Superintegrable Systems on Moduli Spaces of Flat Connections

The main result of this paper is the construction of a family of superintegrable Hamiltonian systems on moduli spaces of flat connections on a principle $G$-bundle on a surface. The moduli space is a Poisson variety with Atiyah-Bott Poisson structure. Among particular cases of such systems are spin generalizations of Ruijsenaars-Schneider models.


Introduction
Let G be a simple linear algebraic group over C. In this paper we construct superintegrable Hamiltonian systems on moduli spaces of flat G-connections over any oriented surface with nonempty boundary. Hamiltonians of such systems are traces of holonomies along non intersecting non self-intersecting curves. The construction naturally works in the same way for various real forms of G, for example for compact simple Lie groups or for split real forms.
The fact that Hamiltonian integrable systems appear in the context of gauge theories is not new, with Hitchin systems [H87] being one of the well known examples. Calogero-Moser type systems were put in the context of gauge theory earlier 1990's, see [GN94], [GN95] and references therein. See [FK13][AO19] for more recent devolopments and references.
The main result of this paper involves the notion of a superintegrable Hamiltonian system which we review in section 1, and of graph functions which we review in section 2.2. Denote the moduli space of flat G-connections on Σ as M G Σ . Our main result is the following theorem. Theorem 1. Let Σ be an oriented surface of genus g with b > 0 boundary components and G be a simple linear algebraic group over C. For each disjoint union of pairwise nonhomotopic simple closed curves C = C 1 ⊔ · · · ⊔ C k in Σ, none of which is homopotic to a boundary component, the following inclusions of Poisson algebras define an affine superintegrable system Here Z ∂Σ is the algebra of G-invariant functions on holonomies around boundary components, B C is a subalgebra of the coordinate ring O[M G Σ ] generated by graph functions F Γ,V,c with Γ being contractible to C ⊔ ∂Σ. Similarly, J Σ\C is a subalgebra generated by graph functions F Γ,V,c with Γ ⊂ (Σ\C).
We also introduce the notion of a refinement of one superintegrable systems by the other. The refinement defines a partial order on the space of superintegrable systems.
For a torus with one puncture these systems are spin Ruijsenaars systems (also known as relativistic spin Calogero-Moser systems) [FGNR00] [Res16]. For a torus with two punctures they are relativistic deformations of two sided spin Calogero-Moser systems [Res18] corresponding to symmetric pair G ⊂ G × G where G is embedded diagonally. A particular case of our systems on a torus with n punctures and rank one conjugation orbits of SL n is closely related to the systems studied in [CF18]. In this case the systems are actually Liouville integrable and they admit a superintegrable refinement. The maximal integrability in this case is achieved by a superintegrable system with r = rank G independent Poisson commuting integrals. The minimal integrability is a Liouville integrability with b rank G Poisson commuting integrals. In our framework this correspond to a different choice of cycles C.
The mapping class group acts on the moduli space of flat connections by Poisson automorphisms. If C and C ′ are collections of cycles on surface Σ which belong to the same orbit of the Mapping Class Group action, the corresponding superintegrable systems are isomorphic. An example of such isomorphism in the genus one case for involution exchanging the equator and meridian of the torus gives rise to a self-duality of the Ruijsenaars system [FGNR00,FR99,FK11]. For a spin version of this duality see [Res16].
The plan of this paper is as follows. In the first section we recall the definition of superintegrable (degenerately integrable systems). There we also define the notion of an affine superintegrable system in the algebro-geometrical setting. The second section is an overview of basic notions about moduli spaces of flat connections on a surface. In this section we also recall the definition of graph functions and the description of Poisson brackets between two such functions. In the third section we describe the main result, the construction of a family of Hamiltonian systems defined by a choice of non intersecting, non selfinersection cycles and prove their superintegrability. At the end of this section we introduce the notion of a a partial order on such systems. In section 4 we explain how solutions to equations of motion of these superintegrable systems can be solved using the projection method. Section five has some genus one examples. In the conclusion we define a conjecturally superintegrable system on the space of chord diagrams, discuss the case of non-generic conjugation orbits and some further directions.
Here we recall briefly basic definitions and introduce the notion of an affine superintegrable system.
The notion of a superintegrable Hamiltonian systems were introduced in [Nek72] (where he called them degenerate integrable systems) as a generalization of the Liouville integrable systems. Superintegrable system on a 2n dimensional smooth symplectic manifold has m > n independent 1 first integrals, however only k of them are in involution and r + k = 2n. A particular case when m = k = n corresponds to Liouville integrable systems. First examples of such systems appeared earlier [Pa26][F35] [FMSW65]. A family of examples of superintegrable systems associated to Lie groups was given in [MF78]. More recent examples include charateristic systems on simple Poisson-Lie groups [Res03a], spin Calogero-Moser and Ruijsenaars systems [Res03b] and their relativistic counterparts [Res16].
Geometrically, a superintegrable system of rank k on a symplectic manifold M 2n consists of a Poisson manifold P 2n−k , a manifold B k 2 (considered as a Poisson manifold with trivial symplectic structure) and Poisson projections where J = C ∞ (P 2n−k ) and B = C ∞ (B k ). The subalgebra B is a Poisson commutative subalgebra and J is the Poisson centralizer of B, i.e. J is the maximal subalgebra in A such every element from J Poisson commutes with every element from B.
Elements of subalgebra B define derivations of A Because B is Poisson commutative, these derivations commute Derivation D b is the Lie derivative along the Hamiltonian vector field on M 2n generated by b. We refer to elements of B as Hamiltonians of a superintegrable system (2). On the other hand, elements of subalgebra J are precisely the first integrals of the Hamilton flow generated by b, i.e. they are constant on flow lines of the Hamiltonian vector field generated by b.
Fix b 1 , . . . , b k ∈ B, a choice of independent Hamiltonians. Corresponding vector fields D b1 , . . . , D b k ∈ T M 2n are independent at every point x ∈ M 2n and hence form a basis on the k-dimensional subspace of T x M 2n defined by the level set of J. Algebraically, these vector fields D b1 , . . . , D b k ∈ Der J (A) define derivations of A relative to J. Balance of dimensions in (1) is equivalent to the fact that Der J (A) is generated as an A-module by Hamiltonian vector fields of the form D b for b ∈ B.
1 On a smooth manifold M (complex or real) we call functions f 1 , . . . , f k ∈ C ∞ (M) independent if the corresponding differentials df 1 , . . . , df k ∈ Γ(T * M) are linearly independent at every point in M.
2 Typically B k is not smooth but stratified by smooth strata with one smooth higher dimensional stratum. In is frequently an orbifold.
is an affine superintegrable system if where J = Spec J, B = Spec B, Z = Spec Z stand for the spectrum of prime ideals of J, B, Z respectively.
Inclusions (4) are equivalent to the chain of dominant maps of affine schemes which preserve the Poisson bracket Definition 2 implies that general fibers of p 2 are symplectic leaves of J 4 , while general fibers of the composition map p are symplectic leaves of A. As a corollary, open subset M 2n (z) of irreducible component of a general fiber p −1 (z), z ∈ Z can be viewed as a phase space of a superintegrable system. M 2n (z) comes equipped with two poisson projections of the form (1). Thus, as we already mentioned, affine superintegrable systems should be regarded as families of superintegrable systems.
1.3. Refinement of a superintegrable system. Here we will introduce the notion of a refinement of a superintegrable system. It gives a partial order on all superintegrable systems on a given Poisson algebra A. A minimal object in this partial order is known as a maximally superintegrable system. Maximal superintegrability means that all invariant tori are one dimensional [SInt]. Let be two superintegrable systems on a Poisson algebra A with the Poisson center Z.
Definition 3. The first superintegrable system is a refinement of the second if the following chain of Poisson inclusions hold: Clearly a refinement defines a partial order on the set of integrable systems. Another important relation is an equivalence of integrable systems Definition 4. Two integrable systems (6) are equivalent if there is a Poisson automorphism ϕ : A → A such that J 2 = ϕ(J 1 ) and B 1 = ϕ(B 2 ).

Moduli Spaces of Flat Connections
2.1. Character Variety of the Fundamental Group. Let Σ g,b be an oriented surface of genus g with b boundary components. We will consider only surfaces with b > 0. The fundamental group π 1 (Σ, p) with the base point p of such surface is generated by elements x 1 , · · · , x g , y 1 , · · · , y g , z 1 , · · · z b with one defining relation: Here x i and y i are fundamental cycles for Σ and z i is the cycle for i-th boundary component. 5 The character variety of π 1 (Σ g,b ) in a simple complex linear algebraic group G is the categorical quotient of the space of group homomorphisms from π 1 (Σ g,b ) to G with respect to conjugations by G: 6 We can choose this isomorphism as the mapping which assigns to a group homomorphism ρ : π 1 (Σ g,b ) → G an element (ρ(x 1 ), . . . , ρ(x g ), ρ(y 1 ), . . . , ρ(y g ), ρ(z 1 ), . . . , ρ(z b−1 )) ∈ G ×(2g+b−1) and then projects it to the conjugacy class in G ×(2g+b−1) / / G.

2.2.
Moduli space of surface graph connections. Denote by V (Γ) and E(Γ) the set of vertices and edges of an oriented graph Γ. Graph connection is an assignment of a parallel transport g e ∈ G to each oriented edge e ∈ E(Γ).
The gauge group . where t(e) is the target vertex for an oriented edge e and s(e) is its source vertex.
Assume that Γ ⊂ Σ is an oriented embedded graph such that Σ\Γ is a disjoint collection of disks and b annuli corresponding to the boundary components of Σ = Σ g,b . We will call such graphs simple. Hereinafter we allow graphs to have multiple edges and loops.
Let F (Γ) be the set of contractible faces, i.e. the set of disks in Σ g,b \Γ. For a disc D with boundary as the holonomy of graph connection along the boundary of a disk. This holonomy is the product of elements g ǫj ej ∈ G associated to each edge e j ∈ ∂D in a cyclic order induced by the orientation of the surface 7 . The relative orientation ǫ j is +1 when the orientations of e j and ∂D coincide and −1 otherwise.
A graph connection over Γ ⊂ Σ is called flat if g ∂D = 1 for all D ∈ F (Γ). Define the space of flat graph G-connections A (Σ,Γ) on Γ ⊂ Σ as the space of such flat connections. 5 Throughout the text we always assume that the composition of paths is read from right to left. For example, [x 1 , y 1 ]z 1 z 2 corresponds to the contractible path along the boundary of a disc in Σ 1,2 on Figure 1. 6 Note that despite its name, M G Σ is not a variety but rather an affine scheme. Throughout the text we will always think of M G Σ in terms of its coordinate ring, which is precisely the G-invariant subring of the coordinate ring of representation variety. 7 The total order is irrelevant when we pass to the moduli space of flat connections. When G is a linear algebraic group, the space of flat graph connections is an algebraic subset of G |E(Γ))| equipped with a regular action of G(V ). The moduli space of flat graph connections is the categorical quotient Let e ∈ E(Γ) be an edge which connects two distinct vertices, denote as Γ e an embedded graph obtained from Γ by contracting an edge e. Similarly, for any contractible face D ∈ F (Γ) which has an edge e ∈ ∂D that appears only once in ∂D we can define an ebedded graph Γ e = Γ\e. Recall that Γ was simple, i.e. its complement in Σ is the union of disks and annuli. It is clear that (Σ\Γ e ) is also simple. The flatness condition g ∂D = 1 guarantees the isomorphism of schemes.
Theorem 5. The following schemes are isomorphic: Proof. Resolving one of the relations g ∂D = 1 we obtain an isomorphism of and therefore M G (Σ,Γ) ≃ M G (Σ,Γ e ) . Now let us prove the first isomorphism in (7). Let e 0 be an edge of Γ with adjacent vertices v = s(e 0 ) to w = t(e 0 ). Here s(e), t(e) stand for the source and target of e respectively. Contracting of e 0 maps vertices v and w to one vertex w 0 of the new graph Γ e0 which is the contraction of Γ. Denote this mapping by π e0 .
The contraction of Γ to Γ e0 defines a functor between the fundamental grouppoids which we will denote by the same letter π e0 : Π(Γ) → Π(Γ e0 ). It is clear that this is a projection. Choose a section φ : Π(Γ e0 ) → Π(Γ) of π e0 , so that π e0 φ = id. It is clear that φ does not have to be unique, but it always exists. This is illustrated on Figure 2.
The functors π e0 and φ define the projection p e0 : Corollary 6. For any oriented embedded graph Γ ∈ Σ g,b such that Σ g,b \Γ is a disjoint union of discs and b annuli, the moduli space of graph connections is isomorphic to the G-character variety Corollary 7. The dimensions of moduli spaces of graph connections on surface Σ g,b with b > 0 are: 2.3. Colored Immersed Graphs. Let Γ be an oriented graph.
Definition 8. An edge coloring of Γ is an assignment of a finite-dimensional G-representation V ǫ to each edge e ∈ Γ of the oriented graph. We will denote edge colored graph as (Γ, V ).
We say that an oriented graph Γ is immersed in a surface Σ g,b if the inclusion mapping I : Γ → Σ g,b is locally an embedding 8 . If the intersection of Γ on Σ g,b are transversal, we will call it transversally immersed.
Orientation of the surface induces a cyclic order on edges adjacent to a given vertex. For each vertex v ∈ V (Γ) ⊂ Σ let us choose a total ordering e 1 , . . . , e n of the adjacent edges which agrees with the cyclic ordering induced by the orientation of Σ. We will refer to this as edge ordering of Γ ⊂ Σ. For a finite dimensional vector space V define V + = V and V − = V * . For each vertex of edge ordered graph define the space Here ǫ i = +1 is oriented outward from v, i.e. v = s(e i ) and ǫ = −1 otherwise, i.e. when v = t(e) 9 .
Definition 9. A vertex coloring of (Γ, V ) is an assignment of a vector c v ∈ V (v) G to each vertex v of an edge ordered oriented graph Γ.
We define colored graph Γ = (Γ, V, c) as an edge ordered oriented immersed graph Γ equipped with an edge coloring V and a vertex coloring c.

Invariant Functions on Connections associated to Immersed Graphs.
2.4.1. For each colored graph (Γ, V, c) embedded into Σ one can associate a function F Γ,V,c on connections on the principal G-bindle over Σ g,b [reference] as follows.
For a graph Γ with an edge coloring V define the following G-module Here we assume a choice of a linear ordering on vertices. The space V (v) are defined in (9). Note that V (v) can also be written as Here S(v) is the star of vertex v 10 and ǫ(e, v) = + if e is outgoing and ǫ(e, v) = − if it is incoming. By changing the order of factors in tensor product we obtain a natural isomorphism of G-modules The space V(Γ) comes equipped with a symmetric bilinear form , defined on pure tensors u, w ∈ V by the following rule u e,t(e) (w e,s(e) ) w e,t(e) (u e,s(e) ), Here u e,s(e) , w e,s(e) ∈ V e and u e,t(e) , w e,t(e) ∈ V * e 8 Immersed graphs may intersect on a surface and intersections does not have to be transversal. 9 Note that because the category of G-modules is symmetric, there exists a canonical isomorphism between V (v) for different choices of edge ordering. Still, it will be convenient for us to assume that we have chosen a total order of edges adjoint to the given vertex. 10 In our case this is the set of edges adjacent to v.
2.4.2. It is clear that a principal G-bundle E Σ over Σ defines a principal G-bundle E Γ over vertices of Γ ⊂ Σ. A connection A on E Σ defines a graph connection on E Γ with the parallel transport along the edge e being the parallel transport along e ⊂ Σ with respect to connection A. The evaluation of the parallel transport along e in the representation V e we will denote by P A e : V e → V e . Define the vector π A ∈ V in the G-module (10) as Here we assume a choice of total ordering on the set of edges and the identification with the tensor product with the reordered tensor product as in (11). The holonomy map P e we regard as a vector in V e ⊗ V * e . From the coloring of vertices we we get another vector c Γ ∈ V(Γ) Here Definition 10. Define graph functions using the symmetric bilinear form (12) as Proposition 11. Graph functions have the following properties: where A g is a flat connection A after gauge transformation g. If the connection is flat, the function F Γ,V,c (A) depends only on the isotopy class [Γ] of Γ.
Proof. By Definition 9, c Γ p ∈ V (p) G is an invariant vector for each vertex p of the graph. Hence, each function whenever an immersed graph Γ 1 can be deformed to Γ 2 by the regular homotopy (i.e. homotopy through graph immersions).
As a result, for each triple ([Γ], V, c) where [Γ] is the isotopy class of an immersed oriented ordered graph Γ, V is an edge coloring and c is a vertex coloring we have a function F [Γ],V,c = F Γ,V,c on the moduli space of flat connections on Σ g,b . As a consequence, F is a function of the moduli space of flat connections.
,V,c span the coordinate ring (7) of the moduli space of G-representations of the fundamental group π 1 (Σ g,b , p) .
Proof. By an algebraic analogue of Peter-Weyl theorem, the coordinate ring where W λ stands for a finite-dimensional irreducible representation with highest weight λ.
In particular, (15) is an isomorphism of G-modules, where G acts by conjugation. Now let Q ∈ O[G N ] G be a G-invariant polynomial, by (15) it can be decomposed as where each Q j is a graph function associated to the ribbon graph with a single vertex and N = 2g + b − 1 morphisms corresponding to free generators of π 1 (Σ g,b , p). The edge coloring of edges corresponding to generators of the fundamental group is given by λ j,1 , . . . , λ j,N , while Q j is an invariant vector which defines the coloring of the single vertex p.
Thinning a pair of neighbouring edges Lemma 13. [FR99], [KRWY] Let Γ 1 , Γ 2 be a pair of colored immersed oriented graphs which coincide away from a disc shown on any of Figures 3a-3d. Then there exists a vertex coloring of Γ 2 , such that the two colored immersed graph give rise to the same invariant function on graph connections: Proof. Consider a flat connection A on a principle G-bundle over Σ g,b and let be a pair of vectors in and as in (13) defined by the parallel transport along edges of the two immersed graphs. In each of the four parts we are going to prove that for an appropriate choice of vertex coloring.
(B) Let Γ 1 and Γ 2 be a pair of colored oriented immersed graphs which coincide away from a disc shown on Figure 3b. Suppose that the vertex coloring of Γ 1 at v is given by Because the holonomy of flat connection along contractible loop e 0 is trivial, we get the following contribution to holonomy vector (16) (C) Assume further that the vertex coloring of vertices a and b is given by a pair of G-invariant vectors Let c v be a G-invariant vector defined as follows: are functions on gauge equivalence classes of flat connections, without loss of generality we can assume that holonomy of A in a contractible disc on Figure 3c is trivial. Using the same reasoning as in (18) and (19) we conclude that (17) holds. (D) For the remaining part, again without loss of generality we can assume that connection is trivial inside contractible disk shown on Figure 3d. For ϕ : V 0 ∼ − → V 1 ⊗ V 2 , assign G-invariant vectors at both vertices induced by the isomorphism ϕ and its inverse, the statement (17) then follows by tautology.
Corollary 14. Let Γ ∈ Σ be an oriented immersed graph with a single vertex p and edges corresponding to free generators of π 1 (Σ, p). For any oriented immersed graph (Γ 1 , W, d) we can present the associated graph function as for an appropriate choice of coloring of simple oriented surface graph Γ ⊂ Σ.
Proof. Because F [Γ1,W,d] depends only on homotopy (through immersed graphs) equivalence type of Γ 1 , w.l.o.g. we can assume that Γ ∪ Γ 1 has only double transverse intersection points. So by Lemma 13 we can turn Γ ∪ Γ 1 first into an embedded graph, then collapse all vertices of Γ 1 into vertices of Γ 2 and, finally, by a sequence of moves (3b), (3c), (3d) eliminate all vertices of Γ 1 .  Definition 15. Let p ∈ Γ ∩ Γ ′ be a transverse double intersection point of two colored immersed graphs Consider a small neighborhood of p and denote by e ∈ Γ and e ′ ∈ Γ ′ the corresponding edges of graphs containing the intersection point p ∈ e ∩ e ′ . Take union of immersed graphs Γ ∪ Γ ′ and add two extra vertices a, b which divide edges e, e ′ into e 1 , e 2 and e ′ 1 , e ′ 2 respectively as shown on Figure 4b. Then add an edge e ′′ connecting a and b colored by adjoint representation V ′′ e ′′ = g of G. 11 Define vertex coloring Casimir. Finally, assume that edge and vertex coloring of the remaining graph is inherited from Γ and Γ ′ .
where the sum is taken over the intersection points p ∈ Γ ∩ Γ ′ and ǫ(p) = ±1 stands for the sing of an intersection point.

Main Construction
The following theorem describes superintegrable systems on moduli spaces flat connections. Poisson commuting Hamiltonians of these systems are invariant functions of holonomies along a systems of nonintersecting simple curves on the surface.
Denote by Z ∂Σ the Poisson center of the algebra of functions on the moduli space of flat connection The subalgebra Z ∂Σ is spanned by G-invariant functions on holonomies around boundary components.
Let C = C 1 ⊔ · · · ⊔ C k in Σ be a disjoint union of pairwise nonhomotopic simple closed curves none of which is homopotic to a boundary component. Define B C ⊂ A as the subalgebra generated by graph functions F Γ,V,c with Γ ⊂ C. Similarly, define J Σ\C as the subalgebra generated by graph functions F Γ,V,c with Γ ⊂ (Σ\C).
Theorem 17. The system of subalgebras below defines an affine superintegrable system In the rest of this section we will prove this theorem. In order to prove it we should prove the balance of dimensions. , where Γ is a colored graph with a single vertex and a single edge homotopic to one of the boundary components of Σ.
] be a pair of isotopy equivalence classes of colored immersed graphs such that [Γ 1 ] is homotopic to one of the boundary components. Then there exist members Γ 1 and Γ 2 with a trivial intersection. From (21) we get By Lemma 18 we know that Z is a subalgebra of the Poisson center of A. On the other hand, by results of [GHJW97,BG93] we know that the general fiber of Superintegrable system associated to non-separating curve. Let Σ be an oriented surface of genus g with b ≥ 1 boundary components. There exists no nonseparating curves on genus zero surface, so throught this section we can assume without loss of generality that g > 0. Let C = C 1 ⊔ C 2 ⊔ · · · ⊔ C r be a nonseparating curve which is a disjoint union of r, 1 r g simple closed curves on Σ (see Figure 5).
Proof. By construction, all algebras involved are finitely generated and from (21) On the other hand, since any representation of π 1 (Σ, p | C) is determined by a 2g + b − 1 elements of G Y i , Y j , X k , Z l , 1 i g, 1 j r, r + 1 k g, 1 l b − 1, such that Y j ∼ Y j belong to the same conjugacy class for all j, 1 j r, we obtain Combining (28) with (29) we conclude that balance of dimensions (5) holds.
3.3. Product of Surface Superintegrable systems associated with gluing. Let Σ be an oriented surface with b > 0 boundary components and C = C 1 ⊔ · · · ⊔ C r be a disjoint union of r simple closed curves in Σ which cuts the surface into two parts Σ = Σ L # C Σ R . As in the statement of Theorem 1 we assume that none of the simple curves is homotopic to a boundary and all C 1 , . . . , C r are pairwise nonhomotopic. As a corollary, neither Σ L , nor Σ R can be a cylinder. Further, because surface Σ has nontrivial boundary ∂Σ = ∅, at least one of the surfaces Σ L , Σ R has a boundary component not in C. Without loss of generality we can assume that ∂Σ L C.
Because Σ L and Σ R both have at least one boundary component, their fundamental groups are free, say with m and n generators respectively. It will be convenient for us to choose the following presentation Here a 1 , . . . , a m are free generators of π 1 (Σ L ). Because ∂Σ L C, we can choose the first r generators a j , 1 j r to be homotopic to C j as shown on Figure 7. At the same time, for Σ R we will choose the presentation as follows: Let b 2 , . . . , b n+1 correspond to closed paths from q to q which freely generate π 1 (Σ R , q). Without loss of generality we can assume that b 2 , . . . , b r are homotopic to C 2 , . . . , C r . Now, let b 1 be the closed path homotopic to C 1 , it can be expressed as a word in other generators b 1 = F (b 2 , . . . , b n+1 ). 12 Denote by π 1 (Σ, p, q) the full subcategory of the fundamental grouppoid π 1 (Σ) with two objects p, q ∈ Σ. As a grouppoid π 1 (Σ, p, q) is generated by the following set of morphisms π 1 (Σ, p, q) = a 1 , . . . , a m , e 1 , . . . , e r , b 1 , . . . , b n , b n+1 a 1 = e −1 1 b 1 e 1 , . . . , a r = e −1 r b r e r , b subject to r + 1 relations.
3.3.2. Coordinate rings of G-representations and relations between them. Any G-representation of π 1 (Σ, p, q) is defined by m + n + r + 1 elements subject to r + 1 relations: This makes the set Hom(π 1 (Σ, p, q), G) of all G-representations of π 1 (Σ, p, q) into an algebraic subset of G ×(m+n+r+1) . Moreover, Hom(π 1 (Σ, p, q), G) comes equipped with a G × G action: Of course, any G-representation of π 1 (Σ, p, q) also defines G-representations of π 1 (Σ L , p) and π 1 (Σ R , q). Explicitly, for our choice of generators we obtain a diagram of G × G-equivariant projections (34a) Hom(π 1 (Σ, p, q), G) Hom(π 1 (Σ R , q), G) where  Proof. Note that the neighbourhood of C is a disjoint union of r annuli. Recall that H ≃ O[G] G ⊗r is the tensor product of r copies of the coordinate ring of the moduli space of G-representations of an annulus. It is naturally equipped with the trivial Poisson structure and with two natural injective Poisson homomorphisms to the Poisson center of the coordinate rings of G-character varieties of π 1 (Σ L , p) and π 1 (Σ R , q) respectively. As a result, is a Poisson embedding into the Poisson center of J. To finalize the proof, note that is a subalgebra of the full coordinate ring of the moduli space of G-representations of π 1 (Σ, p, q).
Proposition 24. Suppose that we have a pair of affine superintegrable systems on Σ L and Σ R defined by some choice of nonseparating cycles C L ⊂ Σ L and C R ⊂ Σ R : where Z L and Z R denote Poisson centers of A L and A R respectively. Then Krull dimensions of algebras B and J are determined as follows 13 Now denote by g L and b L the genus and number of boundary components of Σ L . Similarly, let g R and b R denote the genus and number of boundary components of Σ R . The resulting surface Σ then has a genus g = g L + g R + (r − 1) and number of boundary components b = b L + b R − 2r. Note that we have assume that neither Σ L nor Σ R is a cylinder (i.e. (g L , b L ) = (0, 2) = (g R , b R )). From (45) On the other hand, because both systems (42) are superintegrable, by (5) we conclude that Finally, using explicit formula (8) for Krull dimensions of A L , A R and (23) for Z L , Z R we obtain Hence, condition (5) on Krull dimensions of subalgebras (43) is satisfied and the resulting system is superintegrable.
3.4. The proof of Theorem 17. Now we can prove our main result.
Proof. Let Σ\C = Σ 1 ⊔ · · · ⊔ Σ m , where Σ 1 , . . . , Σ m stand for connected components. We will prove the statement of the theorem by induction in the number m of connected components.
The base case m = 1 follows by Proposition 20. Now assume that the statement of the theorem holds for all (Σ ′ , C ′ ) s.t. Σ ′ \C ′ has at most m connected components. Then for each pair (Σ, C) s.t. Σ\C has m + 1 connected components we can present Σ = Σ L ⊔ Σ R , where both Σ L \C and Σ R \C have at most m connected components. Define Note that C = C L ⊔ C LR ⊔ C R . By the inductive assumption we have two superintegrable systems . Now, applying Proposition 24 we finalize the proof.
The following proposition is an immediate consequence. 13 These two formulas is an algebraic version of a statement that for a pair of projections of irreducible affine varieties X → Z and Y → Z over C, the dimension of the corresponding fiber product is dim( Note that in order for (45) to be valid it is crucial that all algebras involved are finitely generated over C and have no zero divisors. Identities (45) then follow from the fact that transcendence degree of a field extension is additive w.r.t. the composition of field extensions.
Proposition 25. Let Σ be an oriented closed surface with b > 0 boundary components and C = C 1 ⊔· · ·⊔C k and C ′ = C ′ 1 ⊔ · · · ⊔ C ′ m be two collections of simple closed curves none of which is homotopic to a boundary component. Denote by I C and I C ′ the two superintegrable systems associated to C and C ′ as in Theorem 17.
• If C ⊂ C ′ is a subcollection of C ′ , the integrable system I C is a refinement of I C ′ .
• If an element of the Mapping Class Group of surface Σ bring the collection C to C ′ , it induces an equivalence of corresponding suerintegrable systems.
Remark 26. It is natural to expect that for an affine superintegrable system , and this is true in the compact case. Note that this implies that J Σ\C is the full Poisson centralizer of B C .
3.5. Beyond generic orbits. The main result of the paper, formulated in Theorem 1, states that a collection of simple closed curves C = C 1 ⊔· · ·⊔C r on an oriented surface Σ defines a family of superintegrable systems. This family is parameterized by generic values of Casimir functions, in other words, by fixing holonomies around boundary components at regular conjugation orbits. We have left details of the case of nongeneric values of Casimir functions outside of the scope of the current paper.
We expect to prove the following stronger statement in a sequel publication. Consider a superintegrable system from Theorem 1 and let be an associated chain of dominant maps of affine schemes preserving the Poisson bracket. Then for all z ∈ Spec Z, every irreducible component of the fiber p −1 (z) contains an open subset M 2n such that is a superintegrable system. Here π 1 , π 2 are maps induced by p 1 and p 2 respectively. Example of systems with lower rank orbits were studied in [CF18]. In that case G = SL m , the surface Σ is a torus with n punctures, the conjugacy class of the monodromy around each puncture is fixed and is assumed to be of rank 1. It is easy to see that the dimension of the moduli space in this case is 2(m − 1)n. The choice of C = C 1 , . . . C n being separating cycles cutting the torus into n cylinders with one puncture in each cylinder produces a Liouville integrable system with (m − 1)n Poisson commuting integrals. For any C i , the choice C = C i gives a superintegrable refinement of this Liouville integrable system.

Hamilton flows on the full coordinate ring
The Poisson bracket (21) on the coordinate ring O[Hom(π 1 (Σ, p 1 , . . . , p m ))] G of the moduli space M G Σ can be obtained from the bivector field on R = O[Hom(π 1 (Σ, p 1 , . . . , p m ))] using two essentially different methods. One way is the "r-matrix approach" developed in [FR93], the other way is the "Quasi Poisson" bracket approach developed in [AMM98, AK00, AKM02].
In the r-matrix approach one defines Poisson bracket on the full coordinate ring which coincides with the Goldman bracket on the In the Quasi Hamiltonian approach the bivectorfield on Hom(π 1 (Σ, p 1 , . . . , p m )) is not Poisson. The bracket {, } quasi : R ⊗ R → R satisfies the Jacobi identity only up to a fixed trivector field vanishing on R G . As in the r-matrix approach, quasi Poisson bracket induces the Goldman bracket on R G Here we will focus on the r-matrix approach. 14 The bracket (46) makes R into a left Lie module over R G . In other words, the G-invariant part acts on the full coordinate ring by derivations Action (47) allows one to integrate Hamilton flows described in Theorem 17. Moreover, with the appropriate choice of generators of the fundamental group, this action acquires a remarkably simple form presented in Theorem 29 below. This can be viewed as an analogue of (degenerate) separation of variables in the context of character varieties.
In subsection 4.1 for a given collection of simple closed curves as in Theorem 17 we define a particular choice of marked points and generators of π 1 (Σ, p 1 , . . . , p m ). Next, in subsection 4.2 we calculate action (47) via the r-matrix. 4.1. Choice of generators of π 1 . Let Σ be, as before, an oriented surface with nonempty boundary ∂Σ = D 1 ⊔ · · · ⊔ D b and let C = C 1 ⊔ · · · ⊔ C k be a disjoint collection of simple closed curves as in Theorem 17. Namely, we require that all C j are pairwise nonhomotopic and none of them are homotopic to the boundary. Let Σ\C = Σ 1 ⊔ · · · ⊔ Σ m , where Σ 1 , . . . , Σ m stand for connected components. For the purpose of this section we further assume that ∂Σ j ∩ ∂Σ = ∅ for all j ∈ {1, . . . , m}. This assumption can be made with no loss generality when we describe Hamilton flows, because initial conditions with trivial monodromy about the boundary component effectively correspond to absence of the boundary component.
For any connected component Σ j we choose a single marked point p j ∈ ∂Σ j ∩ ∂Σ on the common boundary with Σ. Each surface Σ j is an oriented surface of genus g j with b j boundary components. Its boundary can be presented as a disjoint union ∂Σ j = D βj,1 ⊔ · · · ⊔ D βj,r j ⊔ C αj,r j +1 ⊔ · · · ⊔ C α j,b j of r j connected components D βj,1 , . . . , D βj,r j ⊂ ∂Σ of the boundary of the full surface Σ, as well as b j − r j connected components C αj,r j +1 , . . . , C α j,b j ⊂ C of a curve C. Without loss of generality we can assume that p j ∈ D βj,1 .
Denote by the subcollection of simple closed curves in C which belong to Σ j , but do not belong to ∂Σ j . Such subcollection is nonseparating by definition.
We will choose a set of generators of π 1 (Σ, p 1 , . . . , p n ) as follows: (1) First, for each simple closed curve C i ∈ ∂Σ j ∩ ∂Σ k which separates a pair of distinct connected components Σ j = Σ k of Σ\C we choose a simple arc e i ∈ (Σ j \H j ) ∪ (Σ k \H k ) from p j to p k passing once through C j as shown on Figure 8. Note that without loss of generality we can assume that different e j intersect only at the endpoints. This can be achieved by sliding all intersection points between e i and e i ′ in Σ j to p j followed by left multiplication by an element of π 1 (Σ j , p j ). Similarly, one can get rid of the intersection points in Σ k . Figure 9. An example of a total order of half-edges adjacent to p j ∈ ∂Σ j )left to right).
(2) For each arc e i chosen on the previous step we pick a simple closed curve a i which starts and ends at p j and homotopic to C i as shown on Figure 7. Again, we can assume that none of the a i , e l intersect away from the endpoints. This can be achieved by choosing a i sufficiently close to e i ∪ C i . Also, each a i chosen on this step cuts Σ j into two connected components, one of them is a cylinder given by the neighbourhood of C i , the other is equivalent to the original surface. In what follows we assume that all further curves never enter into the cylinder.
(3) For each nonseparating simple closed curve C γ j,k from (48) we choose a pair of simple closed curves x γ j,k and y γ j,k , where y γ j,k is homotopic to C γ j,k , while x γ j,k passes once through C γ j,k as shown on Figure 6. (4) At the last step we consider Σ j \C. This surface has genus g j −s j and b j +2s j boundary components.
Our choice of arcs already generates the paths around r j + 2s j boundary components of Σ j \C. We can choose the remaining 2g j + b j − r j free generators of π 1 (Σ j \C) arbitrarily.
Collection of marked points and free generators of π 1 (Σ, p 1 , . . . , p m ) chosen above defines an ordered oriented ribbon graph with vertices p 1 , . . . , p m and edges corresponding to generators. The total order of half-edges adjacent to the given vertex is defined by an orientation of the surface. An example of the total order of half-edges is given on Figure 9.
Data of a ribbon graph can be encoded in m ordered sets of half-edges adjacent to p j S j = {h j,1 , . . . , h j,nj }, 1 j m.

4.2.
The r-matrix approach. Let g be a Lie algebra of G and r ∈ g ⊗ g be a classical r-matrix, i.e. a a solution to the Yang-Baxter equation [r 12 , r 13 ] + [r 12 , r 23 ] + [r 13 , r 23 ] = 0.
We will be interested in solutions corresponding to factorizable Lie bialgebras, i.e. classical r-matrices with the following symmetric part (with respect to the exchange of tensor components) given by the Killing form: Here {e J } is a basis in g which is orthonormal with respect to the Killing form and σ(x ⊗ y) = y ⊗ x. Such solutions are classified in [BD82].
The choice of free generators of π 1 (Σ, p 1 , . . . , p m ) fixes a natural isomorphism Hom(π 1 (Σ, p 1 , . . . , p m ), G) ≃ G N where N = 2g + b + m − 2. For an ordered ribbon graph V. Fock and A. Rosly [FR93] introduced a Distinct arcs. Basic building block of brackets corresponds to the case shown on Figure 10a when the two oriented arcs share the source but have distinct targets different from their common source. In this case the Fock and Rosly Poisson bracket between matrix elements 15 of holonomies along these arcs reads One can easily calculate the Poisson bracket between matrix elements of powers and products of matrices. For example, by Leibnitz identity applied to

we immediately obtain Poisson brackets for other three cases shown on Figures 10b-10d
Here r 21 = σ(r).
Bracket between matrix elements of a general distinct pair of arcs now can be computed by simply adding up to four terms of the form (51),(52) corresponding to brackets between pairs of half edges whenever they are adjacent to the same vertex.
Self-brackets. There are only three cases of relative position of half edges of the same arc as shown on Figure  11. Corresponding brackets are given by the following formulae where r a = 1 2 (r 12 − r 21 ) is the antisymmetric (with respect to the exchange of tensor components) part of r-matrix and we write r 12 for r. Here X ∈ g and (., .) is the Killing form.
Proof. It is enough to consider H(A) = Tr A k where trace is taken over a finite dimensional representation (which can be different from V ). Poisson bracket between matrix elements of A and E can be calculated as a sum of two terms: first term of the form (51a) corresponds to the ordered pair of half edges (a, e) adjacent to the same vertex; second term of the form (52c) corresponds to the ordered pair of half edges (e, a −1 ). As a result, we have Hence, by Leibnitz Identity for all k ∈ N we get Taking partial trace with respect to the first component we obtain (54a) as Poisson bracket between matrix elements of A and G can be calculated as a sum of two terms: first term of the form (51a) corresponds to ordered pair of half edges (a, g); second term of the form (51b) corresponds to ordered pair of half edges (a −1 , g). Hence, we have By Leibnitz Identity we get that for all k ∈ N Taking partial trace with respect to the first component we get Poisson bracket between matrix elements of A and F is determined by (52a) and (52c) and has the following form Note that r 21 is on the same side from F in both terms of (55), so as in the previous case we obtain for all k ∈ N {Tr A k , F } = 0.
Finally, Poisson bracket between matrix elements of A is given by (53b) By Leibnitz Identity we get for all k ∈ N Taking the partial trace with respect to the first component we have From the invariance of I and from the cyclic invariance of the trace we get Corollary 28. Let (y, x) and (y, a) be pairs of generators with relative order of half edges as shown on Figure 13 and H and π V as in the previous proposition. Denote by A, X, Y ∈ Hom(π 1 (Σ, p 1 , . . . , p m ), G) the holonomies along the corresponding closed oriented arcs. We have Proof. Indeed, the Poisson bracket between Tr Y k and X ij gets two contributions of the form (54a) and (54b) which correspond to the outgoing and the ingoing half edges of X respectively. As a result, we obtain (57a).
On the other hand, the Poisson bracket between Tr Y k and A ij is trivial due to (54b) because none of the half edges of A appear in between half edges of Y . Similar logic applies to the bracket between Tr A k and Y ij .
Note that the Hamilton flow given by H(Y ) preserves matrix element functions of the group commutator π V (XY X −1 Y −1 ).
Theorem 29. Let Σ be an oriented surface with b > 0 boundary components and C = C 1 ⊔ · · · ⊔ C k be a disjoint union of pairwise nonintersecting simple closed curves in Σ as in Theorem 17, H : G → C be a central function and π V : G → End(V ) be a finite dimensional representation. Fix a choice of marked points p 1 , . . . , p m ∈ ∂Σ and generators of π 1 (Σ, p 1 , . . . , p m ) as in Section 4.1 and denote by A i , E i , X l , Y l , Z o ∈ Hom(π 1 (Σ, p 1 , . . . , p m ), G) the holonomies along arcs a i , e i , x l , y l , z o , then Particular cases of Theorem 29 already appeared in the literature. For example, these brackets were computed in [FR99] for G = SL(N, C) and a torus with one boundary component.
Corollary 30. Fix i, 1 i r and a central function H : G → C. Then the Hamiltonian flow generated by the function H evaluated on a cycle C i ∈ Σ is given by the following expressions • When C i ∈ ∂Σ a ∩ ∂Σ b separates a pair of distinct connected components of Σ\C we have • Similarly, when C l is a nonseparating cycle which belongs to the boundary of a single connected component of Σ\C the Hamiltonian flow generated by H evaluated at the holonomy along C l is given by X s (t) = X s exp (t∇H(Y l )) , l = s, Remark 31. Let ψ t is the evolution on the algebra of functions on Hom(π 1 (Σ, p 1 , . . . , p m ), G) generated by H, i.e.
If the functions H 1 and H 2 Poisson commute then ψ H1 t1 ψ H2 t2 = ψ H2 t2 ψ H1 t1 . If H 1 , . . . , H k is a complete set of independent functions on holonomies along C, their joint flow lines generate ange variables on their level surfaces. . Let Σ 1,1 denote the torus with one boundary component. Choose a marked point p 1 ∈ ∂Σ 1,1 . The fundamental group π 1 (Σ 1,1 , p 1 ) is freely generated by a pair of arcs X, Y along the equator and meridian of the torus. Let Γ be the ribbon graph associated to such choice of generators. The data of Γ is encoded by an ordered set

Particular Cases
of half-edges adjacent to p 1 as shown on Figure 13a. In (58) we have labelled half-edges by generators X, Y of π 1 (Σ 1,1 , p 1 ) and their inverses X −1 , Y −1 according to the convention introduced in the last paragraph of Section 4.1. Namely, we label an outgoing half-edge by the first power of the corresponding generator, while we label an ingoing edge by the inverse of the corresponding generator.
Moduli Space. The moduli space of flat SL(2, C)-connections on a once punctured torus has dimension n = 3. We will use notation τ A = Tr(A) for the trace of a matrix A. The coordinate ring of the moduli space in this example is a free commutative algebra with three generators where X, Y ∈ SL(2, C) are elements representing standard a and b cycles. The Poisson bracket between these coordinate functions are: This is a rank 2 Poisson structure. ] is generated by a single Casimir function Superintegrable system. Choose cycle C defining the system to be Y and the Hamiltonian H = τ Y . From (25) we obtain a chain of subgroups Here and below we write a 1 , . . . , a n for a free group generated by a 1 , . . . , a n . The algebra of first integrals is then generated by three elements, two of which coincide As a result, we obtain the following chain of subalgebras of the coordinate ring ].
In this case the algebra of Hamiltonians coincide with teh algebra of first integrals and therefore the system is Liouville integrable.
Mapping Class Group action. Consider the mapping class group M od(Σ 1,1,0 ) (relative to the boundary) of the torus with one boundary component and no punctures. It contains two left Dehn twists along the X and Y cycles which satisfy the braid relation and are acting on our generators as follows:

This defines a pair of Poisson automorphisms
Note that action of M od(Σ 1,1,0 ) on the character variety factors through the action of the Mapping Class Group of a torus with one puncture M od(Σ 1,0,1 ) ≃ SL(2, Z).
Poisson automorphisms (61) define a family of isomorphic integrable systems associated to nonseparating cycles on Σ 1,1 . 5.2. Torus with two boundary components, G = SL(2, C). The coordinate ring of the moduli space has dimension n = 6 and the coordinate ring is generated by 7 polynomials subject to the single relation .
Here τ A stands for Tr (A) and Poisson brackets between generators are summarized in Table 1 ]. The Poisson center Z of the coordinate ring has Krull dimension dim Z = 2 and has two algebraically independent Casimir elements corresponding to traces of monodromies around each of the boundary components and two more first integrals 5.2.2. Separating cycle. Now consider a superintegrable system given by a separating cycle homotopic to XY X −1 Y −1 . As in the previous case we have two Casimir elements z 1 and z 2 . Their generic level sets form a 2-parametric family of 4-dimensional symplectic manifolds. Choose the Hamiltonian on these phase spaces as Σ0,3 ]. Here J R is Poisson commutative, while J L is equipped with a Poisson bracket of generic rank 2 (see first three rows of Table 1). Algebra J has Krull dimension 5. An example of maximal algebraically independent subset is z 1 , z 2 , H, g 1 = τ X , g 2 = τ Y , i.e. generators of the Poisson center, the Hamiltonian and two additional first integrals.
5.3. Torus with one boundary component, G = SL(3, C). The moduli space of flat SL(3, C)-connections on a once punctured torus has dimension n = 8, the coordinate ring is generated by 9 polynomials subject to a single relation ,1 ] has a Poisson center Z of Krull dimension dim Z = 2. Two algebraically idependent Casimir elements: z 1 = τ Y XY −1 X −1 and z 2 = τ XY X −1 Y −1 . The formula expressing z 1 in terms of generators is given in (66) 5.3.1. A superintegrable system. Here we consider a superintegrable system defined by a single cycle homotopic to X. By (25) we have a chain of inclusions of subgroups of the fundamental group π 1 (C) = X ⊂ π 1 (Σ 1,1 , p | C) = X, Y XY −1 ⊂ π 1 (Σ 1,1 , p) = X, Y . These two Poisson commuting Hamiltonians define a superintegrable system with the Poisson algebra of first integrals J defined above. We can choose a maximal algebraically idependent subset in J to be z 1 , z 2 , H 1 , H 2 with two more integrals 6. Conclusion 6.1. The algebra of chord diagrams on a surface and "universal superintegrable systems". The algebra of chord diagrams is a universal model for the Poisson algebra of functions on moduli spaces of flat connections [AMR96], see also [T91] for the case of loop algebras on a surface, i.e. skein modules. Chord diagrams first appeared in the setting of finite type invariants of knots [V90] and in perturbative Chern-Simons invariants [BN91] [K93]. They became an important tool in the related theory of Vassiliev invariants. Let us recall some basic definitions. A chord diagram is a homotopy class of a graph on a surface which consists of solid lines and chords. Solid lines are oriented, cords are not oriented. They satisfy the analogue of Reidemeister relations (see [AMR96] for details) together with the so-called 4T-relation shown on Figure  14.
The To define a conjecturally superintegrable system on Ch(Σ) choose a system of simple curves on the surface. Define the subalgebra of Hamiltonians B(C) and the subalgebra of chord diagrams which are contractible to C and the subalgebra of first integrals J(C) and the subalgebra of chord diagrams which can be separated from C. We have a natural inclusion of Poisson subalgebras: We conjecture that this chain of inclusions define a superintegrable systems. In this infinite dimensional setting the superintegrability means that the space of all nontrivial Poisson derivations Der J(C) (Ch(Σ)) of Ch(Σ) which act trivially on J(C) is generated as an Ch(Σ)-module by Hamilton derivations corresponding to elements from B(C).
6.2. Quantization and the algebra of links in a cylinder. The natural question about any integrable system is how to quantize it. In the case of superintegrable systems on chord diagrams there is a natural quantization.
There is a natural associative algebra that can be naturally associated to a surface. The his is the algebra of links, or linked graphs, in Σ × I where I is an interval. This algebra, which we will denote A(Σ) is the space of Z-linear combinations of links in Σ × I. The natural associative multiplication on this space is the "placing of one link on the top of the other": where in the right side of this equation we assume that L 1 ⊂ Σ[1, 1/2] and L 2 ⊂ Σ × [1/2, 0]. here [L] is the topological link which the isotopy class of a geometrical link L : S 1 ×n → Σ × I.
The algebra A(Σ) has a natural filtration with Ch(Σ) being isomorphic to its associated graded algebra. In this sense A(Σ) is the quantization of Ch(Σ).
Finite dimensional representations of the algebra A(Σ) can be easily constructed using [RT90]. Such representations use representation theory of U q (g) at roots of unity. We will present details in a separate publication.
For generic q, the quantization can be done by quantizing Fock and Rosly brackets by R-matrices, see for example [AGS95,AS96]. It is worth noting that a closely related question for the extended moduli space with the choice of flags at marked points on the boundary can be studied by means of Cluster Algebras [FG06], see for example [SS17,GSV18,CMR17] and references therein.