double reduced word j = (j_{1}, ..., j_{l(u)+l(v)}) for (u, v) and 0 ≤ k ≤ l(u) + l(v) there is a regular open embedding

x_{j,k} : H × 𝔾_{m}^{l(u)+l(v)} ↪ G^{u,v}
(h; t_{1}, ..., t_{l(u)+l(v)}) ⟼ x_{i_{1}}(t_{1})...x_{i_{k}}(t_{i_{k}})hx_{i_{k+1}}(t_{i_{k+1}})...x_{i_{l(u)+l(v)}}(t_{l(u)+l(v)})

b) The collection of all regular open embeddings x_{j,k} provide a regular positive structure on the double Bruhat cell G^{u,v}.

Indeed, if k = 0 this is exactly what stated in loc. cit. Moving h we do not change the positive structure since the characters of H are positive by definition. It follows from Proposition 5.2 that dimG^{u,v} = r + l(u) + l(v).

The generalized minors and double Bruhat cells [FZ99]. Let i = (i_{1}, ..., i_{n}) be a reduced decomposition of w_{0}. Set

\overline{\overline{s}}_{i} := φ_{i} ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) , \overline{\overline{w}}_{0} := \overline{\overline{s}}_{i_{1}}...\overline{\overline{s}}_{i_{n}}

Then \overline{\overline{w}}_{0} does not depend on choice of a reduced decomposition. Using both reduced decompositions (i_{1}, ..., i_{n}) and (i_{n}, ..., i_{1}) one immediately sees that

\overline{\overline{w}}_{0}\overline{w}_{0} = e, \overline{\overline{w}}_{0} = s_{G}\overline{w}_{0}, Ψ(\overline{\overline{w}}_{0}) = \overline{w}_{0}, Ψ(\overline{w}_{0}) = \overline{\overline{w}}_{0}

Let G_{0} := U^{−}HU. Any element x ∈ G_{0} admits unique decomposition x = [x]_{−}[x]_{0}[x]_{+} with [x]_{−} ∈ U_{−}, [x]_{0} ∈ H, [x]_{+} ∈ U_{+}. Let i ∈ [1, ..., r]. Then according to [FZ99], the generalized minor Δ_{uω_{i},vω_{i}} is the regular function on G whose restriction to the open subset \overline{u}G_{0}\overline{v}^{−1} is given by

Δ_{uω_{i},vω_{i}}(x) = ω_{i}([\overline{u}^{−1}x\overline{v}]_{0})  (35)

It was shown in loc. cit. that it depends only on the weights uω_{i}, vω_{j}. For example if G = SL_{r+1} then for permutations u, v ∈ S_{r+1} the function Δ_{uω_{i},vω_{i}} is given by the i × i minor whose rows (columns) are labeled by the elements of the set u([1, ..., i]) (resp., v([1, ..., i])). Let i be a reduced decomposition of w_{0} and k ∈ {[1, l(u) + l(v)]. Following ([BFZ03], Section 2.3), we set

u_{≤k} = u_{≤k}(i) = ∏_{l=1,...,k, ε(i_{l})=−1} s_{|i_{l}|}  (36)

v_{>k} = v_{>k}(i) = ∏_{l=l(u)+l(v),...,k+1, ε(i_{l})=+1} s_{|i_{l}|}  (37)

where the index l is increasing (decreasing) in (36) (resp., (37)). For k ∈ −[1, ..., r] we set u_{≤k} = e and v_{>k} = v^{−1}. Finally, for k ∈ −[1, r] ∪ [1, l(u) + l(v)], set

Δ_{k;i} := Δ_{u_{≤k}ω_{|i_{k}|},v_{>k}ω_{|i_{k}|}}

Let us set

F (i) = {Δ(k; i) : k ∈ −[1, r] ∪ [1, l(u) + l(v)]}

Let

G^{u,v}(i) := {g ∈ G^{u,v} : Δ(g) ≠ 0 for all Δ ∈ F (i)}

The lemma below is a part of Lemma 2.12 in [BFZ03], and follows directly from [FZ99].

Lemma 5.3 The map G^{u,v} → 𝔸^{r+l(u)+l(v)} given by g ⟼ (Δ(g))_{Δ∈F (i)} restricts to an isomorphism

G^{u,v}(i) ⟶^{∼} 𝔾_{m}^{r+l(u)+l(v)}  (38)

Let

Δ_{i} : 𝔾_{m}^{r+l(u)+l(v)}↪G^{u,v}  (39)

be the regular open embedding provided by Lemma 5.3. The theorem below is a reformulation of the results in [FZ99].

Theorem 5.4 The collection of embeddings {Δ_{i}}, when i runs through the set of all reduced decompositions of (u, v), provides a positive regular structure on the double Bruhat cell G^{u,v}. It is compatible with the positive regular structure defined in Proposition 5.2.

3. A positive regular structure on the intersection of two opposite open Schubert cells in B. Recall the two opposite Borel subgroups B^{−}, B^{+}. They determine the two open Schubert cells B_{−} and B_{+} in the flag variety B, parametrising the Borel subgroups of G in generic position to B^{−} and B^{+}. One has

B_{−} := {u_{−}B^{+}u_{−}^{−1}}, u_{−} ∈ U^{−};  B_{+} := {u_{+}B^{−}u_{+}^{−1}}, u_{+} ∈ U^{+}  (40)

The formula (40) provides isomorphisms α^{−} : U^{−} ⟶^{∼} B_{−}, α^{+} : U^{+} ⟶^{∼} B_{+}. Let us set

B_{∗} := B_{−} ∩ B_{+}, U_{∗}^{+} := U^{+} ∩ B^{−}w_{0}B^{−}, U_{∗}^{−} := U^{−} ∩ Bw_{0}B

Lemma 5.5 a) There are canonical isomorphisms

α_{∗}^{+} : U_{∗}^{+} ⟶^{∼} B_{∗}, u_{+} ⟼ u_{+}B^{−}u_{+}^{−1}, α_{∗}^{−} : U_{∗}^{−} ⟶^{∼} B_{∗}, u_{−} ⟼ u_{−}B^{+}u_{−}^{−1}

b) Ψ : U_{∗}^{−} ⟶ U_{∗}^{+} is an isomorphism.

Proof. a) The map α_{∗}^{+} is evidently injective. Let us check the surjectivity. Given a pair (u_{+}, u_{−}) as in (40), such that u_{+}B^{−}u_{+}^{−1} = u_{−}B^{+}u_{−}^{−1}, let us show that u_{−}w_{0}B^{−} ∩ U^{+} is non empty, and hence is a point. Indeed, conjugating B^{−} by u_{−}w_{0}B^{−} we get u_{−}B^{+}u_{−}^{−1}. By the assumption conjugation of B^{−} by u_{+} gives the same result. Thus u_{+} ∈ u_{−}w_{0}B^{−}. The part b) is obvious. The lemma is proved.

Lemma 5.5 implies that there is an isomorphism

ϕ : U_{∗}^{−} ⟶^{∼} U_{∗}^{+}, ϕ := (α_{∗}^{+})^{−1}α_{∗}^{−}

Proposition 5.6 a) For every reduced decomposition i of w_{0} there are regular open embeddings

α_{∗}^{−} ∘ ψ_{i} : (𝔾_{m})^{n} ↪ B_{+};  α_{∗}^{+} ∘ ψ\overline{i} : (𝔾_{m})^{n} ↪ B_{−}  (41)

b) The maps {α_{∗}^{−} ∘ ψ_{i}} provide a positive regular structure on B_{∗}. Similarly the maps {α_{∗}^{+} ∘ ψ_{i}} provide another positive regular structure on B_{∗}.

c) The map ϕ is positive. Therefore the two positive structures from the part b) are compatible, and thus provide a positive regular structure on B_{∗} dominating them.

Proof. a) According to Theorem 1.2 in [FZ99] there is a regular open embedding

x_{i} : H × 𝔾_{m}^{n} → G^{e,w_{0}} := B^{+} ∩ B^{−}w_{0}B^{−}, x_{i}(h; t_{1}, ..., t_{n}) := hx_{i_{1}}(t_{1}) · ... · x_{i_{n}}(t_{n})

Putting h = e we land in U^{+} ∩ B^{−}w_{0}B^{−}.

b) Follows from Proposition 5.1.

c) In s. 3.2 of [L2] there is an algorithm for computation of a map ϕ′ : U^{−} ⟶ U^{+}, such that

uB^{+}u^{−1} = ϕ′(u)B^{−}ϕ′(u)^{−1}, u ∈ U^{−}  (42)

Such a map is unique. It follows from loc. cit. that it is a positive rational map for the positive structures on U^{+} and U^{−} given by Proposition 5.1. Its restriction to U_{∗}^{−} is our ϕ. The proposition is proved.

4. A positive regular variety V. Let V be the quotient of the affine variety U_{∗} by the action of H:

V := U_{∗}/H := Specℚ[U_{∗}]^{H}  (43)

Let us define a coordinate system on V corresponding to a reduced decomposition i of w_{0}. Let α_{1}, ..., α_{r} be the set of all simple roots. Each i_{k} ∈ i is labeled by a simple root, providing a decomposition

i = i(α_{1}) ∪ ... ∪ i(α_{r})

Precisely, j ∈ i(α_{p}) if and only if i_{j} = p. Let l_{k} := |i(α_{k})|. So there is a positive regular open embedding

ψ_{i} : 𝔾_{m}^{i(α_{1})} × ... × 𝔾_{m}^{i(α_{r})} ↪ U_{∗}

The group H acts on the factor 𝔾_{m}^{i(α_{k})} through its character χ′_{k}. Since the center of G is trivial, there is an isomorphism (χ′_{1}, ..., χ′_{r}) : H ⟶^{∼} 𝔾_{m} × ... × 𝔾_{m}. So H acts freely on the product. The map ψ_{i} transforms this action to the action of H on U_{∗} by conjugation. Thus we get a regular open embedding

\overline{ψ}_{i} : 𝔾_{m}^{i(α_{1})}/𝔾_{m} × ... × 𝔾_{m}^{i(α_{r})}/𝔾_{m} ↪ U_{∗}/H

where the group 𝔾_{m} acts on 𝔾_{m}^{i(α_{k})} diagonally. Let a_{α_{k},1}, ..., a_{α_{k},l_{k}} be the natural coordinates on 𝔾_{m}^{i(α_{k})}. Then the natural coordinates on its quotient by the diagonal action of 𝔾_{m} are

t_{α_{k},j} := \frac{a_{α_{k},j+1}}{a_{α_{k},j}} , 1 ≤ j ≤ l_{k} − 1  (44)

Lemma 5.7 The collection of regular open embeddings \{\overline{ψ}_{i}\}, where i run through the set of all reduced expressions of w_{0}, provides a regular positive structure on V.

Proof. Follows immediately from the very definition and Proposition 5.1.

Similarly, replacing U by U^{−} we get a positive variety V^{−}. It is handy to denote V by V^{+}. The map Ψ evidently commutes with the action of H. It follows from the construction of the map ϕ given in s. 3.2 of [L2] that ϕ also commutes with the action of H. Thus they provide isomorphisms

ϕ : V^{−} ⟶ V^{+}, Ψ : V^{−} ⟶ V^{+}  (45)

Abusing the notation we denote them the same way as the original maps.

Lemma 5.8 The maps ϕ and Ψ in (45) are isomorphisms of positive regular varieties.

Proof. For Ψ this is clear. For ϕ this follows from Proposition 5.6. The lemma is proved.

5. A positive structure on configurations of triples of flags. We say that a pair of flags is in generic position if they are conjugated to the pair (B^{+}, B^{−}), or, equivalently, project to the open G-orbit in B × B. Recall that configurations of n–tuples of flags are by definition the G–orbits on B^{n}. Denote by ∼ the equivalence relation provided by G–orbits. We say that a configuration of flags is generic if every two flags of the configuration are in generic position. The moduli space Conf_{n}(B) of generic configurations of n > 2 flags is a variety over ℚ. Indeed, a generic configuration can be presented as

(B^{−}, B^{+}, u_{1}B^{+}u_{1}^{−1}, ..., u_{k}B^{+}u_{k}^{−1}), u_{k} ∈ U_{∗}^{−}

Thus they are parametrised by a Zariski open part of (U_{∗}^{−} × ... × U_{∗}^{−})/H.

Let us define a positive structure on the configuration space Conf_{3}(B). Let pr : U_{∗}^{±} → V^{±} be the canonical projection. Consider a regular open embedding

V^{−} ↪ Conf_{3}(B),  v ⟼ (B^{−}, B^{+}, vB^{+}v^{−1}) := (B^{−}, B^{+}, uB^{+}u^{−1}), pr(u) = v  (46)

Since H stabilizes the pair (B^{−}, B^{+}), this map does not depend on a choice of u ∈ U_{∗}^{−} lifting v ∈ V^{−}.

Definition 5.9 A positive structure on Conf_{3}(B) is defined by the map (46) and Lemma 5.7.

According to Lemma 5.8 the positive structure is compatible with the one given by

V^{+} ↪ Conf_{3}(B),  v ⟼ (B^{−}, B^{+}, vB^{−}v^{−1}) := (B^{−}, B^{+}, uB^{−}u^{−1}), pr(u) = v  (47)

Yet another natural positive regular structure on Conf_{3}(B) is given by a regular open embedding

V^{+} ↪ Conf_{3}(B),  v ⟼ (B^{−}, v^{−1}B^{−}v, B^{+})  (48)

Recall that the map Ψ acts on the flag variety B as well as on the configurations of flags.

Theorem 5.10 Let G be a split reductive group with trivial center. Consider a positive atlas on Conf_{3}(B) provided by (46). Then one has:

a) The cyclic shift (B_{1}, B_{2}, B_{3}) ⟼ (B_{2}, B_{3}, B_{1}) is a positive isomorphism on Conf_{3}(B). Precisely:

(B^{−}, B^{+}, uB^{−}u^{−1}) ∼ (B^{+}, cB^{−}c^{−1}, B^{−}), u ∈ U_{∗}^{+}, c := (ϕΨ)^{2}(u)  (49)

b) Ψ : (B_{1}, B_{2}, B_{3}) ⟼ (ΨB_{1}, ΨB_{2}, ΨB_{3}) provides a positive isomorphism on Conf_{3}(B).

c) The two positive regular structures on Conf_{3}(B) given by (46) and (48) are compatible.

d) The reversing map (B_{1}, B_{2}, B_{3}) ⟼ (B_{3}, B_{2}, B_{1}) is a positive isomorphism on Conf_{3}(B). So the positive structure on Conf_{3}(B) is invariant under the action of the symmetric group S_{3}.

Proof. a) Consider a configuration

(B^{−}, B^{+}, u_{+}B^{−}u_{+}^{−1}),  pr(u_{+}) ∈ V_{+}  (50)

Let us compute the composition Ψ ∘ Ψ applied to this configuration. Conjugating (50) by u_{+}^{−1} we get (u_{+}^{−1}B^{−}u_{+}, B^{+}, B^{−}). It is, of course, equivalent to the initial configuration (50). Then applying Ψ we get

(u_{−}B^{+}u_{−}^{−1}, B^{−}, B^{+}),  u_{−} = Ψ(u_{+}) ∈ U_{∗}^{−}, u_{+} ∈ U_{∗}^{+}  (51)

According to (42) it can be written as

(v_{+}B^{−}v_{+}^{−1}, B^{−}, B^{+}) for some v_{+} ∈ U_{∗}^{+}, v_{+} = ϕΨ(u_{−})

Conjugating by v_{+}^{−1} we get (B^{−}, v_{+}^{−1}B^{−}v_{+}, B^{+}). Applying Ψ again we obtain

(B^{+}, v_{−}B^{+}v_{−}^{−1}, B^{−}), v_{−} := Ψ(v_{+}) ∈ U_{∗}^{−}

We can write it as

(B^{+}, c_{+}B^{−}c_{+}^{−1}, B^{−}), c_{+} := (ϕΨ)^{2}(u_{+}) ∈ U_{∗}^{+}

This configuration is the cyclic shift of a configuration (50). On the other hand, since Ψ^{2} = Id, it coincides with (50). It remains to use positivity of the maps ϕ (Proposition 5.6c)) and Ψ. The statement a) is proved.

b) The configuration (51) is a cyclic shift of the configuration (50). On the other hand, as the proof of a) shows, it is equivalent to Ψ(B^{−}, B^{+}, u_{+}B^{−}u_{+}^{−1}). The cyclic shift is positive by a). The statement b) is proved.

c) Conjugating (50) by u_{+}^{−1} we get (u_{+}^{−1}B^{−}u_{+}, B^{+}, B^{−}). Applying the cyclic shift and using a) we get c).

d) By the parts c) and a) the following equivalences provide positive maps:

(B^{−}, B^{+}, u_{+}B^{−}u_{+}^{−1}) ∼ (B^{−}, v_{+}^{−1}B^{−}v_{+}, B^{+}) ∼ (v_{+}^{−1}B^{−}v_{+}, B^{+}, B^{−})

Thus it remains to check that map (B^{+}, B^{−}, v_{+}^{−1}B^{−}v_{+}) ⟼ (B^{+}, B^{−}, v_{+}B^{−}v_{+}^{−1}) induced by the inversion v ⟼ v^{−1} is a positive map on Conf_{3}(B). Although the inversion map is not positive on U, it induces a positive map on the quotient U_{∗}/H. Indeed, inversion changes the order of factors in a reduced decomposition, which amounts to changing the reduced decomposition, and replaces each factor x_{i}(t) by x_{i}(−t). However this does not affect the ratios t_{i}/t_{j} used to define the positive atlas on U_{∗}/H, and changing a reduced decomposition we get a positive map. The part d) is proved.

The last assertion of the theorem is a corollary of a) and d). The theorem is proved.

6. A positive structure on Conf_{4}(B). Consider a triangulation of a convex 4-gon with the flags B_{i} assigned to its vertices, as on the picture. Choose in addition an orientation of the internal edge, from B_{1} to B_{3}. Such a data, denoted by \overline{T}, is determined by an ordered configuration of 4 flags (B_{1}, B_{2}, B_{3}, B_{4}).

Figure 11: A triangulated quadrilateral with flags assigned to vertices

We will employ the following notation for the group action by conjugation on the flag variety:

u · B := uBu^{−1}

Lemma 5.11 The map

(u_{+}, u_{−}) ⟼ (B^{−}, u_{+}^{−1} · B^{−}, B^{+}, u_{−} · B^{+})  (52)

provides a regular open embedding

i\overline{T} : Conf_{4}(B) ↪ (U_{∗}^{+} × U_{∗}^{−})/H  (53)

Proof. Let us identify (B_{1}, B_{3}) with the standard pair (B^{−}, B^{+}) and choose one of the pinnings related to (B^{−}, B^{+}). Then the element

(B_{1}, B_{2}, B_{3}, B_{4}) ∈ Conf_{4}(B)  (54)

can be written as the right hand side of (52). The action of the group H by conjugation on the pinnings with the given pair (B_{1}, B_{3}) is a simple transitive action. So the element (54) is described uniquely by an element of (U_{∗}^{−} × U_{∗}^{+})/H. The lemma is proved.

Just like in Section 5.2, there is a positive regular structure on (U_{∗}^{+} × U_{∗}^{−})/H. Precisely, take a pair of reduced words (i_{1}, i_{2}) for w_{0}. Using i_{1} to define a positive coordinate system on U_{∗}^{+} and i_{2} for U_{∗}^{−}, we get a regular open embedding

\overline{ψ}_{(i_{1},i_{2})} : 𝔾_{m}^{i_{1}(α_{1})+i_{2}(α_{1})}/𝔾_{m} × ... × 𝔾_{m}^{i_{1}(α_{r})+i_{2}(α_{r})}/𝔾_{m} ↪ (U_{∗}^{−} × U_{∗}^{+})/H

Similar to (44), we can introduce homogeneous coordinates on each of the factors 𝔾_{m}^{i_{1}(α_{k})+i_{2}(α_{k})}/𝔾_{m}. Introducing the homogeneous coordinates one must take into account the fact that 𝔾_{m} acts on the factor 𝔾_{m}^{i_{1}(α_{k})} through the character χ′_{k}, while on the factor 𝔾_{m}^{i_{2}(α_{k})} via its inverse.

Theorem 5.12 a) The rational map

i−1 ; \overline{T} : (U_{∗}^{+} × U_{∗}^{−})/H ⟶ Conf_{4}(B)

transforms the positive structure on (U_{∗}^{+} × U_{∗}^{−})/H to a positive structure on the variety Conf_{4}(B). b) The cyclic shift on Conf_{4}(B) is a positive rational map.

c) The reversion (B_{1}, B_{2}, B_{3}, B_{4}) ⟼ (B_{4}, B_{3}, B_{2}, B_{1}) is a positive rational map on Conf_{4}(B). So the positive structure on Conf_{4}(B) has a natural dihedral symmetry.

Proof. a). The map i_{T} is not an isomorphism. Its image consists of the elements such that the corresponding pair of flags (u_{+}^{−1}B^{−}u_{+}, u_{−}B^{+}u_{−}^{−1}) is in generic position. So to prove a) it remains to show that the complement to the image of i_{T} is contained in a positive divisor.

Let \overline{T}′ be another triangulation of the 4-gon on Picture (11), obtained from \overline{T} by changing the oriented edge B_{1} → B_{3} to the one B_{2} → B_{4}. It provides another regular open embedding

i\overline{T}′ : Conf_{4}(B) ↪ (U_{∗}^{+} × U_{∗}^{−})/H  (55)

We need the following proposition.

Proposition 5.13 The rational map

i\overline{T} ∘ i_{\overline{T}′}^{−1} : (U_{∗}^{+} × U_{∗}^{−})/H ↪ (U_{∗}^{+} × U_{∗}^{−})/H

as well as its inverse are positive rational maps.

Proof. Applying Proposition 5.2 to j_{1} := (i, \overline{i}), k = n and j_{2} := (\overline{i}, i), k = n we get

Lemma 5.14 The map

η : U_{∗}^{+} × H × U_{∗}^{−} ⟶ U_{∗}^{−} × H × U_{∗}^{+}, η : (u_{−}, h, u_{+}) ⟼ (a_{+}, t, a_{−}) if u_{−}hu_{+} = a_{+}ta_{−}

is a positive rational map.

Let (u_{+}, u_{−}) be as in (52). Then thanks to Corollary 5.14 there exists a positive rational map

(u_{+}, u_{−}) ⟶ (a_{−}, t, a_{+}), a_{±} ∈ U_{∗}^{±}, t ∈ H such that u_{+}u_{−} = a_{−}a_{+}t

Set g := a_{−}^{−1}u_{+} = a_{+}t^{−1}u_{−}^{−1}. Conjugating (52) by g we get

(a_{+} · B^{−}, B^{−}, a_{−}^{−1} · B^{+}, B^{+}) = (ϕ(a_{+}) · B^{+}, B^{−}, ϕ(a_{−})^{−1} · B^{−}, B^{+})

Since ϕ is a positive isomorphism, this proves that the rational map i_{T} ∘ i_{T′}^{−1} is positive. Applying this argument three times we show that the rational map induced by the cube of the cyclic shift is positive. This is equivalent to positivity of the inverse of i_{T} ∘ i_{T′}^{−1}. Proposition 5.13 is proved.

Let us return to the proof of Theorem 5.12. The map i\overline{T}′ ∘ i−1 ; \overline{T} is certainly non regular at the subvariety of configurations of flags (B_{1}, B_{2}, B_{3}, B_{4}) determined by the condition that all pairs except (B_{2}, B_{4}) are in generic position. But according to Proposition 5.13 this subvariety is contained in a positive divisor. The part a) of the theorem is proved. The part b) follows immediately from Proposition 5.13. The part c) follows from the part d) of Theorem 5.10 and the following observation. The theorem is proved.

7. Another description of Conf_{4}(B). Recall the triangulation T of a convex 4-gon with the flags B_{i} assigned to its vertices as on the picture and the internal edge oriented from B_{1} to B_{3}.

Proposition 5.15 There exists a canonical birational isomorphism

φ_{T} : Conf_{4}(B) ⟶^{∼} Conf_{3}(B) × H × Conf_{3}(B) = V^{+} × H × V^{−}  (56)

Proof. Let (B_{1}, B_{2}, B_{3}, B_{4}) ∈ Conf_{4}(B). Then there are projections

p_{1} : Conf_{4}(B) ⟶ Conf_{3}(B), (B_{1}, B_{2}, B_{3}, B_{4}) ⟼ (B_{1}, B_{2}, B_{3});
p_{2} : Conf_{4}(B) ⟶ Conf_{3}(B), (B_{1}, B_{2}, B_{3}, B_{4}) ⟼ (B_{1}, B_{3}, B_{4})

corresponding to the two triangles of the triangulation T.

We are going to define a rational projection

p_{e} : Conf_{4}(B) ⟶ H  (57)

corresponding to the internal oriented edge e of the triangulation. Unlike the constructions in Section 5.4, it will not depend on the choice of a reduced decomposition i of w_{0}.

Recall the map (53). Recall that we assume that G has trivial center. Let R_{U} be the maximal abelian quotient of the unipotent radical U of a Borel subgroup B. The torus H_{U} := B/U acts on R_{U}. Let R_{U}^{∗} be the unique dense H_{U}-orbit in R_{U}. It is a principal homogeneous H_{U}–space. Let π : U → R_{U} be the canonical projection. It provides a rational map

p_{e} : (U_{∗}^{+} × U_{∗}^{−})/H ⟶ (R_{U^{+}}^{∗} × R_{U^{−}}^{∗})/H = H, (v_{+}, u_{−}) ⟼ π(Ψ(v_{+}))/π(u_{−})

Here the right hand side has the following meaning. For generic (v_{+}, u_{−}) both Ψ(v_{+}) and u_{−} lie in the H-torsor R_{U^{−}}^{∗}, so their ratio is a well defined element of H.

Proposition 5.16 The birational isomorphism (56) provides a positive atlas on the variety Conf_{4}(B). It is equivalent to the one defined in Section 5.4.

Proof. It is an immediate corollary of the following lemma.

Lemma 5.17 Let (s_{1}, ..., s_{m}, t_{1}, ..., t_{n}) be the natural coordinates on 𝔾_{m}^{n+m}. Set

p_{i} = \frac{s_{i+1}}{s_{i}} , i = 1, ..., m − 1, q_{j} = \frac{t_{j+1}}{t_{j}} , j = 1, ..., n − 1, X = \frac{t_{1}}{s_{m}} , Y = \frac{s_{1} + ... + s_{m}}{t_{1} + ... + t_{n}}

Then the two coordinate systems ({p_{i}}, {q_{j}}, X) and ({p_{i}}, {q_{j}}, Y) on the quotient of 𝔾_{m}^{n+m} by the diagonal action of 𝔾_{m} are related by positive transformations.

Proof. Follows immediately from the formula

\frac{1 + p_{1} + p_{1}p_{2} + ... + p_{1}...p_{m−1}}{1 + q_{1} + q_{1}q_{2} + ... + q_{1}...q_{n−1}} = p_{1}...p_{m−1}XY

Proposition 5.15 is proved.

8. A remark. According to Theorem 5.12, positive configurations of four flags have a dihedral symmetry. The following lemma shows that they do not have S_{4}-symmetry. Recall the H-invariant p_{e}(B_{1}, B_{2}, B_{3}, B_{4}) of a configuration of four flags defined by (57).

Lemma 5.18 If p_{e}(B_{1}, B_{2}, B_{3}, B_{4}) ∈ H(ℝ_{>0}), then p_{e}(B_{1}, B_{3}, B_{2}, B_{4}) ∈ H(ℝ_{<0}).

Proof. One has

(B_{1}, B_{2}, B_{3}, B_{4}) ∼ (B^{−}, v_{−}^{−1}B^{+}v_{−}, B^{+}, u_{+}B^{−}u_{+}^{−1}) ∼ (B^{−}, B^{+}, v_{−}B^{+}v_{−}^{−1}, v_{−}u_{+}B^{−}(v_{−}u_{+})^{−1})

where v_{−} ∈ U^{−}(ℝ_{>0}) and u_{+} ∈ U^{+}(ℝ_{>0}). Writing v_{−}u_{+} = u′_{+}hv′_{−}, where u′_{+}, h, v′_{−} are also positive, we get

(B_{1}, B_{3}, B_{2}, B_{4}) ∼ (B^{−}, v_{−}B^{+}v_{−}^{−1}, B^{+}, u′_{+}B^{−}(u′_{+})^{−1})

Recall the canonical projection π : U^{−} → R_{U^{−}} := U^{−}/[U^{−}, U^{−}]. Elements of the abelian group R_{U^{−}} are represented uniquely as ∏ y_{α}(t_{α}), where α runs through all simple positive roots. If v_{−} ∈ U^{−}(ℝ_{>0}) then these t_{α}-coordinates of the element π(v_{−}) are in ℝ_{>0}, while for π(v_{−}^{−1}) they are in ℝ_{<0}. Now the lemma follows immediately from the definition of the H-invariant. The lemma is proved.

9. A positive atlas on Conf_{5}(B) and its dihedral symmetry. Consider a convex pentagon with vertices labeled by a configuration (B_{1}, ..., B_{5}). Let T be its triangulation. The two diagonals of the triangulation share a vertex. We will assume the diagonals are oriented out of the vertex. Then there is a birational isomorphism

φ_{T} : Conf_{5}(B) ⟶^{∼} Conf_{3}(B)^{3} × H^{2}  (58)

where the factors Conf_{3}(B) correspond to triangles with ordered vertices of the triangulation T, and the factors H are assigned to the oriented diagonals of the triangulation. Precisely, assuming that the diagonals share the vertex B_{1}, the map is given by

(B_{1}, ..., B_{5}) ⟼ (B_{1}, B_{2}, B_{3})×(B_{1}, B_{3}, B_{4})×(B_{1}, B_{4}, B_{5})×p_{e_{13}}(B_{1}, B_{2}, B_{3}, B_{4})×p_{e_{14}}(B_{1}, B_{3}, B_{4}, B_{5})

The decomposition (58) provides a positive atlas of the space Conf_{5}(B). We are going to show that it is invariant under the natural action of the dihedral group D_{5}. So we have to show that it is invariant under the flips and reversing the order of the flags. The claim that a dihedral symmetry map followed by projection to a factor Conf_{3}(B) is positive immediately reduced to dihedral symmetry of positive atlases on the spaces Conf_{n}(B) for n = 3, 4. So it remains only to check similar claims for projections onto the H-factors. More specifically, the only statement not covered by the dihedral symmetry of positive atlases on the spaces Conf_{n}(B) for n = 3, 4 is this: a flip on the quadrilateral (B_{1}, B_{2}, B_{3}, B_{4}) of the triangulation on Figure 12, followed by projection to the H-invariant assigned to the diagonal B_{1}B_{4}, is a positive map.

It is convenient to use the following parametrisation of the space Conf_{5}(B). Consider the map

\frac{U^{−} × U^{+} × U^{+}}{H} ⟶ Conf_{5}(B);  (u_{−}, v_{+}, w_{+}) ⟼ (B^{−}, v_{+}^{−1} · B^{−}, v_{+}^{−1}w_{+}^{−1} · B^{−}, B^{+}, u_{−} · B^{+}) (59)

This map is evidently a birational isomorphism. The positive atlases on U^{±} provide in the standard way a positive atlas on the quotient on the left, and hence on Conf_{5}(B).

Figure 12: A triangulated pentagon with flags assigned to vertices

Lemma 5.19 a) The positive atlas on Conf_{5}(B) given by the birational isomorphism (59) is equivalent to the standard one corresponding to the triangulation shown on Figure 12.

b) The flip at the punctured diagonal on Figure 12 is a positive automorphism of Conf_{5}(B). Thus the cyclic shift is a positive automorphism of Conf_{5}(B).

c) The reversion (B_{1}, ..., B_{5}) ⟼ (B_{5}, ..., B_{1}) is a positive automorphism of Conf_{5}(B).

Proof. a) It is obvious that, taking into account the S_{3}-symmetry of the positive atlas on Conf_{3}(B), the configurations of triples of flags assigned to the three triangles of the triangulation shown on Figure 12 are parametrised by u_{−}, v_{+}, w_{+}, considered modulo the H-conjugation. So it remains to investigate the H-invariants assigned to the edges.

The two quadrilaterals of the triangulation provide two configurations of 4 flags. The first is

(B_{1}, B_{2}, B_{3}, B_{4}) ∼ (v_{+} · B^{−}, B^{−}, w_{+}^{−1} · B^{−}, B^{+}) = (ϕ(v_{+}) · B^{+}, B^{−}, w_{+}^{−1} · B^{−}, B^{+})

Applying the cyclic shift we recover the standard positive atlas on the configurations (B_{1}, B_{2}, B_{3}, B_{4}). Further, the configuration corresponding to the second quadrilateral is

(B_{1}, B_{2}, B_{4}, B_{5}) = (B^{−}, v_{+}^{−1} · B^{−}, B^{+}, u_{−} · B^{−})

is already in the standard form.

b) After the flip at the diagonal B_{1}B_{3} we get

(B_{1}, B_{3}, B_{4}, B_{5}) ∼ (B^{−}, (w_{+}v_{+})^{−1} · B^{−}, B^{+}, u_{−} · B^{−})

c) Follows immediately from the properties of Conf_{n}(B) for n = 3, 4. The lemma is proved. Remark. The configuration (B_{1}, B_{3}, B_{4}, B_{5}) is parametrised by the pair (w_{+}v_{+}, u_{−}) considered

modulo the H-conjugation. So the group product in U^{+} appears naturally here.

10. Positive configurations of K-flags: two points of view. Given a semifield P there are the P -parts U_{P}^{±}, H_{P}, G_{P} of the corresponding positive varieties. They are monoids. One has G_{P} = U_{P}^{+}H_{P} U_{P}^{−} = U_{P}^{−}H_{P} U_{P}^{+}.

Let K_{>0} ⊂ K be a semifield in a field K. Then there are two ways to define K_{>0}-positive configurations of flags. The first, preferable one, is this: we furnish the space Conf_{n}(B) with a positive atlas and take its K_{>0}-points. For n = 3, 4, 5 the positive atlas has been defined above, and the general case is deduced from this in Section 6. This method works for an arbitrary semifield. Below we elaborate another, more naive approach, where the fact that K_{>0} is a semifield in a field K is important. We show that the two approaches lead to two a priori different notions, which nevertheless coincide if K_{>0} = ℝ_{>0}.

Recall Conf_{n}(B(K)) := B(K)^{n}/G(K).

Definition 5.20 a) A configuration (B_{1}, B_{2}, B_{3}) ∈ Conf_{3}(B(K)) is K_{>0}-positive if

(B_{1}, B_{2}, B_{3}) ∼ (B^{−}, B^{+}, u · B^{+}) for some u ∈ U_{K_{>0}}^{−}

b) A configuration (B_{1}, B_{2}, B_{3}, B_{4}) ∈ Conf_{4}(B(K)) is K_{>0}-positive if

(B_{1}, B_{2}, B_{3}, B_{4}) ∼ (B^{−}, v^{−1} · B^{−}, B^{+}, u · B^{+}) for some v ∈ U_{K_{>0}}^{+} and u ∈ U_{K_{>0}}^{−}

The set of K_{>0}-positive configurations of flags for n = 3, 4 is denoted by Conf_{n}^{+}(B(K)). Repeating the proves of Theorems 5.10 and 5.12 we get the following:

Corollary 5.21 Let K_{>0} be a semifield in a field K. Let G be as in Theorem 5.10. Then the sets Conf_{n}^{+}(B(K)) for n = 3, 4 have a natural dihedral symmetry.

Definition 5.22 Choose a triangulation T of a convex n-gon whose vertices are labeled by flags (B_{1}, ..., B_{n}). A configuration (B_{1}, ..., B_{n}) ∈ Conf_{n}(B(K)) is K_{>0}-positive if the four flags at the vertices of each quadrilateral of the triangulation T form a K_{>0}-positive configuration of flags.

Corollary 5.21 shows that only dihedral order of vertices of each quadrilateral matters. Any two triangulations of a convex n-gon are related by a sequence of flips. Using Corollary 5.21 and the proof of Lemma 5.19 we see that the notion of K_{>0}-positivity does not change under a flip. So we get the following proposition, telling us that Definition 5.22 does not depend on choice of a triangulation:

Proposition 5.23 If a configuration (B_{1}, ..., B_{n}) ∈ Conf_{n}(B(K)) is K_{>0}-positive with respect to one triangulation, it is positive with respect to any triangulation of the n-gon.

Let us denote by Conf_{n}^{+}(B(K)) the set of all K_{>0}-positive configurations of K-flags.

Corollary 5.24 The set Conf_{n}^{+}(B(K)) is invariant under the natural dihedral group action.

Proof. Although the dihedral group action changes the triangulation used to define K_{>0}-positive configurations, by Proposition 5.23 it does not affect K_{>0}-positivity. The corollary is proved.

For an arbitrary semifield K_{>0} one may have

Conf_{n}^{+}(B(K)) ≠ Conf_{n}(B)(K_{>0})  (60)

Example. The configuration space of three distinct points on P^{1} is defined as follows:

Conf_{3}(P^{1}) := (P^{1} × P^{1} × P^{1} − diagonals)/SL_{2}

It is a single point. However for a field F the set of SL_{2}(F)-orbits on P^{1}(F)^{3} −diagonals is F^{∗}/(F^{∗})^{2}. Indeed, under the action of SL_{2}(F) the first two points can be made 0 and ∞, and then the subgroup diag(t, t^{−1}) of diagonal matrices in SL_{2}(F) stabilizing these two points acts on z ∈ F^{∗} by z ⟼ t^{2}z. For example when F = ℝ there are 2 orbits. Indeed, three ordered distinct points on P^{1}(ℝ) provide an orientation of P^{1}(ℝ). Moreover Conf_{n}^{+}(P^{1}(K)) = Conf_{n}(P^{1})(K_{>0}) if and only if (K_{>0})^{2} = K_{>0}, which may not be the case.

Here is a general set up. Assume that we have a map of F -varieties f : X → Z, and an F -group G acts on X from the right. Assume that G acts on fibers of f, and this action is transitive on the fibers (over \overline{F}). Let z ∈ Z(F). Set Y := f^{−1}(z). It is a right homogeneous space for G. Let us assume that there exists a point y ∈ Y (F). Then, setting H := Stab_{G}(y), we have Y = H\G. So by ([Se], Cor 1 of Prop. 36 in I-5.4) there is canonical isomorphism

Y (F)/G(F) = Ker(H^{1}(F, H) ⟶ H^{1}(F, G))

The question whether there exists an F -point in Y is non trivial if Y is not a principal homogeneous space for G, and, as pointed out to us M. Borovoi, is treated for p-adic and number fields in the Subsection 7.7 of [Bor]. We will not need this in our story.

So if Z := X/G is the quotient algebraic variety, there are two different sets: Z(F) and X(F)/G(F). The canonical map X(F)/G(F) ⟶ Z(F) can be neither surjective nor injective.

However for positive configurations of real flags the situation is as nice as it could be:

Lemma 5.25 For n = 3, 4, 5, one has Conf_{n}^{+}(B(ℝ)) = Conf_{n}(B)(ℝ_{>0}), and there is a canonical inclusion

Conf_{n}(B)(ℝ_{>0}) ↪ Conf_{n}(B(ℝ))

Proof. It boils down to the following two facts: V_{∗}(ℝ_{>0}) = U_{∗}(ℝ_{>0})/H(ℝ_{>0}) and similarly ((U_{∗}^{−} × U_{∗}^{+})/H)(ℝ_{>0}) = U_{∗}^{−}(ℝ_{>0}) × U_{∗}^{+}(ℝ_{>0})/H(ℝ_{>0}). They follow from (ℝ_{+}^{∗})^{m} = ℝ_{+}^{∗}. The lemma is proved.

A positive atlas on the space Conf_{n}(B) is defined in Section 6. Then it is straitforward to check that Lemma 5.25 remains valid for all n > 2. So the real positive configurations of flags in the sense of Definition 5.22 always coincide with the ℝ_{>0}-points of the positive space Conf_{n}(B), and are embedded into the configuration space of n real flags in generic position modulo the G(ℝ)-action.

6 A positive structure on the moduli space X_{G,Ŝ}

In this Section we give three different proofs, in Sections 6.1, 6.3 and 6.5, of the decomposition of the moduli space X_{G,Ŝ} according to a triangulation. This provides a positive atlas on X_{G,Ŝ}.

0. Basic properties of positive atlas on Conf_{n}(B) for n = 3, 4, 5. Let us summarize the properties of the positive atlases on the spaces Conf_{n}(B), where n = 3, 4, 5, which are used below. The proves are given in Theorem 5.10, Theorem 5.12, and Lemma 5.19.

1) The space Conf_{3}(B) has a positive atlas invariant under the action of the group S_{3}.

2) There is a birational isomorphism

φ : Conf_{4}(B) ⟶^{∼} Conf_{3}(B) × H × Conf_{3}(B)  (61)

where projections to the left and right factors Conf_{3}(B) are given by the maps

(B_{1}, B_{2}, B_{3}, B_{4}) ⟼ (B_{1}, B_{2}, B_{3}) and (B_{1}, B_{2}, B_{3}, B_{4}) ⟼ (B_{1}, B_{3}, B_{4})

3) The birational isomorphism (61) provides a positive atlas on the space Conf_{4}(B). This positive atlas is invariant under the cyclic shift and reversion maps given by

(B_{1}, B_{2}, B_{3}, B_{4}) ⟶ (B_{4}, B_{1}, B_{2}, B_{3}), (B_{1}, B_{2}, B_{3}, B_{4}) ⟶ (B_{4}, B_{3}, B_{2}, B_{1})

If we picture a configuration (B_{1}, B_{2}, B_{3}, B_{4}) at the vertices of a 4-gon as on Figure 11, then the cyclic invariance boils down to the invariance of the positive atlas on Conf_{4}(B) under a flip, and hence the invariance under the change of orientation of the diagonal e. The invariance under the reversion map shows the invariance of the positive atlas under the change of orientation of the plane where the 4-gon is located.

4) Consider the convex pentagon whose vertices are labeled by a configuration (B_{1}, ..., B_{5}). Let T_{1} be its triangulation by the diagonals B_{1}B_{3} and B_{1}B_{4}. Then there is a birational isomorphism

φ_{T_{1}} : Conf_{5}(B) ⟶^{∼} (Conf_{3}(B))^{3} × H^{2}  (62)

where the factors Conf_{3}(B) correspond to the triangles of the triangulation T_{1}, and the factors H are assigned to the diagonals B_{1}B_{3} and B_{1}B_{4} of the triangulation.

The positive atlas of the space Conf_{5}(B) provided by (62) is invariant under the natural action of the dihedral group D_{5}.

1. A positive atlas on X_{G,Ŝ}. Let us spell in detail the definition of the map π_{T} outlined in Section 1.2. Let Ŝ be a marked hyperbolic surface. Let us choose a pair T given by

T = {an ideal triangulation T of Ŝ, a choice of orientation of all internal edges of T }  (63)

Let assume first that the triangulation is regular in the sense of Section 4.8. Denote by tr(T) the set of the triangles of T, and by ed_{i}(T) be the set of the internal edges of T. Denote by X_{G,t̂} the moduli space of the framed G–local systems on a triangle t, considered as a disc with three marked points on the boundary given by certain three points inside of the edges, one point per each edge. Let (L, β) ∈ X_{G,Ŝ}. We can view a triangle t of the ideal triangulation as a disc with three marked points at the boundary, as we just explained. Restricting the framed local system (L, β) ∈ X_{G,Ŝ} to such a triangle t we get an element of X_{G,t̂}. Indeed, if a vertex s of the triangle t belongs to a boundary arc on Ŝ then there is a flag at this point. If s is a puncture then we take the monodromy invariant flag at the intersection of the triangle with a little disc around the puncture. So we get a projection p_{t} : X_{G,Ŝ} ⟶ X_{G,t̂}. Further, each internal edge e of the triangulation T determines a pair of triangles t_{1}, t_{2} sharing e. Their union t_{1} ∪ t_{2} is a 4-gon with four marked points on the boundary, one per each side. Thus restricting (L, β) to t_{1} ∪ t_{2} we get an element of X_{G,4}. Its H–invariant corresponding to the oriented edge e is given by (57). It provides a rational projection p_{e} : X_{G,Ŝ} ⟶ H. The collection of projections {p_{t}}, {p_{e}} provides a rational map

π_{T} : X_{G,Ŝ} ⟶ ∏_{t∈tr(T)} X_{G,t̂}× ∏_{e∈ed_{i}(T)} H  (64)

If the triangulation T is special, the above construction works unless we consider the edge given by the loop of the virus. The construction of the H-invariant for this edge is obtained by going to the 2 : 1 cover of S described in Section 4.8, and then using the above construction of the H-invariant. The same trick, described in detail in the Section 9.6 below, is used in the proof of the theorems in this Section in the part related to this edge.

Theorem 6.1 Let G be a split semisimple algebraic group with trivial center and Ŝ a marked hyperbolic surface. Then the map π_{T} is a birational isomorphism. The collection of rational maps {π_{T}^{−1}}, where T run through all datas (63), provides a Γ_{S}–equivariant positive atlas on X_{G,Ŝ}.

Proof. Let us prove first the theorem in the special case when Ŝ = D̂_{n} is a disc with n marked points on the boundary. We proceed by the induction on n. For n = 3 this is given the property 1) above. To make the inductive step from n to n + 1 consider a convex (n + 1)-gon P_{n+1} with the vertices b_{1}, ..., b_{n+1} taken in the order compatible with the orientation of the disc. We assign a configuration of flags (B_{1}, ..., B_{n+1}) to the vertices of P_{n+1}, so that B_{i} is assigned to the vertex b_{i}. Let us triangulate this (n + 1)-gon so that (b_{1}, b_{2}, b_{3}) is one of the triangles of the triangulations. Let P_{n} be the n-gon obtained by cutting out the triangle (b_{1}, b_{2}, b_{3}). By the induction assumption a generic data on the right hand side of (64) determines a configuration (B_{1}, B_{3}, B_{4}, ..., B_{n+1}) ∈ Conf_{n}(B) assigned to the vertices of the n-gon P_{n}. Adding the triangle (b_{1}, b_{2}, b_{3}) we add to this data the element β_{123} ∈ Conf_{3}(B) corresponding to this triangle, and an H-invariant h_{13} ∈ H corresponding to the edge (b_{1}, b_{3}). Let (b_{1}, b_{3}, b_{k}) be the triangle of the triangulation adjacent to the edge (b_{1}, b_{3}), and β_{13k} ∈ Conf_{3}(B) is the invariant assigned to this triangle. According to the property 2) there is a unique configuration (B′_{1}, B′_{2}, B′_{3}, B′_{k}) of four flags with the invariant (β_{123}, h_{13}, β_{13k}). Observe that the configurations (B′_{1}, B′_{3}, B′_{k}) and (B′_{1}, B_{3}, B_{k}) are the same. Since G has trivial center, it acts without stable points on the space of generic triples of flags in G. Therefore there exists unique configuration (B_{1}, B_{2}, ..., B_{n+1}) such that (B′_{1}, B′_{2}, B′_{3}, B′_{k}) = (B_{1}, B_{2}, B_{3}, B_{k}) corresponding to the data provided by the triangulation of the (n + 1)-gon. So, given a triangulation T of the (n + 1)-gon, the map (64) is a birational isomorphism for the disc D̂_{n+1}. Thus it provides a positive atlas on Conf_{n+1}(B). It follows from 3) that the positive atlas corresponding to the triangulation T′ obtained from T by a flip is compatible with this positive atlas. The role of the mapping class group in this case is played by the dihedral group generated by the cyclic shift and reversion maps. This proves the theorem for the disc.

Figure 13: The quadrilateral corresponding to the triangle b_{1}b_{2}b_{3}

Now let us consider the general case. It is sufficient to show that, given an arbitrary field F, a generic point x of the right hand side of (64) provides a F -point y of the right hand side such that π(x) = y. Let us define a representation ρ_{x} : π_{1}(S) ⟶ G(F) corresponding to x. Let T be an ideal triangulation of S. Lifting it to a universal cover S̃ of S we obtain a triangulation T̃. The triangulated surface (S̃, T̃) is isomorphic to the hyperbolic disc H equipped with the Farey triangulation. Let (a, b, c) be a triangle of T̃. Let γ ∈ G(F). Take a finite polygon P_{γ} of the triangulation T̃ containing the triangles (a, b, c) and (a′, b′, c′) := γ(a, b, c). The polygon P_{γ} inherits a triangulation. So by the first part of the proof, the data x, restricted to the triangulated polygon, gives rise to a unique element y(P_{γ}) ∈ Conf_{n}(B(F)) corresponding to it. Let (B_{a}, B_{b}, B_{c}) be the corresponding configuration of flags attached to the vertices (a, b, c) of the polygon. Then there exists an element g_{γ} ∈ G(F) such that

(B_{a′}, B_{b′}, B_{c′}) = g_{γ}(B_{a}, B_{b}, B_{c})

Such an element is unique since G has trivial center. It is easy to see that γ → g_{γ} is a representation of π_{1}(S, p). and it does not depend on the choice of the polygon. We leave construction of the framing as an easy exercise. The theorem is proved.

In the next subsection we work out an explicit construction of the framed local system corresponding to the generic data in (64).

2. The six canonical pinnings determined by a generic triple of flags in G. Let us choose two reduced words i^{+} and i^{−} of the element w_{0}. Set

U_{∗}^{−}(i^{−}) := ψ_{i^{−}}(𝔾_{m}^{n}) ⊂ U_{∗}^{−}, U_{∗}^{+}(i^{+}) := ψ_{i^{+}}(𝔾_{m}^{n}) ⊂ U_{∗}^{+}

The group H acts freely on each of these spaces. So the H-orbit of any element u ∈ U_{∗}^{−}(i^{−}) intersects the subvariety

\overline{U}_{∗}^{−}(i^{−}) := ψ_{i^{−}}(e × 𝔾_{m}^{i^{−}(α_{1})−1} × e × 𝔾_{m}^{i^{−}(α_{2})−1} × ... × e × 𝔾_{m}^{i^{−}(α_{r})−1}) ⊂ U_{∗}^{−}(i^{−})  (65)

in a single point, called the normalized representative \overline{u}. Similarly we define the normalized representative \overline{v} of the H-orbit of an element v ⊂ U_{∗}^{+}(i^{+}) := ψ_{i^{+}}(𝔾_{m}^{n}) as the intersection point of the orbit with the subvariety

\overline{U}_{∗}^{+}(i^{+}) := ψ_{i^{+}}(𝔾_{m}^{i^{+}(α_{1})−1} × e × 𝔾_{m}^{i^{+}(α_{2})−1} × e × ... × 𝔾_{m}^{i^{+}(α_{r})−1} × e) ⊂ U_{∗}^{+}(i^{+})  (66)

Let (B_{1}, B_{2}, B_{3}) be a sufficiently generic triple of flags in G. Then one has

(B_{1}, B_{2}, B_{3}) = g_{−} · (B^{−}, B^{+}, \overline{u}_{−}B^{+}\overline{u}_{−}^{−1}), u_{−} ∈ U_{∗}^{−}(i^{−})
(B_{1}, B_{2}, B_{3}) = g_{+} · (B^{−}, \overline{v}_{+}^{−1}B^{−}\overline{v}_{+}, B^{+}), v_{+} ∈ U_{∗}^{+}(i^{+})

for the uniquely defined elements g_{−}, g_{+} ∈ G. (“Sufficiently generic”, by definition, means that the above two formulas hold). Therefore such a triple (B_{1}, B_{2}, B_{3}) of flags in G determines the following two pinnings for G:

p_{(B_{1},B_{2},B_{3})}^{+}(B_{1}, B_{2}) := g_{−} · (B_{−}, B_{+}), p_{(B_{1},B_{2},B_{3})}^{−}(B_{1}, B_{3}) := g_{+} · (B_{−}, B_{+})

Using the cyclic shifts we define the rest of the pinnings. So we get the six pinnings for G, where the indices i are modulo 3:

p_{(B_{1},B_{2},B_{3})}^{+}(B_{i}, B_{i+1}), p_{(B_{1},B_{2},B_{3})}^{−}(B_{i}, B_{i−1}),  (67)

These pinnings are determined by the cyclic configuration of the three flags (B_{1}, B_{2}, B_{3}) and the choice of two reduced words i^{−}, i^{+} for w_{0}. Observe that p^{±} corresponds to g^{∓}. The + and − in (67) indicate whether the corresponding edge is oriented accordantly to the cyclic order of (B_{1}, B_{2}, B_{3}) or not. It is handy to picture the six pinning related to a triple (B_{1}, B_{2}, B_{3}) ∈ Conf_{3}^{∗}(B) by arrows as on the picture below. Say that U is a positive Zariski open subset of a positive variety X if X − U

Figure 14: Arrows encode the six pinnings assigned to a configuration of three flags

is contained in a positive divisor of X. The discussion above proves the following lemma.

Lemma 6.2 There exists a positive Zariski open subset Conf_{3}^{∗}(B) ⊂ Conf_{3}(B) such that for any configuration (B_{1}, B_{2}, B_{3}) ∈ Conf_{3}^{∗}(B) the construction above determines the six pinnings (67) for G. Changing a standard pinning for G amounts to a shift of each of our six pinnings by a common element in G.

We leave a precise description of Conf_{3}^{∗}(B) as the intersection of several explicitly given positive subvarieties of Conf_{3}(B) as a straitforward exercise.

The edge invariant. Given a generic configuration (B_{1}, B_{2}, B_{3}, B_{4}) of flags in G, such that both (B_{1}, B_{2}, B_{3}) and (B_{1}, B_{3}, B_{4}) are in Conf_{3}^{∗}(B), there exists unique element

h_{(i^{−},i^{+})}(B_{1}, B_{2}, B_{3}, B_{4}) ∈ H

such that

p_{(B_{1},B_{3},B_{4})}^{+}(B_{1}, B_{3}) = h_{(i^{−},i^{+})}(B_{1}, B_{2}, B_{3}, B_{4}) · p_{(B_{1},B_{2},B_{3})}^{−}(B_{1}, B_{3})  (68)

The element h_{(i^{−},i^{+})}(B_{1}, B_{2}, B_{3}, B_{4}) ∈ H in (68) is called the edge invariant of the configuration of flags (B_{1}, B_{2}, B_{3}, B_{4}) corresponding to the oriented edge (B_{1}, B_{3}).

Figure 15: The edge invariant is obtained by comparison of the two pinnings shown by arrows

3. The positive structure of X_{G,Ŝ}. To describe the rational inverse to this map we need a finer data assigned to a triangulation T of Ŝ.

Definition 6.3 Denote by 𝕋 the following data on Ŝ:

i) An ideal triangulation T of a marked hyperbolic surface Ŝ.

ii) For each internal edge of the triangulation T a choice of orientation of this edge.

iii) For each triangle t of the triangulation T a choice of a vertex of this triangle.

iv) A pair of reduced decompositions (i^{−}, i^{+}) of the maximal length element w_{0} ∈ W.

We can visualize a choice described in iii) as a choice of an angle for each of the triangles t.

Theorem 6.4 Let G be a split semisimple algebraic group with trivial center. Let Ŝ be a marked hyperbolic surface and 𝕋 a data assigned to an ideal triangulation T of Ŝ, as in Definition 6.3. Then

a) There exists a regular open embedding

ψ_{𝕋} : Conf_{3}^{∗}(B)^{{triangles of T }} × H^{{edges of T }} ↪ X_{G,Ŝ}  (69)

b) The collection of regular open embeddings {ψ_{𝕋}}, when 𝕋 runs through all possible datas from Definition 6.3 assigned to Ŝ, provides a Γ_{S}–equivariant positive structure on X_{G,Ŝ}.

c) The positive structure defined in Theorem 6.1 is compatible with the one defined in b).

Proof. a) Recall the six pinning assigned according to Lemma 6.2 to an element of C ∈ Conf_{3}^{∗}(B). Their dependence on the choice of a standard pinning (B^{−}, B^{+}) boils down to a shift by a common element in G. Further, let us suppose that we have two elements C_{1}, C_{2} ∈ Conf_{3}^{∗}(B) and an element h ∈ H. Then there exists unique configuration (B_{1}, B_{2}, B_{3}, B_{4}) ∈ Conf_{4}(B) such that

C_{1} = (B_{1}, B_{2}, B_{3}), C_{2} = (B_{1}, B_{3}, B_{4}), h_{(i^{−},i^{+})}(B_{1}, B_{2}, B_{3}, B_{4}) = h  (70)

Moreover if we choose one of the six pinnings corresponding to the configuration (B_{1}, B_{2}, B_{3}), then there is unique way to normalize the six pinnings corresponding to the configuration (B_{1}, B_{3}, B_{4}) so that formula (68) holds with h_{(i^{−},i^{+})}(B_{1}, B_{2}, B_{3}, B_{4}) = h. Therefore given (70) and one of the 12 pinnings shown by arrows on the picture we determine uniquely the rest of the pinnings. Let Γ be

Figure 16: The twelve pinnings assigned to a quadrilateral

a graph on a marked hyperbolic surface Ŝ dual to an ideal triangulation T of Ŝ. Consider a closed path p on the graph. We are going to attach an element g_{p} ∈ G to this path which will give rise to a homomorphism π_{1}(S) → G. Let us consider the triangles of the triangulation corresponding to the vertices of the path p, as on the picture. Let us choose one of these triangles and denote it t_{1}. The data 𝕋 allows to assign to each triangle t a configuration of flags (B_{1}, B_{2}, B_{3}) sitting at the vertices of t, so that B_{1} is assigned to the distinguished vertex (provided by the condition iii) in definition of 𝕋), and the orientation of the triangle t given by the order of B_{i}’s is the same as the one induced by the orientation of the surface S (clockwise on our pictures). It follows from the remarks made about pinnings above that given one of the six canonical pinnings associated to the flags sitting at the vertices of the triangle t_{1}, we determine uniquely the 6-tuples of pinnings corresponding to the triangles following t_{1} along the path p, till we return to the triangle t_{1}. Thus for a closed path p

Figure 17: Constructing the local system

there are two 6-tuples of canonical pinnings assigned to the triangle t_{1}: the original one, and the one obtained after traveling along the loop p. These two 6-tuples of pinnings differ by a shift in G. So we pick up an element g_{p} ∈ G transforming the original 6-tuple of pinnings to the final one. Clearly the map p ⟼ g_{p} is multiplicative: p_{2}p_{1} ⟼ g_{p_{2}}g_{p_{1}}. Moreover p^{−1} ⟼ g_{p}^{−1}. Since p is a path on the graph, its homotopy class is determined uniquely by the path itself. So we get a group homomorphism π_{1}(S, x) → G where x is a base point on Γ inside of the triangle t_{1}.

Let us define an framed structure on this representation. Let s be a puncture on S. Consider all triangles of the triangulation sharing the vertex s. Let us choose a path p_{s} from the base point x to

Figure 18: The link of a puncture s

a point inside of certain triangle t_{s} sharing the vertex s. Then given a pinning P_{x} in the triangle t_{x} containing x we get the six pinning assigned to the triangle T_{s}. Let P_{s} be one of them corresponding to an arrow going from the vertex s. Let (B, B′) be the pair of Borel subgroups underlying this pinning, so that B is assigned to the vertex s. Going around s we define pinnings corresponding to the other triangles sharing s. Observe that all pinnings corresponding to the arrows starting at s have the same ”initial” Borel subgroup, B. Therefore the monodromy of the local system corresponding to our representation around a small loop l_{s} going around s belongs to B. So if P_{s} = gP_{x}, then the element p_{s}^{−1}l_{s}p_{s} leaves the Borel subgroup g^{−1}Bg invariant. This provides an framed structure of our representation at the puncture s.

We have defined the map ψ_{𝕋}. Let us show that it is injective. Let us define a left inverse map

π_{𝕋} : X_{G,Ŝ} ⟶ ∏_{t∈tr(T)} X_{G,t̂}× ∏_{e∈ed_{i}(𝕋)} H

Here ed_{i}(𝕋) is the set of internal edges of the triangulation T oriented as prescribed by the data 𝕋. So the second product is over the set of all internal edges of T. The ∏_{t∈tr(T)} X_{G,t̂}–component of the map π_{𝕋} is the same as for the map π_{T}. The H-component corresponding to an oriented internal edge e is the edge invariant from (68) assigned to e. Notice that it is different from the one used in π_{T}. Then by the very construction we have π_{𝕋} ∘ ψ_{𝕋} = Id. Thus the map ψ_{𝕋} is injective.

Lemma 6.5 Let f : X ⟶ Y and g : Y ⟶ X be maps of irreducible algebraic varieties of the same dimension. Then g ∘ f = Id_{X} implies that both f and g are birational isomorphisms.

Proof. The conditions of the lemma imply that the maps f, g are dominant, so there are maps of fields f^{∗} : ℚ(Y) → ℚ(X), g^{∗} : ℚ(X) → ℚ(Y). Since f^{∗}g^{∗} = Id, the lemma follows.

Lemma 6.6 Let Ŝ be a marked hyperbolic surface. Denote by t(T) and e_{i}(T) the number of triangles and internal edges of an ideal triangulation T of Ŝ. Then one has

dimX_{G,Ŝ} = e_{i}(T)dimH + t(T)(dimU − dimH)  (71)

This Lemma, of course, follows from the proof given in Section 6.1. However we will give an independent proof for completeness of the arguments.

Proof. Assume for a moment that the number k of marked boundary points on Ŝ is equal to zero, i.e. Ŝ = S and e(T) = e_{i}(T) is the number of all edges. Then χ(S) = −e(T)+t(T), 3e(T) = 2|t(T)|. Therefore

e(T) = −3χ(S), t(T) = −2χ(S)

It is well known that dimL_{G,S} = χ(S)dimG. The canonical map X_{G,S} → L_{G,S} is a finite map at the generic point. This follows from the well known fact that the monodromies of a generic G-local system on S around the pictures are regular conjugacy classes in G. This gives dimX_{G,S} = −χ(S)dimG. The equality (71) in the case Ŝ = S follows immediately from these remarks. To check (71) in the general case observe that adding a marked point we increase the dimension by dim(G/B) = dimU. On the other hand adding a marked point we add by one the number of internal edges and triangles of a triangulation. So it remains to check the statement in the case when S is a disc or cylinder to have the base for the induction. The lemma is proved.

It follows from Lemma 6.6 both spaces in (69) are of the same dimension. Thus ψ_{𝕋} is a birational isomorphism. The part a) of the theorem is proved.

b) It is sufficient to show that if T and T′ are two triangulations of Ŝ related by a flip, as on the picture below, and 𝕋 and 𝕋′ are two datas refining these triangulations, then the positive structures defined by ψ_{𝕋} and ψ_{𝕋′} are compatible. This follows from Theorem 5.12and Lemma 5.19. The part b) of the theorem is proved.

c) It follows from Proposition 5.16. The theorem is proved.

4. The higher Teichmüller X -space and the X -lamination space related to G.

Definition 6.7 Let Ŝ be a marked hyperbolic surface and G a split reductive algebraic group with trivial center.

a) The Teichmüller space X_{G,Ŝ}^{+} is the ℝ_{>0}-positive part X_{G,Ŝ}(ℝ_{>0}) of the positive space X_{G,Ŝ}.

b) Let 𝔸^{t} be one of the tropical semifields ℤ^{t}, ℚ^{t}, ℝ^{t}. The set of X^{𝔸}-laminations on Ŝ corresponding to G is the set X_{G,Ŝ}(𝔸^{t}) of points of the positive space X_{G,Ŝ} with values in the tropical semifield 𝔸^{t}.

According to Section 4.5 projectivisation of the set of real X -laminations can be considered as the Thurston type boundary of the space X_{G,Ŝ}^{+}.

Applying the definition of the regular part of a positive variety (Section 4.1) in our case we get the regular part X_{G,Ŝ}^{reg} ↪ X_{G,Ŝ}. The higher Teichmüller space is a connected component in X_{G,Ŝ}^{reg}(ℝ).

Recall from Section 2.1 the set X̃_{G,Ŝ}(ℝ)/G(ℝ) parametrising framed representations π_{1}(S, x) → G(ℝ) modulo G(ℝ)-conjugation. If D̂_{3} is a disc with three marked points, then ordering these points we get an isomorphism X̃_{G,D̂_{3}}(ℝ) = B(ℝ)^{3}. Let T be an ideal triangulation of Ŝ. Then each triangle t of the triangulation provides a restriction map X̃_{G,Ŝ}(ℝ)/G(ℝ) → X̃_{G,t̂}(ℝ)/G(ℝ). We employ quadrilaterals of the triangulation in a similar way.

An element of X̃_{G,Ŝ}(ℝ)/G(ℝ) is called positive if its restriction to every quadrilateral of a given triangulation T belongs to Conf_{4}^{+}(B(ℝ)).

Corollary 6.8 The set of positive elements in X̃_{G,Ŝ}(ℝ)/G(ℝ) does not depend on the choice of triangulation T. It is isomorphic to X_{G,Ŝ}^{+}. So there is an embedding X_{G,Ŝ}^{+} ↪ X̃_{G,Ŝ}(ℝ)/G(ℝ).

Proof. Any two ideal triangulations of a hyperbolic marked surface are related by a sequence of flips. Corollary 5.21 plus Lemma 5.19 imply that a flip does not change the set of positive configurations. The second claim follows from Lemma 5.25 and the proof of Theorem 6.4.

5. A constructive proof of Theorem 6.1. Here is a convenient way to spell the definition of the framed local system given by the map ψ_{𝕋}, which we will use to study its monodromy. To simplify the exposition let us assume that Ŝ = S. To describe the answer we use the following picture. Starting from an ideal triangulation T of S, we construct a graph Γ′_{T} embedded into the surface by drawing small edges transversal to each side of the triangles and inside each triangle connect the ends of edges pairwise by three more edges, as shown on the picture. The edges of the little oriented triangles are called t-edges. The edges dual to the edges of the triangulation T are called e-edges.

Figure 19: The graph Γ′_{T}

Given a left principal homogeneous G-space P, a point p_{0} ∈ P, and a configuration (p_{1}, p_{2}) of two pints in P (that is, a pair of points in P considered up to the left diagonal action of G), there is an element g_{p_{0}}(p_{1}, p_{2}) ∈ G defined as follows:

g_{p_{0}}(p_{1}, p_{2}) := g_{1}^{−1}g_{2} ∈ G,  where p_{1} = g_{1}p_{0}, g_{2} = g_{2}p_{0}  (72)

We will apply this construction to the principal homogeneous space of pinnings for G. Recall that we choose a standard pinning P_{0} = (B^{−}, B^{+}) for G. Every edge of the graph Γ′_{T} provides a configuration of two pinnings in G, assigned naturally to the vertices of this edge. Indeed, the vertices of the graph on the right of the picture match the pinnings assigned to the triangulation on the left, as shown on the picture. We use only the pinnings compatible with the orientation of S. Therefore we can assign to every oriented edge a of the graph Γ′_{T} an element M(a) ∈ G. Namely, it is the element assigned by (72) to the configuration of two pinnings attached to the oriented edge a, and the standard pinning P_{0}. It follows from the very definitions that these elements enjoy the following properties:

1) M(a)M(\overline{a}) = Id, where \overline{a} denotes the edge as a taken with the opposite orientation.

2) M(t_{1})M(t_{2})M(t_{3}) = Id, where the oriented edges t_{1}, t_{2}, t_{3} make a small oriented triangle on the graph Γ′_{T}.

Thanks to 1) the elements {M(a)} define a G-local system on the graph Γ′_{T}. Thanks to 2) it is descended to a G-local system on the graph Γ. Since Γ is homotopy equivalent to S, we got a G-local system on S.

6. Proof of Theorem 1.11. Given a loop on S, let us shrink it to a loop on Γ′_{T}. We may assume that this loop contains no consecutive t-edges: indeed, a composition of two t-edges is a t-edge. Thus we may choose an initial vertex on the loop so that the edges have the pattern et...etet. Let t and e be consecutive t- and e-edges of the graph Γ′_{T}. Then there is a unique way to orient them so that e goes after t, i.e. the oriented edge e starts at the end of the one t. We say that a t-edge is oriented clockwise if it is compatible with the clockwise orientation of the corresponding triangle. Therefore the monodromy is computed as a product of elements of type M(e)M(t), where e follows after t.

This construction defines the universal framed G-local system on Ŝ. Recall the semifield 𝔽_{G,S}^{+} of positive rational functions on the moduli space X_{G,S} for the positive atlas defined above.

Proposition 6.9 Let t and e be two consecutive t- and e-edges of the graph Γ′_{T}. Let us orient them so that e goes after t. Then

M(e)M(t) ∈ \{ \begin{matrix}B^{+}(𝔽_{G,S}^{+}) & if t is oriented clockwise \\ B^{−}(𝔽_{G,S}^{+}) & if t is oriented counterclockwise \end{matrix}

Proof. We have to consider separately the cases when t is oriented clockwise, or counterclockwise, i.e. t = t_{1} or t = t_{2} where t_{1} and t_{2} are as on Figure 20.

Figure 20: The oriented edges t_{1} and t_{2} on the graph Γ′_{T}

We denote by [B_{1}, B_{2}, B_{3}] triples of flags, to distinguish from configurations of flags, and by g · [B_{1}, B_{2}, B_{3}] the result of the action of g ∈ G on the triple [B_{1}, B_{2}, B_{3}].

Case 1: t = t_{1}. Let u_{+} = h_{+}(\overline{u}_{+}) and u_{−} = h_{−}(\overline{u}_{−}), where h_{±} ∈ H acts on U^{±} by conjugation. Then we have

[u_{+}^{−1}B^{−}u_{+}, B^{+}, B^{−}] = u_{+}^{−1} · [B^{−}, B^{+}, u_{+}B^{−}u_{+}^{−1}] = u_{+}^{−1}h_{+} · [B^{−}, B^{+}, \overline{u}_{+}B^{−}\overline{u}_{+}^{−1}]
= u_{+}^{−1}h_{+} · [B^{−}, B^{+}, ϕ(\overline{u}_{+})B^{+}ϕ(\overline{u}_{+})^{−1}]

On the other hand

[B^{−}, B^{+}, u_{−}B^{+}u_{−}^{−1}] = h_{−} · [B^{−}, B^{+}, \overline{u}_{−}B^{+}\overline{u}_{−}^{−1}]  (73)

Let h(ϕ(\overline{u}_{+}))(\overline{ϕ(\overline{u}_{+})}) = ϕ(\overline{u}_{+}). Then

M(e)M(t_{1}) = h(ϕ(\overline{u}_{+}))^{−1}h_{+}^{−1}u_{+}h_{−} = h(ϕ(\overline{u}_{+}))^{−1}\overline{u}_{+}h_{+}^{−1}h_{−} ∈ B^{+}(𝔽_{G,S}^{+})

Indeed, h_{+}^{−1}h_{−} is the H-invariant, and h(φ(\overline{u}_{+}))^{−1} is a positive rational function by positivity of ϕ. Case 2: t = t_{2}. We have

[B^{−}, u_{+}^{−1}B^{−}u_{+}, B^{+}] = [B^{−}, ϕ(u_{+})^{−1}B^{−}ϕ(u_{+}), B^{+}] = ϕ(u_{+})^{−1} [B^{−}, B^{+}, ϕ(u_{+})B^{+}ϕ(u_{+})^{−1}] =

ϕ(u_{+})^{−1} [B^{−}, B^{+}, u_{+}B^{−}u_{+}^{−1}] = ϕ(u_{+})^{−1}h_{+} [B^{−}, B^{+}, \overline{u}_{+}B^{−}\overline{u}_{+}^{−1}]

Comparing with (73), and using the fact that u_{+} = h_{+}\overline{u}_{+}h_{+}^{−1} implies ϕ(u_{+}) = h_{+}ϕ(\overline{u}_{+})h_{+}^{−1}, we get

M(e)M(t_{2}) = h_{+}^{−1}ϕ(u_{+})h_{+}h_{+}^{−1}h_{−} = ϕ(\overline{u}_{+})h_{+}^{−1}h_{−} ∈ B^{−}(𝔽_{G,S}^{+})

Indeed, h_{+}^{−1}h_{−} is the H-invariant, and φ(\overline{u}_{+}) is positive. The proposition is proved.

Let us return to the proof of the theorem. A loop on S contains elements of just one kind, B^{+} or B^{−}, if and only if it is homotopic to the loop around a boundary component. Therefore the monodromy around any non-boundary loop is obtained as a product of elements both from B^{−}(𝔽_{G,S}^{+}) and B^{+}(𝔽_{G,S}^{+}). Thus it lies in G(𝔽_{G,S}^{+}). The theorem is proved.

Example. Let G = P GL_{2}, u_{+} = ( \begin{matrix}1 & y \\ 0 & 1 \end{matrix} ), u_{−} = ( \begin{matrix}1 & 0 \\ x & 1 \end{matrix} ). Then

(B^{−}, u_{+}^{−1}B^{−}u_{+}, B^{+}, u_{−}B^{−}u_{−}^{−1}) = (∞, −y^{−1}, 0, x) = (∞, −1, 0, xy)

The edge invariant equals xy, and coincides with the cross-ratio of the corresponding 4 points on ℙ^{1}. 7. The universal higher Teichmüller X -space related to G.

Definition 6.10 A set C is cyclically ordered if its any finite subset is cyclically ordered, and any inclusion of finite subsets preserves the cyclic order.

To define a cyclic order on a set C it is sufficient to specify cyclic orders for all triples and quadruples of elements of C so that the following compatibility condition holds: forgetting any element of any cyclically ordered quadruple we get a triple whose induced cyclic order is the same as the prescribed one. Reversing a cyclic order on C we get a new one.

According to Theorems 5.10 and 5.12 the set of positive n-tuples of flags in B(ℝ) is invariant under the operation of reversion of the cyclic order.

Definition 6.11 i) A dihedral structure on C is a cyclic order considered up to reversion.

ii) A map β from a set C with given dihedral structure to the flag variety B(ℝ) is positive if it maps every cyclically ordered quadruple (a, b, c, d) to a positive quadruple of flags (β(a), β(b), β(c), β(d)).

Remarks. 1. Positive maps are necessarily injective. We say that C is a positive subset of the flag variety B(ℝ), if there exists a cyclically ordered set C′ and a positive map β : C′ → B(ℝ) whose image is C.

2. There exist quadruples of flags in ℝℙ^{2} such that forgetting any of them we get a positive triple, while the quadruple itself is not positive. For example take four points (x_{1}, x_{2}, x_{3}, x_{4}) going clockwise on a circle, and consider the tangent flags to the points (x_{1}, x_{3}, x_{2}, x_{4}).

Consider the dihedral structure on ℙ^{1}(ℚ) provided by the embedding ℙ^{1}(ℚ) ↪ ℙ^{1}(ℝ).

Definition 6.12 The universal Teichmüller space X_{G}^{+} is the quotient of the space of positive maps

β : ℙ^{1}(ℚ) ⟶ B(ℝ)  (74)

by the natural action of the group G(ℝ) on it.

In other words, X_{G}^{+} is the space of configurations of real flags in G(ℝ) parametrised by ℙ^{1}(ℚ). Canonical decomposition of X_{G}^{+}. To parametrise X_{G}^{+}, let us choose three distinct points on

ℙ^{1}(ℚ), called 0, 1, ∞. Recall the Farey triangulation, understood as a triangulation of the hyperbolic disc with a distinguished oriented edge (= flag). Then we have canonical identifications

ℙ^{1}(ℚ) = ℚ ∪ ∞ = {vertices of the Farey triangulation}  (75)

The distinguished oriented edge connects 0 and ∞. So a point of X_{G}^{+} gives rise to a positive map

{vertices of the Farey triangulation} ⟶ B(ℝ)

considered up to the action of G(ℝ). Assigning to every Farey triangle the corresponding configuration of the three flags at the vertices, and to every Farey geodesic, oriented in a certain way, the H-invariant of the associated quadruple of flags, we get a canonical map

φ_{G} : X_{G}^{+} ⟶ X_{G,3}^{+}{Farey triangles} × H(ℝ_{>0})^{{Farey diagonals}}  (76)

Theorem 6.13 The map (76) is an isomorphism.

Proof. It is completely similar to the one of Theorem 6.1, and thus is omitted.

Example. When G = P SL_{2} our universal Teichmüller space is essentially the one considered by L. Bers and R. Penner.

Recall the Thompson group 𝕋 of all piece-wise linear automorphisms of ℙ^{1}(ℚ). By definition, for every element g ∈ 𝕋 there is a decomposition of ℙ^{1}(ℚ) into a union of finite number of segments, which may overlap only at the ends, so that restriction of g to each segment is given by an element of P GL_{2}(ℚ) (depending on the segment). It acts on X_{G}^{+} in an obvious way. Each element of the Thompson group can be presented as a composition of flips.

Let Γ ∈ P SL_{2}(ℤ) be a torsion free finite index subgroup. Denote by H the hyperbolic disc equipped with the Farey triangulation. Set S_{Γ} := H/Γ. The surface S_{Γ} inherits an ideal triangulation T_{Γ}. The higher Teichmüller space X_{G,S_{Γ}}^{+} is embedded into the universal one X_{G}^{+} as the subspace of Γ-invariants. Moreover, let

φ_{G,S_{Γ}} : X_{G,S_{Γ}}^{+} ⟶^{∼} X_{G,3}^{+}{triangles of T_{Γ}} × H(ℝ_{>0})^{{diagonals of T_{Γ}}}  (77)

be the decomposition of the higher Teichmüller space according to a triangulation T_{Γ} of S, where we assume an orientation of the edges of T_{Γ} is chosen. This isomorphism is obtained by taking the ℝ_{>0}- points of the spaces in (64). Then the isomorphism (77) can be obtained by taking the Γ-invariants of the isomorphism (76).

When G = P SL_{m}, there is a more geometric way to describe points of the space X_{G}^{+}, by (parametrised by ℙ^{1}(ℚ)) convex curves in ℝℙ^{m−1}, see Section 8.10.

Remark. The universal Teichmüller space X_{G}^{+} admits a natural quantization, equivariant with respect to the action of the Thompson group. The quantum universal Teichmüller space provides a combinatorial approach to the W-algebra related to the group G. In the case G = P SL_{m} the quantization is especially explicit, and follows from the results of Chapter 11 and [FG2]. The case G = P SL_{3} is discussed in [FG3].

8. Discreteness of positive representations: Proof of theorem 1.9. Take a positive framed G(ℝ)-local system on a hyperbolic surface S. Let T be an ideal triangulation of S. Consider a universal cover S̃ and the corresponding triangulation T̃. The triangulated surface (S̃, T̃) is isomorphic to the hyperbolic disc H equipped with the Farey triangulation. The vertices of the Farey triangulation are identified with ℚℙ^{1}. Choose a point p̃ on S̃ and a trivialization of the local system at p̃. Let V (T̃) be the set of vertices of T̃. Let us construct a map

β : V (T̃) = ℚℙ^{1} → B(ℝ)

Let ṽ be a vertex of T̃ and α̃ be a path from p̃ to ṽ. Let p, v and α be the corresponding data on S. Let β be a small counterclockwise loop on S surrounding the point v and intersecting with α at one point r. Consider the flag at the fiber of the local system over r provided by the enhanced structure. The part of the path α between p and r maps it to the fiber over p, thus giving the desired map β.

The map β has the following properties.

1. Changing the base point p̃ or trivialization at p̃ results in a change β → hβ for certain h ∈ G(ℝ).

2. The map β is π_{1}(S)-equivariant. Precisely, let γ ∈ π_{1}(S, p) and ρ : π_{1}(S, p) → G(ℝ) be the monodromy representation corresponding to our local system and the chosen trivialization over p. Then ρ(γ)β(ṽ) = β(γṽ), where γ acts on vertices of T̃ by deck transformations.

3. β is a positive map.

The first two properties follow directly from the construction. The third one follows from the definition of a positive framed local system if the triple of vertices of T̃ constitute vertices of a triangle or quadrilateral. If they do not, there exists a finite polygon of the triangulation T̃ containing all these vertices. One can change the triangulation of this polygon by a finite number of flips so that the original triple/quadruple of points become vertices of one triangle/quadrilateral of the new triangulation. According to Theorem 5.21 if a local system is positive with respect to a triangulation, it is positive with respect to any other one, hence the property.

Let us fix a pair (B^{−}, B^{+}) of the opposite Borel subgroups in G(ℝ). Consider the following two subsets in the real flag variety:

B^{+} := {u_{−}B^{+}u_{−}^{−1}}, B^{−} := {v_{−}^{−1}B^{+}v_{−}}, u_{−}, v_{−} ∈ U^{−}(ℝ_{>0})

They are open in B(ℝ). Let \overline{B}^{+} be the closure of B^{+} in B(ℝ).

Lemma 6.14 The sets \overline{B}^{+} and B^{−} are disjoint.

Proof. One has u_{−}B^{+}u_{−}^{−1} = v_{−}^{−1}B^{+}v_{−} if and only if u_{−} = v_{−}^{−1}, i.e. u_{−}v_{−} = 1. Let us show that this is impossible. Recall the homomorphism π : U^{−} → R to the maximal abelian quotient. Then π(u_{−}) = ∏_{i} x_{i}(t_{i}) and π(v_{−}) = ∏_{i} x_{i}(s_{i}), where t_{i} > 0, s_{i} ≥ 0, and the product is over all simple positive roots. Thus 0 = π(u_{−}v_{−}) = ∏_{i} x_{i}(t_{i} + s_{i}). But t_{i} + s_{i} > 0. This contradiction proves the lemma.

Now fix a triangle of T̃ and let a, b, c be its vertices. The deck transformation along a loop γ sends the triangle (abc) to a triangle (a′b′c′) without common interior points with (abc). Without loss of generality we can assume that the vertices of the triangle (a′b′c′) are at the ark between a and c which does not contain b. However the point b′ may coincide with a or c. Let us assume first this does not happen. Then, since β is positive, we have a positive quadruple of flags (β(a), β(b), β(c), β(b′)), and β(b′) = ρ(γ)β(b). We may assume that β(a) = B^{−}, β(c) = B^{+}. Then β(b) ∈ B^{−} and β(b′) ∈ B^{+}. Since B^{−} is open, there exists an open neighborhood O of the unit in G(ℝ) such that gβ(b) ∈ B^{−} for any g ∈ O. Since B^{+} is disjoint with B^{−}, ρ(γ) ∉ O. In the case when b′ coincides with a or c, we may assume without loss of generality that it coincide with c. Then the argument is the same except that in the very end we use the fact that B^{−} does not contain B^{+}, which is a special case of the lemma. The theorem is proved.

Proof of Theorem 1.14. It follows immediately from the above constructions.

9. Positive curves in B(ℝ) are G(ℝ)-opers on S^{1}. A canonical distribution on B. The flag variety B is equipped with a G-invariant non-integrable distribution D whose dimension is equal to the rank of the group G. It is defined as follows. Let x be a point of B. We may assume that the corresponding Borel subgroup is B^{−}. Then the tangent space T_{x}B is identified with the Lie algebra Lie(U^{+}). Consider the subspace of Lie(U^{+}) given by the direct sum of the one dimensional subspaces corresponding to the simple roots. Transforming it to the tangent space T_{x}B, we get the fiber D_{x} of the distribution at the point x. It is easy to see that this definition is independent of the choices involved.

Proposition 6.15 Let C be a positive differentiable curve of the flag variety B(ℝ). Then C is an integral curve of the canonical distribution D on B(ℝ).

Proof. Let c(s) ∈ B(ℝ) be a positive curve. We may assume that c(0) = B^{−}, c(1) = B^{+}. So c(s) = x(s)B^{−} where x(s) ∈ U^{+}(ℝ_{>0}). Choose a reduced decomposition w_{0} = w_{i_{1}}...w_{i_{n}}. Then the positivity of x(s) implies that we can write it as x(s) = ∏_{i_{k}} x_{i_{k}}(t_{k}(s)), where t_{k}(0) = 0, t_{k}(s) > 0. Since [x_{i}(s)x_{j}(t)] = O(st), linearizing x(s) at s = 0 we get ∑_{k} t′_{i_{k}}(0)E_{i_{k}} ∈ Lie(U^{+}), where x_{i}(t) = exp(tE_{i}). The proposition is proved.

A smooth map from the circle S^{1} to B(ℝ) tangent to the canonical distribution is nothing else as a G(ℝ)-oper on S^{1} in the terminology of Beilinson and Drinfeld. So Proposition 6.15 asserts that positive differentiable loops in B(ℝ) are G(ℝ)-opers on S^{1}.

Describing P GL_{n+1}(ℝ)-opers on real curves. A C^{n−1}-smooth (meaning n − 1 times continuously differentiable) curve K in ℝℙ^{n} gives rise to osculating curve K̃ in the real flag variety B_{n+1}(ℝ) for P GL_{n+1}. Namely, K̃ is formed by the osculating flags to the curve at variable point p. It is an easy and well known fact that any differentiable P GL_{n+1}(ℝ)-oper is the osculating curve for certain sufficiently smooth curve K in ℝℙ^{n}. The curve K is recovered by applying the canonical projection π. The G-opers for other classical groups have a similar description.

We will show in Section 8.11 that positive P GL_{n+1}(ℝ)-opers on S^{1} come from convex curves in ℝℙ^{n}. So positive G(ℝ)-opers on real curves can be considered as a generalization of convex curves ℝℙ^{n} to the case of an arbitrary split semi-simple group G.

10. Semi-continuity of positive subsets of real flag varieties. Let φ : C → B(ℝ) be a positive map from a cyclically ordered set C. Changing the cyclic order of C to the opposite one we get a cyclically ordered set \overline{C}. By Theorem 5.12 the map φ, considered as a map from \overline{C}, provides a positive map \overline{φ} : \overline{C} → B(ℝ).

The group G(ℝ) acts on the set of maps from C to B(ℝ). If a map φ : C → B(ℝ) is positive then for any g ∈ G(ℝ) the map gφ is also positive. The G(ℝ)-orbits on the set of positive maps are called positive configurations of flags in B(ℝ).

Definition 6.16 Let C be a cyclically ordered set. A map φ : C → B(ℝ) is semi-continuous from the left if for any sequence of distinct points c_{0}, ..., c_{n}, ... in C, whose order is compatible with the cyclic order of C, the limit lim_{n→∞} φ(c_{n}) exists.

A positive configuration in B(ℝ) is semi-continuous from the left if any map φ : C → B(ℝ) representing it is semi-continuous from the left.

Changing the cyclic order to the opposite one we define the maps semi-continuous from the right.

Theorem 6.17 Any positive map φ : C → B(ℝ) is semi-continuous from the left and from the right.

Proof. Set B_{i} := φ(c_{i}). The positivity of φ means that for every cyclically ordered quadruple i < j < k < l < i the configuration of flags (B_{i}, B_{j}, B_{k}, B_{l}) is positive. Any positive configuration of 4 flags is equivalent to a configuration

(B^{−}, v · B^{−}, vu · B^{−}, B^{+}), v, u ∈ U^{−}(ℝ_{>0})  (78)

Lemma 6.18 The 4-tuple of flags (78) representing a positive configuration is well defined up to the action of the positive tori H(ℝ_{>0}). A similar statement is valid for the triple (B^{−}, u · B^{−}, B^{+}).

Proof. Suppose we have a 4-tuple of flags (B^{−}, v′ · B^{−}, v′u′ · B^{−}, B^{+}) representing the same configuration, with v′, u′ ∈ U(ℝ_{>0}). Then there exists an element of H(ℝ) conjugating one to the other. The projection of u to the maximal abelian quotient U^{+}/[U^{+}.U^{+}] has natural positive coordinates u_{α} parametrised by the simple roots α. The element t ∈ H(ℝ) acts on u_{α} by multiplication on χ_{α}(t). Thus χ_{α}(t) > 0 for every simple root α. Since G has trivial center, this implies that t ∈ H(ℝ_{>0}). The lemma is proved.

It follows that the sequence of flags (B_{2}, B_{3}, B_{4}, ..., B_{0}, B_{1}) can be written as

(B^{−}, v_{3} · B^{−}, v_{4} · B^{−}, ..., v_{n} · B^{−}, ..., v_{0} · B^{−}, B^{+}); v_{k} = u_{3}u_{4}...u_{k}, u_{i} ∈ U^{+}(ℝ_{>0}); v_{0} ∈ U^{+}(ℝ_{>0})

(79)

Moreover we may assume that for every n ≥ 3 one has v_{0} = v_{n}v′_{n} with v′_{n} ∈ U^{+}(ℝ_{>0}).

To proceed further we need the following fact. There exists a finite dimensional representation ρ of G(ℝ) and a basis in it such that the elements of U(ℝ_{>0}) act in this basis by unipotent upper triangular matrices with positive entries above the diagonal, and the map ρ provides an isomorphism of the flag variety B(ℝ) onto its image. Indeed, take the adjoint representation. It follows from [L1] that the canonical basis in any finite dimensional representation has this property.

Then the values of each of the matrix elements in the sequence ρ(v_{i}), i = 3, 4, ..., provide a monotonously increasing sequence of real numbers bounded by the value of the corresponding matrix element of ρ(v_{0}). Thus there exists the limit lim_{i→∞} ρ(v_{i}). Since the map B(ℝ) → ρ(B(ℝ)) is an isomorphism, and the image is closed, the sequence of Borel subgroups {B_{i}} has a limit. The Theorem is proved.

11. Concluding remarks: quantum universal Teichmüller spaces and W -algebras. The space of G(ℝ)-opers on the parametrised circle S^{1} has a natural Poisson structure, the Gelfand-Dikii bracket, defined by Drinfeld and Sokolov in [DS]. On the other hand, since P^{1}(ℚ) is a subset S^{1}, a positive G(ℝ)-oper provides a point of the universal Teichmüller space X_{G}^{+}. The universal Teichmüller space X_{G}^{+} has a natural Poisson structure. For instance if G = P GL_{m} it is defined in the canonical coordinates by the function ε_{pq} from Definition 1.16 applied to the m-triangulation of the Farey triangulation. One can show that the Gelfand-Dikii bracket is compatible with the Poisson bracket on X_{G}^{+}. The Poisson structure X_{G}^{+} admits a natural quantization equivariant under the action of the Thompson group (see [FG3] the exposition of the G = P GL_{3} case). Therefore we suggest that the quantum universal Teichmüller space can be considered as a combinatorial version of the W -algebra corresponding to G, divided by G.

7 A positive structure on the moduli space A_{G,Ŝ}

In this section we assume that G is a simply connected and connected.

We define a positive structure on the moduli space A_{G,Ŝ}. In particular if Ŝ is a disc with n marked points at the boundary we get a positive structure on the variety Conf_{n}(A) of configurations of n affine flags in generic position in G. We start with a definition of a positive structure on the varieties Conf_{n}(A) for n = 2 and n = 3. The definition in general is given by using an ideal triangulation of a hyperbolic marked surface Ŝ. We show that it does not depend on the choice of an ideal triangulation.

1. Generic configurations of pairs of affine flags. We say that a pair of affine flags is in generic position if the underlying pair of flags is in generic position. An n-tuple of affine flags is said to be in generic position if every two of them are in generic position. Denote by Conf_{n}(A) the variety of configurations of n affine flags in generic position in G. Similarly one defines a generic pair (A_{1}, B_{2}) where A_{1} is an affine flag and B_{2} is a flag. The generic pairs (A_{1}, B_{2}) form a principal homogeneous G–space.

Recall the element \overline{w}_{0} ∈ N(H). Recall that [g]_{0} := h if g = u_{−}hu_{+}, where u_{±} ∈ U^{±}, h ∈ H. Let (A_{1}, A_{2}) = (g_{1}U^{−}, g_{2}\overline{w}_{0}U^{−}) ∈ Conf_{2}(A). We set

α_{2}(g_{1}U^{−}, g_{2}\overline{w}_{0}U^{−}) := [g_{1}^{−1}g_{2}]_{0} ∈ H

We get a map α_{2} : Conf_{2}(A)→H, (A_{1}, A_{2}) ⟼ α_{2}(A_{1}, A_{2}).

Recall that the Cartan group H acts naturally from the right on the affine flag variety: A ⟼ A·h. Since H is commutative we can also consider it as a left action.

Proposition 7.1 The map α_{2} provides an isomorphism

α_{2} : Conf_{2}(A) ⟶^{∼} H, (A_{1}, A_{2}) ⟼ α_{2}(A_{1}, A_{2})  (80)

It has the following properties:

α_{2}(A_{1} · h_{1}, A_{2} · h_{2}) = h_{1}^{−1}w_{0}(h_{2}) · α_{2}(A_{1}, A_{2}),  α_{2}(A_{2}, A_{1}) = s_{G} · w_{0} (α_{2}(A_{1}, A_{2})^{−1}) , (81)

Proof. Bruhat decomposition shows that an arbitrary generic pair of affine flags (A_{1}, A_{2}) can be written uniquely as (h · U^{−}, \overline{w}_{0}U^{−}). This provides the first claim of the proposition. To check the first of the properties (81) notice that α_{2}(h_{1}U^{−}, \overline{w}_{0}h_{2}U^{−}) = h_{1}^{−1}w_{0}(h_{2}). The second of the properties (81) follows immediately from this one. The proposition is proved.

Observe that since i_{B_{1}} = w_{0}i_{B_{2}}w_{0}^{−1} one has (A_{1} · h, A_{2} · w_{0}(h)) = h(A_{1}, A_{2}).

Recall the twisted cyclic shift map from the Definition 2.8.

Lemma 7.2 The twisted cyclic shift map is a positive automorphism of Conf_{2}(A).

Proof. Follows immediately from (81).

Remark. If s_{G} ≠ e then the cyclic shift is not a positive automorphism of Conf_{2}(A).

2. A positive structure on Conf_{3}(A). Let us define a map

γ : U_{∗}^{+} ⟶ H, u^{+} ⟼ [u^{+}\overline{w}_{0}]_{0} ∈ H  (82)

Lemma 7.3 The map γ is a positive map

Proof. Using the isomorphism 𝔾_{m}^{I} = H provided by ω_{1}, ..., ω_{r} we have γ(u_{+}) := {Δ_{ω_{i},w_{0}(ω_{i})}}_{i∈I} ∈ 𝔾_{m}^{I}. It follows from Theorem 5.4 that it is a positive map. The lemma is proved.

Consider the regular open embeddings

β_{1}^{−} : B^{−} ↪ G/U^{−}, b_{−} ⟼ b_{−}\overline{\overline{w}}_{0}U^{−}
β_{2}^{−} : B^{+} ↪ G/U^{−}, b_{+} ⟼ b_{+}U^{−}

The Borel subgroups B^{−} and B^{+} have the standard positive structures. They provide G/U^{−} with two positive structures. Similarly the regular open embeddings

β_{1}^{+} : B^{+} ↪ G/U^{+}, b_{+} ⟼ b_{+}\overline{w}_{0}U^{+}
β_{2}^{+} : B^{−} ↪ G/U^{+}, b_{−} ⟼ b_{−}U^{+}

provide G/U^{+} with two positive structures. Let Φ : B^{−} → B^{+} be a rational map such that β_{2}^{+} = β_{1}^{+}Φ on their common domain of the definition.

Lemma 7.4 a) The two positive structures on G/U^{−} provided by the maps β_{1}^{−}, β_{2}^{−} are compatible. b) The two positive structures on G/U^{+} provided by the maps β_{1}^{+} and β_{2}^{+} are compatible.

c) The maps Φ and Φ^{−1} are positive rational maps.

Proof. a) <=> b). Indeed, consider the involutive automorphism θ : G ⟼ G which is uniquely determined by

θ(h) := h^{−1}, θ(x_{i}(t)) := y_{i}(t), θ(y_{i}(t)) := x_{i}(t)

Observe that θ(\overline{\overline{w}}_{0}) = \overline{w}_{0} and θ(\overline{w}_{0}) = \overline{\overline{w}}_{0}. Applying θ we immediately deduce b) from a) since θ(B^{+}) = B^{−}, θ(B^{−}) = B^{+}, θ(\overline{\overline{w}}_{0}) = \overline{w}_{0}. Similarly b) implies a).

Let us prove b). Recall that b_{+}\overline{w}_{0}U^{+} = b_{−}U^{+}. We have to show that the maps b_{+} → b_{−} and b_{−} → b_{+} are positive. Consider the first map. Write b_{+} = h_{+}u_{+}, b_{−} = h_{−}u_{−}. We may assume without loss of generality that h_{+} = 1. Then h_{−} = γ(u_{+}), so it is a positive map by Lemma 7.3. The map u_{+} ⟼ u_{−} is the map ϕ^{−1}, and hence it is positive. The claim is proved.

c) Follows from b). The lemma is proved.

3. Positive configurations of affine flags. Let (A_{1}, A_{2}, A_{3}) be a generic triple of affine flags in G. Denote by B_{i} the flag corresponding to A_{i}. So every two of the flags B_{1}, B_{2}, B_{3} are in generic position. Let Conf(A_{1}, B_{2}, B_{3}) be the configuration space of generic triples (A_{1}, B_{2}, B_{3}). Then there is an isomorphism

Conf(A_{1}, B_{2}, B_{3}) ⟶^{∼} U_{∗}^{+}  (83)

Indeed, let (U^{−}, w_{0}B^{−}) be a standard pair. Then there exists unique u ∈ U_{∗}^{+} such that

(A_{1}, B_{2}, B_{3}) ∼ (U^{−}, w_{0}B^{−}, u_{+}B^{−});  u_{+} ∈ U_{∗}^{+}  (84)

Lemma 7.5 There is a canonical isomorphism

α_{3} : Conf_{3}(A) ⟶^{∼} Conf_{2}(A) × Conf_{2}(A) × Conf(A_{1}, B_{2}, B_{3})  (85)
(A_{1}, A_{2}, A_{3}) ⟼ (A_{1}, A_{2}) × (A_{2}, A_{3}) × (A_{1}, B_{2}, B_{3})

Proof. The affine flags A_{2} and A_{3} are determined by the elements α_{2}(A_{1}, A_{2}) and α_{2}(A_{1}, A_{3}) in H. The lemma is proved.

The isomorphism α_{3} combined with the isomorphisms (80) and (83) leads to an isomorphism

α_{3} : Conf_{3}(A) ⟶^{∼} H × H × U_{∗}^{+},  (86)

Lemma 7.6 The inverse to the isomorphism α_{3} is defined as follows:

α3^{′} : H × H × U_{∗}^{+} ⟶ Conf_{3}(A), (h_{2}, h_{3}, u_{+}) ⟼ (U^{−}, h_{2}\overline{w}_{0}U^{−}, h_{2}w_{0}(h_{3})s_{G}u_{+}U^{−})

Proof. Consider the natural projection

e_{ij} : Conf_{3}(A) ⟶ H; (A_{1}, A_{2}, A_{3}) ⟼ (A_{i}, A_{j}), 1 ≤ i < j ≤ 3

It follows from the definition of the map α_{2} that e_{12}α3^{′} is the projection on the first H-factor. Let us check that e_{23}α3^{′} is the projection on the second H-factor. One has

(h_{2}\overline{w}_{0}U^{−}, h_{2}w_{0}(h_{3})s_{G}u_{+}U^{−}) ∼ (U^{−}, h_{3}s_{G}\overline{w}_{0}^{−1}u_{+}\overline{w}_{0}\overline{w}_{0}^{−1}U^{−}) = (U^{−}, h_{3}\overline{w}_{0}^{−1}(u_{+})\overline{w}_{0}U^{−})

Since \overline{w}_{0}^{−1}(u_{+}) := \overline{w}_{0}^{−1}u_{+}\overline{w} ∈ U^{−}, the statement follows. The similar claim about the U_{∗}^{+}-factor is obvious. The lemma is proved.

Definition 7.7 A positive regular structure on the variety Conf_{3}(A) is given by the isomorphism (86) and the standard positive regular structures on U_{∗}^{+} and H.

Lemma 7.8 The map e_{13} : Conf_{3}(A) ⟶ H is a positive rational map.

Proof. Recall the map γ, see (82). Using the isomorphism (86) we have

e_{13}α′_{3}(h_{2}, h_{3}, u_{+}) = (U^{−}, h_{2}w_{0}(h_{3})s_{G}u_{+}U^{−}) = (U^{−}, h_{2}w_{0}(h_{3})u_{+}\overline{w}_{0}\overline{w}_{0}U^{−})

Comparing this with (35) we observe that this is equal to h_{2}w_{0}(h_{3})γ(u_{+}). The lemma is proved.

Proposition 7.9 The cyclic shift is a positive rational map on the variety Conf_{3}(A).

Proof. To simplify the calculation we may assume without loss of generality that h_{2} = h_{3} = e. So we start from the configuration (U^{−}, \overline{w}_{0}U^{−}, u_{+}s_{G}U^{−}). Multiplying from the left by u_{+}^{−1} and then using \overline{w}_{0}^{−1}u_{+}^{−1}\overline{w}_{0} ∈ U^{−} we get

(U^{−}, \overline{w}_{0}U^{−}, u_{+}s_{G}U^{−}) ∼ (u_{+}^{−1}U^{−}, \overline{w}_{0}U^{−}, s_{G}U^{−})

Applying the inversion map g ⟼ g^{−1} we get

(U^{−}u_{+}, U^{−} \overline{w}_{0}^{−1}, U^{−}s_{G})

Applying the antiautomorphism Ψ we get

(u_{−}U^{+}, Ψ(\overline{w}_{0}^{−1})U^{+}, s_{G}U^{+}), u_{−} := Ψ(u_{+})

Since Ψ(\overline{w}_{0}^{−1})^{−1}u_{−}Ψ(\overline{w}_{0}^{−1}) ∈ U^{+}, this configuration can be written as

(u_{−}U^{+}, Ψ(\overline{w}_{0}^{−1})U^{+}, s_{G}U^{+})

According to Lemma 7.4b) one has u_{−}U^{+} = v_{+}h\overline{w}_{0}U^{+}. Here v_{+}h = Φ(u_{−}) and v_{+} = ϕ(u_{−}). The map Φ is a positive map. Thus, since v_{+}^{−1}U^{+} = U^{+}, we can write our configuration as

(v_{+}h\overline{w}_{0}U^{+}, Ψ(\overline{w}_{0}^{−1})U^{+}, s_{G}U^{+}) ∼ (h\overline{w}_{0}U^{+}, v_{+}^{−1}Ψ(\overline{w}_{0}^{−1})U^{+}, s_{G}U^{+})

Applying the inversion map we write it as

(U^{+}\overline{w}_{0}^{−1}h^{−1}, U^{+}Ψ(\overline{w}_{0})v_{+}, U^{+}s_{G})

Then applying Ψ we get

(h^{−1}Ψ(\overline{w}_{0}^{−1})U^{−}, Ψ(v_{+})\overline{w}_{0}U^{−}, s_{G}U^{−})

Applying Lemma7.4a) we can write Ψ(v_{+})\overline{\overline{w}}_{0}U^{−} = h′ϕΨ(v_{+})U^{−}, where the map Ψ(v_{+}) ⟼ h′ is positive. So we get

(h^{−1}\overline{w}_{0}U^{−}, s_{G}h′(ϕΨ)^{2}(u_{+})U^{−}, s_{G}U^{−})

We conclude that

(U^{−}, \overline{w}_{0}U^{−}, u_{+}s_{G}U^{−}) ∼ (h^{−1}\overline{w}_{0}U^{−}, s_{G}h′(ϕΨ)^{2}(u_{+})U^{−}, s_{G}U^{−})  (87)

Since the maps ϕΨ, u_{+} ⟼ h^{−1} and u_{+} ⟼ h′ are positive, formula (87) implies that the twisted cyclic shift is a positive map on Conf_{3}(A). The proposition is proved.

Let us consider the following a regular open embedding map

H^{3} × U_{∗}^{+} × U_{∗}^{−} ↪ Conf′_{4}(A)
(h_{2}, h_{3}, h_{4}, u_{+}, v_{−}) ⟼ (U^{−}, u_{+}^{−1}h_{2}U^{−}, h_{3}\overline{w}_{0}U^{−}, v_{−}h_{4}\overline{w}_{0}U^{−})  (88)

Here Conf′_{4}(A) is the moduli space of configurations of four flags (B_{1}, ..., B_{4}) such that all flags except the pair (B_{2}, B_{4}) are in generic position.

We are going to define a positive structure on the variety Conf_{4}(A) by using the birational map

H^{3} × U_{∗}^{+} × U_{∗}^{−} ⟶ Conf_{4}(A)  (89)

and the standard positive structures on H, U_{∗}^{−} and U_{∗}^{−}. We have to check however that this map is defined on a complement to a positive rational divisor.

Proposition 7.10 a) The map (89) is defined on a complement to a positive rational divisor. b) The twisted cyclic shift provides a positive automorphism of the variety Conf_{4}(A).

Proof. Let us define a rational map

U^{+} × U^{−} × H × H ⟶ U^{−} × U^{+} × H, (u_{+}, v_{−}, h_{2}, h_{4}) ⟼ (a_{−}, a_{+}, t)  (90)

by solving the equation h_{2}^{−1}u_{+}h_{4}v_{−} = a_{−}a_{+}t. It is a positive rational map. Let us set

g := a_{−}^{−1}h_{2}^{−1}u_{+} = a_{+}th_{4}^{−1}v_{−}^{−1}

Then multiplying the configuration (88) by g from the left we get

(a_{+}th_{4}^{−1}U^{−}, U^{−}, a_{−}^{−1}h_{2}^{−1}h_{3}\overline{w}_{0}U^{−}, t\overline{w}_{0}U^{−})

Applying to it the twisted cyclic shift we get

(U^{−}, a_{−}^{−1}h_{2}^{−1}h_{3}\overline{w}_{0}U^{−}, t\overline{w}_{0}U^{−}, a_{+}th_{4}^{−1}s_{G}U^{−})  (91)

We can write the configuration (91) in the form of (88):

(91) = (U^{−}, u_{+}^{−1}h̃_{2}U^{−}, h̃_{3}\overline{w}_{0}U^{−}, ṽ_{−}h̃_{4}\overline{w}_{0}U^{−})

In other words, one must have

i) a_{−}^{−1}h_{2}^{−1}h_{3}\overline{w}_{0}U^{−} = u_{+}^{−1}h̃_{2}U^{−}, ii) t = h̃_{3}, iii) a_{+}th_{4}^{−1}s_{G}U^{−} = ṽ_{−}h̃_{4}\overline{w}_{0}U^{−}  (92)

Lemma 7.11 The rational map (h_{2}, h_{3}, h_{4}, u_{+}, v_{−}) ⟼ (h̃_{2}, h̃_{3}, h̃_{4}, ũ_{+}, ṽ_{−}) provided by equations (92) is a positive rational map.

Proof. For h̃_{3} and the equation ii) this follows from the positivity of the map (90). For the equation iii) and ṽ_{−}, h̃_{4} this follows immediately from Lemma 7.4a) since s_{G}\overline{w}_{0} = \overline{\overline{w}}_{0}. To handle i) means to show that the rational map B^{−} → B^{+}, x_{−} ⟼ x_{+} determined from the equation x_{−}^{−1}\overline{w}_{0}U^{−} = x_{+}^{−1}U^{−} is a positive rational map. This equation is equivalent to x_{+} = ΨΦΨ(x_{−}). Indeed, applying the inversion antiautomorphism to this equation we get U^{−}\overline{\overline{w}}_{0}x_{−} = U^{−}x_{+}. Then applying Ψ we get Ψ(x_{−})\overline{w}_{0}U^{+} = Ψ(x_{+})U^{+}. So Ψ(x_{+}) = Φ^{−1}Ψ(x_{−}), and hence x_{+} = ΨΦ^{−1}Ψ(x_{−}). By Lemma 7.4 the rational map ΨΦ^{−1}Ψ : B^{+} → B^{−} is positive. So the lemma follows. Both statements of the proposition follows immediately from this lemma. The proposition is proved.

Let us consider the canonical projection (the edge projection)

e = (e_{12}, e_{23}, e_{13}) : Conf_{3}(A) ⟶ H × H × H  (93)

It follows from Lemma 7.6 and proof of Lemma 7.8 that one has

eα′_{3}(h_{2}, h_{3}, u_{+}) = (h_{2}, h_{3}, h_{2}w_{0}(h_{3})γ(u_{+}))  (94)

Therefore the map e is surjective. Let us define the variety V_{G} as its fiber over (e, e, e):

V_{G} := e^{−1}(e, e, e) ⊂ Conf_{3}(A)  (95)

Proposition 7.12 a) The variety V_{G} has a natural positive structure.

b) The positive structure on the variety V_{G} is invariant under the cyclic shift map.

Proof. a) We deduce this from Theorem 5.4 applied to the double Bruhat cell G^{e,w_{0}}. Observe that G^{e,w_{0}} can be identified with the configurations space Conf(A_{1}, B_{2}, A_{3}). Given a reduced decomposition i of w_{0}, the collection of the generalized minors F (i) = {Δ(k; i), k ∈ [1, r]∪[1, l(w_{0})]} provides a positive coordinate system on G^{e,w_{0}}. The minors Δ(k; i), k ∈ [1, r] are the ones Δ_{ω_{i},w_{0}(ω_{i})}, so they give the map γ. On the other hand among these minors there are Δ_{ω_{i},ω_{i}} for every i ∈ I = {1, ..., r}. To see this write the sequence i = (i_{1}, ..., i_{l(w_{0})}). Then for each p ∈ {1, ..., r} take the very right among all i_{a}’s such that i_{a} = p. Denote it by i_{a(p)}. Then Δ(i_{a(p)}; i) = Δ_{ω_{p},ω_{p}}. The rest of the generalized minors, that is the minors

F′(i) = {Δ(k; i), k ∈ [1, l(w_{0})] − {a(1), ..., a(r)}}

give a positive coordinate system on V_{G}. The part a) is proved.

b) Follows easily from a) and Proposition 7.9. The proposition is proved.

4. A positive structure on the moduli space A_{G,Ŝ}. Let Ŝ be a marked hyperbolic surface. Choose an ideal triangulation T of Ŝ. Denote by A_{G,t̂} the moduli space of the twisted decorated G-local systems on the triangle t, considered as a disc with three marked points on the boundary. These marked points are located on the sides of the triangle, one point per each side. The punctured tangent space to a triangle t of the triangulation T sits in the punctured tangent space T′S. So restricting the local system L on T′S representing an element of A_{G,Ŝ} to T′t we get an element of A_{G,t̂}. So we get a projection q′_{t} : A_{G,Ŝ} ⟶ A_{G,t̂}. Similarly, each edge e of the triangulation T can be thickened a bit to became a disc with two distinguished points on its boundary provided by the vertices of e. They cut the boundary of the disc into two arcs. Let ê be this disc with two marked points on the boundary, one on each of these arcs.

(96)

The restriction provides a map q_{e} : A_{G,Ŝ} ⟶ A_{G,ê}. All together, they provide a map

φ_{T} : A_{G,Ŝ} ⟶ ∏_{t∈tr(T)} A_{G,t̂}× ∏_{e∈ed(T)} A_{G,ê}  (97)

However the target space of this map is too big. Indeed, let e be an edge of a triangle t. Then there is a natural projection q_{t̂,ê} : A_{G,t̂} → A_{G,ŝ}, and q′ ; t̂q_{t̂,ê} = qê. The subvariety in the right hand side of (97) defined by these conditions is the right target space. It contains the following open subset.

Definition 7.13 Let T be an ideal triangulation of Ŝ. The subvariety

A_{G,T} ⊂ ∏_{t∈tr(T)} { \atop Conf}_{3}A × ∏_{e∈ed(T)} { \atop Conf}_{2}A

is defined by the conditions q′ ; t̂q_{t̂,ê} = qê for every pair (e, t) where e is an edge of the triangle t of T.

Theorem 7.14 Let G be a split simply-connected semisimple algebraic group. Let Ŝ be a marked hyperbolic surface and T is an ideal triangulation of Ŝ. Then there exists a regular open embedding

ν_{T} : A_{G,T} ↪ A_{G,Ŝ}  (98)

such that φ_{T} ν_{T} is the identity.

Proof. Let us introduce a convenient way to think of local systems on the punctured tangent bundle T′S having the monodromy s_{G} around the circle T′_{x}S. Let Γ be a graph on S dual to an ideal triangulation T. Let us define a collection of points in T′S as follows. For every edge e of the graph Γ let us choose an internal point x(e) on this edge, and a pair of non zero tangent vectors v_{1}(e) and v_{2}(e) at this point looking in the opposite directions from e, as on the right Picture (99).

(99)

We are going to define a groupoid G_{Γ} with the objects given by the points {v_{i}(e)}, where e runs through all the edges of Γ and i ∈ {1, 2}, which is equivalent to the fundamental groupoid of T′S. Therefore local G(F)-systems on T′S can be understood as functors G_{Γ} → G(F). Here is a system of generators of the groupoid G_{Γ}.

1). In the torsor of paths in T′_{x(e)}S connecting the vectors v_{1}(e) and v_{2}(e) there are two distinguished elements, p_{+} and p_{−}, where p_{+} moves the vector v_{1}(e) clockwise, and p_{−} counterclockwise towards the vector v_{2}(e). By definition p_{±} ∈ Mor_{G_{Γ}}(v_{1}(e), v_{2}(e)).

2). Let v be a vertex of Γ. Choose a sufficiently small neighborhood D_{v} of v containing the three edges sharing v. Then the space of nonzero tangent vectors at the points of Γ ∩ D_{v} which are not tangent to Γ is a union of three connected simply connected components. Therefore if we denote by e_{1}, e_{2}, e_{3} the three edges sharing v, each connected component has exactly two of the six vectors v_{i}(e_{j}). By definition the unique path connecting each pair belongs to the morphisms in G_{Γ}.

The arcs on the left Picture (99) illustrate the six generators of the groupoid near a vertex of Γ. They form a hexagon H_{v}. The arcs illustrate how the tangent vectors move. So for instance the left ark on Picture (99) intersecting Γ moves the tangent vector looking up clockwise to the one looking down. Putting the hexagons corresponding to all vertices of Γ together we get the set of the generators of the groupoid G_{Γ}. On Picture (100) the objects of the groupoid G_{Γ} are pictured by fat points, and the generators by arcs.

(100)

Here are two simple but crucial observations about this picture.

Lemma 7.15 a) The composition of the six paths forming the hexagon H_{v} on the left picture in (99) amounts to rotation of a tangent vector by 4π.

b) The composition p_{+}p_{+} as well as p_{−}p_{−} amounts to rotation of a tangent vector by 2π.

Proof. A simple exercise.

Let us consider a triangle t of the triangulation T dual to a vertex v of Γ. We may assume it intersects the edges e_{j} of Γ sharing the vertex v at the chosen points x(e_{j}). Then there is a canonical homotopy class of path in T′S going from the tangent vectors v_{i}(e_{j}) along the corresponding side of the triangle t to proximity of the corresponding vertex of the triangle t, as on Picture (101).

(101)

Using this path and the decoration at this vertex of t we picking up an affine flag over the tangent vector v_{i}(e_{j}). Let q be a side of the hexagon H_{v} which which does not intersect Γ on Picture 100 (i.e. sits in one of the connected components discussed in 2)). Then the affine flags sitting at the endpoints of q are obtained one from the other by the parallel transport along q.

Lemma 7.15a) implies that the monodromy around the hexagon H_{v} near a vertex v is the identity. So the restriction of the local system L to the hexagon H_{v} is trivial. Therefore we can transform all the affine flags into a fiber over one point, getting a configuration of three affine flags (A_{1}, A_{2}, A_{3}) in G corresponding to the vertex v, as on Picture (102). The affine flags are parametrised by the vertices of the triangle t. Denote by B_{1}, B_{2}, B_{3} the corresponding flags. Then we can attach the pairs (A_{i}, B_{j}) to the vertices of the hexagon H_{v} as on Picture (102).

(102)

For each oriented edge a of the hexagon H_{v} corresponds an element g_{a} of G defined as follows. Consider a triple of flags (A_{1}, A_{2}, A_{3}) representing our configuration. Let (A_{i}, B_{j}) and (A_{i′}, B_{j′}) be the pairs assigned to the first and second vertices of the oriented edge a. There exists unique element g ∈ G transforming (A_{i}, B_{j}) to the standard pair (U^{−}, B^{+}). Set (A′_{1}, A′_{2}, A′_{3}) := (gA_{1}, gA_{2}, gA_{3}). We define g_{a} as the unique element of G such that transforming the pair (A′_{i′}, B′_{j′}) = g_{a}(U^{−}, B^{+}).

There are two types of edges of the graph on Picture (100): the ones intersecting the graph Γ on Picture (100), and the ones which do not intersect it. Each edge of the first type determines an edge of the triangulation T, and each edges of the second type determines a vertex of T. If p (resp. q) is an edge of the first (resp. second) type, then

g_{p} ∈ \overline{w}_{0}H, g_{q} ∈ U^{−}  (103)

Each edge of the graph on Picture (100) is assigned to a unique hexagon H_{v}. Applying the above construction to the oriented edges a of these hexagons we get a collection of elements {g_{a}} of G. It has the following properties:

i) Reversing orientation of an edge amounts to replacing the corresponding element by its inverse. ii) The product of the elements corresponding to a path around the hexagon is the identity. iii) The product of the elements corresponding to a simple path around a two-gon is s_{G}. iv) Let q (resp. p) be an oriented edge corresponding to a vertex (resp. edge) of the triangulation

T. Then one has (103)

By Lemma 7.15b) the monodromy around each of the two-gons on Picture (100) must be s_{G}. This agrees with iii).

We can invert this construction.

Lemma 7.16 Let {g_{a}} be an arbitrary collection of elements of G(F) assigned to oriented edges of the graph on Picture (100), which satisfy the conditions i) - iv). Then there exists a twisted decorated G(F)-local system on Ŝ such that the collection of elements g_{a} constructed for it coincide with the original collection.

Proof. Thanks to the properties i) - iii) there exists unique functor G_{Γ} → G(F) determined by the condition a ⟼ g_{a}. Here a_{1}a_{2} means that the path a_{1} follows a_{2}. It provides a twisted G(F)-local system on Ŝ. The condition iv) is used to define a decoration. The lemma is proved.

Recall that a point of the variety A_{G,T} is given by a collection of twisted cyclic configurations (A_{1}^{t}, A_{2}^{t}, A_{3}^{t}) of affine flags in generic position, one for each triangle t of T, compatible over the internal edges of T. Given such a configuration (A_{1}^{t}, A_{2}^{t}, A_{3}^{t}) attached to a vertex v of Γ we assign to the oriented edges of the hexagon H_{v} the elements g_{a} as above. They satisfy the conditions i), ii), iv). Finally, the compatibility over the edges implies the condition iii). It remains to apply the construction from Lemma 7.16. The theorem is proved.

It remains to define a positive structure on the variety A_{G,T} and prove that the positive structures corresponding to different ideal triangulations T are compatible.

Consider { \atop Conf}_{3}A as a subvariety of A_{G,t̂}. Then choosing a vertex of the triangle t, we have

{ \atop Conf}_{3}A = V_{G} × ({ \atop Conf}_{2}A)^{3}

Recall that a positive structure on V_{G} is given by the Proposition 7.12. The positive coordinate systems are parametrized by reduced decompositions i of the element w_{0}. A positive structure on { \atop Conf}_{2}A is given by Proposition 7.1 and Lemma 7.2. This suggests the following definition.

Definition 7.17 Denote by T the following data on Ŝ:

i) An ideal triangulation T of a marked hyperbolic surface Ŝ.

ii) For each triangle t of the triangulation T a choice of a vertex of this triangle.

iii) A reduced decomposition i of the maximal length element w_{0} ∈ W.

So given such a data T and using Theorem 7.14 we have an open embedding

ν_{T} : +V_{G}^{{triangles of T }} × H^{{edges of T }} ↪ A_{G,Ŝ}  (104)

Theorem 7.18 Let G be a split simply-connected semisimple algebraic group. Let Ŝ be a marked hyperbolic surface. Then the collection of regular open embeddings {ν_{T}}, when T runs through all the datas from the Definition 7.17 assigned to Ŝ, provides a Γ_{S}–equivariant positive structure on A_{G,Ŝ}.

Proof. We have to check that we get a compatible positive structure by changing any of the three components in the data T from the definition 7.17. For iii) this follows from Lusztig’s theory [L1]. For ii) this follows from Proposition 7.9. To prove the statement for for i) we need to check that a flip as on Figure 21 provides a positive transformation. This follows from the Proposition 7.10. The theorem is proved.

Figure 21: Flip

Definition 7.19 Let Ŝ be a hyperbolic marked surface and G a split simply-connected semisimple algebraic group.

a) The decorated Teichmüller space A_{G,Ŝ}^{+} is the set of all ℝ_{>0}-points of the positive space A_{G,Ŝ}.

b) Let 𝔸^{t} be one of the tropical semifields ℤ^{t}, ℚ^{t}, ℝ^{t}. The set A_{G,Ŝ}(𝔸^{t}) of points of the positive variety A_{G,Ŝ} with the values in the tropical semifield 𝔸^{t} is called the set of A^{𝔸}-laminations on Ŝ corresponding to G.

The space of real A-laminations serves as the Thurston boundary of the Teichmüller space A_{G,Ŝ}^{+}.

The decorated Teichmüller space A_{G,D̂_{n}}^{+} is identified with the space of positive twisted cyclic

configurations of n affine flags in G(ℝ) on a disc D̂_{n} with n marked points on the boundary.

Corollary 7.20 The space A_{G,D̂_{n}}^{+} is invariant under the twisted cyclic shift.

5. The universal Teichmüller A-space for G. Let ℙ̃^{1}(ℝ) be a two-sheeted covering of ℙ^{1}(ℝ), and ℙ̃^{1}(ℚ) the subset of its rational points. Let s be the nontrivial automorphism of the covering.

Let {p_{1}, ..., p_{n}} be a cyclically ordered subset of ℙ^{1}(ℚ), i.e. its order is compatible with one of the orientations of ℙ^{1}(ℝ). Then a choice of an initial point p_{1} plus its lift p̃_{1} determines a lift {p̃_{1}, ..., p̃_{n}} of the set {p_{1}, ..., p_{n}} to ℙ̃^{1}(ℚ): we lift an oriented loop started at p_{1}, whose orientation agrees with the cyclic order of the points p_{i}, to a path starting at p̃_{1}, and lift the points p_{1}, ..., p_{n} using this path. We say {p̃_{1}, ..., p̃_{n}} is a coherent lift of the cyclically ordered set {p_{1}, ..., p_{n}}.

Definition 7.21 Let G be a split semisimple simply-connected algebraic group. A map

α : ℙ̃^{1}(ℚ) ⟶ A(ℝ)  such that α(sp) = s_{G}α(p)  (105)

is positive if for any cyclically ordered n-tuple of points {p_{1}, ..., p_{n}} on ℙ^{1}(ℚ), and a coherent lift {p̃_{1}, ..., p̃_{n}} of these points, the configuration of affine flags (α(p̃_{1}), ..., α(p̃_{n})) is positive.

The universal decorated Teichmüller space A_{G}^{+} is the quotient of the space of positive maps (105) by the action of the group G(ℝ).

Observe that if the configuration of affine flags (α(p̃_{1}), ..., α(p̃_{n})) is positive for a certain coherent lift of the cyclically ordered set {p_{1}, ..., p_{n}}, it is positive for any such a coherent lift. So the positivity is a property of {p_{1}, ..., p_{n}}. Further, it follows from the main results of this Section that is sufficient to check the positivity for n = 2, 3 only.

Recall the positive space V_{G}, see (95). Let V_{G}^{+} be its positive part. Identifying ℙ^{1}(ℚ) with vertices of the Farey triangulation, we have a decomposition theorem for the universal Teichmüller space A_{G}^{+}:

Theorem 7.22 There exists a canonical isomorphism

A_{G}^{+} = V_{G}^{+}^{{Farey triangles}} × H(ℝ_{>0})^{{Farey diagonals}}

To fix an isomorphism in this theorem, we have to choose a vertex for each of the Farey triangles.

The higher Teichmüller spaces A_{G,S}^{+} are embedded into the universal one just in the same fashion as for the X -space.

8 Special coordinates on the A and X spaces when G is of type A_{m}

Let Γ be a marked trivalent ribbon graph. Then there are varieties X_{G,Γ} and A_{G,Γ} parametrising framed and decorated unipotent G-local systems on Γ. If Γ is of type Ŝ, they are isomorphic to X_{G,Ŝ} and A_{G,Ŝ}. The corresponding isomorphisms depend on a choice of an isotopy class of embedding of Γ to Ŝ. They form a principal homogeneous space over the mapping class group.

In this chapter we introduce some natural functions {Δ_{i}} and {X_{j}} on the varieties A_{SL_{m},Γ} and X_{PGL_{m},Γ}. The combinatorial data parametrising these functions has been described already in the Section 1.5, and will be briefly recalled below.

1. The sets I_{m}^{Γ} and J_{m}^{Γ} parametrising the canonical coordinates. Let Γ be a marked trivalent graph on S of type Ŝ. Then there is an ideal triangulation T_{Γ} of Ŝ dual to Γ. It is determined up to an isomorphism by the ribbon structure of Γ. Precisely, let Γ_{1}, Γ_{2} be two graphs as above such that the corresponding marked ribbon graphs \overline{Γ}_{1} and \overline{Γ}_{2} are isomorphic. Then an isomorphism \overline{Γ}_{1} → \overline{Γ}_{2} determines uniquely an element g of the mapping class group of S such that gΓ_{1} is isotopic to Γ_{2}, and hence gT_{Γ_{1}} is isotopic to T_{Γ_{2}}.

Summarizing, a marked ribbon trivalent graph Γ of type Ŝ determines an ideal triangulation T = T_{Γ} of a marked surface Ŝ_{Γ} isomorphic to Ŝ. The isotopy class of an isomorphism Ŝ_{Γ} → Ŝ is determined by an isotopy class of an embedded graph Γ. In this chapter we work with the marked ribbon trivalent graph Γ, the corresponding triangulation T = T_{Γ}, and surface Ŝ = Ŝ_{Γ}. Recall the m-triangulation of an ideal triangulation T_{Γ} of the surface Ŝ_{Γ} defined in the Section 1.5. It provides the oriented graph described there and denoted T_{m}. Recall the two sets, I_{m}^{T} and J_{m}^{T}, attached to the triangulation. Since the triangulation T and the corresponding graph Γ determine each other, we can use the upper script Γ in the notations. So

I_{m}^{Γ} := {vertices of the m–triangulation of T_{Γ}} - {vertices at the punctures of S}  (106)
J_{m}^{Γ} := I_{m}^{Γ} − { the vertices at the boundary of S }

Remark. If Γ ⟼ Γ′ is a flip then there is canonical bijection I_{m}^{Γ} ⟶ I_{m}^{Γ′}. However the oriented graphs T_{m} and T′_{m} are different.

Examples. 1. The set J_{2}^{Γ} is identified with the set of all internal (= not ends) edges of Γ.

2. Ŝ = S if and only if I_{m}^{Γ} = J_{m}^{Γ} is the same set.

3. The picture below illustrates the 3-triangulation of a disk with 5 marked points on the boundary. The disk is given by a pentagon. The ideal triangulation of the disk is given by triangulation of the pentagon. The boldface points illustrate the elements of the set I_{3}^{Γ}. Adding to them the points visualized by little circles we get the set J_{3}^{Γ}. So |I_{3}^{Γ}| = 7 and |J_{3}^{Γ}| = 17.

A parametrisation of the sets I_{m}^{Γ} and J_{m}^{Γ}. Choose a vertex v of Γ. It is dual to a triangle of the triangulation T. The inner vertices of the m–triangulation of this triangle are described by triples of nonnegative integers with sum m

(a, b, c),  a + b + c = m, a, b, c ∈ ℤ_{≥0}  (107)

such that at least two of the numbers a, b, c are different from zero.

We say that a triple (a, b, c) is of vertex type if a, b, c > 0, and picture the integers a, b, c at the pieces of faces of Γ sharing the vertex v:

Figure 22: A triple (a, b, c) of vertex type

If one (and hence the only one) of the integers a, b, c is zero, the other two determine an edge e containing v. In this case we say that the triple (a, b, c) is of edge type:

Figure 23: A pair (a, b) of the edge type

The vertex/edge type elements correspond to the vertices of the m–triangulation located inside of the triangles/edges of the dual triangulation Γ̂.

2. The regular functions Δ_{i} on the space A_{SL_{m},Γ}. They are parametrised by i ∈ I_{m}^{Γ}.

We use the following notation. Let V be a vector space of dimension n and ω ∈ detV^{∗} a volume form in V. Let v_{1}, ..., v_{n} be vectors of V. Set

Δ(v_{1}, ..., v_{n}) := Δ_{ω}(v_{1}, ..., v_{n}) :=< v_{1} ∧ ... ∧ v_{n}, ω >

Let i ∈ I_{m}^{Γ}. Then there is a triangle t_{v} of the triangulation T containing i, dual to a vertex v of Γ. This triangle is unique if i is of vertex type. Otherwise there are two such triangles.

The vertex functions. Let L_{v} be the fiber of the local system L at a vertex v. Then, by definition, it carries a monodromy invariant volume form ω. There are three monodromy operators M_{v,α}, M_{v,β} and M_{v,Γ} acting on L_{v}. They correspond to the three face paths α, β, Γ starting at v. Given a decorated SL_{m}–local system L on Γ we get three affine flags

X = (x_{1}, ..., x_{m}); Y = (y_{1}, ..., y_{m}); Z = (z_{1}, ..., z_{m})

at L_{v}. Each of the flags X, Y and Z is invariant under the corresponding monodromy operator:

M_{v,α}(X) = X, M_{v,β}(Y) = Y, M_{v,Γ}(Z) = Z

Recall that i is parametrised by a solution (a, b, c) of (107), and that the numbers (a, b, c) match the flags X, Y, Z. Put x_{(a)} := x_{1} ∧ ... ∧ x_{a}. The vertex function Δ_{i} = Δ_{a,b,c}^{v} is defined by

Δ_{i}(L) := Δ_{a,b,c}^{v}(X, Y, Z) := Δ_{ω}(x_{(a)} ∧ y_{(b)} ∧ z_{(c)}) = Δ_{ω}(x_{1} ∧ ... ∧ x_{a} ∧ y_{1} ∧ ... ∧ y_{b} ∧ z_{1} ∧ ... ∧ z_{c})

Figure 24: The data giving rise to a vertex function

The edge functions. Let L_{e} be the fiber of the local system L at an edge e. The SL_{m}–structure provides a volume form ω in L_{e}. There are two affine flags

X = (x_{1}, ..., x_{m}); Y = (y_{1}, ..., y_{m})

at L_{e} invariant under the monodromies around the two face paths α and β sharing the edge e: M_{v,α}(X) = X, M_{v,β}(Y) = Y. We picture each of the flags located at the corresponding face. We define the edge functions by

Δ_{i}(L) = Δ_{a,b}^{e}(X, Y) := Δ_{ω}(x_{(a)} ∧ y_{(b)}) = Δ_{ω}(x_{1} ∧ ... ∧ x_{a} ∧ y_{1} ∧ ... ∧ y_{b})

3. The rational functions X_{j} on the space X_{P GL_{m},Γ}. Let i ∈ J_{m}^{Γ}. Take all the vertices j connected with i by an edge in the graph T_{m}. Then there is one (if i of vertex type) or two (if i is of

Figure 25: The data giving rise to an edge function

edge type) triangles of T containing all these j’s. Consider the invariant flags attached to the vertices of these triangles. There are three such flags if i is vertex type, and four if it is of the edge type, see the pictures below. We can arrange these flags to sit in the same vector space L_{v} (respectively L_{e}). For each of these flags choose an affine flag dominating it. We denote by F̃ the affine flag dominating flag F. Then for each j as above we have the Δ_{j}–coordinate calculated for these affine flags. We denote it Δ̃_{j}. We set

X_{i} = ∏_{j∈I_{m}^{Γ}}(Δ̃_{j})^{ε_{ij}}  (108)

The right hand side evidently does not depend on the choice of dominating affine flags. Below we work out this definition for the vertex and edge type i’s.

The vertex functions X_{a,b,c}^{v}. They are given by

X_{a,b,c}^{v}(X, Y, Z) := −\frac{Δ_{a−1,b+1,c}(X̃, Ỹ, Z̃)Δ_{a,b−1,c+1}(X̃, Ỹ, Z̃)Δ_{a+1,b,c−1}(X̃, Ỹ, Z̃)}{Δ_{a+1,b−1,c}(X̃, Ỹ, Z̃)Δ_{a,b+1,c−1}(X̃, Ỹ, Z̃)Δ_{a−1,b,c+1}(X̃, Ỹ, Z̃)}

The structure of this formula is illustrated on the picture:

Figure 26: The vertex function

The edge functions X_{a,b}^{e}. They are given by

X_{a,b}^{e}(X̃, Ỹ, Z̃, T̃) := −\frac{Δ_{a,b−1,1}(X̃, Z̃, T̃)Δ_{a−1,1,b}(X̃, Ỹ, Z̃)}{Δ_{a−1,b,1}(X̃, Z̃, T̃)Δ_{a,1,b−1}(X̃, Ỹ, Z̃)}

The structure of this formula is illustrated on the picture:

Recall the canonical projection

p : A_{SL_{n},Ŝ} ⟶ X_{P GL_{n},Ŝ}  (109)

Proposition 8.1 One has

p^{∗}X_{i}^{Γ} = ∏_{j∈I_{m}^{Γ}}(Δ_{j}^{Γ})^{ε_{ij}^{Γ}}  (110)

Figure 27: The edge function

Proof. This is a reformulation of the formula (108). The proposition is proved.

It is easy to see using lemma 6.6 that the total number of the X–functions equals to the dimension of the space of all P GL_{m}-local systems on S.

4. The cross-ratio and the triple ratio. The space of orbits of the group P GL_{2} acting on the 4-tuples of distinct points on P^{1} is one dimensional. There is a classical coordinate on the orbit space, called the cross-ratio of four points x_{1}, ..., x_{4}. To define it we present P^{1} as the projectivisation of a two dimensional vector space V, and choose four vectors v_{1}, ..., v_{4} in V projecting to the given points x_{1}, ..., x_{4} in P (V). Then

r^{+}(x_{1}, ..., x_{4}) := \frac{Δ(v_{1}, v_{2})Δ(v_{3}, v_{4})}{Δ(v_{1}, v_{4})Δ(v_{2}, v_{3})}  (111)

One has

r^{+}(x_{1}, x_{2}, x_{3}, x_{4}) = r^{+}(x_{2}, x_{3}, x_{4}, x_{1})^{−1}; r^{+}(x_{1}, x_{2}, x_{3}, x_{4}) = −1 − r^{+}(x_{1}, x_{3}, x_{2}, x_{4})

To check the second equality we use the Plücker relation

Δ(v_{1}, v_{2})Δ(v_{3}, v_{4}) − Δ(v_{1}, v_{3})Δ(v_{2}, v_{4}) + Δ(v_{1}, v_{4})Δ(v_{2}, v_{3})

We also need a triple ratio of three flags in P^{2} as defined in s. 3.5 of [G2]. The space of the orbits of the group P GL_{3} acting on the triples of flags F_{1}, F_{2}, F_{3} in P^{2} is one dimensional. The triple ratio r_{3}^{+}(F_{1}, F_{2}, F_{3}) is a natural coordinate on this space. To define it, let us present P^{2} as a projectivisation of a three dimensional vector space V. Choose three affine flags A_{1}, A_{2}, A_{3} dominating the flags F_{1}, F_{2}, F_{3} in P (V). We represent the flags A_{i} as the pairs (v_{i}, f_{i}) where v_{i} ∈ V, f_{i} ∈ V^{∗} and < v_{i}, f_{i} >= 0. Then set

r_{3}^{+}(F_{1}, F_{2}, F_{3}) := \frac{< f_{1}, v_{2} >< f_{2}, v_{3} >< f_{3}, v_{1} >}{< f_{1}, v_{3} >< f_{2}, v_{1} >< f_{3}, v_{2} >}

This definition obviously does not depend on the choice of the pair (v, f) representing a flag F. It is useful to rewrite this definition using the presentation of the affine flags given by

A_{1} = (x_{1}, x_{1} ∧ x_{2}), A_{2} = (y_{1}, y_{1} ∧ y_{2}), A_{3} = (z_{1}, z_{1} ∧ z_{2})

Then

r_{3}^{+}(F_{1}, F_{2}, F_{3}) := \frac{Δ(x_{1}, x_{2}, y_{1})Δ(y_{1}, y_{2}, z_{1})Δ(z_{1}, z_{2}, x_{1})}{Δ(x_{1}, x_{2}, z_{1})Δ(y_{1}, y_{2}, x_{1})Δ(z_{1}, z_{2}, y_{1})}  (112)

The cross and triple ratios provide isomorphisms

r^{+} : {4–tuples of distinct points of ℙ^{1}}/P GL_{2} ⟶^{∼} ℙ^{1} − {0, −1, ∞}

r_{3}^{+} : {generic triples of flags of ℙ^{2}}/P GL_{3} ⟶^{∼} ℙ^{1} − {0, −1, ∞}

Let F_{i} = (x_{i}, L_{i}) be a flag in P^{2}, so x_{i} is a point on a line L_{i}, and i = 1, 2, 3. Consider the triangle formed by the lines L_{1}, L_{2}, L_{3}. Denote by y_{1}, y_{2}, y_{3} its vertices, so y_{i} is opposite to the line L_{i}. Let x̂ be the point of intersection of the line L_{3} and the line through the points x_{1} and x_{2}. Then one has

r_{3}^{+}(F_{1}, F_{2}, F_{3}) = r^{+}(y_{1}, y_{2}, x̂, x_{3})

Remark. A yet different definition of the triple ratio one can find in Section 4.2 of [G5].

Lemma 8.2 A triple (F_{1}, F_{2}, F_{3}) of real flags in ℝP^{2} is positive if and only if the points x_{i} are at the boundary of a convex domain bounded by the lines of the flags, as on Figure 28.

Figure 28: A positive triple of flags

Proof. Any triple of flags satisfying the condition of the lemma can be presented as follows: v_{1} = (0, 1, 1); v_{2} = (1, 0, 1); v_{3} = (1, t, 0); f_{1} = x_{1}, f_{2} = x_{2}, f_{3} = x_{3}, where t > 0. Its triple ratio is t.

Remark. It is often useful to change the sign in the definition of the cross- and triple ratios: r := −r^{+} and r_{3} := −r_{3}^{+}.

5. Expressing the X-functions via the cross- and triple ratios. The vertex functions. Recall that F_{k} is the k-dimensional subspace of a flag F. For generic configuration of flags (X, Y, Z) the quotient

\frac{L_{v}}{X_{a−1} ⊕ Y_{b−1} ⊕ Z_{c−1}}  (113)

is a three dimensional vector space. It inherits a configuration of three flags (\overline{X}, \overline{Y} , \overline{Z}). Namely, the flag \overline{X} is given by the projection of the flag (X_{a}/X_{a−1} ⊂ X_{a+1}/X_{a−1} ⊂ ...) onto (113), and so on. The vertex X–function X_{a,b,c} equals to their triple ratio:

X_{a,b,c}(X, Y, Z) = r_{3}^{+}(\overline{X}, \overline{Y} , \overline{Z})

The edge X–functions. For a generic configuration of flags (X, Y, Z, T) in L_{e} then the quotient

\frac{L_{e}}{X_{a−1} ⊕ Y_{b−1}}

is a two dimensional vector space. It inherits a configuration (\overline{X}_{a}, \overline{Y}_{1}, \overline{Z}_{b}, \overline{T}_{1}) of four distinct one dimensional subspaces: \overline{X}_{a} := X_{a}/X_{a−1} and so on. Their cross–ratio is the edge X-function X_{a,b}:

X_{a,b}(X, Y, Z, T) = r^{+}(\overline{X}_{a}, \overline{Y}_{1}, \overline{Z}_{b}, \overline{T}_{1})

6. The main result. Let us choose an isotopy class Γ of a marked trivalent graph on Ŝ isomorphic to a marked ribbon graph Γ. It provides an isomorphism of the moduli spaces f_{Γ} : X_{P GL_{m},Γ} ⟶ X_{P GL_{m},Ŝ}. Thus we get the rational functions X_{j}^{Γ} := f_{Γ}^{∗}X_{j} on the moduli spaces X_{P GL_{m},Ŝ}. Similarly there are regular functions Δ_{i}^{Γ}

Theorem 8.3 a) The collections of rational functions {X_{j}^{Γ}}, j ∈ J_{m}^{Γ}, where Γ runs through the set of all isotopy classes of marked trivalent graphs on Ŝ of type Ŝ, provide a positive regular atlas on X_{P GL_{m},Ŝ}.

b) The collections of regular functions {Δ_{i}^{Γ}}, i ∈ I_{m}^{Γ}, provide a positive regular atlas on A_{SL_{m},Ŝ}.

Plan of the proof. In Sections 8.7 and 8.8 we give a yet another definition of the X–coordinates on the configuration space of triples and quadruples of flags in V_{m}. We show in Lemmas 8.4, 8.6, and 8.8 that it gives the same X-coordinates as in Section 8.2. The results of Section 8.8 make obvious that they are related to the general definition for a standard reduced decomposition of w_{0}, see Lemma 8.8. Similarly our Δ-coordinates on the space A_{SL_{m},Γ} are evidently related to the A-coordinates from Section 7 for a standard reduced decomposition of w_{0}. However our special coordinates have some remarkable features which are absent in the general case even for SL_{m} and nonstandard reduced decompositions of w_{0}. For example the defined in Section 8.2-8.3 coordinates on the configuration space of triples of (affine) flags in V_{m} are manifestly invariant under the cyclic (twisted cyclic) shift.

So the positivity statements in Theorem 8.3 are special cases of the ones in Theorem 1.5. On the other hand these statements follow immediately from the results of Section 9. This way we get simpler proofs of the positivity results, which are independent from the proof of the Theorem 1.5.

We show that our X-functions provide a regular atlas in Section 8.9. A similar statement about the Δ-coordinates is reduced to the one about the X-coordinates using the fact that the projection A_{SL_{m},Ŝ} ⟶ X_{P GL_{m},Ŝ} is obtained by factorization along the action of a power of the Cartan group.

Let us start the implementation of this plan.

7. Projective bases determined by three flags. A projective basis in a vector space V is a basis in V up to a multiplication by a common scalar. Projective bases in V form a principal homogeneous space for P GL(V). Let A, B, C be three flags at generic position in an m-dimensional vector space V. Below we introduce several different projective bases in V related to these flags, and compute the elements of P GL_{m} transforming one of these bases to the other.

Consider the m–triangulation of a triangle shown on the picture. Each side of the triangle carries m vertices of the m–triangulation. There are two types of the triangles, the triangles looking down, and the triangles looking up. We picture them as the white and grey triangles. Let us assign the flags A, B, C to the vertices of the big triangle. The side across the A-vertex is called the A–side, etc.. The vertices of the m–triangulation are parametrized by non negative integer solutions of the equation a + b + c = m − 1. The (a, b, c)–coordinates of a vertex show the distance from the vertex to the A, B and C sides of the triangle. A vertex A of the big triangle provides two arrows which start at A and look along one of the two sides sharing A. A snake is a path in the one-skeleton of the m–triangulation from one of the vertices of the big triangle to the opposite side, which each time goes in the direction of one of the two arrows assigned to the vertex.

Figure 29: A snake

A snake provides a projective basis in V. Indeed, let F^{a} := F_{m−a} be the codimension a subspace of a flag F. We attach to each vertex (a, b, c) a one–dimensional subspace V^{a,b,c} := A^{a} ∩ B^{b} ∩ C^{c}. The one–dimensional subspaces V_{1}, V_{2}, V_{3} attached to the vertices of any grey triangle t span a two– dimensional subspace V (t). The subspaces V_{1}, V_{2}, V_{3} provide six projective bases in V (t), assigned to the oriented sides of the triangle t. Namely, we assume that the triangle t is counterclockwise oriented. Then if the side V_{1}V_{2} is oriented according to the orientation of t, we choose vectors v_{1} ∈ V_{1} and v_{2} ∈ V_{2} so that v_{1} + v_{2} ∈ V_{3}. If not, we choose vectors v_{1} ∈ V_{1} and v_{2} ∈ V_{2} so that v_{1} − v_{2} ∈ V_{3}, as shown on the picture.

Remark. This definition of the six projective bases agrees with the one given in Section 5.2. Each edge of the triangulation is a boundary of just one grey triangle. Using this observation,

Figure 30: Defining two projective bases in a two dimensional space

let us construct a projective basis corresponding to a snake. Let e_{1}, ..., e_{m−1} be the edges of the snake in their natural order, so the A–vertex belongs to e_{1}. Denote by t_{1}, ..., t_{m−1} the grey triangles containing these edges. Each edge e_{i} determines a projective basis in the subspace V (t_{i}). A nonzero vector v_{1} ∈ V^{m−1,0,0} determines a natural basis v_{1}, ..., v_{m} parametrized by vertices of the snake. Indeed, v_{1} plus a projective basis in V_{t_{1}} corresponding to e_{1} determines a vector v_{2} in V (t_{1}). The vector v_{2} plus a projective basis in V (t_{2}) corresponding to e_{2} provides v_{3}, and so on. All together they provide a projective basis.

8. Transformations between different projective bases related to three flags. Let us compute the element of P GL_{m} transforming the projective basis corresponding to a snake to the one corresponding to another snake. We assume that the group P GL_{m} acts on projective bases from the left. First, let us do it for the snakes sharing the same vertex of the big triangle, say the vertex A. An elementary move of a snake is one of the following two local transformations on Figure 31:

Figure 31: Elementary moves of the snake

We can transform one snake to another using a sequence of elementary moves. An example for m = 3 see on Figure 32. In general such a sequence of elementary moves is not unique.

Figure 32: Moving the snake in the case m = 3

Let us calculate the elements of P GL_{2} and P GL_{3} corresponding to the elementary moves I and II. For the elementary move I we get

( \begin{matrix}v_{1} \\ v_{2} \end{matrix} ) ⟼ ( \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} ) ( \begin{matrix}v_{1} \\ v_{2} \end{matrix} ) = ( \begin{matrix}v_{1} \\ v_{1} + v_{2} \end{matrix} )

Here the vectors v_{1}, v_{2}, v_{1} + v_{2} in V (t) correspond to the vertices of a grey triangle t, so (v_{1}, v_{2}) is the projective basis related to the left edge of t, and (v_{1}, v_{1} + v_{2}) is the one related to the bottom edge of t.

The vertex X–coordinates are assigned to the white triangles. Namely, let α, β, γ be the distances from a white triangle to the sides of the triangle, α + β + γ = m − 3. Then the flags A, B, C induce the flags \overline{A}, \overline{B}, \overline{C} in the three dimensional quotient

\frac{V}{A_{α} ⊕ B_{β} ⊕ C_{γ}}  (114)

The triple ratio of these three flags is the X–coordinate corresponding to the white triangle.

Now consider an elementary move of type II corresponding to a white triangle. Let us calculate the corresponding element of P GL_{m}, and show that it depends only on the X-coordinate assigned to this white triangle. Denote by a a vector spanning the one dimensional subspace \overline{A}_{1} of the flag \overline{A}. There are two choices for the vectors spanning \overline{B}_{2} ∩ \overline{C}_{2}: the vector r_{1} is defined via the first snake, and the vector r_{2} defined via the second one on the picture describing the elementary move II. Then

r_{2} = Xr_{1}  (115)

Lemma 8.4 The number X in (115) is the X–coordinate of the flags \overline{A}, \overline{B}, \overline{C}.

Proof. A choice of vector a provides the vectors b, c, p, q spanning the subspaces \overline{B}_{1}, \overline{C}_{1}, \overline{A}_{2}∩\overline{B}_{2}, \overline{A}_{2} ∩ \overline{C}_{2} respectively, and defined using the left (for b and p) and the right (for c and q) snakes. The three grey triangles provide the relations

r_{1} = p + b, q = a + p, c = q + r_{2}  (116)

Therefore, as easily follows from (115) and (116), one has

X = \frac{Δ(p, q, r_{2})}{Δ(p, q, r_{1})} = −\frac{Δ(p, q, c)}{Δ(p, q, b)}

It is easy to check that the second fraction equals to the X-coordinate of the flags \overline{A}, \overline{B}, \overline{C}. The lemma is proved.

Example. Let us compute the transformation between the projective bases corresponding to the two snakes on Figure 32. We present it as a composition: (a, p, b) ⟼ (a, p, r_{1}) ⟼ (a, q, r_{1}) ⟼ (a, q, r_{2}) ⟼ (a, q, c). So thanks to (116) and Lemma 8.4 the corresponding matrix is

( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 1 \end{matrix} ) ( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & X \end{matrix} ) ( \begin{matrix}1 & 0 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) ( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 1 \end{matrix} ) = ( \begin{matrix}1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 + X & X \end{matrix} ) ∈ P GL_{3};  (117)

Let us generalize this computation. Let φ_{i} : SL_{2} ↪ GL_{m} be the canonical embedding corresponding to the i-th root λ_{i} − λ_{i+1}. Set

F_{i} = φ_{i} ( \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} ) , H_{i}(x) := diag(\underbrace{1, ..., 1}_{i}, x, ..., x)

So H_{i} : 𝔾_{m} ⟶ H is the coroot corresponding to the root λ_{i} − λ_{i+1}. So H_{i}(x) commutes with F_{j} and E_{j} if i ≠ j. So the left hand side of (117) can be written as F_{2}H_{2}(X)F_{1}F_{2} = F_{2}F_{1}H_{2}(X)F_{2}.

Let X_{a,b,c}, where a + b + c = m − 3, a, b, c ≥ 0, be the vertex coordinates of the configuration of flags (A, B, C). Let M_{AB→AC} be the element transforming the projective basis corresponding to the snake AB to the one for the snake AC.

Proposition 8.5 a) One has

M_{AB→AC} = ∏_{j=1}^{m−1} (( ∏_{i=j}^{m−2} H_{i+1}(X_{m−i−2,i−j,j−1})F_{i})F_{m−1})  (118)

b) The element M_{AB→BA} is the transformation S : e_{i} ⟼ (−1)^{i}e_{m−i}.

c) One has

M_{CA→BA} = SM_{AB→AC}^{−1}S^{−1}  (119)

For example for m = 2 we get M_{AB→AC} = F_{1}, and for m = 3 and m = 4 we get

M_{AB→AC} = F_{2}H_{2}(X_{000})F_{1}F_{2}, M_{AB→AC} = F_{3}H_{3}(X_{001})F_{3}F_{3}H_{2}(X_{100})F_{1}H_{3}(X_{010})F_{2}F_{3}

Remark. In the formula (118) between every two elements F_{i}, i > 1, which have no F_{i}’s in between, there is unique element H_{i}(X) between them. Its precise location between the two F_{i}’s is not essential since H_{i}(X) commutes with all the F ’s located between the two F_{i}’s.

Proof. a) Let us define a sequence of elementary moves transforming the snake AB to the snake AC. The crucial intermediate steps of this sequence are shown on the picture below. We show the snakes which go i steps towards C, and then m − i − 1 steps in the direction AB, for i = 1, ..., m − 2. In between we use the obvious elementary moves transforming the i-th snake to the the (i + 1)-st.

Figure 33: Moving the snake

Now, just like in the above example, our formulas and Lemma 8.4 give the formula (118).

b) Let (A, B, C) be a cyclically ordered triple of distinct one dimensional subspaces in a two dimensional vector space. Let [v_{1}, v_{2}] (respectively [u_{1}, u_{2}]) be the projective basis determined by the snake AB (respectively BA). This means that v_{1} + v_{2} ∈ C (respectively u_{1} − u_{2} ∈ C). It follows that u_{1} = v_{2}, u_{2} = −v_{1}. This proves the claim in the two dimensional case. The general case immediately reduces to this.

c) This is obvious. The proposition is proved.

Remark. Since SF_{i}^{−1}S^{−1} = E_{m−i}, there is a formula for M_{CA→BA} similar to (118). In this formula we use the other standard reduced decomposition of w_{0} encoding the sequence of E_{i}’s then the one encoding the sequence of F_{i}’s in (118).

Consider a configuration of four flags (A, B, C, D). We assign them to vertices of a quadrilateral triangulated by the diagonal AC. There are two projective bases related to the snake AC: the one, P_{−}(AC), comes from the triangle ABC, and the other, P_{+}(AC), comes from the triangle ACD.

Lemma 8.6 Let x_{a} = X_{a,m−2−a} be the coordinates attached to the edge AC. Then

M_{P_{−}(AC)→P_{+}(AC)} = diag(1, x_{0}, x_{0}x_{1}, ..., x_{0}x_{1}...x_{m−2})  (120)

Proof. Follows from the SL_{2} case, which was considered in the Example in the end of Section 6.7.

9. Comparison with the definitions of Sections 4 and 5. Recall the subsets \overline{U}_{∗}^{−}(i) ⊂ U^{−} and \overline{U}_{∗}^{+}(j) ⊂ U^{+} defined by reduced decompositions i and j of w_{0}, see (65) and (66). Elements \overline{u}_{+} ∈ \overline{U}_{∗}^{+}(i) and \overline{u}_{−} ∈∈ \overline{U}_{∗}^{+}(i) have canonical decompositions:

\overline{u}_{+} = ∏_{j∈j} x_{j}(s_{j}), \overline{u}_{−} = ∏_{i∈i} y_{i}(t_{i}),  s_{j}, t_{i} ∈ F^{∗}  (121)

The elements of the sets i and j are labeled by the simple roots α_{1}, ..., α_{r}, providing the decompositions

i = i(α_{1}) ∪ ... ∪ i(α_{r}) j = j(α_{1}) ∪ ... ∪ j(α_{r})

Our convention was the following: let α be a simple root, j_{α}^{+} is the right element of j(α), i_{α}^{−} is the left element of i(α). Then for each simple root α one has

x_{j_{α}^{+}}(s_{j_{α}^{+}}) = 1,  y_{i_{α}^{−}}(t_{i_{α}^{−}}) = 1,  (122)

Consider the following two reduced decomposition of the maximal length element in S_{m}:

i = (m, m − 1, m, ..., 2, ..., m, 1, 2, ..., m), j = (m, ..., 1, m, ..., 2, ..., m, m − 1, m)

Consider the m-triangulation of the triangle ABC, as on Figure 29. Its grey triangles are presented as a union of m strips going parallel to the side BC. We enumerate the strips so that the k-th strip contains k the grey triangles. These numbers match the simple roots: k corresponds to λ_{k} − λ_{k+1}. Then the string i provides an enumeration of the grey triangles from the top to the bottom, going in each layer parallel to the line AC from A to C. Let t be a white triangle and t′ the grey triangle sharing a side with t, obtained from t by going towards the vertex C. Recall that our special Xcoordinates are naturally attached to the white triangles. We assign the X-coordinate attached to the white triangle t, to the corresponding grey triangle t′. Finally, we assign 1 to the grey triangles neighboring the side AB.

Lemma 8.7 One has

M_{AB→AC} = \overline{u}_{AB→AC}^{−}h_{AB→AC}^{−}, \overline{u}_{AB→AC}^{−} ∈ U_{∗}^{−}(i), h_{AB→AC}^{−} ∈ H

The canonical coordinates (121) of the element \overline{u}_{AB→AC}^{−} are given by the X-coordinates assigned as above to the grey triangles, provided the grey triangles are enumerated by the elements of the set i as was explained above.

The element h_{AB→AC}^{−} is the product of the elements H_{i_{t}}(X_{t}) assigned to the grey triangles t.

Proof. Follows from the very definition and formulas (121), (118) and (122).

A projective basis in V_{m} is the same thing as a pinning for P GL_{m}.

Lemma 8.8 The six projective bases corresponding to the six oriented edges of the triangle ABC coincide with the six canonical pinnings attached to the triple of flags (A, B, C) in V_{m} in Section 5.2.

Proof. Lemma 8.7 implies that the projective basis assigned to the snake AB coincides with the one assigned to the side AB of the triangle whose vertices are labeled by the configuration of flags

(A, B, C) = (B^{−}, B^{+}, u_{−}B^{+}u_{−}^{−1})

The case of the snakes BC and CA is similar. To treat the case of the snake BA we use the remark in Section 8.8. The lemma is proved.

10. Constructing a framed local system with arbitrary given non-zero coordinates. Let Γ be a trivalent graph on Ŝ, of the type of Ŝ. In Section 8.2 we defined the X-functions on the moduli space X_{P GL_{m},Ŝ} corresponding to Γ. Recall that they are parametrized by the set J_{m}^{Γ}. Below we are going to show how to invert this construction. Precisely, let F be a field. Then given X_{i} ∈ F^{∗}, i ∈ J_{m}^{Γ} we will construct a framed P GL_{m}-local system on Ŝ with the given X-coordinates {X_{i}}.

Consider a decomposition of Ŝ into hexagons and rectangles obtained as follows. Replace the edges of Γ by rectangles and the vertices by hexagons, as on the left picture. Let Δ be the 1–skeleton of this decomposition.

Figure 34: The graph Δ

Let G a split reductive algebraic group. Then, given a framed G-local system on Ŝ, every vertex v of Γ provides a configuration of three flags in the vicinity of v. Therefore there are the six canonical pinning assigned to v. These pinnings match the vertices of the hexagon of Δ corresponding to v.

To construct a framed G(F)-local system on Ŝ we assign to every oriented edge e of Δ an element M(e) ∈ G(F). Let p := e_{1}e_{2}e_{3}... be a path on Δ, where ef means e follows f. Then p gives rise to M(p) := M(e_{1})M(e_{2})M(e_{3})....

Lemma 8.9 Suppose that elements M(e) ∈ G(F) satisfy the following conditions:

i) If e does not intersect the graph Γ then M(e) ∈ B(F).

ii) Let \overline{e} be the edge e equipped with the opposite orientation. Then M(e)M(\overline{e}) = Id.

iii) The element assigned to a path around a rectangle or hexagon is the identity. Then the elements M(p) depend only on the homotopy class of the path p, providing a G(F)-local system on S. Moreover it has a natural framing on Ŝ.

Proof. The properties ii) and iii) guaranty that we get a G(F)-local system on S, and i) provides a natural framing. The lemma is proved.

Let us construct such elements M(e). The graph Δ has three types of edges: the edges intersecting Γ, the edges of the rectangles which do not intersect Γ, and the edges of the hexagons which do not intersect Γ. We denote them, respectively, by a, b and c. We will assume that the c-edges of the hexagons are oriented counterclockwise, i.e. in the opposite way to the cyclic order of the edges and hence the flags in the vicinity of v. The following definition is suggested by Proposition 8.5 and Lemma 8.6.

a) Set M(a) := S.

b) Let b be an edge of a rectangle which goes along an oriented edge e of Γ. Let x_{a} = X_{a,m−2−a}^{e} be the coordinates attached to the edge e. Then M(b) is given by the right hand side of (120).

c) Let {X_{a,b,c}}, where a + b + c = m − 3 and a, b, c ≥ 0, be the coordinates assigned to a vertex v of Γ. Then integers a, b, c match the edges of the hexagon surrounding v which do not intersect Γ. Let c be such an edge assigned to a, counterclockwise oriented. Then M(c) is given by the right hand side of (118).

Theorem 8.10 Let F be a field, and we are given X_{i} ∈ F^{∗}, where i ∈ J_{m}^{Γ}. Then

a) The elements M(a), M(b), M(c) ∈ P GL_{m}(F) satisfy the conditions of Lemma 8.9. So they give rise to a framed local system on Ŝ.

b) Let L be a framed P GL_{m}(F)–local system on Ŝ. Then the framed P GL_{m}(F)–local system L(X) provided by the X–coordinates of L is isomorphic to L.

Proof. Follows immediately from Proposition 8.5 and Lemma 8.6. The theorem is proved.

Let H_{m} (respectively V_{m}) be the algebraic torus parametrising the internal edge (respectively vertex) X–coordinates. Then H_{m} is isomorphic to the Cartan group of P GL_{m}, and V_{m} = 𝔾_{m}^{(m−2)(m−1)/2}.

The part a) of Theorem 8.10 provides a regular map

ψ_{Γ} : H_{m}^{{internal edges of Γ}} × V_{m}^{{vertices of Γ}} = 𝔾_{m}^{J_{m}^{Γ}} ⟶ X_{P GL_{m},Ŝ}

Corollary 8.11 The map ψ_{Γ} is a birational isomorphism, and is an injective regular map.

Proof. The part b) of Theorem 8.10 plus the dimension count imply that all conditions of Lemma 6.5 are valid for the map ψ_{Γ}. So it is a birational isomorphism. Since ψ_{Γ} is regular, it is injective. The corollary is proved.

Definition 8.12 The atlas given by the collection of embeddings {ψ_{Γ}}, when Γ run through all isotopy classes of marked trivalent graphs on Ŝ of type Ŝ, is called special atlas on X_{P GL_{m},Ŝ}.

By Lemma 8.7 the special atlas is a subatlas of the atlas defined in Section 5. This implies that the special atlas is a positive atlas. An elementary proof of this claim is contained in Section 9. This positivity claim plus Theorem 8.10 gives a complete proof of Theorem 8.3.

In the next subsection we investigate the monodromy properties of the universal P GL_{m}-local system on S with respect to the special positive atlas on X_{P GL_{m},Ŝ}.

11. Positive Laurent property of the monodromy of universal P GL_{m}-local system. A special good positive Laurent polynomial on X_{P GL_{m},S} is a rational function which in every coordinate system {T_{i}} from the special positive atlas on X_{P GL_{m},S} is a Laurent polynomial with positive integral coefficients. We say that a matrix with entries in the field 𝔽_{P GL_{m},S} is a special good totally positive (integral) Laurent matrix if in every special positive coordinate system, every minor of this matrix is a good positive integral Laurent polynomial. There is a similar definition for the upper/lower triangular matrices.

Theorem 8.13 The monodromy of the universal P GL_{m}-local system L_{m} around a non-boundary loop on S is conjugated to a special good totally positive Laurent matrix. The monodromy around a boundary loop is conjugated to an upper/lower triangular special good totally positive Laurent matrix.

Proof. It is similar to the proof of Theorem 1.11. Starting from the connection on the graph Δ given by the elements M(a), M(b), M(c) we define a connection on the corresponding graph Γ′_{T}, shown on Figure 20, as follows. Observe that the oriented t-edges of Γ′_{T} match the oriented c-edges of Δ. Further, the b-edges of Δ have canonical orientations: the ones compatible with the clockwise orientation of the holes. So oriented e-edges of Γ′_{T} match the b-edges of Δ. So we set

M(t) := SM(c) = M(a)M(c); M(e) := M(b)S = M(b)M(a)

assuming that the t and e-edges of Γ′_{T} match the corresponding edges of Δ. Then M(e)M(t) is a lower triangular special totally positive integral Laurent matrix. Indeed, it is obtained by a product of the matrices F_{j}, H_{i}(x) and M(b). Similalry, if e follows \overline{t}, then M(e)M(\overline{t}) = M(e)M(t)^{−1} = M(b)SM(c)S^{−1} is an upper triangular special totally positive integral Laurent matrix: see Remark after Proposition 8.5. The rest is as in the proof of Theorem 1.11. The theorem is proved.

Let ρ : P GL_{m} → Aut(V) be a finite dimensional representation. Consider the associated local system L_{ρ} := L_{m} ×_{P GL_{m}} V on S × X_{P GL_{m}}.

Corollary 8.14 a) Let M_{α} be the monodromy of the universal local system around a loop α. Then, for any integer n > 0, tr(M_{α}^{n}) is a special good positive Laurent polynomial on X_{P GL_{m},S}.

b) For any finite dimensional representation ρ of P GL_{m} the monodromy of the universal local system L_{ρ} around any loop on S is conjugated to a matrix whose entries are special good positive rational Laurent polynomials on X_{P GL_{m},Ŝ}.

c) The trace of the monodromy of L_{ρ} around any loop is a special good positive rational Laurent polynomial on X_{P GL_{m},Ŝ}.

Proof. a) Follows immediately from Theorem 8.13.

b) Take the canonical basis in V. Then the elements E_{i} and F_{i} of P GL_{m} act by matrices with non-negative ratinal coefficients, and the elements H_{i}(X) act by diagonal matrices whose coefficients are monomials in X^{±1}. c) follows from b). The corollary is proved.

Conjecture 8.15 tr(M_{α}^{n}) can not be presented as a sum of two non-zero elements of 𝕃_{+}(X_{P GL_{m},S}):

tr(M_{α}^{n}) ∈ E(X_{P GL_{m},S})

12. Convex curves in ℝℙ^{n} and positive curves in the flag variety for P GL_{n+1}(ℝ). A curve K in ℝℙ^{n} is convex if any hyperplane contains no more then n points of K. This definition goes back to Schoenberg [Sch]. It is interesting that every convex algebraic curve is projectively equivalent to the Veronese curve given by (x_{0} : x_{1}) ⟼ (x_{0}^{n} : x_{0}^{n−1}x_{1} : ... : x_{1}^{n}).

We start from a useful inductive criteria of positivity of configurations of flags in ℝℙ^{n}.

Lemma 8.16 A configurations of m flags in ℝℙ^{2} is positive if and only if the lines of the flags form a convex m-gon, and the points of the flags are at the boundary of this m-gon, but not at the vertices, see Figure 35.

Proof. For m = 3 this is Lemma 8.2. The general case is deduced from it, see Section 1 of [FG3].

Figure 35: A positive configuration of six flags

Denote by B_{n+1}(ℝ) the flag variety for P GL_{n+1}(ℝ). There are two natural projections

π : B_{n+1}(ℝ) ⟶ ℝℙ^{n}, π̂ : B_{n+1}(ℝ) ⟶ ℝ̂ℙ^{n}

Let (F_{1}, ..., F_{m}) be a generic configuration of flags in ℝℙ^{n}. Set x_{k} := π(F_{k}). We define a configuration of flags (x_{k}|F_{1}, ..., F_{m}) in ℝℙ^{n−1} by projection with the center at the point x_{k}. Namely, let F be a flag in ℝℙ^{n} whose (n − 2)-dimensional subspace does not contain x_{k}. Then F provides a flag in the projective space of all lines passing through x_{k}: its p-dimensional subspace consists of the lines through x_{k} passing through the similar subspace of F. This way the flags F_{i}, i ≠ k, give rise to the flags \overline{F}_{i}. Finally, the p-dimensional subspace of the the flag F_{k} provides (p − 1)-dimensional subspace of the flag \overline{F}_{k}. Now we set (x_{k}|F_{1}, ..., F_{m}) := (\overline{F}_{1}, ..., \overline{F}_{m}).

Figure 36: Projecting a configuration of five flags

Lemma 8.17 Let n ≥ 3. A configuration of flags (F_{1}, ..., F_{m}) in ℝℙ^{n} is positive if and only if all configurations (x_{1}|F_{1}, ..., F_{m}), ..., (x_{m}|F_{1}, ..., F_{m}) are positive.

Proof. Let us prove the implication <=. Take a convex m-gon P_{m} with vertices labelled by the flags F_{1}, ..., F_{m}. Observe that, assuming n ≥ 3, given a triangulation of the polygon P_{m}, the corresponding canonical cooordinates of the configuration (F_{1}, ..., F_{m}) are defined as follows: we take a triple or quadruple of flags assigned to a triangle or quadrilateral of the triangulation, project them to the flags in ℝℙ^{2}, and then take the canonical coordinates of the obtained configuration. It follows from this that positivity of the configurations (x_{1}|F_{1}, ..., F_{m}), ..., (x_{m}|F_{1}, ..., F_{m}) implies positivity of all the coordinates of the original configuration (F_{1}, ..., F_{m}) with respect to given triangulation.

Let us prove that if the configuration (F_{1}, ..., F_{m}) is positive, then the one (x_{m}|F_{1}, ..., F_{m}) is also positive. Since the property of positivity of configurations of flags is cyclically invariant, this would imply the implication =>. Consider the triangulation of the polygon P_{m} by the diagonals from the vertex F_{m}. Consider the triple (F_{k}, F_{k+1}, F_{m}) assigned to the vertices of one of the triangles. Then, assuming n ≥ 3, the set of the coordinates corresponding to the triple (x_{m}|F_{k}, F_{k+1}, F_{m}) is a subset of the set of coordinates of the triple (F_{k}, F_{k+1}, F_{m}). A similar statement is true for the coordinates assigned to a diagonal (F_{k}, F_{m}) of the polygon. The claim follows immediately from this. The lemma is proved.

Remark. The implication <= of Lemma 8.17 is false for n = 2.

Let K be an oriented continuous convex curve in ℝℙ^{n}. Let x_{1}, x_{2} be two points of K. We say that x_{1} < x_{2} if x_{2} follows x_{1} according to the orientation of the curve K. Given a collection of points X = x_{1} < x_{2} < ... < x_{n} on K, we associate to it a flag

F_{X} := (x_{1}, [x_{1}x_{2}], ..., [x_{1}...x_{n}])

where [x_{1}...x_{k}] denotes the subspace spanned by the points x_{1}, ..., x_{k}. Convexity of K implies that dim[x_{1}...x_{k}] = k − 1, so F_{X} is indeed a flag. More generally, given n-tuples of points

X(1) := {x_{1}(1) < ... < x_{n}(1)}, ... , X(m) := {x_{1}(m) < ... < x_{n}(m)}  (123)

we get the flags F_{X(1)}, ..., F_{X(m)}. We say that X(a) < X(b) if x_{i}(a) < x_{j}(b) for all 1 ≤ i, j ≤ n.

Proposition 8.18 Let K be an oriented continuous convex curve in ℝℙ^{n}. Assume that X(1) < X(2) < ... < X(m) in (123). Then the configuration of flags (F_{X(1)}, ..., F_{X(m)}) is positive.

Proof. Lemma 8.16 implies Proposition 8.18 for n = 2. So we may assume that n > 2. The projection of a convex curve from a point on this curve is a convex curve. So Lemma 8.17 implies Proposition 8.18 by descending induction on n. The proposition is proved.

The osculating curves. Let K be an oriented continuous convex curve in ℝℙ^{n}. Choose a point x ∈ K, and consider an n-tuple of points x < x_{2} < ... < x_{n} on K. It provides a flag. We claim that when the points x_{i} approach the point x keeping their order, the corresponding family of flags has a limit. We prove this claim below. The limit is denoted F_{r}(x) and called the right osculating flag at x. The map x ⟼ F_{r}(x) provides a map β_{r} : K → B_{n+1}(ℝ), which, as one easily sees using the proof of Proposition 8.18, is positive. The curve K̃_{r} := β_{r}(K) is called the right osculating curve to K. Similarly starting from the points x > x_{2} > ... > x_{n} we define the left osculating curve K̃_{l} to K.

To prove the claim, consider a sequence of (n − 1)-tuples of points X(k) := x_{2}(k) < ... < x_{n}(k), k = 1, 2, ..., such that x < x_{i}(k) and x_{i}(k) converges to x as k → ∞. Let us assume that X(a) < X(b). Projecting K from x we get a convex curve in ℝℙ^{n−1}, and a sequence of flags F_{X(a)} on it. This sequence is positive by Proposition 8.18. Thus by Theorem 6.17 it has a limit \overline{F}_{r}(x). The limit evidently does not depend on the choice of sequence X(k). Then F_{r}(x) is the unique flag containing x and projecting to the flag \overline{F}_{r}(x).

Theorem 8.19 a) Let C be a positive curve in the flag variety B_{n+1}(ℝ). Then π(C) and π̂(C) are convex curve of ℝℙ^{n} and ℝ̂ℙ^{n}.

b) Let K be a continuous convex curve in ℝℙ^{n}. Then the osculating curves K̃_{l} and K̃_{r} are positive curves in B_{n+1}(ℝ). So if K is a C^{n−1}-smooth, then K̃_{l} = K̃_{r} = K̃ is a positive curve in B_{n+1}(ℝ).

c) The rule K ⟼ K̃ gives rise to a bijective correspondence

C^{n}-smooth convex curves in ℝℙ^{n} < − > C^{1}-smooth positive curves in B_{n+1}(ℝ)  (124)

Proof. a) It follows immediately from the following crucial proposition:

Proposition 8.20 Let (F_{1}, ..., F_{n+1}) be a positive configuration of flags in ℝℙ^{n}. Let (x_{1}, ..., x_{n+1}) be the corresponding configuration of points in ℝℙ^{n}, i.e. π(F_{i}) = x_{i}. Then these points are not contained in a hyperplane.

Proof. Observe that a positive configuration of flags consists of the flags in generic position. Lemma 8.17 implies that (x_{n+1}|F_{1}, ..., F_{n}) is a positive configuration of flags. Further, the claim of Proposition 8.20 for the configuration of flags (x_{n+1}|F_{1}, ..., F_{n}) in ℝℙ^{n−1} evidently implies the one for the initial flags F_{1}, ..., F_{n+1}. So arguing by the induction on n ≥ 2 we reduce the Proposition to the case n = 2. Let us prove the Proposition when n = 2.

Let V_{3} be a three dimensional vector space. Choose a volume form ω in V_{3}. For any two vectors a, b we define the cross-product a × b ∈ V_{3}^{∗} by < a × b, c >:= Δ(a, b, c). The volume form ω defines the dual volume form in V_{3}^{∗}, so we can define Δ(x, y, z) for any three vectors in V_{3}^{∗}. The following fact is Lemma 5.1 from [G4].

Lemma 8.21 For any 6 vectors in generic position a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3} in V_{3} one has

Δ(a_{1}, a_{2}, b_{2}) · Δ(a_{2}, a_{3}, b_{3}) · Δ(a_{3}, a_{1}, b_{1}) − Δ(a_{1}, a_{2}, b_{1}) · Δ(a_{2}, a_{3}, b_{2}) · Δ(a_{3}, a_{1}, b_{3}) =
Δ(a_{1}, a_{2}, a_{3}) · Δ(a_{1} × b_{1}, a_{2} × b_{2}, a_{3} × b_{3})

Proof. Any configuration of six generic vectors in V_{3} is equivalent to the following one:

\begin{matrix}a_{1} & a_{2} & a_{3} & b_{1} & b_{2} & b_{3} \\ − & − & − & − & − & − \\ 1 & 0 & 0 & x_{1} & y_{1} & z_{1} \\ 0 & 1 & 0 & x_{2} & y_{2} & z_{2} \\ 0 & 0 & 1 & x_{3} & y_{3} & z_{3} \end{matrix}

Then both the left and right hand sides are equal to y_{3}z_{1}x_{2} − x_{3}y_{1}z_{2}. The lemma is proved.

Let (A, B, C) be a positive configuration of three flags in ℝℙ^{2}. The flags A, B, C can be defined by pairs of vectors (a_{1}, a_{2}), (b_{1}, b_{2}), (c_{1}, c_{2}) in V_{3}, so that the flag A is determined by the affine flag (a_{1}, a_{1} ∧ a_{2}), and so on. Recall the triple ratio (112). Using Lemma 8.21 we have

1 + r_{3}^{+}(A, B, C) = 1 + \frac{Δ(a_{1}, a_{2}, b_{1})Δ(b_{1}, b_{2}, c_{1})Δ(c_{1}, c_{2}, a_{1})}{Δ(a_{1}, a_{2}, c_{1})Δ(b_{1}, b_{2}, a_{1})Δ(c_{1}, c_{2}, b_{1})} =

\frac{Δ(a_{1}, b_{1}, c_{1})Δ(a_{1} × a_{2}, b_{1} × b_{2}, c_{1} × c_{2})}{Δ(a_{1}, a_{2}, c_{1})Δ(b_{1}, b_{2}, a_{1})Δ(c_{1}, c_{2}, b_{1})}

Therefore Δ(a_{1}, b_{1}, c_{1}) ≠ 0 if r_{3}^{+}(A, B, C) > 0. Proposition 8.20, and hence the part a) of Theorem 8.19 are proved.

b) Observe that projection with the center at a point x ∈ K maps the osculating flags on K to the ones on the projected curve. So the part b) follows immediately from Lemma 8.17.

c) By Proposition 6.15 a differentiable positive curve C in B_{n+1}(ℝ) is tangent to the canonical distribution, and hence is the osculating curve for the n times differentiable curve π(C) in ℝℙ^{n}. This curve is convex by the part a). The converse statement is the part b) of the theorem. The theorem is proved.

9 The pair (X_{P GL_{m},Ŝ}, A_{SL_{m},Ŝ}) corresponds to an orbi-cluster ensemble

In this Section we formulate precisely and prove Theorems 1.18 and 1.19.

1. A dictionary. We assume that the reader is familiar with the notion of cluster ensemble as defined in [FG2]. A cluster ensemble is determined by the same data as the cluster algebras of Fomin-Zelevinsky [FZI]: it consists of a finite set I and a skew-symmetrizable function E = (ε_{ij}) on I×I. There is an I-marked graph Tr_{I} whose vertices v parametrize the A- and X - coordinate systems, and whose edges adjacent to a given vertex are in bijective correspondence with the set I. For a given vertex v we have the A-coordinates {A_{i}^{v}} parametrized by the set I. Further, there is a subset J ⊂ I. The A-coordinates parametrized by the complement I − J are called tropical A-coordinates. The set J parametrizes the X -coordinates {X_{j}^{v}}.

Here is a dictionary relating the ingredients of the cluster ensemble structure to the data assigned to the (A_{SL_{n},Ŝ}, X_{P GL_{n},Ŝ}) moduli spaces. Let us choose a marked trivalent graph Γ on S, of type Ŝ, and take the corresponding m-triangulation T_{m} = T_{m}(Γ) of S. We understood T_{m} as an oriented graph. Recall that an oriented graph G determines a skew-symmetric function ε_{pq}(G) on the set of pairs of its vertices, see (14).

i) The set I of the cluster ensemble is the set I_{m}^{Γ} of vertices of the graph T_{m}(Γ) minus the punctures of S, and the subset J is given by J_{m}^{Γ}. The sets I_{m}^{Γ} as well as J_{m}^{Γ} are canonically identified for different Γ’s.

ii) The cluster function E = E_{SL_{m},Ŝ} is the function ε_{pq} = ε_{pq}(T_{m}).

iii) The oriented graph T_{m}(Γ) corresponds to a vertex v of the cluster complex C_{E}.

iv) The coordinates Δ_{i}^{Γ} are the cluster ensemble coordinates A_{i}^{v}.

v) The coordinates X_{j}^{Γ} are the cluster ensemble coordinates X_{j}^{v}.

vi) Formula (110) shows that the function X_{j}^{Γ} and Δ_{i}^{Γ} are related under the projection p the same way as the cluster ensemble coordinates X_{j}^{v} and A_{i}^{v}, see [FG2].

However an interpretation of a flip in the cluster language is a non trivial problem, which we address now. We are going to prove that a flip can be presented as a composition of mutations.

Describing a flip and its action on the cluster function. Let Γ′ be a graph obtained from Γ by a flip at an edge E. Denote by F the new edge of Γ′. Let {A, B, C, D} (resp. {A′, B′, C′, D′}) be the set of edges of the Γ (resp. Γ′) adjacent to E (resp. F). Then there is canonical bijection {A, B, C, D} ⟶ {A′, B′, C′, D′} such that X ⟼ X′. Let us number by 1, 2, 3, 4 the pieces of faces in the vicinity of the edge E. They match the ones in the vicinity of F. We assume that their order is compatible with the cyclic order provided by the ribbon structure of Γ, see Figure (37). Denote by T′_{m} the m-triangulation corresponding to Γ′.

Figure 37: Describing a flip

The claim that a flip can be obtained as a composition of mutations means the following:

i) The composition of mutations transforms the function ε_{pq}(T_{m}) to ε_{pq}(T′_{m})

ii) The transformation rules for the A- and X -coordinates under a flip coincide with the ones provided by the sequence of mutations.

2. A sequence of mutations representing a flip at an edge E. We present it as a composition of m−1 steps. Consider the diamond with the vertices 1, 2, 3, 4 on Figure (37). One can inscribe into the m-triangulation of this diamond rectangles of the size i×(m−i), where i = 1, ..., m−1, centered at the center of the diamond. Here i (resp. (m − i)) is the length in direction of the edge F (resp. E). We subdivide this rectangle into i(m − i) equal squares. Figure (39) illustrates this.

Step i. Do mutations in the direction of the elements of the set I := I_{m} corresponding to the centers of little squares of the i × (m − i) rectangle.

For example at the step 1 (resp. m − 1) we do mutations corresponding to the m − 1 vertices located inside of the vertical (resp. horizontal) diagonal of the diamond.

Proposition 9.1 The described sequence of mutations transforms the cluster function ε_{pq}(T_{m}) to the one ε_{pq}(T′_{m}).

Figure 38:

Proof. Consider an m × n rectangle where m, n ≥ 1 are integers. Subdivide it into mn equal squares, mark the vertices and centers of the squares. Connect the marked points by arrows as on Figure (38). Denote by R_{m,n} the obtained oriented graph. Observe that the right picture on Figure (38) is obtained from the left one by inverting directions of the arrows at each square center, and replacing the arrows at the sides of the rectangle by the ones at the top and bottom, keeping the counterclockwise orientation.

Lemma 9.2 Consider the cluster function ε_{pq} corresponding to the oriented graph on the left of Figure (38). Let us perform a sequence of flips at all centers of the mn squares, in any order. Then the obtained cluster function ε′_{pq} is described by the picture on the right of Figure (38).

Proof. Follows immediately from the definitions.

Example. The two pictures (39) and (40) illustrate the case m = 6. We show a part of the mutated graph after the steps 1 and 2. On each picture we show the distinguished rectangles corresponding to this and previous steps.

Contemplation will convince the reader that after m − 1 steps we get the graph of the function ε_{pq}(T′_{m}). The proposition is proved.

3. Hypersimplices and a decomposition of a simplex. Recall [GGL] that hypersimplex Δ^{k,l}, where k + l = n − 1, is a section of the n-dimensional cube 0 ≤ x_{i} ≤ 1 by the hyperplane ∑ x_{i} = n − 1 − k. The hypersimplex Δ^{k,l} can be obtained as the convex hull of the centers of k-dimensional faces of an (n − 1)–dimensional simplex.

Example. Δ^{1,1} is an octahedron. It is the convex hull of the centers of the edges of a tetrahedron. Let m be a positive integer. Consider the simplex

Δ_{(m)}^{n} := {(x_{0}, ..., x_{n}) ∈ ℝ^{n} | ∑_{i=0}^{n} x_{i} = m, x_{i} ≥ 0}  (125)

Lemma 9.3 The hyperplanes x_{i} = k where k is an integer cut the simplex (125) into a union of hypersimplices.

Figure 39:

Figure 40:

Proof. Consider decomposition of ℝ^{n} into unit cubes with the faces x_{i} = k where k is an integer. The hyperplane ∑ a_{i} = m intersect each of these cubes either by an empty set, or by a hypersimplex. The lemma is proved.

We call it the hypersimplicial decomposition of the simplex Δ_{(m)}^{n}.

Example. When n = 2 we get an m–triangulation of a triangle defined in Section 1.6.

Consider the three dimensional simplex Δ_{(m)}^{3}(E) given by

x_{1} + x_{2} + x_{3} + x_{4} = m, x_{i} ≥ 0,  (126)

whose vertices are labeled by 1, 2, 3, 4, as on Figure (41). Its edges are dual to the A, B, C, D, E, F edges, and get the same labels. The faces match the vertices of the edges E and F. The two faces matching the vertices of E are called the e–faces, and the other two are called the f–faces.

Figure 41:

The hypersimplicial decomposition of Δ_{(m)}^{3}(E) induces the m–triangulation of its faces. The faces of Δ_{(m)}^{3}(E) are identified with the corresponding triangles of T or T′, so that the m–triangulations of the faces match the ones of the triangles. Thus to make a flip we glue the simplex Δ_{(m)}^{3}(E) so that its e–faces match the corresponding two triangles of T. Then the f–faces of Δ_{(m)}^{3}(E) provide the two triangles of T′.

4. The action of a flip on the A-coordinates. Let us attach to the integral points inside of the simplex (126) functions on the configurations space of four affine flags in an m-dimensional vector space V_{m} with a volume form ω. Given an integral solution (a, b, c, d) of (126) and 4-tuple of affine flags (X, Y, Z, T) we set

Δ_{a,b,c,d}(X, Y, Z, T) := Δ_{ω}(x_{(a)} ∧ y_{(b)} ∧ z_{(c)} ∧ t_{(d)})

where x_{(a_{1})} = x_{1} ∧ ... ∧ x_{a_{1}} and so on. If one of the numbers a, b, c, d is zero we get the functions defined in Section 8.2. The Plücker relations allow to derive some identities between these functions which we are going to describe now.

The hypersimplicial decomposition of the three dimensional simplex (126) is a decomposition into tetrahedrons of type Δ^{2,0}, octahedrons Δ^{1,1}, and tetrahedrons of type Δ^{0,2}. The octahedrons play an important role in our story. They are parametrized by the solutions (a_{1}, a_{2}, a_{3}, a_{4}) of the equation

a_{1} + a_{2} + a_{3} + a_{4} = m − 2, a_{i} ∈ ℤ, a_{i} ≥ 0  (127)

The collection of vertices of these octahedrons coincide with the set of the non negative integral solutions of the equation (126). The vertices of the (a_{1}, a_{2}, a_{3}, a_{4})–octahedron have the coordinates

(a_{1}, a_{2}, a_{3}, a_{4}) + (ε_{1}, ε_{2}, ε_{3}, ε_{4}), ε_{i} ∈ {0, 1}, ∑_{i=1}^{4} ε_{i} = 2  (128)

We will use a shorthand Δ_{b_{1},b_{2},b_{3},b_{4}} := Δ_{b_{1},b_{2},b_{3},b_{4}}(X, Y, Z, T).

Lemma 9.4 For any solution \overline{a} = (a_{1}, a_{2}, a_{3}, a_{4}) of (127) one has

Δ_{\overline{a}+(1,1,0,0)}Δ_{\overline{a}+(0,0,1,1)} + Δ_{\overline{a}+(1,0,0,1)}Δ_{\overline{a}+(0,1,1,0)} = Δ_{\overline{a}+(1,0,1,0)}Δ_{\overline{a}+(0,1,0,1)}  (129)

Proof. Let X = (x_{1}, x_{1}∧x_{2}, x_{1}∧x_{2}∧x_{3}, ...) be an affine flag in V_{m}. Let X_{1} ⊂ X_{2} ⊂ ... ⊂ X_{m} = V_{m} be the corresponding flag. The affine flag X induces an affine flag in each of the quotients V_{m}/X_{i}. The four affine flags (X, Y, Z, T) in generic position in V_{m} induce four affine flags in the quotient

\frac{V_{m}}{X_{a_{1}} ⊕ Y_{a_{2}} ⊕ Z_{a_{3}} ⊕ T_{a_{4}}}  (130)

Since a_{1} + a_{2} + a_{3} + a_{4} = m − 2, this quotient is two dimensional, and the induced affine flags provide a configuration of vectors (\overline{x}_{a_{1}+1}, ..., \overline{t}_{a_{4}+1}). We define a volume form ω′ in the space (130) by

ω′(v_{1} ∧ v_{2}) := Δ(x_{(a_{1})} ∧ y_{(a_{2})} ∧ z_{(a_{3})} ∧ t_{(a_{4})} ∧ v_{1} ∧ v_{2})  (131)

Set (v_{1}, v_{2}, v_{3}, v_{4}) := (\overline{x}_{a_{1}+1}, ..., \overline{t}_{a_{4}+1}) in the two dimensional space (130). One easily checks that multiplying (129) by (−1)^{a_{2}+a_{4}} we get the Plücker relation

ω′(v_{1} ∧ v_{2})ω′(v_{3} ∧ v_{4}) + ω′(v_{1} ∧ v_{4})ω′(v_{2} ∧ v_{3}) = ω′(v_{1} ∧ v_{3})ω′(v_{2} ∧ v_{4})

The lemma is proved.

One can interpret it as follows. The six vectors (ε_{1}, ε_{2}, ε_{3}, ε_{4}) from (128) match the edges of the graphs Γ and Γ′ as shown on Figure 42. For instance (1, 1, 0, 0) matches the edge A on the Picture (37) and so on. Set Δ_{\overline{a},A} := Δ_{\overline{a}+(1,1,0,0)} and so one. Then we have

Δ_{\overline{a},A}Δ_{\overline{a},C} + Δ_{\overline{a},B}Δ_{\overline{a},D} = Δ_{\overline{a},E}Δ_{\overline{a},F}  (132)

The defined in Section 9.2 sequence of mutations can be described geometrically as follows. The octahedrons of the hypersimplicial decomposition of Δ_{(m)}^{3}(E) form m − 1 horizontal layers, if we assume that the edges E and F parallel to a horizontal plane, and on the i-th step we add the octahedrons on the i-th level.

Recall that a mutation in the direction of an edge e_{k} : v → v′ changes just one A-coordinate, namely A_{k}^{v} to A_{k}^{v′}, and the new coordinate is defined from the exchange relation

A_{k}^{v}A_{k}^{v′} = ∏_{ε_{ki}>0}(A_{i}^{v})^{ε_{ki}} + ∏_{ε_{ki}<0}(A_{i}^{v})^{−ε_{ki}}

In our situation the mutation corresponding to an octahedron replaces Δ_{\overline{a},E} by Δ_{\overline{a},F}, so the exchange relation is given by (132).

5. The action of a flip on the X-coordinates. Recall the canonical projection p : A_{SL_{n},Ŝ} ⟶ X_{P GL_{n},Ŝ}. Observe that in the special case when Ŝ is a disc with n marked points the canonical projection p : Conf_{n}(A) → Conf_{n}(B) is a map onto. Thus to prove any identity between the functions X_{i}^{Γ} on Conf_{n}(B) it is sufficient to prove them for the pull backs p^{∗}X_{i}^{Γ}. Since we already proved the that the A-coordinates and the cluster functions transform under a flip the same way they transform under the composition of mutations corresponding to a flip, and since we have the cluster type formula (110) for p^{∗}X_{i}^{Γ}, the X-coordinates also transform under a flip the same way as under the corresponding composition of mutations.

The functions corresponding to the octahedrons of hypersimplicial decomposition. The cross–ratio of the vectors \overline{x}_{a_{1}+1}, ..., \overline{t}_{a_{4}+1} provides a function on configurations of 4–tuples of flags in V_{m}:

R_{a_{1},a_{2},a_{3},a_{4}}(X, Y, Z, T) := r^{+}(\overline{x}_{a_{1}+1}, \overline{y}_{a_{2}+1}, \overline{z}_{a_{3}+1}, \overline{t}_{a_{4}+1})  (133)

These functions correspond to the octahedrons of the hypersimplicial decomposition of the simplex Δ_{(m)}^{3}(E). They are determined by the part of the graph Γ in the vicinity of E. A flip Γ → Γ′ at the edge E changes these functions to their inverses.

Proof of Theorem 1.18. Since the Farey triangulation is not special in the sense of Section 3.8, the results obtained above imply the proof of Theorem 1.18i). To get Theorem 1.18ii) we use in addition Lemma 1.17.

To formulate precisely and prove Theorem 1.19 it remains to consider the special triangulations. We do it below.

6. Treatment of the special graphs. Recall the special trivalent ribbon graphs defined in Section 3.8. By definition, they contain a part given by either an eye or a virus. There are well defined X - and A-coordinates assigned to edges and triangles of an eye. For the X -coordinates assigned to an edge E of an eye this deserves a comment. Namely, let Q_{E} be the rectangle having E as an edge, see the middle graph on Figure 7. The two vertices opposing the edge E are glued. However the four flags assigned to the vertices of Q_{E} are generic: the flags assigned to the two glued vertices differ by a monodromy transformation. Thus the corresponding X -coordinates on the edge E are well defined. Contrary to this, to define the X -coordinates corresponding to the loop of a virus we have to go to the 2 : 1 covering of S pictured on Figure 9 and described below. So it is no surprise that we face problems comparing the effects at this loop of cluster and geometric transformations, and that these problems are resolved by going to the cover.

Let us consider first the case G = P GL_{2}. Figure 43 shows how we obtain a virus by making a flip on an eye at the edge E, and tells us how the coordinates on the moduli space X_{P GL_{2},Ŝ} are transformed by this flip. Figure 44 tells us the effect of the cluster transformation, mutation, for the cluster ensemble corresponding to the same moduli space, under the same flip. Observe that the only difference is for the X -coordinate A assigned to the edge obtained by gluing the two edges shown by arrows on the Figure. In the geometric case the transformation is A ⟼ AX, while in the cluster case the coordinate A does not change. Since X_{P GL_{2},Ŝ} is embedded into X_{P GL_{m},Ŝ}, the cluster rule for the transformation of the X -coordinates for G = P GL_{2} also does not agree with the geometric one. There is a similar problem for the A-coordinates. Below we are going to show how to resolve this problem.

Figure 43: A virus is obtained from an eye by the flip at the edge E.

Figure 44: Cluster transformation of coordinates for the flip at E making a virus from an eye.

Let v be the vertex of triangulation shown by the fat point on Figure 43. Let us construct a 2 : 1 cover S̃ → S ramified at the single point v of the compactification of S. Then there is an embedding i : X_{G,S} ↪ X_{G,S̃}. Its image is identified with the set of the stable points of the involution σ of the covering.

The coordinate system on either A- or X -space corresponding to a trivalent graph Γ on S is described by a torus T_{S,Γ}^{∗}, where ∗ stays either for A or for X. A flip Γ → Γ′ at an edge E provides a birational automorphism φ_{E}^{∗} : T_{S,Γ}^{∗} → T_{S,Γ′}^{∗}, The coordinate system on S̃ corresponding to the lifted graph Γ̃ on S̃ is described by the torus T_{S̃,Γ̃}^{∗}. One has

T_{S,Γ}^{∗} = (T_{S̃,Γ̃}^{∗})^{σ=Id}  (134)

Lifting the trivalent graph on S to the cover S̃, we get the graph on the left of Figure 45. The involution σ interchanges the edges marked by the same letter, and acts as the central symmetry with respect to the 4-valent vertex. We will make the flip at the edge E_{1}, followed by the flip at the edge E_{2}. These two flips are illustrated on Figure 45.

We claim that, using the identification (134), the geometric coordinate transformation corresponding to the flip at the edge E on Figure 43 equals to the composition of two flips on the cover S̃. Each of these flips is a product of mutations for the cluster ensemble (X_{P SL_{m},S̃}, A_{P SL_{m},S̃}).

Proposition 9.5 i) The composition of the flips at the edges E_{1} and E_{2}, restricted to the subtorus (134), is equivalent to the flip at the edge E on S, i.e. the following diagram is commutative:

T_{S,Γ}^{∗}  ⟶^{φ_{E}^{∗}}  T_{S,Γ′}^{∗}
↓=  ↓=
(T_{S̃,Γ̃}^{∗})^{σ=Id} ⟶^{φ_{E_{2}}^{∗}∘φ_{E_{1}}^{∗}} (T_{S̃,Γ̃′}^{∗})^{σ=Id}

The flips at E_{1} and E_{2} commute.

ii) Each of the flips at the edges E_{1} and E_{2} is a product of mutations in the cluster ensemble defined using the trivalent graph on S̃ (partly shown on Figure 45) and the function ε_{ij} related to it.

This proposition allows to deduce many properties of the coordinate transformations of the flips from the an eye to a virus and back from the similar properties of mutations on the cover S̃.

Proof. i) This boils down to the very definitions.

ii) It is instructive to consider first the P GL_{2} case. Then the coordinate transformations for the X -coordinates are as follows:

B_{1} = B(1 + X^{−1})^{−1}, C_{1} = C(1 + X), \overline{A}_{1} = A(1 + X), A_{1} = A(1 + X^{−1})^{−1}
\overline{A}_{2} = A(1 + X)(1 + X^{−1})^{−1} = AX, A_{2} = A(1 + X^{−1})^{−1}(1 + X) = AX

Thus the composition of two mutations on Figure 45 commutes with the automorphism σ of the cov-

Figure 45: A flip on an eye is a composition of two mutations on a 2 : 1 cover.

ering, and thus corresponds to a transformation on S. The obtained transformation is the geometric transformation of coordinates corresponding to the flip at the edge E, see Figure 43.

Now let us address the general case, when G = P GL_{m}/SL_{m}. We say that an edge E is generic if for the quadrilateral Q_{E} of the triangulation having E as the diagonal both maps A_{G,Ŝ} → A_{G,Q_{E}} and X_{G,Ŝ} → X_{G,Q_{E}} provided by the restriction to Q_{E} are surjective at the generic point. For the A-space a flip at a generic edge is a composition of mutations. Indeed, the A-coordinates on a quadrilateral are defined using the four affine flags assigned to the vertices of the quadrilateral. The quadrilaterals corresponding to the edges E_{1} on the left picture and E_{2} on the middle picture of Figure 45 are generic. This gives the claim ii) for the A-coordinates.

The X -coordinates in the internal part of the quadrilateral Q_{E} also depend only on the four flags assigned to the vertices of Q_{E}. However the X -coordinates at an external edge F depend also on the fifth flag, and so we have to take into account the pentagon P_{E,F} containing as the internal diagonals the edge E and the external edge F. The natural projection X_{G,Ŝ} → X_{G,P_{E,F}} provided by the restriction to the pentagon P_{E,F} is surjective at the generic point. This is clear for the edge E_{1}. To check this for the edge E_{2} observe that although the pentagon P_{E,F} has two vertices identified (at the vertex w on Figure 46), the flags at these vertices differ by the monodromy around the puncture v, and thus they are generic. The claim ii) for the X -coordinates follows from this. The proposition, and hence Theorem 1.19 are proved.

Figure 46: The two vertices of the shaded pentagon P_{E,F} are identified at the vertex w

7. Integral points of higher Teichmüller spaces. The following proposition is due to Grothendieck. We will prove it in a slightly different way.

Proposition 9.6 There is canonical bijection between the ribbon graphs and finite index torsion free subgroups of SL_{2}(ℤ) modulo conjugation.

Construction. Recall that ribbon graphs encodes pairs (S, T), where S is a surface with punctures, and T an ideal triangulation of S, considered modulo diffeomorphisms of surfaces. A torsion free finite index subgroup Δ ⊂ SL_{2}(ℤ) provides a surface S_{Δ} := H/Δ with triangulation given by the image of the Farey triangulation of H. Vice versa, given a pair (S, T) we consider the point of the Teichmüller space whose coordinates with respect to the triangulation T are equal to 1 at every edge of T. According to, say, Section 8.10, the monodromy representation of such a local system provides a subgroup Δ_{(S,T)} of P SL_{2}(ℤ), well defined up to a conjugation. One checks that the triangulated surface corresponding to Δ_{(S,T)} is isomorphic to the pair (S, T). Indeed, to compute the coordinate assigned to an edge E of the triangulation T, we take the two triangles sharing E to the universal cover, which is the hyperbolic plane with the Farey triangulation. There we get the two triangles (∞, −1, 0) and (∞, 0, 1). So the coordinate of E is r^{+}(∞, −1, 0, 1) = 1. The proposition is proved.

Definition 9.7 A point of the Teichmüller space X_{G,S}^{+} is integral if the corresponding monodromy representation can be represented by a homomorphism π_{1}(S) ↪ G(ℤ).

So the pairs (S, T) considered modulo Diff_{0}(S) describe the integral points of the Teichmüller space X_{P GL_{2},S}^{+}. Here is a hypothetical generalization.

Consider a cluster coordinate system on X_{P GL_{m},S}, and take a faithful representation π_{1}(S) ↪ P GL_{m}(ℚ) corresponding to the point with all cluster X -coordinates equal to 1.

Question 9.8 Is it true that the above construction provides an inclusion π_{1}(S) ↪ P GL_{m}(ℤ), and every integral point of the Teichmüller space X_{P GL_{m},S}^{+} appears this way?

Example. The cluster coordinate system assigned to an ideal triangulation of S provides a map given as a composition π_{1}(S) ↪ P SL_{2}(ℤ) ↪ P SL_{m}(ℤ), where the first map comes from Proposition 9.6, and the second is given by the irreducible m-dimensional representation of SL_{2}.

10 The classical Teichmüller spaces

1. Proof of Theorem 1.10a). Let p be an point of X_{P GL_{2},S}^{+}. Let us assign to p the corresponding representation ρ_{p} : π_{1}(S) → P SL_{2}(ℝ). By Theorem 1.9 we get a discrete faithful representation of π_{1}(S), and hence a point of the classical Teichmüller space. A framed structure for ρ_{p} is equivalent to choice of orientations of all holes in S. Indeed, a framed structure near a hole is given by choice of a point on ℝP^{1} invariant under the monodromy M_{hole} around the hole. To see this, we identify ℝP^{1} with the absolute of H. Then the M_{hole}-invariant points are the ends of the unique M_{hole}-invariant geodesic covering the boundary geodesic on the surface H/ρ_{p}(π_{1}(S)). (An element M_{hole} ∈ P SL_{2}(ℝ) is defined up to a conjugation by the image of π_{1}(S)). So choosing one of them we orient the geodesic towards the chosen point. So we get an inclusion

A : X_{P GL_{2},S}^{+} ↪ T_{S}^{+}

Similarly there is a tautological inclusion B left inverse to A:

B : T_{S}^{+} ↪ X_{P GL_{2},S}(ℝ), B ∘ A = Id

It remains to show that A is surjective or, equivalently, the image of the map B lies in X_{P GL_{2},S}(ℝ_{>0}). So we have to show that our coordinates on X_{P GL_{2},S}(ℝ) take positive values on the image of B.

Let S be a surface equipped with a hyperbolic metric with geodesic boundary. Choose a trivalent graph Γ on S and cut S along the sides of the dual graph Γ^{∨} into hexagons, as on Picture (135). The two hexagons sharing an edge form an octagon. It has four geodesic sides surrounding holes. Let us lift the octagon to the hyperbolic plane and continue the geodesic sides till the absolute ℝP^{1}.

(135)

The orientations of the holes induce the orientations of the geodesics on H, and hence the endpoints of these four geodesics. Therefore we get four points p_{1}, ..., p_{4} on the absolute ℝP^{1}. The orientation of S induces a cyclic order of geodesic sides of the octagon on S, and hence a cyclic order of the points p_{1}, ..., p_{4}. The key fact is that this cyclic order is compatible with the orientation of the absolute. Therefore the cross-ratio r^{+}(p_{1}, ...., p_{4}) is positive. Finally, if we assume that p_{1} is the point incident to the diagonal E of the octagon, then r^{+}(p_{1}, ...., p_{4}) is the restriction of our coordinate function X_{E}^{Γ} to X_{S}^{+}. The part a) of the theorem is proved.

Before we start the proof of the part b) of Theorem 1.10, it is instructive to consider the following toy example.

2. Describing the decorated Teichmüller space for disc with n marked points. Let V_{2} be a real two dimensional vector space with a symplectic form ω. There is canonical map s : V_{2} ⟶ S^{2}V_{2}, v ⟼ v · v. A symplectic structure in V_{2} determines a quadratic form (·, ·) in S^{2}V_{2}:

(v_{1} · v_{2}, v_{3} · v_{4}) := \frac{1}{2}(ω(v_{1}, v_{3})ω(v_{2}, v_{4}) + ω(v_{1}, v_{4})ω(v_{2}, v_{3}))

In particular

(s(v_{1}), s(v_{2})) = ω(v_{1}, v_{2})^{2}  (136)

It is of signature (2, 1). The vectors s(v) are isotropic and lie on one half of the isotropic cone in S^{2}V_{2}, denoted Q_{+}. We identify Q_{+} with the quotient of V_{2} − 0 by the reflection v ⟼ −v.

Observe that the projectivisation P Q_{+} of the cone Q_{+} is identified with the projective line P V_{2} = ℝP^{1}. So an orientation of the vector space V_{2} induces an orientation of P Q_{+}. A point q on the cone Q_{+} projects to the point \overline{q} ∈ P Q_{+}. A configuration (q_{1}, ..., q_{n}) of n points on the cone Q_{+} is positive if the corresponding projective configuration of points (\overline{q}_{1}, ..., \overline{q}_{n}) in P Q_{+} = P V_{2} is positive, i.e. its cyclic order coincides with the one induced by the orientation of the projective line P V_{2}.

Denote by Conf_{n}^{+}(Q_{+}) the space positive configurations of n points (q_{1}, ..., q_{n}) on the cone Q_{+}, and by CConf_{n}^{+}(Q_{+}) the corresponding space of cyclic positive configurations.

Recall that the space { \atop Conf}_{n}(V_{2}) of twisted cyclic configurations of n vectors in the vector space V_{2}. Let { \atop Conf}_{n}^{+}(V_{2}) be the set of ℝ_{>0}-points for the positive atlas defined in Section 7.

Proposition 10.1 There is canonical isomorphism

CConf_{n}^{+}(Q_{+}) ⟶^{∼} { \atop Conf}_{n}^{+}(V_{2})  (137)

Proof. Let us interpret the space CConf_{n}^{+}(Q_{+}) in terms of the real vector space V_{2}. Since Q_{+} = (V_{2} −{0})/±1, a configuration of n points in Q_{+} is the same thing as a P SL_{2}(ℝ)-configuration (±v_{1}, ..., ±v_{n}) of n non-zero vectors in V_{2}, each considered up to a sign. Now starting from such a cyclic P SL_{2}(ℝ)-configuration (±v_{1}, ..., ±v_{n}) we make a twisted cyclic SL_{2}(ℝ)-configuration of n vectors in V_{2} as follows. Consider a one dimensional subspace L in V_{2} which does not contain any of the vectors ±v_{i}. Let L′ be one of the two connected components of V_{2} − L. Let v′_{i} be the one of the two vectors ±v_{i} located in L′. We may assume that (v′_{1}, ..., v′_{n}) is the order of the vectors v′_{i} in L′: otherwise we change cyclically the numerations of the vectors v_{i}. Then the correspondence

(±v_{1}, ..., ±v_{n}) ⟶ (v′_{1}, ..., v′_{n})

provides map (137). To check that it is well defined notice that if we move the line L so that it crosses the line spanned by the vector v_{1}, but not v_{2}, then we will get a configuration (v′_{2}, ..., v′_{n}, −v′_{1}) which is twisted cyclic equivalent to (v′_{1}, ..., v′_{n}). The proposition is proved.

The isomorphism (137) transforms the real coordinates |Δ(±v_{i}, ±v_{j})|, i < j, on the left to the positive coordinates Δ(v′_{i}, v′_{j}), i < j, on the right.

3. Proof of Theorem 1.10b). Let us realize the hyperbolic plane as the sheet of the hyperboloid

−x_{1}^{2} − x_{2}^{2} + x_{3}^{2} = 1, x_{3} > 0

Denote by < x, x′ > the symmetric bilinear form corresponding to the quadratic form −x_{1}^{2} − x_{2}^{2} + x_{3}^{2}. The horocycles on the hyperbolic plane are described by vectors x on the cone Q_{+}, defined by the equation < x, x >= 0 in the upper half space x_{3} > 0. Namely, such a vector x provides a horocycle

h_{x} := {ξ | < ξ, x >= 1, < ξ, ξ >= 1, ξ_{3} > 0}

The distance ρ(h_{x}, h_{x′}) between the two horocycles h_{x} and h_{x′} is defined as follows. Take the unique geodesic perpendicular to both horocycles. The horocycles cut out on it a geodesic segment of finite length. Then ρ(h_{x}, h_{x′}) is the length of this segment taken with the sign plus if the segment is outside of the discs inscribed into the horocycles, and with the sign minus otherwise, see the picture.

Observe that < x, x′ > is positive unless x is proportional to x′. The distance between the two horocycles can be computed by the formula

ρ(h_{x}, h_{x′}) = \frac{1}{2} log < x, x′ >  (138)

Figure 47: The distance between the horocycles can be any real number

Now take a hyperbolic topological surface S with n punctures equipped with a complete hyperbolic metric g. Its universal cover is the hyperbolic plane. The punctures are described by π_{1}(S)–orbits of the arrows on the cone < x, x >= 0, x_{3} > 0.

According to Penner [P1] a decoration of (S, g) is given by choice of a horocycle h_{i} for each of the punctures s_{i}. Let A_{S}^{+} be the moduli space of decorated complete hyperbolic surfaces of given topological type S. Penner’s coordinates a_{E}^{Γ} on it are defined using a trivalent graph Γ on S and parametrised by the set {E} of edges of Γ. Namely, consider the ideal triangulation of S dual to the graph Γ. For each side of the triangulation there is a unique geodesic between the punctures isotopic to the side. The coordinate a_{E}^{Γ} is the length of the geodesic γ_{E} of the triangulation dual to E.

We can interpret this definition as follows. Choose a lift γ̃_{E} of the geodesic γ_{E} on the hyperbolic plane. It is defined up to the action of π_{1}(S). Let l_{1} and l_{2} be the arrows on the cone < x, x >= 0 corresponding to the boundary of γ̃_{E}. So the plane spanned by l_{1} and l_{2} intersects the hyperboloid by a geodesic γ̃_{E}. The arrows l_{1}, l_{2} project to the punctures s_{1}, s_{2} bounding γ_{E}. A choice of horocycle(s) at the puncture s_{i}, i = 1, 2, is equivalent to the choice of vector(s) x_{i} on the arrow l_{i}.

We need to show that the coordinates a_{i}^{Γ} coincide with our Δ–coordinates. We can spell the definition of the coordinates a_{i}^{Γ} as follows. Choose an edge E of the ribbon graph Γ. Let F_{1} and F_{2} be the two face paths sharing the edge E. Then the decoration on a local P SL_{2}(ℝ)–system L on S provides vectors ±v_{1} and ±v_{2}, each defined only up to a sign, in the fiber L_{E} of the local system L over the edge E. The vectors ±v_{i} are invariant up to a sign under the monodromy along the face path F_{i}. Then we assign to the edge E the coordinate

Δ_{E}^{Γ} := |Δ(v_{1}, v_{2})|

Observe that (136) implies Δ(v_{1}, v_{2})^{2} = (s(v_{1}), s(v_{2})). So thanks to (138)

a_{E}^{Γ} = log |Δ(v_{1}, v_{2})| = log |Δ_{E}^{Γ}|

It is well known that the monodromy along the conjugacy classes determined by a punctures are unipotent. The theorem is proved.

11 Laminations and canonical pairings

In this Section we give a geometric definition of two types of laminations, the A- and X -laminations, with integral, rational and real coefficients, on a punctured surface S. Our rational A-laminations are essentially the same as Thurston’s laminations [Th], which are usually considered for a surface S without the boundary. Our definition of A-laminations differs in the treatment of the holes, and more convenient for our purposes.

Our main results about the laminations are the following:

i) The rational and real lamination spaces are canonically isomorphic to the points of the positive moduli spaces A_{SL_{2},S} and X_{P SL_{2},S} with values in the tropical semifields ℚ^{t}, ℝ^{t}. The same is true for the integral X -laminations, while the set of integral A-laminations contains A_{SL_{2},S}(ℤ^{t}) as an explicitly described subset.

ii) The real A-lamination space is canonically isomorphic to the space of Thurston’s transversally measured laminations [Th], [PH].

iii) We define canonical pairing between the real A- (resp. X -) laminations and the X - (resp. A-) Teichmüller spaces.

iv) We define canonical map from the integral A- (resp. X -) laminations to functions on the X - (resp. A-) moduli spaces. Its image consists of functions on the moduli space which are Laurent polynomials with positive integral coefficients in every of our coordinate systems. We conjecture that the image of this map provides a canonical basis in the space of such functions, and that the canonical map admits a natural q-deformation.

It is worth mentioning that the definitions of laminations are similar to the definitions of the singular homology groups, and in a sense is an unoriented version of the latter. There are two different ways to define the notion of laminations for surfaces with boundary, which are analogous to the definition of homology group with compact and closed support respectively.

Throughout this section S is a surface with n > 0 holes, Γ is a trivalent graph embedded into S in such a way that S is retractable onto it, and ε_{ij}, where i and j run through the set of edges of Γ, is a skew-symmetric matrix given by

ε_{ij} = (number of vertices where the edge i is to the left from the edge j) −
− (number of vertices where the edge i is to the right from the edge j) .

The entries of ε_{ij} take five possible values: ±1, ±2 or 0.

1. A- and X -laminations and tropicalisations of the A- and X - moduli spaces.

Definition 11.1 Rational A-lamination on a surface S is a homotopy class of a collection of finite number of self- and mutually nonintersecting unoriented closed curves with rational weights, subject to the following conditions and equivalence relations.

1. Weights of all curves are positive, unless a curve surrounds a hole.

2. A lamination containing a curve of zero weight is equivalent to that with this curve removed.

3. A lamination containing two homotopy equivalent curves of weights a and b is equivalent to a lamination with one of these curves removed and with the weight a + b on the other.

The set of all rational A-laminations on a surface S is denoted by A_{L}(S, ℚ). It contains a subset A_{L}(S, ℤ) of integral A-laminations, defined as rational A-lamination with positive integral weights.

Any rational A-lamination can be represented by a collection of 3g − 3 + n curves. Any integral A-lamination can be represented by a finite collection of curves with unit weight.

Definition 11.2 Rational X -lamination on a surface S is a pair consisting of

a) A homotopy class of a collection of finite number of nonselfintersecting and pairwise nonintersecting curves either closed or connecting two boundary components (possibly coinciding) with positive rational weights assigned to each curve and subject to the following equivalence relations:

1. A lamination containing a curve retractable to a boundary component is equivalent to the one with this curve removed.

2. A lamination containing a curve of zero weight is equivalent to the one with this curve removed.

3. A lamination containing two homotopy equivalent curves of weights a and b is equivalent to the lamination with one of these curves removed and with the weight a + b on the other.

b) A choice of orientations of all boundary components but those nonintersecting with curves of the lamination.

Denote the space of rational X -laminations on S by X_{L}(S, ℚ). It contains a subset X_{L}(S, ℤ) of integral X -laminations, defined as rational X -laminations representable by collections of curves with integral weights.

Any rational X -lamination can be represented by a collection of no more than 6g − 6 + 2n curves (for the Euler characteristic reasons). Any integral X -lamination can be represented by a finite collection of curves with unit weights.

Theorem 11.3 There are canonical isomorphisms

A_{SL_{2},S}(ℚ^{t}) =^{∼} A_{L}(S, ℚ), X_{P SL_{2},S}(ℚ^{t}) =^{∼} X_{L}(S, ℚ)

The proof of this theorem will occupy the rest of this subsection. The idea is this. We are going to introduce natural coordinates on the space of A- and X -laminations corresponding to a given isotopy class of a trivalent graph Γ on S, which is homotopy equivalent to S. Our construction is a modification of Thurston’s ”train tracks” ([Th], section 9 and [PH]), and borrows from [F] with some modifications. Then we observe that the transformation rules for this coordinates under the flips are precisely the tropicalisations of the corresponding transformation rules for the canonical coordinates on the A- and X -moduli spaces. This proves the theorem. Let us start implementation of this plan.

Construction of coordinates for A-laminations. We are going to assign to a given rational Alamination rational numbers on the edges of the graph Γ and show that these numbers are coordinates on the space of laminations.

Retract the lamination to the graph in such a way that each curve retracts to a path without folds on edges of the graph, and no curve goes along an edge and then, without visiting another edge, back. Assign to each edge i the sum of weights of curves, going through it (Picture (139)), divided by two. The collection of these numbers, one for each edge of Γ, is the desired set of coordinates.

(139)

Remark. We divided the natural coordinates of A-laminations by two to simplify some formulas for the multiplicative canonical pairing, see below.

Reconstruction of A-laminations. We need to prove that these numbers are coordinates indeed. For this purpose we describe an inverse construction which gives a lamination starting from the numbers on edges.

First of all note that if we are able to reconstruct a lamination corresponding to a set of numbers {a_{i}}, we can do this as well for the set {ra_{i}} and {a_{i} + t} for any rational r ≥ 0 and t. Indeed, multiplication of all numbers by r can be achieved by multiplication of all weights by r and adding t is obtained by adding loops with weight t/2 around each hole. Therefore we can reduce our problem to the case when {a_{i}} are positive integers and any three numbers a_{1}, a_{2}, a_{3} on the edges incident to each given vertex satisfy the following triangle and parity conditions

|a_{1} − a_{2}| ≤ a_{3} ≤ a_{1} + a_{2}  (140)
a_{1} + a_{2} + a_{3} is an integer  (141)

Now the reconstruction of the lamination is almost obvious. Draw a_{i} lines on the i-th edge and connect these lines at vertices in an nonintersecting way (Picture (142)), which can be done unambiguously.

Graph change for A-laminations. The flip of the edge k changes the coordinates according to the formula:

a_{i} → \{ \begin{matrix}a_{i} & if & k ≠ i, \\ −a_{k} + max( ∑_{j|ε_{kj}>0} ε_{kj}a_{j}, − ∑_{j|ε_{kj}<0} ε_{kj}a_{j}) & if & k = i, \end{matrix}

Or in a graphical form:

[figure: a_{1}; a_{2}; a_{0}; a_{3}; a_{4}; a_{4}; a_{3}; a_{2}; a_{1}; max(a_{1} + a_{3}, a_{2} + a_{4}) − a_{0}]  (143)

(Only part of the graph is shown here, the numbers on the other edges remain unchanged.)

Construction of coordinates for X -laminations. We are going to assign for a given point of the space X_{L}(S, ℚ) a set of rational numbers on edges of the graph Γ, and show that these numbers are global coordinates on this space.

Straightforward retraction of an X -lamination onto Γ is not good because some curves may shrink to points or finite segments. To avoid this problem, let us first rotate each oriented boundary component infinitely many times in the direction prescribed by the orientation as shown on Picture (144).

(144)

The resulting lamination can be retracted on Γ without folds. However we can get infinitely many curves going through an edge. Now assign to the edge the sum with sign of weights of curves that goes diagonally, i.e., such that being oriented turn to the left at one end of the edge and to the right at another one (in this case we choose the plus sign), or first to the right and then to the left (in this case we choose the minus sign), as shown on Picture (145).

(145)

The collection of such rational numbers, one for each edge, is the desired coordinate system on X_{L}(S, ℚ). Note that the number of curves going diagonally is always finite and therefore the numbers assigned to an edge are well defined. Indeed, consider the curves retracted on the graph. We can mark a finite segment of each non-closed curve in such a way that each of two unmarked semi-infinite rays goes only around a single face and therefore never goes diagonally along the edges. Therefore only the finite marked parts of curves contribute to the numbers on the edges.

Reconstruction of X -laminations. We need to prove that these numbers are coordinates indeed. We will do it by describing an inverse construction. Note that if we are able to construct a lamination corresponding to the set of numbers {x_{i}}, we can equally do it for the set {rx_{i}} for any rational r > 0. Therefore we can reduce our task to the case when all numbers on edges are integral. Now draw ℤ-infinitely many lines along each edge. In order to connect these lines at vertices we need to split them at each of the two ends into two ℕ-infinite bunches to connect them with the corresponding bunches of the other edges. Let us make it at the i-th edge, such that x_{i} ≤ 0 (resp. x_{i} ≥ 0), in such a way that the intersection of the right (resp. left) bunches at the both ends of the edge consist of x_{i} lines (resp. −x_{i} lines). Here the left and the right side are considered from the centre of the edge toward the corresponding end. The whole procedure is illustrated on Picture (146). The resulting collection of curves may contain infinite number of curves surrounding holes, which should be removed in accordance with the definition of an unbounded lamination.

(146)

Note that though we have started with infinite bunches of curves the resulting lamination is finite. All these curves glue together into a finite number of connected components and possibly infinite number of closed curves surrounding punctures. Indeed any curve of the lamination is either closed or goes diagonally along at least one edge. Since the total number of pieces of curves going diagonally I = | ∑_{i} x_{i}| is finite the resulting lamination contain no more than this number of connected components. (In fact the number of connected components equals I provided all numbers x_{i} are all non-positive or all nonnegative.)

Graph change for X -laminations. The flip of the edge k changes the coordinates according to the formula

x(i) → \{ \begin{matrix}−x_{i} & if & i = k, \\ x_{i} & if & ε_{ki} = 0 and k ≠ i, \\ x_{i} + max(0, x_{k}) & if & ε_{ki} > 0, \\ x_{i} − max(0, −x_{k}) & if & ε_{ki} < 0, \end{matrix}

or, in graphical form:

[figure: x_{1}; x_{2}; x_{0}; x_{3}; x_{4}; x_{4} − min(x_{0}, 0); x_{3} + max(x_{0}, 0); x_{2} − min(x_{0}, 0); x_{1} + max(x_{0}, 0); −x_{0}]  (147)

(Only part of the graph is shown here, the numbers on the other edges remain unchanged.)

End of the proof of the theorem 11.3. Recall that the positive structure on the moduli spaces A_{SL_{2},S} and X_{P SL_{2},S} has been defined using the coordinate systems corresponding to trivalent ribbon graphs Γ embedded to S. The coordinate systems on the lamination spaces are defined using the same graphs. The transformation rules (143) and (147) are tropical limits of the ones for the respective moduli spaces A_{SL_{2},S} and X_{P SL_{2},S}, as one sees comparing the pictures (143) and (147) with the pictures (148) and (149) where these rules are exhibited. The theorem is proved.

[figure: A_{1}; A_{2}; A_{0}; A_{3}; A_{4}; A_{4}; A_{3}; A_{2}; A_{1}; \frac{A_{1}A_{3}+A_{2}A_{4}}{A_{0}}]  (148)

[figure: X_{1}; X_{2}; X_{0}; X_{3}; X_{4}; X_{4}(1 + (X_{0})^{−1})^{−1}; X_{3}(1 + X_{0}); X_{2}(1 + (X_{0})^{−1})^{−1}; X_{1}(1 + X_{0}); (X_{0})^{−1}]  (149)

Relations and common properties of X_{L}(S, ℚ) and A_{L}(S, ℚ). 1. Since the transformation rules for coordinates (143) and (147) are continuous w.r.t. the standard topology of ℚ^{n} the coordinates define a natural topology on the lamination spaces. One can now define the spaces of real laminations as a completion of the corresponding spaces of rational laminations. These spaces are denoted as A_{L}(S, ℝ) and X_{L}(S, ℝ), respectively. Of course we have the coordinate systems on these spaces automatically. Evidently there are canonical isomorphisms

A_{SL_{2},S}(ℝ^{t}) =^{∼} A_{L}(S, ℝ), X_{P SL_{2},S}(ℝ^{t}) =^{∼} X_{L}(S, ℝ)

The constructed spaces of real laminations coincide with the spaces of transversally measured laminations introduced by W.Thurston [Th]. We’ll prove this statement below.

2. The space of A-laminations (resp. X) is covariant (resp. contravariant) functor on the category of embeddings of surfaces.

3. An X -lamination is integral if and only if it has integral coordinates. However an A-lamination is integral if and only if it has half-integral coordinates and their sum is an integer. So we have

X_{P SL_{2},S}(ℤ^{t}) =^{∼} X_{L}(S, ℤ), A_{SL_{2},S}(ℤ^{t}) ⊂ A_{L}(S, ℤ),

4. There is a canonical map

p : A_{L}(S, ℚ) → X_{L}(S, ℚ)

which removes all the curves retractable to boundary components and provide each of the boundary components with canonical orientation induced by the orientation of the surface. In coordinates this map is given by

x_{i} = ∑_{j} ε_{ij}a_{j},  (150)

5. For any given X -lamination and any given puncture one can calculate the sum

∑_{i∈γ} x_{i},  (151)

where γ is the set of edges surrounding the puncture. The absolute value of this sum gives the total weight of the curves entering the puncture, and its sign gives the orientation of the puncture.

6. For any given A-lamination and any given puncture consider the expression

max_{i∈γ}(a_{i} + a_{i′} − a_{i′′}),  (152)

where i′ is the edge next to i in counterclockwise direction and i′′ is the edge between i and i′. This expression gives the total weight of the horocycles surrounding the puncture.

2. Additive canonical pairing and intersection pairing. Additive canonical pairing is a function of two arguments. One argument is a rational (or, later on, real) lamination, which can be either A- or X -lamination, and the other is a point of the opposite type Teichmüller space, X^{+} := X_{P SL_{2},S}^{+} or A^{+} := A_{SL_{2},S}^{+}:

I : A_{L}(ℚ) × X^{+} → ℝ, I : X_{L}(ℚ) × A^{+} → ℝ,

We will also define the intersection pairing

I : A_{L}(ℚ) × X_{L}(ℚ) → ℝ

which should be thought of as a degeneration of additive pairing. Abusing notation, we shall denote the first two of them by a single letter I. We shall also denote the canonical maps p : A^{+} → X^{+} and p : A_{L} → X_{L} by one letter p.

Let us define these pairings. Below by the length of a curve on a surface with metric of curvature −1 we always mean the length of the unique geodesic isotopic to this curve.

Definition 11.4 1. Let l ∈ X_{L} be a single closed curve. Then I(l, m) is its length w.r.t the hyperbolic metric on S defined by m.

2. Let l ∈ A_{L} be a single closed curve. Then I(l, m) is equal to ± its length w.r.t the hyperbolic metric on S defined by m. The sign is + unless l surrounds a negatively oriented hole.

3. Let l be a curve connecting two punctures and m ∈ A^{+}. Then I(l, m) is the signed length of the curve l between the horocycles.

4. Let l_{1} and l_{2} be nonintersecting collections of curves. Then I(αl_{1} + βl_{2}, m) = αI(l_{1}, m) + βI(l_{2}, m), and similarly for I.

5. Let l_{1} and l_{2} be two curves. Then I(l_{1}, l_{2}) is the minimal number of intersection points between l_{1} and l_{2}.

In the proposition below by coordinates we always mean the coordinates corresponding to a certain given trivalent graph Γ such that S is retractable on Γ. All the statements are assumed to be valid for every such graph Γ. Recall the logarithms x_{i} := log X_{i} and a_{i} := log A_{i} of the canonical positive coordinates on the X - and A- Teichmüller spaces. The coordinates on the lamination spaces are also denoted by x_{i} and a_{i}.

Proposition 11.5

1. The pairings I and I are continuous.

2. I(l, p(m)) = I(p(l), m), where l ∈ A_{L} and m ∈ A^{+}.

3. If a point l of the space X_{L} has positive coordinates x_{1}, …, x_{n} and a point m of the Teichmüller A-space has coordinates a_{1}, …, a_{n}, then I(l, m) = ∑_{α} a_{α}x_{α}.

4. Let m be a point of a Teichmüller space with coordinates u_{1}, …, u_{n} with respect to a graph Γ. Let C be a positive real number. Denote by C · m a point with coordinates Cu_{1}, …, Cu_{n}. Let m_{L} be a lamination with the coordinates u_{1}, …, u_{n}. Then lim_{C→∞} I(l, C · m)/C = I(l, m_{L}).

Observe that the last property implies the properties 2 and 3 for I.

Proof. The property 2 follows from the definition and the property 1. The property 3 follows from the observation that if all coordinates of an X -laminations are positive w.r.t. a given graph Γ, all its leaves can be deformed to paths across the edges of Γ. The proof of the property 4 will be postponed to our consideration of the multiplicative pairing. Since the definition describes explicitly the value of the pairing if one argument is a rational lamination, to prove existence one needs to only to check that it is continuous.

Let us first prove the continuity of the pairing between A-laminations and the X -Teichmüller space. To do this it suffices to prove it for laminations without curves with negative weights. Indeed, if we add such curve to a lamination the length obviously changes continuously. We are going to show that the length of integral lamination is a convex function of its coordinates, i.e. that

I(l_{1}, m) + I(l_{2}, m) ≥ I(l_{1} +_{Γ} l_{2}, m)  (153)

were by l_{1} +_{Γ} l_{2} we mean a lamination with coordinates being sums of the respective coordinates of l_{1} and l_{2}. Taking into account the homogeneity of I, one sees that the inequality (153) holds for all rational laminations and therefore can be extended by continuity to all real laminations.

Let us prove the inequality (153). Draw both laminations l_{1} and l_{2} on the surface and deform them to be geodesic. These laminations in general intersect each other in finite number of points. Then retract the whole picture to the ribbon graph in such a way that no more intersection points appear and the existing ones are moved to the edges. Now it becomes obvious that at each intersection point we can rearrange our lamination cutting both intersecting curves at the intersection point and gluing them back in another order in a way to make the resulting set of curves homotopically equivalent to a nonintersecting collection. We can do it at each intersection point in two different ways, and we use the retraction to the ribbon graph to choose one of them.

Indeed, connect them as shown on Picture (154).

(154)

One can easily see that the numbers on edges, corresponding to the new lamination l are exactly the sums of the numbers, corresponding to l_{1} and l_{2}. On the other hand, the lamination on the original surface is no longer geodesic, because the curves of it have breaks. But its length is exactly I(l_{1}, m) + I(l_{2}, m). When we deform l to a geodesic lamination, its length can only decrease, what proves the inequality (153).

The proof of the continuity when the first argument is an X -lamination and the second is a point of the A-Teichmüller space is is similar. Call a pair of laminations l_{1} and l_{2} similar if at each hole both laminations l_{1} and l_{2} have the same orientation or at least one of them does not touch the hole. Once the inequality (153) is proved for similar laminations l_{1} and l_{2}, the continuity follows since the set of laminations similar to a given one is a closed subset in the space X_{L}. Conjecturally the inequality (153) holds for any pair l_{1} and l_{2}. But for a pair of similar laminations the arguments used above works: one can first forget about orientations and move the leaves in order to minimize the number of intersection points. Then one can first spiralize the leaves according to the orientations, retract the resulting collection of curves to the graph and then resolve the intersection points just as above (Picture (154)). The inequality (153) follows.

Using 3 we deduce for this the convexity of the pairing I.

The continuity implies the following

Corollary 11.6 Our real A-laminations are canonically identified with Thurston’s transversally measured laminations.

Proof. Thurston proved that rational laminations are dense in the space of all transversally measured laminations w.r.t. the topology given by the length function. Our rational A-laminations coincide with Thurston’s. Our canonical pairing of A-laminations with the Teichmüller space is nothing but Thurston’s length function. Therefore the continuity shows that our topology coincides with Thurston’s. The corollary is proved.

Corollary 11.7 The additive canonical pairings and the intersection pairing provide continuous pairings between the real lamination and Teichmuller/lamination spaces:

I : A_{L}(ℝ) × X → ℝ, I : X_{L}(ℝ) × A → ℝ, I : A_{L}(ℝ) × X_{L}(ℝ) → ℝ

We say that an element of X_{P SL_{2},S}(ℂ) is quasi-Fuchsian if the corresponding representation of π_{1}(S) is quasi-Fuchsian. The set of all quasi-Fuchsian elements form an open domain of X_{P SL_{2},S}(ℂ) which we call the quasi-Fuchsian domain.

Conjecture 11.8 Let l be a real A-lamination. Then the function I(l, ∗) is analytically continuable exactly to the quasi-Fuchsian domain of X_{P SL_{2},S}(ℂ).

3. Multiplicative canonical pairing. The multiplicative canonical pairing 𝕀 is a pairing between integral A_{SL_{2},S}-laminations and X_{P SL_{2},S}-moduli space or between integral X_{P SL_{2},S}-laminations and A_{SL_{2},S}-moduli space. These pairings are better understood as the maps

𝕀_{A} : A_{SL_{2},S}(ℤ^{t}) ⟶ ℚ(X_{P SL_{2},S}), 𝕀_{X} : X_{P SL_{2},S}(ℤ^{t}) ⟶ ℚ(A_{SL_{2},S}),

Before defining the map 𝕀_{A} we will define an auxiliary map 𝕀̃_{A} : A_{SL_{2},S}((ℤ^{t})/2) ⟶ { \atop ℚ(X_{P SL_{2},S})}, where the hat denotes certain covering of the space X_{P SL_{2},S}. Afterwards we shall check that the restriction of this pairing to A_{SL_{2},S}(ℤ^{t}) comes from X_{P SL_{2},S}, thus defining the desired pairing.

Here is our strategy. Let us say that a Laurent polynomial is sign definite if all its non zero coefficients are of the same sign. Given a sign definite Laurent polynomial we can multiply it by ±1 to make a positive Laurent polynomial out of it. We are going to define first the related maps 𝕀′_{A} and 𝕀′_{X}. Then we will show that 𝕀′_{A}(l) and 𝕀′_{X} (l) are sign definite Laurent polynomials. By definition the elements 𝕀_{A} and 𝕀_{X} are the corresponding positive Laurent polynomials. We will produce our sign definite Laurent polynomials as products of sign definite Laurent polynomials, so we allow them to be well defined only up to a sign. For example a trace of a matrix from P SL_{2}(F) is a defined up to a sign function.

Recall that a curve l connecting two punctures on S provides a well defined regular function Δ(l) on the space A_{SL_{2},S}. Further, let l be a closed curve surrounding a puncture on S. Then the monodromy operator for an framed local system L on S around l has a distinguished eigenvalue λ_{l} determined by the framed structure: the eigenvector with this eigenvalue λ_{l} generates the invariant flag assigned to this puncture.

Definition 11.9 1. Let l be a single closed curve with the weight k which does not surround a puncture. Then

i) the value of the function ±𝕀̃′_{A}(l) on the decorated P SL_{2}-local system L is the trace of the k-th power of the monodromy of L along l.

ii) the value of the function 𝕀′_{X} (l) on the framed SL_{2}-local system L is the trace of the k-th power of the monodromy of L along l.

2. Let l be a closed curve surrounding a puncture on S, and k · l the A-lamination determined by this curve and a weight k. Then 𝕀̃′_{A}(k · l) = λ_{l}^{k}.

3. Let l be a curve connecting two punctures with weight k. Let us assume that orientations of these punctures are induced by the orientation of the surface. Then 𝕀′_{X} (l) = Δ(l)^{k}.

4. Let l_{1} and l_{2} be nonintersecting collections of curves, such that no isotopy class of a curve enters both l_{1} and l_{2}. Then 𝕀′_{∗}(l_{1} + l_{2}) = 𝕀′_{∗}(l_{1})𝕀′_{∗}(l_{2}), where ∗ stays for A or X, and 𝕀̃′ is needed in the A-case.

Here we assume that we have reduced the lamination to the form without equivalent curves.

Observe that in this definition the map 𝕀′_{A}(l) is defined for all integral laminations. In the definition of the map 𝕀′_{X} (l) we are missing the laminations corresponding to curves connecting two punctures in the case when at least one of the punctures is oriented differently then the surface S. We do not know how to define the map 𝕀′_{X} (l) for those laminations.

In the theorem below by coordinates we always mean the coordinates corresponding to a certain fixed trivalent graph Γ such that S is retractable on Γ. And all the statements are assumed to be valid for every such graph Γ. The function 𝕀′_{X} (l) is restricted to those laminations l where it has been defined.

Theorem 11.10

1. Let a_{1}, …, a_{n} ∈ ℤ be coordinates of an integral A-lamination l. Then the function 𝕀̃′_{A}(l) is a sign definite Laurent polynomial with integral coefficients in the coordinates X_{1}, …, X_{n} on X. Its highest term is equal to ∏_{i} X_{i}^{a_{i}} and the lowest one is ∏_{i} X_{i}^{−a_{i}}.

2. The function 𝕀′_{X} (l) is a sign definite Laurent polynomial with integral coefficients in the coordinates A_{1}, …, A_{n} on A.

3. Let C ∈ ℤ. Then lim_{C→∞} log 𝕀(l, C · m)/C = I(l, m_{L}), where the definitions of C · m and m_{L} are the same as for the property 4 in the additive case.

4. 𝕀_{A}(l, p(m)) = 𝕀_{X} (p(l), m) where l ∈ A_{SL_{2},S}(ℤ^{t}) and m ∈ A_{SL_{2},S}.

5. 𝕀_{∗}(l_{1})𝕀_{∗}(l_{2}) = ∑_{l} c_{∗}(l_{1}, l_{2}; l)𝕀_{∗}(l), where ∗ stays for either X or A, and c_{∗}(l_{1}, l_{2}; l) are integral (and conjecturally positive) constants, such that the sum is always finite.

Observe that the property 1 implies that if an A-lamination is integral, the corresponding function is lifted from X_{P SL_{2},S}, the map 𝕀_{A}(l) is defined.

Proof. The statement 3 immediately follows from the analogous additive statement and the fact that for a curve l connecting punctures exp(I(l, m)) = 𝕀(l, m) and for a closed curve l we have cosh(I(l, m)) = 𝕀(l, m).

As a corollary one deduces the property 4 of the proposition 11.5. Indeed, the properties 1 and 3 shows that the identity lim_{C→∞} I(l, C ·m)/C = I(l, m) holds for an integral lamination l. By continuity it can be extended to any laminations.

The statement 4 is obvious if l is a loop: if this loop does not surround a puncture, both sides of the equality are equal to the trace of the monodromy of the corresponding local system; if l surrounds a puncture, it is 1 for both sides. The general case follows by the multiplicativity.

The properties 1 and 2 for the X -moduli space follow essentially from Theorem 8.13 and Corollary 8.14, valid for P SL_{m}. However for the sake of completeness of this Section, we prove them below.

Before presenting the proof of the properties 1 and 2 recall the construction of monodromy matrices from coordinates of our moduli spaces given in Sections 7 and 8, specified to the case of G = P GL_{2} and G = SL_{2}.

Let us do it first for the X -moduli space. Replace the graph Γ by another one Γ′ by gluing small triangles into each vertex and define a graph connection on it as shown on Picture (155) (The graph Γ is shown by the thin lines).

[figure: k; f_{k}; g; g; g; g; g; g]  (155)

Here f_{k} and g are given by

f_{k} = ( \begin{matrix}0 & −X_{k}^{1/2} \\ X_{k}^{−1/2} & 0 \end{matrix} ) , g = ( \begin{matrix}1 & 1 \\ −1 & 0 \end{matrix} ) .

Here X_{k}^{±1/2} are natural coordinates on a covering. The monodromy around small triangles is the unity since g^{3} = −1 in SL_{2}, i.e. g^{3} = 1 in P SL_{2}. The orientations of the edges with the matrices f_{i} do not play any role since f_{i} = f_{i}^{−1} in P SL_{2}.

Here is a similar construction for the A-moduli space. Replace the graph Γ by another one by replacing each vertex by a hexagon and define a graph connection on it as shown on Picture (156). (The graph Γ is shown by the thin lines).

[figure: k; i; j; d_{k}; d_{i}; d_{j}; u_{ijk}; u_{jki}; u_{kij}]  (156)

Here the matrices d_{k} and u_{ijk} are given by:

d_{i} = ( \begin{matrix}0 & −A_{i} \\ \frac{1}{A_{i}} & 0 \end{matrix} ) , u_{ijk} = ( \begin{matrix}1 & 0 \\ \frac{A_{k}}{A_{i}A_{j}} & 1 \end{matrix} ) .

The monodromy around the hexagons is the unity since d_{j}u_{jki}d_{k}u_{kij}d_{i}u_{ijk} = 1 in SL_{2}. The mon-

odromy around the 2-gons is −1. The monodromy along a face path fixes vector ( \begin{matrix}0 \\ 1 \end{matrix} ).

Now proceed with the proof. Due to the multiplicativity it suffices to prove statements 1 and 2 for a connected curve l. Consider first the statement 1. In this case l must be a loop. Let α be a loop on the graph Γ′ homotopic, on S, to the loop l. We may assume that the loop α contains no consecutive edges of little triangles on Γ′. Observe that

f_{k}g = ( \begin{matrix}X_{k}^{1/2} & 0 \\ X_{k}^{−1/2} & X_{k}^{−1/2} \end{matrix} ) ,  f_{k}g^{−1} = ( \begin{matrix}X_{k}^{1/2} & X_{k}^{1/2} \\ 0 & X_{k}^{−1/2} \end{matrix} )  (157)

So calculation of the monodromy around such a loop reduces to taking products of matrices of type (157), just one of them per each edge of the original graph Γ which appears in the loop α. These matrices are positive and Laurent. Evidently the trace of such monodromy operator is a positive Laurent polynomial, and the coefficients at the highest and the lowest monomial equal to 1.

Remark. In general the coordinates of an A-lamination l are half-integers a_{1}, ..., a_{n} ∈ ℤ/2, and the function 𝕀̃′_{A}(l) is defined only on a covering of the space X. However if a_{i} ∈ ℤ, the proof shows that it is defined on the space X itself. So we get the function 𝕀′_{A}(l).

Consider now the property 2. Call a matrix with Laurent-polynomial coefficients sign definite if each its entry is a Laurent polynomials with all positive or all negative coefficients. Let the type of a sign definite matrix M be a matrix |M| of the same size as M with entries equal to +, − or 0 indicating the sign of the monomials of the corresponding entries of M. Define a partial product on the set of types by the requirement, that ab = c if for any two matrices M_{1} and M_{2}, such that |M_{1}| = a and |M_{2}| = b, we have |M_{1}M_{2}|.

Consider a path l connecting two holes, retracted to the graph. The desired intersection index is given by the bottom left element of the monodromy matrix. The sequence of matrices along the path is composed out of the matrices u, u^{−1} and d. Since it connects two holes, we may assume that it starts and ends with a d. The path can be deformed to make the sequence to avoid the subsequences uu^{−1}, u^{−1}u, udu, d^{2} and u^{−1}du^{−1}. To investigate the sign properties of the product, let us replace every matrix in the sequence by its type and investigate the type of the product. Let

D := |d_{i}| = ( \begin{matrix}0 & − \\ + & 0 \end{matrix} ) , U^{+} = |u_{ijk}| = ( \begin{matrix}+ & 0 \\ + & + \end{matrix} ) , U^{−} = |(u_{ijk})^{−1}| = ( \begin{matrix}+ & 0 \\ − & + \end{matrix} )

Now observe that (U^{+})^{n} = U^{+} and (U^{−})^{n} = U^{−} for any n ∈ ℕ, and replace the sequences of U^{+} and U^{−} by single letters. the resulting sequence has one of the following four forms:

DU^{−}D(U^{+}DU^{−}D)^{k}U^{+}D, D(U^{+}DU^{−}D)^{k}U^{+}D, DU^{−}D(U^{+}DU^{−}D)^{k}, D(U^{+}DU^{−}D)^{k}

The crucial observation is this:

the products U^{+}DU^{−}D and DU^{−}DU^{+} are of the type −P = ( \begin{matrix}− & − \\ − & − \end{matrix} )  (158)

In particular the type of (U^{+}DU^{−}D)^{k} is either (−1)^{k}P. Our problem now is reduced to the verification that in each of the four cases the lower left entry is sign definite. Since DP = P D = D, this reduces to the cases D, DU^{+}D, DU^{−}D, D, which is an easy exercise.

Let us prove the statement for a closed loop l. In this case 𝕀′_{X} (l) is the trace of a cyclic word of type ...DU^{±}DU^{∓}DU^{±}DU^{∓}DU^{±}. We will assume that its length is minimal possible for a given loop l. So the following cases may occur:

i) The word is U^{+}, U^{−} or D. These are trivial cases.

ii) The cyclic word is DU^{+} or DU^{−}. One easily checks that in this case the trace is sign-definite. iii) The cyclic word has the length greater then two. Then the length is even: otherwise the

length of the cyclic word can be reduced. Thus the word contains U^{+}. Since the word is cyclic, we may assume that it starts from U^{+}. Now the claim follows from (158) and ii).

Let us prove the property 5. We start with the the second part of the property. For this we introduce the notion of quasi-lamination. A quasi-lamination is a collection of curves, connecting punctures or closed. Unlike the laminations, they can intersect or have self-intersections. One can define the canonical multiplicative pairings for quasi-laminations by the same rules as for laminations for single curves and extend it for the collections of curves by multiplicativity, i.e., by definition 𝕀_{∗}(l_{1})𝕀_{∗}(l_{2}) = 𝕀_{∗}(l), where l is a quasi-lamination given by the union of l_{1} and l_{2}. The functions on the moduli spaces corresponding to quasi-laminations satisfy the so-called linear skin relations. Using these relations one can represent any function corresponding to a quasi-lamination as an integral linear combination of similar functions corresponding to laminations, thus proving the property. Let us describe the relations. Let x be an intersection or self-intersection point of a quasi-lamination l. The lamination in the vicinity of x looks like a cross . Let l_{1} and l_{2} be the quasi-laminations obtained by replacing this cross by [figure] and [figure] respectively. The relation is

I(m, l) ± I(m, l_{1}) ± I(m, l_{1}) = 0,

for some choice of signs depending on l, l_{1}, l_{2}. The verification of these relations is a straightforward corollary of the identities for 2 × 2 matrices:

trAtrB = trAB + trAB^{−1}
trAC B_{21} = (ABC)_{21} + (AB^{−1}C)_{21}
(AC)_{21}(BD)_{21} = (AD)_{21}(BC)_{21} − (AB^{−1})_{21}(D^{−1}C)_{21}

Here A_{21} is the left bottom matrix element of a 2 × 2 matrix A.

Since l_{1} as well as l_{2} have one intersection point less than l has, repeating this procedure one can express the pairing with any quasi-lamination by a linear combinations of pairings with quasilaminations without intersection points. Now to reduce the pairing to the pairing with a lamination we need to remove contractible curves and replace equivalent curves by one curve with multiplicity. The first procedure can be done by the obvious statement that the pairing with a closed curve is equal to the constant 2. The second procedure is a corollary of the fact that trA^{k} = P_{k}(trA), where P_{k} is the k-th Tchebychev polynomial.

The proof of the second part of the property is absolutely similar and even simpler, since we don’t need to consider curves connecting punctures. The theorem is proved.

5. Conjectures. Let L be a set. Denote by ℤ_{+}{L} the abelian semigroup generated by L. Its elements are expressions ∑_{i} n_{i}{l_{i}} where l_{i} ≥ 0, the sum is finite, and {l_{i}} is the generator corresponding to an element l_{i} ∈ L. Similarly denote by ℤ{L} the abelian group generated by L.

Let X be a positive space. Recall from Section 4.3 the ring 𝕃[X] of all good Laurent polynomials, the semiring 𝕃_{+}[X] of positive good Laurent polynomials and the set of extremal elements E(X).

It follows from Definition 11.9 and Proposition 11.10 that the canonical multiplicative pairings provide canonical maps

𝕀_{A}^{+} : ℤ_{+}{A_{SL_{2},S}(ℤ^{t})} ⟶ 𝕃_{+}[X_{P SL_{2},S}],  𝕀_{X}^{+} : ℤ_{+}{X_{P SL_{2},S}(ℤ^{t})} ⟶ 𝕃_{+}[A_{SL_{2},S}],  (159)

and hence

𝕀_{A} : ℤ{A_{SL_{2},S}(ℤ^{t})} ⟶ 𝕃[X_{P SL_{2},S}],  𝕀_{X} : ℤ{X_{P SL_{2},S}(ℤ^{t})} ⟶ 𝕃[A_{SL_{2},S}],  (160)

which are multiplicative in the sense that property 4. of Definition 11.9 holds. Conjecturally the polynomials provided by laminations in both cases are minimal, i.e., any difference of such polynomials have at least one negative coefficient. Moreover, we conjecture the following.

Conjecture 11.11 The maps (159) and (160) are isomorphisms. They provide the isomorphisms

A_{SL_{2},S}(ℤ^{t}) = E(X_{P SL_{2},S}), X_{P SL_{2},S}(ℤ^{t}) = E(A_{SL_{2},S})

In particular this means that, both in A and X cases, any good Laurent polynomial is a linear combination with integer coefficients of the images of the corresponding type laminations under the canonical map. For positive good Laurent polynomials all the coefficients are positive.

Observe that if G is simply-connected then the Langlands dual ^{L}G always has trivial center, so the corresponding moduli space X_{^{L}G,S} has the positive structure defined in Section 6.

Conjecture 11.12 Let G be a connected, simply-connected, split semisimple algebraic group. Then a) There exist canonical additive pairings between the real lamination and Teichmüller spaces

I : A_{G,S}(ℝ^{t}) × X_{^{L}G,S}^{+} → ℝ, I : X_{^{L}G,S}(ℝ^{t}) × A_{G,S}^{+} → ℝ,

as well as the intersection pairing between the real lamination spaces

I : A_{G,S}(ℝ^{t}) × X_{^{L}G,S}(ℝ^{t}) → ℝ,

All of them are continuous.

b) There exist canonical isomorphisms

𝕀_{A}^{+} : ℤ_{+}{A_{G,S}(ℤ^{t})} ⟶ 𝕃_{+}[X_{^{L}G,S}],  𝕀_{X}^{+} : ℤ_{+}{X_{^{L}G,S}(ℤ^{t})} ⟶ 𝕃_{+}[A_{G,S}],  (161)
𝕀_{A} : ℤ{A_{G,S}(ℤ^{t})} ⟶ 𝕃[X_{^{L}G,S}],  𝕀_{X} : ℤ{X_{^{L}G,S}(ℤ^{t})} ⟶ 𝕃[A_{SL_{2},S}],  (162)

They provide the isomorphisms

A_{G,S}(ℤ^{t}) = E(X_{^{L}G,S}), X_{^{L}G,S}(ℤ^{t}) = E(A_{G,S})

When G = SL_{2} we have ^{L}G = P SL_{2}, so Conjecture 11.12 reduces to Conjecture 11.11.

Quantum canonical map. We use notations from [FG2]. So X_{P SL_{2},S}^{q} denotes the q-deformation of the moduli space X_{P SL_{2},S}, defined by gluing the quantum tori given by the generators X_{i} and relations

q^{−ε_{ij}}X_{i}X_{j} = q^{−ε_{ji}}X_{j}X_{i}

Conjecture 11.13 There exists quantum canonical map

𝕀̂^{q} : A_{SL_{2},S}(ℤ^{t}) ⟶ X_{P SL_{2},S}^{q}

In other words for any positive integral A-lamination l one can associate a Laurent polynomial 𝕀̂^{q}(l) of the variables X_{1}, …, X_{n}, satisfying the following properties:

1. 𝕀̂^{1}(l) = 𝕀_{A}(l)

2. The highest term of 𝕀̂^{q}(l) is q^{− ∑_{i<j} ε_{ij}a_{i}a_{j}} ∏_{i} X_{i}^{a_{i}}.

3. All coefficients of 𝕀̂^{q}(l) are positive Laurent polynomials of q.

4. Let ∗ be the canonical involutive antiautomorphism on the fraction field ℚ(X_{P SL_{2},S}^{q}). Then ∗𝕀̂^{q}(l) = 𝕀̂^{q}(l).

5. 𝕀̂^{q}(l_{1})𝕀̂^{q}(l_{2}) = ∑_{l} c^{q}(l_{1}, l_{2}; l)𝕀̂^{q}(l), where the c^{q}(l_{1}, l_{2}; l) are Laurent polynomials of q with positive integral coefficients and such that the sum is finite.

Of course, we expect a similar conjecture for an arbitrary G. The most general and precise conjectures are formulated in the set up of cluster ensembles, see Section 4 of [FG2]. However if S is any hyperbolic surface but the punctured torus, the positive atlases defined in this paper for the pair of moduli spaces are different then the ones for the cluster ensemble, and therefore the corresponding conjectures describe different objects. All conjectures formulated in Section 4 of [FG2] in the framework of cluster ensembles are easily rephrased in the framework of the pair of positive spaces (X_{G,S}, A_{G,S}), and expected to be valid in this framework.

The exists a generalization of these conjectures where S is replaced by Ŝ.

12 Positive coordinate systems for X -laminations

0. Extended positive atlas on X_{G,S}.

1. Positive coordinate systems: definitions and the main result. Let C_{S} be the set of pairs (an ideal triangulation of S considered modulo isotopy, a choice of orientations of boundaries of all holes of S). The direct product of the mapping class group of S and the group (ℤ/2ℤ)^{n} acts naturally on the set C_{S}: the factor (ℤ/2ℤ)^{n} acts by changing orientations of holes. The coordinate systems on the space of X -laminations on S are parametrized by the set C_{S}. So there are 2^{n} coordinate systems corresponding to an ideal triangulation of S.

Definition 12.1 Let l be an X -lamination on S. A coordinate system corresponding to an ideal triangulation of S is positive (non-negative) for the lamination l, if all the coordinates of l in this coordinate system are > 0 (respectively ≥ 0).

Theorem 12.2 a) There exists a dense (resp. dense open) subset in the set of all real X -laminations on S such that every lamination from it admits a non-negative (resp. positive) coordinate system.

b) Let T be an ideal triangulation of S corresponding to a non-negative coordinate system for a finite real X -lamination l. Then the isotopy class of the collection of edges of T with > 0 coordinates is isotopic to l, and hence uniquely determined by l.

c) A positive coordinate system for an X -lamination is unique if exists.

To get the part a) of Theorem 12.2 we need the following result: for generic real X -lamination all curves of the lamination go between the punctures. For this we use some results from the theory of Morse-Smale foliations.

2. Morse-Smale foliations and real laminations. Let us recall the notion of a Morse-Smale foliation on S and some results about them.

Given a foliation on a punctured disc, the winding number of the foliation at the puncture is an integer given by the the rotation number of the fibers of the foliation restricted to a small circle near the puncture. For example the radial foliation has the winding number +1, and the foliation tangent to the family of concentric circles also has the winding number +1.

Let S be a surface with holes. A Morse-Smale foliation on S is a foliation F on S minus a finite number of points, called singular points of the foliation, with the following structures/properties:

i) F is equipped with a real transversal measure.

ii) The fibers of F are transversal to the boundary of S.

iii) The winding number of the foliation at each singular point is < 0.

A fiber of a Morse-Smale foliation on S is singular if it ends at a singular point of the foliation. The condition iii) implies that the number of singular fibers of a Morse-Smale foliation ending in a singular point is finite. The condition ii) easily implies that a Morse-Smale foliation does not have fibers contractible onto the boundary.

Lemma 12.3 The number of homotopy types of the leaves of a Morse-Smale foliation going between the boundary components is finite.

Proof. The above remarks show that we may assume that the fibers can not be contracted to the boundary. Cutting the surface along a non-singular leaf going between the boundary components we decrease either the number of holes, or the genus. The lemma follows.

We need the following result.

Theorem 12.4 There exists an open dense domain U_{MS}(S) in the space of all Morse-Smale foliations on S such that every non-singular curve of a foliation from this domain joins two boundary components.

Now let us replace the homotopic non-singular fibers of a Morse-Smale foliation F by a single curve with the weight provided by the transversal measure. According to a well known theorem of Thurston, this way we get a bijection between the Morse-Smale foliation on S and Thurston’s ℝ-laminations: every ℝ-lamination is representable as a Morse-Smale foliation on S, and this representation is unique: if two laminations correspond to two Morse-Smale foliations, these foliations are isotopic. It follows from Lemma 12.3 that if F was from the set U_{MS}(S), we will get a Thurston ℝ-lamination l_{F} with a finite number of curves.

3. Proof of Theorem 12.2. Obviously c) is a particular case of b). Moreover clearly that in b) not only the set of the edges of T carrying positive coordinates, but also the set of the orientations of the holes serving as the endpoints of these edges is determined by l.

a) We will assume without the loss of generality that an X -lamination l is defined using the orientations of the holes induced by the orientation of the surface S. The general case is reduced to this using the action of the group (ℤ/2ℤ)^{n} changing the orientations of the holes.

A real X -lamination is called finite if it can be presented by a finite collection l of mutually and self non-intersecting curves with real weights on them. Recall that to get an X -lamination out of such a collection of curves with real weights l we have to choose in addition orientations of all holes. As we already said, we do it by taking the orientations induced by the orientation of the surface S, getting an X -lamination l^{+}. To define the X -coordinates of the finite X -lamination l^{+} we have to perform the following procedure at every hole of S: we replace a segment of the lamination going to the hole by a ray winding infinitely many times around the puncture in the direction prescribed by the orientation of S, and then retract the winded lamination to a trivalent graph on S.

Let us call finite real X -laminations such that every leaf of the lamination joins two boundary component special real X -laminations. Theorem 12.4 and Lemma 12.3 imply that the space of special finite real X -laminations is an open dense subset in the space of all real X -laminations on S.

Now, given a special X -lamination l^{+}, let us exhibit a non-negative triangulation of S for it. The curves of the finite lamination l provide a decomposition D_{l} of S. Adding a finite number of curves to this decomposition we obtain a triangulation T of S containing the decomposition D_{l}. Let Γ be the dual graph for this triangulation. We claim that it provides a non-negative coordinate system for the X -lamination l^{+}: all coordinates assigned to the edges of D_{l} are > 0, while the rest of the coordinates equal to zero. Indeed, a curve of the lamination, being retracted onto the graph Γ, will cross the only one edge of Γ, the one dual to the edge of the triangulation T given by this curve. Figure 48 illustrates this claim, by showing the winding procedure for a curve of the lamination on S traveling between the holes. The part a) of the theorem is proved.

Figure 48: Calculating a coordinate of an X -lamination assigned to an edge of the dual graph b) Given a special real X -lamination l, there is the following correspondence

I_{l} ⊂ {edges of Γ} × {connected components of the lamination l}  (163)

Namely, a pair (an edge e of Γ, a connected component α of l) belongs to I_{l} if and only if there exists a part of α which goes diagonally along the edge e. Let us denote by π_{2} the natural projection of the correspondence I_{l} to the second factor in (163):

π_{2} : I_{l} ⟶ {connected components of the lamination l}

Lemma 12.5 Let l be a special real X -lamination on S. Then:

a) The map π_{2} is always surjective.

b) If π_{2}^{−1}(α) is a single edge e, then this edge is isotopic to an edge of the dual triangulation T.

c) If all coordinates of l are ≥ 0, then the map π_{2} is injective.

Proof. a) Take a connected component β of l which is not in the image of π_{2}. This means that there is no edge of Γ such that β goes diagonally along this edge. Thus β can be shrunken into the boundary of a hole. Indeed, β either turn right at every edge of Γ, or it turns left at every edge of Γ, and thus it is isotopic to a curve which does not intersect Γ.

b) If π_{2}^{−1}(α) = e, then α is homotopic to a curve which intersects Γ just once. Indeed, it either goes left before e and right after e, or right before and left after e. The claim follows from this.

c) Assume the opposite. Then there is a connected component α of l such that |π_{2}^{−1}(α)| > 1. But this is impossible: after the curve α crossed diagonally the first edge on its way it can not cross the other edge since all its coordinates are ≥ 0.

The lemma is proved.

Corollary 12.6 Let l be a special real X -lamination on S. Suppose that all coordinates of l in the coordinate system corresponding to an ideal triangulation T are ≥ 0. Then the set of the edges of T which carry > 0 coordinates of l is isotopic to the lamination l.

Proof. Follows immediately from the parts b) and c) of Lemma 12.5.

The part b) of the theorem is an immediate consequence of Corollary 12.6. The theorem is proved.

Corollary 12.7 a) Any rational X -lamination without loops admits a non-negative coordinate system.

b) The set of rational X -laminations without loops is dense in the space X_{L}(S; ℚ) of all rational X -laminations.

Proof. a) It has been proved during the proof of Theorem 12.2.

b) The intersection of the domain described in Theorem 12.2 with the space of all rational X - laminations is described as the set of the ones without closed loops. The corollary is proved.

So a little perturbation destroys loops in a lamination.

It is tempting to avoid the use of the Morse-Smale theory in the proof, making the proof selfcontained. Here is perhaps a first step in this direction.

Lemma 12.8 Let l be a real X -lamination and α be a closed leaf of it. Take any 3-valent graph and let now x_{i} be the coordinates of l and a_{i} be the coordinates of α considered as an A-lamination. Then ∑_{i} x_{i}a_{i} = 0.

Proof. The intersection pairing of α with l equals zero. On the other hand it is computed by the formula ∑_{i} x_{i}a_{i}. The lemma is proved.

This lemma implies that since all a_{i}’s are integers if all x_{i}’s are linearly independent over ℚ, there are no closed curves. However this does not prove yet that for generic real X -lamination all curves of the lamination go between the punctures: we have to take care of the non-closed curves winding inside of S.

4. An application to the canonical map 𝕀_{X}. It follows from the part a) of Corollary 12.7 that restriction of the canonical map 𝕀_{X} to the integral laminations without loops is especially simple. Namely, given such a lamination l, in every non-negative coordinate system for l the element 𝕀_{X} (l) is given by a product of the coordinate functions in this coordinate system. So 𝕀_{X} (l) is in the (orbi)-cluster algebra corresponding to the moduli space A_{SL_{∈},S}. On the other hand, the image of laminations containing loops does not belong to it.

13 A Weil–Petersson form on A_{G,Ŝ} and its motivic avatar

1. The K_{2}-invariant of a G–local system on a compact surface. Let S be a compact smooth oriented surface of genus g > 1. A representation ρ : π_{1}(S, x) → G(F) of π_{1}(S, x) to the group of F –points of a reductive group G induces a map of the classifying spaces ρ_{B} : Bπ_{1}(S, x) → BG(F). Since S is a K(π_{1}(S, x)-space, applying the functor H_{2} we get a map

ρ_{∗} : H_{2}(S) = H_{2}(Bπ_{1}(S, x)) ⟶ H_{2}BG(F) = H_{2}G(F)

According to Suslin’s stabilization theorem [Sus] the natural map H_{2}GL_{2}(F) → H_{2}GL_{n}(F) is an isomorphism for any infinite field F provided n ≥ 2. There is a natural map H_{2}GL_{2}(F) → K_{2}(F) (see e.g. loc. cit.). Thus if n ≥ 2 there is a map π_{n} : H_{2}GL_{n}(F) → K_{2}(F). A finite dimensional representation V of G provides a map BG → BGL(V). Composing it with π_{dimV} we get a homomorphism π_{G,V} : H_{2}G(F) → K_{2}(F). Let

π_{G} : H_{2}G(F) → K_{2}(F)

be the homomorphism corresponding to the adjoint representation of G. Composing it with ρ_{∗} we get a map

π_{G} ∘ ρ_{∗} : H_{2}(S) ⟶ K_{2}(F)

Let [S] ∈ H_{2}(S) be the fundamental class of S.

Definition 13.1 Let G be a split reductive group and F is a field. The K_{2}–invariant

w_{ρ} ∈ K_{2}(F)

of a representation of ρ : π_{1}(S) → G(F) is the element element π_{G} ∘ ρ_{∗}([S]) ∈ K_{2}(F).

Since the mapping class group Γ_{S} preserves the fundamental class of S, w_{ρ} is invariant under the action of Γ_{S}, i.e. w_{g}ρ = w_{ρ} for g ∈ Γ_{S}.

Applying this to the field 𝔽 of rational functions on the moduli space L_{G,S} and the corresponding tautological (or universal) representation π_{1}(S, x) → G(𝔽) we get a Γ_{S}-invariant class W_{G,S} ∈ K_{2}(𝔽). The restriction of this class to any point ρ ∈ L_{G,S} is the class w_{ρ}. Therefore the class W_{G,S} is regular on L_{G,S}, i.e. provides an element

W_{G,S} ∈ H^{2}(L_{G,S}, ℚ_{M}(2))^{Γ_{S}} ⊂ K_{2}(𝔽)^{Γ_{S}}

Applying to it the map d log we a regular 2-form Ω_{G,S} on L_{G,S}.

2. The weight two motivic complex. Let F be a field. Recall the Bloch complex of F

δ : B_{2}(F) ⟶ Λ^{2}F^{∗}; B_{2}(F) := \frac{ℤ[F^{∗}]}{R_{2}(F)}

The subgroup R_{2}(F) is generated by the elements

∑_{i=1}^{5}(−1)^{i}{r(x_{1}, ..., x̂_{i}, ..., x_{5})}, x_{i} ∈ P^{1}(F), x_{i} ≠ x_{j}

where {x} is the generator of ℤ[F^{∗}] corresponding to x ∈ F^{∗}. Let {x}_{2} be the projection of {x} to B_{2}(F). Then δ{x}_{2} := (1 − x) ∧ x. One has δ(R_{2}(F) = 0, (see [G1]).

Let X_{k} be the set of all codimension k irreducible subvarieties of X. The weight two motivic complex Γ(X; 2) of a regular irreducible variety X with the field of functions ℚ(X) is the following cohomological complex

Γ(X; 2) :=  B_{2}(ℚ(X)) ⟶^{δ} Λ^{2}ℚ(X)^{∗} ⟶^{Res} ∏_{Y ∈X_{1}} ℚ(Y)^{∗} ⟶^{div} ∏_{Y ∈X_{2}} ℤ  (164)

where the first group is in the degree 1 and Res is the tame symbol map (10). If Y ∈ X_{1} is normal, the last map is given by the divisor div(f) of f. In Y is not normal, we take its normalization Ỹ, compute div(f) on Ỹ, and then push it down to Y.

3. The second motivic Chern class. According to Milnor the universal G–bundle over the classifying space for an algebraic group G can be thought of as the following (semi)simplicial variety EG_{•}:

⟶^{⟶} G^{4} ⟶^{⟶} G^{3} ⟶^{⟶} G^{2} ⟶^{⟶} G

The group G acts diagonally from the left on EG_{•}, and the quotient BG_{•} is the simplicial model for the classifying space of G. The simplicial structure includes the maps

s_{i}^{n} : G^{n} ⟶ G^{n−1}; (g_{1}, ..., g_{n}) ⟼ (g_{1}, ..., ĝ_{i}, ..., g_{n})

Let F^{•}(X) be a complex of sheaves on X which depends functorially on X for the smooth projections. So a smooth projection f : X → Y provides a morphism of complexes f^{∗} : F^{•}(Y) ⟶ F^{•}(X). Then the hypercohomology H^{∗}(BG_{•}, F^{•}) are the cohomology of the total complex associated with the following bicomplex:

...  ...  ...  ...  ...  ...  ...
↑  ↑  ↑  ↑
⟵^{d} F^{3}(G^{3}) ⟵^{d} F^{3}(G^{2}) ⟵^{d} F^{3}(G) ⟵^{d} F^{3}(∗)
↑ ↑ ↑ ↑
⟵^{d} F^{2}(G^{3}) ⟵^{d} F^{2}(G^{2}) ⟵^{d} F^{2}(G) ⟵^{d} F^{2}(∗)
↑ ↑ ↑ ↑
⟵^{d} F^{1}(G^{3}) ⟵^{d} F^{1}(G^{2}) ⟵^{d} F^{1}(G) ⟵^{d} F^{1}(∗)

The differential d : F^{∗}(G\G^{n−1}) ⟶ F^{∗}(G\G^{n}) is defined by d := ∑_{i=1}^{n}(−1)^{i−1}(s_{i}^{n})^{∗}. So H^{n}(BG_{•}, F^{•}) is represented by cocycles in ⊕_{p+q=n}F^{p}(G^{q}).

It is well known that

H^{∗}(BG_{•}, ℚ_{M}(2)) = H^{∗}(BG_{•}, Γ(X; 2) ⊗ ℚ)

where the right hand side is computed via the bicomplex above. Moreover it is known that if ∗ = 4, this ℚ–vector space can be identified with the one of G–invariant quadratic forms on the Lie algebra of G. In particular the Killing form provides a distinguished cohomology class

c_{2,G}^{M} = c_{2}^{M} ∈ H^{4}(BG_{•}, ℚ_{M}(2))  (165)

A cocycle representing the class c_{2}^{M} is given by the following data:

C_{4} ∈ B_{2}(ℚ(G^{4})^{∗})^{G}, C_{3} ∈ (Λ^{2}ℚ(G^{3})^{∗})^{G}, C_{2} ∈ ( ∏_{Y ∈(G^{2})_{1}} ℚ(Y)^{∗})^{G}  (166)

Lemma 13.2 There exists a cocycle C_{•}^{U} = (C_{4}^{U}, C_{3}^{U}, C_{2}^{U}) with C_{2}^{U} = 0 representing the class c_{2}^{M} such that its component on G^{k} is invariant under the right action of U^{k}, and skew symmetric under the permutations of the factors of G^{k} modulo 2-torsion.

Proof. Let Q(X_{∗}(H))^{W} be the group of W -invariant quadratic forms on the group of cocharacters of the Cartan group H. Recall ([BrD], Section 4; [EKLV], 3.2, 4.7, 4.8) that for a split semi-simple group G there exists canonical isomorphism, functorial in G,

H^{2}(BG, ℤ_{M}(2)) ⟶^{∼} H^{2}(BH, ℤ_{M}(2))^{W} = Q(X_{∗}(H))^{W}  (167)

Remark. The work [BrD] employs the group H^{0}(X, K_{2}), where K_{2} is the sheaf of K_{2} groups in the Zariski topology. The Gersten resolution shows that it is canonically isomorphic to our group H^{2}(X, ℤ_{M}(2)). In ([BrD], 4.1) the isomorphism (167) is formulated for simply-connected G, but it is valid for all G.

If G = GL_{n} a cocycle satisfying all the conditions of the lemma was constructed in [G3]. We will recall its construction below. We may assume that G is simple. In particular it provides a class in H^{2}(BGL_{n}, ℤ_{M}(2)) denoted by w_{GL_{n}}.

Let ρ : G ⟶ GL_{n} be a nontrivial homomorphism. The Killing form on GL_{n} restricts to λ_{ρ} times the Killing form on G. Since λ_{ρ} ≠ 0, the class ρ^{∗}w_{GL_{n}} is not zero. The class ρ^{∗}w_{GL_{n}} is represented by an element in C′_{3} ∈ K_{2}(ℚ(G^{3})). Since ρ(U) lies in a maximal unipotent subgroup U′ of GL_{n}, it is invariant under the right action of U^{3}. Therefore there exists an U^{3}-invariant lift C_{3}^{U} ∈ Λ^{2}(ℚ(G^{3})) of the class ρ^{∗}w_{GL_{n}}. Consider the element

∑_{i=1}^{4}(−1)^{i−1}(s_{i}^{4})^{∗}C_{3}^{U}  (168)

It lies in Λ^{2}ℚ((G/U)^{4})^{∗}. We claim that its projection to K_{2} is zero. This means that there exists an U^{4}-invariant element C_{4}^{U} ∈ B_{2}(ℚ(G^{4})) such that δ(C_{4}^{U}) = ∑_{i=1}^{4}(−1)^{i−1}(s_{i}^{4})^{∗}C_{3}^{U}. It remains to skewsymmetrize it. To check the claim observe that the projection of the element (168) to K_{2} lies in the subspace H^{2}((G/U)^{4}, ℤ_{M}(2)) of the elements of K_{2} with zero tame symbols. On the other hand we know that its pull back to G^{4} is zero. Since H^{2}(X, ℤ_{M}(2)) is homotopy invariant, and the variety U is isomorphic to an affine space, we are done. The lemma is proved.

Observe that G\G^{3}/U^{3} is birationally isomorphic to Conf_{3}(A). So it comes from a class in K_{2} of the field of rational functions on Conf_{3}(A). Set 𝔽_{G,k} := ℚ(Conf_{k}(A)). Lemma 13.2 can be reformulated as follows: there exist elements

C_{4}^{U} ∈ B_{2}(𝔽_{G,4}), C_{3}^{U} ∈ Λ^{2}𝔽_{G,3}^{∗}

which satisfies the following three conditions:

i) ∑_{i=1}^{5}(−1)^{i−1}(s_{i}^{5})^{∗}C_{4}^{U} = 0; ii) ∑_{i=1}^{4}(−1)^{i−1}(s_{i}^{4})^{∗}C_{3}^{U} = δC_{4}^{U}; iii)Res(C_{3}^{U}) = 0

4. The motivic avatar of the Weil–Petersson form. The mapping class group Γ_{S} acts on the space A_{G,Ŝ}. Thanks to Proposition 3.2 we can define the second rational Γ_{S}–equivariant weight two motivic cohomology of A_{G,Ŝ} explicitly by

H_{Γ_{S}}^{2}(A_{G,Ŝ}, ℚ_{M}(2)) := H^{2}(𝔾_{•}(S) ×_{Γ_{S}} A_{G,Ŝ}, ℚ_{M}(2)) := H^{2}(𝔾_{•}(S) ×_{Γ_{S}} A_{G,Ŝ}, Γ_{M}(2))

Remarks. 1. A_{G,Ŝ} and 𝔾_{•}(S) are objects of different nature: the first is a stack whose generic point is an algebraic variety, while the second is a polyhedral complex. Their product is mixed object. To define its hypercohomology with coefficients in a complex of sheaves F^{•} one proceeds similarly to the definition of hypercohomology of a simplicial algebraic variety.

2. Our cocycles for the classes in H_{Γ_{S}}^{2} will sit at the generic point of A_{G,Ŝ}. Therefore we can forget about the complications related to the stack nature of A_{G,Ŝ}.

Theorem 13.3 A choice of W -invariant quadratic form Q on X_{∗}(H) provides a class

𝕎_{G}(S) ∈ H_{Γ_{S}}^{2}(A_{G,Ŝ}, ℚ_{M}(2))

It depends linearly on the form Q.

Here is a refined version of Theorem 13.3. Let 𝔽 be the field of rational functions on A_{G,Ŝ}. So the generic 𝔽–point of the variety A_{G,Ŝ} describes generic decorated unipotent G–local system on Γ.

Theorem 13.4 A cocycle C_{•}^{U} as in Lemma 13.2 representing the class c_{2}^{M} provides a homomorphism of complexes

G_{5}(S) ⟶ G_{4}(S) ⟶ G_{3}(S) ⟶  0
↓  ↓ w_{4}  ↓ w_{3}  ↓
0 ⟶ B_{2}(𝔽) ⟶^{δ} Λ^{2}𝔽^{∗} ⟶^{∂} ∏_{Y ∈A_{m}(S)_{1}} ℚ(Y)^{∗}

Different cocycles lead to homotopic homomorphisms.

Proof of theorem 13.4. A local system on a graph Γ is the following data:

i) a collection {L_{v}} of vector spaces attached to the vertices v of Γ.

ii) For any edge e an orientation of e gives rise to an operator M_{e^{→}} : L_{v_{1}} ⟶ L_{v_{2}}. Here v_{1} and v_{2} are the vertices of e, and the chosen orientation is from v_{1} to v_{2}. Reversing the orientation of e we get the inverse operator.

Let Γ be a graph embedded to S, so that S retracts to Γ. Then there is a natural bijection between the local systems on S and Γ.

We picture a valence k vertex v of a ribbon graph by a little oriented circle S_{v}^{1} with k marked points on it corresponding to the edges shared by v. The orientation of the circle is compatible with the cyclic order at v. The marked points cut the circle into k arcs α_{1}, ..., α_{k}. A decorated unipotent G–local system on a ribbon graph Γ is given by a collection of affine flags X_{v,α_{i}} over v. The affine flag X_{v,α_{i}} is invariant under the monodromy around the face path provided by α_{i}.

Take a generator (Γ, ε_{Γ}) of G_{3}(S). We assign to it an element of Λ^{2}𝔽^{∗} as follows. For each vertex v of Γ there are three cyclically ordered affine flags X_{v}^{1}, X_{v}^{2}, X_{v}^{3} over v. Evaluating the component C_{3}^{U} on them we get an element C_{3}^{U}(X_{v}^{1}, X_{v}^{2}, X_{v}^{3}) ∈ Λ^{2}𝔽^{∗}. Observe that a trivalent ribbon graph has canonical orientation provided by the cyclic structure at the edges. We define the sign sgn(ε_{Γ}) as the ratio of this orientation to the one ε_{Γ}. We define the map w_{3} by setting

w_{3}(Γ, ε_{Γ}) := ∑_{v∈V (Γ)} sgn(ε_{Γ})C_{3}^{U}(X_{v}^{1}, X_{v}^{2}, X_{v}^{3}) ∈ Λ^{2}𝔽^{∗}

Let (Γ, ε_{Γ}) ∈ G_{4}(S). Then Γ is represented by a ribbon graph with exactly one 4–valent vertex v_{0}. Let us enumerate the edges at this vertex in a way compatible with the cyclic order. Then the graph inherits canonical orientation, and we will assume that ε_{Γ} is this orientation. One has

∂ : (Γ, ε_{Γ}) ⟼ (Γ_{1}, ε_{Γ_{1}}) − (Γ_{2}, ε_{Γ_{2}})

where Γ_{1} and Γ_{2} are the two trivalent ribbon graphs shown on Figure 49, and ε_{Γ_{i}} is the canonical orientation of the trivalent ribbon graph Γ_{i}.

A decorated unipotent G–local system on Γ provides us four affine flags X^{1}, ..., X^{4} in the domains near the vertex v_{0}. We enumerate them using the chosen order of the edges at v_{0} and set

w_{4}(Γ, ε_{Γ}) := C_{4}^{U}(X^{1}, X^{2}, X^{3}, X^{4}) ∈ B_{2}(𝔽)

The skew symmetry property of the component C_{4}^{U} guarantees that this definition does not depend on the ordering of the edges at v_{0}. The condition δ ∘ w_{4}(Γ, ε_{Γ}) = w_{3} ∘ δ(Γ, ε_{Γ}) is just equivalent to the cocycle condition ii) for the cycle C_{•}^{U}.

Let (Γ, ε_{Γ}) ∈ G_{5}(S). We assign to (Γ, ε_{Γ}) zero. One needs to check that

w_{4}(∂(Γ, ε_{Γ})) = 0 in B_{2}(𝔽)

Observe that Γ has either one 5–valent vertex v_{0}, or two 4–valent vertices, and all the other vertices are of valence three. Suppose that Γ has a 5–valent vertex v_{0}. Let us choose an order of the edges at v_{0} compatible with the cyclic structure at the vertex. It provides an orientation of Γ, and we can assume that ε_{Γ} is this orientation. Then ∂(Γ, ε_{Γ}) is given by the five generators shown on the picture. Let us denote by X^{1}, ..., X^{5} the five affine flags over the vertex v_{0}, enumerated by using the numeration of the edges, as shown on Figure 50.

Figure 50: The pentagon relation

Then

w_{4}(∂(Γ, ε_{Γ})) = ∑_{i=1}^{5}(−1)^{i−1}C_{4}^{U}(X^{1}, ..., X̂^{i}, ..., X^{5}) = 0

The right hand side is zero thanks to the condition i) for the cocycle C_{•}^{U}.

If Γ has two 4–valent vertices then it is easy to check, using only the skew symmetry of C_{4}^{U}, that w_{4}(∂(Γ, ε_{Γ})) = 0. We conclude that the properties i)–iii) for the cocycle C_{•}^{U} imply that w_{•} is morphism of complexes. The theorem is proved.

Proof of theorem 13.3. It follows the same lines as the proof of Theorem 13.4.

5. The K_{2}–avatar of the degenerate symplectic form on A_{G,Ŝ}.

Corollary 13.5 Let G be a split semi-simple algebraic group. Then a choice of a W -invariant quadratic form Q on X_{∗}(H) provides a class

W_{G,Ŝ} ∈ H^{2}(A_{G,Ŝ}, ℚ_{M}(2))^{Γ_{S}}

The form Ω_{G,Ŝ} := d log(W_{G,Ŝ}) is a regular Γ_{S}-invariant 2-form on the nonsingular part of A_{G,Ŝ}.

Proof. For a field F one has K_{2}(F) = Coker(δ) by Matsumoto’s theorem. Thus for a regular variety X we have H^{2}(X, ℚ_{M}(2)) = H^{2}Γ(X, 2) ⊗ ℚ where the left hand side was defined in (11).

We define the class W_{G,Ŝ} as the image of the class 𝕎_{G,Ŝ} under the canonical projection

H^{2}(𝔾_{•}(S) ×_{Γ_{S}} A_{G,Ŝ}, ℚ_{M}(2)) ⟶ H^{2}(A_{G,Ŝ}, ℚ_{M}(2))^{Γ_{S}}

The properties of the map d log provide the second statement. The corollary is proved.

A direct construction of the class W_{G,Ŝ}. Recall the canonical orientation of a trivalent ribbon graph Γ. Let Γ → Γ′ be a flip defined by an edge e, and \overline{Γ} the ribbon graph obtained by shrinking of the edge e. Then \overline{Γ} has just one 4–valent vertex, the shrinked edge e. According to the proof of Theorem 13.4 we have

w_{3}(Γ) − w_{3}(Γ′) = δw_{4}(\overline{Γ})  (169)

So the projection of w_{3}(Γ) ∈ Λ^{2}𝔽^{∗} to K_{2}(𝔽) does not depend on the choice of Γ. It is the class W_{G,Ŝ}.

In the next subsection we look at the proof of theorem 13.4 from a bit different perspective, getting a more general result.

5. The graph and the affine flags complexes. Let G be a group and X a G–set. Let C_{n}(X) be the coinvariants of the diagonal action of G on ℤ[X^{n}]. So the generators of C_{n}(X) are elements (x_{1}, ..., x_{n}) such that (x_{1}, ..., x_{n}) = (gx_{1}, ..., gx_{n}) for any g ∈ G. The groups C_{n}(X) form a complex C_{•}(X), called the configurations complex, with the differential ∂ given by

∂ : (x_{1}, ..., x_{n}) ⟼ ∑_{i=1}^{n}(−1)^{i−1}(x_{1}, ..., x̂_{i}, ...x_{n})  (170)

Applying this construction to the set of 𝔽-points of the affine flag variety A_{G} for G we get a configuration complex C_{•}(A_{G}(𝔽)). We define the related to Ŝ affine flags complex

AŜ ; •(G) := ... ⟶^{∂} AŜ ; n+2(G) ⟶^{∂} AŜ ; n+1(G) ⟶^{∂} AŜ ; n(G) ⟶^{∂} ...

is its subcomplex generated by configurations of affine flags over 𝔽 in generic position.

Let us define a homomorphism f_{k} : G_{k}(S) ⟶ AŜ ; k(G). If Γ has at least two vertices of valence ≥ 4 we set f_{k}(Γ, ε_{Γ}) = 0. Suppose that Γ has just one vertex v_{0} of valence ≥ 4. Then val(v_{0}) = k. Choose an order of the edges sharing v_{0} compatible with the cyclic order at v_{0}. This order provides an orientation ε̃_{Γ} of Γ. The order of the edges provides an order of the domains near v_{0}, and hence the affine flags X^{1}, ..., X^{k} attached to the domains. We set

f_{k}(Γ, ε_{Γ}) := ε̃_{Γ}/ε_{Γ} · (X^{1}, ..., X^{k})

It clearly does not depend on the chosen order of the edges.

Proposition 13.6 The map f_{•} provides a homomorphism of complexes

⟶ G_{5}(S) ⟶ G_{4}(S) ⟶ G_{3}(S) ⟶  0
↓ f_{5}  ↓ f_{4}  ↓ f_{3}  ↓
⟶ AŜ ; 5(G) ⟶ AŜ ; 4(G) ⟶ AŜ ; 3(G) ⟶ AŜ ; 2(G)

Proof. Let us check that for a generator (Γ, ε_{Γ}) ∈ G_{k}(S) one has

∂ ∘ f_{k−1}(Γ, ε_{Γ}) = f_{k} ∘ ∂(Γ, ε_{Γ})  (171)

Suppose first that k > 3. Then obviously both terms in (171) are zero if either there are three vertices in Γ of valence ≥ 4, or there are two vertices of valence ≥ 5. Suppose there are just two vertices, v_{1} and v_{2}, of valence ≥ 4. We may assume that one of them, say for v_{1}, is of valence 4. Then v_{1} provides two terms to ∂(Γ). They enter with the opposite signs, and hence f_{k−1} kills their sum.

It remains to consider the case when there is just one vertex v of valence k > 3. Then the only component of ∂(Γ) which is not immediately killed by f_{k−1} is the one shown on Figure 51. It matches the i-th term in the formula for ∂ ∘f_{k}(Γ). So (171) holds. If k = 3 then each edge of Γ contributes two

Figure 51:

terms to f_{k} ∘∂(Γ, ε_{Γ}), and they appear with the opposite signs, canceling each other. The proposition is proved.

7. Calculation of the 𝕎-class for G = SL_{m}. Recall that Γ is a ribbon graph embedded to Ŝ, and I_{m}^{Γ} is the set parametrising the corresponding Δ- and X-coordinates, see Section 6. Recall the canonical map p : A_{SL_{m},Ŝ} → X_{P SL_{m},Ŝ}.

Theorem 13.7 The W –class on A_{P SL_{m},Ŝ} is given by

W_{SL_{m},Ŝ} = ∑_{i,j∈I_{m}^{Γ}} ε_{ij}{Δ_{i}^{Γ}, d log Δ_{j}^{Γ}} = ∑_{i∈I_{m}^{Γ}}{Δ_{i}^{Γ}, p^{∗}(X_{α}^{Γ})} ∈ K_{2}(𝔽), 𝔽 = ℚ(A_{SL_{m},Ŝ})  (172)

It does not depend on the choice of the graph Γ.

Corollary 13.8 The Weil-Petersson form on A_{P SL_{m},Ŝ} is given by

Ω_{SL_{m},Ŝ} = ∑_{i,j∈I_{m}^{Γ}} ε_{ij}d log Δ_{i}^{Γ} ∧ d log Δ_{j}^{Γ}

It does not depend on the choice of the graph Γ.

Proof. We are going to apply our construction to the cocycle C_{•}^{U} for the second motivic Chern class c_{2}^{M} defined in [G3]. This cocycle satisfies the conditions of Lemma 13.2. We will get a cocycle representing the 𝕎-class whose Λ^{2}𝔽^{∗}-component is given by the following formula:

∑_{i,j∈I_{m}^{Γ}} ε_{i,j}Δ_{i}^{Γ} ∧ Δ_{j}^{Γ} = ∑_{i∈I_{m}^{Γ}}{Δ_{i}^{Γ}, p^{∗}(X_{i}^{Γ})} ∈ Λ^{2}𝔽^{∗}  (173)

It of course depends on the choice of the graph Γ. Theorem 13.7 follows immediately from this formula. Observe that the equality of the two terms in the formula is a straitforward consequence of the formula ?? for p^{∗}(X_{i}^{Γ}). To prove formula (173) we will give below an explicit construction of the w_{•}-map.

Denote by A_{•}(m) the affine flag complex for P SL_{m}. We will construct the following diagram of morphisms of complexes, where the AŜ ; •(m) and BCŜ ; •(2) are complexes defined using 𝔽-vector spaces, as explained below:

⟶ G_{5}(S) ⟶ G_{4}(S) ⟶ G_{3}(S)
↓ f_{5}  ↓ f_{4}  ↓ f_{3}
⟶ AŜ ; 5(m) ⟶ AŜ ; 4(m) ⟶ AŜ ; 3(m)
↓ g_{5}  ↓ g_{4}  ↓ g_{3}
⟶ BCŜ ; 5(2) ⟶ BCŜ ; 4(2) ⟶ BCŜ ; 3(2)
↓  ↓ h_{4}  ↓ h_{3}
0  ⟶ B_{2}(𝔽) ⟶^{δ} Λ^{2}𝔽^{∗}  ⟶^{Res} ∏_{Y ∈(A_{SLm,Ŝ})_{1}} ℚ(Y)^{∗}

Then w_{•} is the composition w_{•} := h_{•} ∘ g_{•} ∘ f_{•}.

The complexes in the diagram are given by horizontal lines, while morphisms are given by vertical arrows. The top complex is the graph complex on S. The next one is the complex of affine flags in the generic position in an m-dimensional vector space. Then follow the rank 2 bigrassmannian complex ([G1]) and the complex Γ(A_{SL_{m},Ŝ}; 2). The homomorphisms g_{•} and h_{•} were defined in [G3] and [G1] - [G2]. We recall their definition below.

The grassmannian complex. Take X = V_{m}, where V_{m} is an F –vector space. The grassmannian complex

C_{•}(m)_{/F} := ... ⟶^{∂} C_{m+2}(m) ⟶^{∂} C_{m+1}(m) ⟶^{∂} C_{m}(m)

is the subcomplex of the m–truncated complex σ_{≥m}C_{•}(V_{m}) given by the configurations of vectors in the generic position in V_{m}.

The rank two bigrassmannian complex BC_{•}(2) ([G1]). Consider a bicomplex:

...  ...  ...  ...
...  ↓ ∂′  ↓ ∂′  ↓ ∂′
... ⟶^{∂} C_{6}(3) ⟶^{∂} C_{5}(3) ⟶^{∂} C_{4}(3)
... ↓ ∂′ ↓ ∂′ ↓ ∂′
... ⟶^{∂} C_{5}(2) ⟶^{∂} C_{4}(2) ⟶^{∂} C_{3}(2)

The horizontal differential are given by (170). The vertical ones are

∂′ : C_{n}(k) ⟶ C_{n−1}(k − 1); (x_{1}, ..., x_{n}) ⟼ ∑_{i=1}^{n}(−1)^{i−1}(x_{i}|x_{1}, ..., x̂_{i}, ...x_{n})

Here (x_{i}|x_{1}, ..., x̂_{i}, ...x_{n}) denotes the configuration of vectors obtained by projecting x_{j}, j ≠ i, to the quotient V_{k}/ < x_{i} >. The complex BC_{•}(2) is the total complex associated with this bicomplex.

The homomorphism g_{•} ([G3]). Let us define its component

g_{n}^{k} : A_{n}(m) ⟶ C_{n}(k); k ≤ m

Choose non negative integers p_{1}, ..., p_{n} such that p_{1} + ... + p_{n} = m − k. Let X_{•}^{1}, ..., X_{•}^{n} be the flags underlying an n–tuple of affine flags X̃^{1}, ..., X̃^{n}. Consider the quotient

\frac{V_{m}}{X_{p_{1}} ⊕ ... ⊕ X_{p_{n}}}  (174)

It is a vector space of dimension k. Our affine flags induce an n–tuple of the affine flags in this quotient. Taking the first vector of each of these affine flags, we get a configuration of n vectors in (174). The corresponding element of the group C_{n}(k) is, by definition, g_{n}^{k}(X̃^{1}, ..., X̃^{n}). According to the Key Lemma from [G3] this way we get a morphism of complexes.

The homomorphism h_{•}. Consider the following diagram:

C_{5}(2) ⟶^{∂} C_{4}(2) ⟶^{∂} C_{3}(2)
↓ l_{5}  ↓ l_{4}  ↓ l_{3}
0 ⟶ B_{2}(𝔽) ⟶^{δ} Λ^{2}𝔽^{∗}
l_{3} : (l_{1}, l_{2}, l_{3}) ⟼ Δ(l_{1}, l_{2}) ∧ Δ(l_{2}, l_{3}) + Δ(l_{2}, l_{3}) ∧ Δ(l_{3}, l_{1}) + Δ(l_{3}, l_{1}) ∧ Δ(l_{1}, l_{2})
l_{4} : (l_{1}, l_{2}, l_{3}, l_{4}) ⟼ −{r(l_{1}, l_{2}, l_{3}, l_{4})}_{2}

Using the Plücker relation we see that, modulo 2–torsion, it provides a morphism of complexes, (see s. 2.3 in [G1]). The rank two grassmannian complex is a subcomplex of the rank two grassmannian bicomplex. Extending it by zero to the rest of the rank two grassmannian bicomplex we get the map h_{•}. By s. 2.4 of [G1] it is a morphism of complexes.

The formula (173) follows immediately from the construction of the map w_{3} given above. The Theorem 13.7 and hence the Corollary 13.8 are proved.

The class 𝕎_{P SL_{m},Ŝ} is lifted from a similar class on the space U_{P SL_{m},Ŝ}:

Theorem 13.9 a) There exists unique class

𝕎_{P SL_{m},Ŝ}^{U} ∈ H_{Γ_{S}}^{2}(U_{P SL_{m},Ŝ}, ℚ_{M}(2)) such that 𝕎_{SL_{m},Ŝ} = p^{∗}𝕎_{P SL_{m},Ŝ}^{U}

b) The 2-form Ω_{SL_{m},Ŝ}^{U} := d log(W_{P SL_{m},Ŝ}^{U}) is symplectic form on the moduli space U_{P SL_{m},Ŝ}.

Proof. a) The class 𝕎_{SL_{m},Ŝ} has two nontrivial components, with values in B_{2} and Λ^{2}. The B_{2} component comes from the U_{P SL_{m},Ŝ}-space by the very definition. For the Λ^{2}-component this is deduced from the formula (173).

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http://www.msri.org/publications/books/gt3m. A.G.: Department of Math, Brown University, Providence RI 02906, USA; sasha@math.brown.edu V.F.: ITEP, B. Cheremushkinskaya 25, 117259 Moscow, Russia. fock@math.brown.edu