arXiv:1107.5588v2 [math.AG] 12 Nov 2012
Dimers and cluster integrable systems
A.B. Goncharov, R. Kenyon
Abstract
We show that the dimer model on a bipartite graph Γ on a torus gives rise
to a quantum integrable system of special type, which we call a cluster integrable
system. The phase space of the classical system contains, as an open dense subset,
the moduli space LΓ of line bundles with connections on the graph Γ. The sum of
Hamiltonians is essentially the partition function of the dimer model.
We say that two such graphs Γ1 and Γ2 are equivalent if the Newton polygons
of the corresponding partition functions coincide up to translation. We define elementary transformations of bipartite surface graphs, and show that two equivalent
minimal bipartite graphs are related by a sequence of elementary transformations.
For each elementary transformation we define a birational Poisson isomorphism
LΓ1 → LΓ2 providing an equivalence of the integrable systems. We show that it is
a cluster Poisson transformation, as defined in [FG03b].
We show that for any convex integral polygon N there is a non-empty finite set
of minimal graphs Γ for which N is the Newton polygon of the partition function
related to Γ. Gluing the varieties LΓ for graphs Γ related by elementary transformations via the corresponding cluster Poisson transformations, we get a Poisson
space XN . It is a natural phase space for the integrable system. The Hamiltonians
are functions on XN , parametrized by the interior points of the Newton polygon N .
We construct Casimir functions whose level sets are the symplectic leaves of XN .
The space XN has a structure of a cluster Poisson variety. Therefore the algebra
of regular functions on XN has a non-commutative q-deformation to a ∗-algebra
Oq (XN ). We show that the Hamiltonians give rise to a commuting family of quantum Hamiltonians. Together with the quantum Casimirs, they provide a quantum
integrable system. Applying the general quantization scheme [FG07], we get a
∗-representation of the ∗-algebra Oq (XN ) in a Hilbert space. The quantum Hamiltonians act by commuting unbounded selfadjoint operators.
For square grid bipartite graphs on a torus we get discrete quantum integrable
systems, where the evolution is a cluster automorphism of the ∗-algebra Oq (XN )
commuting with the quantum Hamiltonians. We show that the octahedral recurrence, closely related to Hirota’s bilinear difference equation [Hir81], appears this
way.
Any graph G on a torus T gives rise to a bipartite graph ΓG on T. We show
that the phase space X related to the graph ΓG has a Lagrangian subvariety R,
defined in each coordinate system by a system of monomial equations. We identify
it with the space parametrizing resistor networks on G. The pair (X , R) has a large
1
group of cluster automorphisms. In particular, for a hexagonal grid graph we get a
discrete quantum integrable system on X whose restriction to R is essentially given
by the cube recurrence [Miwa82], [CS04].
The set of positive real points XN (R>0 ) of the phase space is well defined. It is
isomorphic to the moduli space of simple Harnack curves with divisors studied in
[KO03]. The Liouville tori of the real integrable system are given by the product of
ovals of the simple Harnack curves.
In the sequel [GK] to this paper we show that the set of complex points XN (C) of
the phase space is birationally isomorphic to a finite cover of the Beauville complex
algebraic integrable system related to the toric surface assigned to the polygon N .
Contents
1 Introduction
1.1 Dimer models and Poisson geometry. . . . . . . . . . . . . . . . . . . . .
1.1.1 Conjugate surface graph and Poisson structure . . . . . . . . . . .
1.1.2 Natural coordinates on LΓ . . . . . . . . . . . . . . . . . . . . . . .
1.1.3 Zig-zag loops and the center of the Poisson algebra of functions on
LΓ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.4 Gluing the Poisson phase space . . . . . . . . . . . . . . . . . . .
1.2 Cluster integrable systems for dimer models on a torus. . . . . . . . . . .
1.2.1 Classical integrable system . . . . . . . . . . . . . . . . . . . . . .
1.2.2 Quantum integrable system . . . . . . . . . . . . . . . . . . . . .
1.2.3 Discrete cluster integrable systems. . . . . . . . . . . . . . . . . .
1.3 Cluster nature of a resistor network model on a torus . . . . . . . . . . .
1.3.1 Dimer model arising from a graph on a torus . . . . . . . . . . . .
1.3.2 Resistor network Lagrangian subvariety . . . . . . . . . . . . . . .
1.4 Toric surfaces, spectral data and the Beauville integrable system . . . . .
1.4.1 The Kasteleyn operator . . . . . . . . . . . . . . . . . . . . . . .
1.4.2 The spectral data. . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.3 A cover of the Beauville integrable system. . . . . . . . . . . . . .
1.4.4 Connection with the dimer integrable system. . . . . . . . . . . .
1.5 Analogies between dimers, Teichmüller theory and cluster varieties. . . .
1.6 Further perspectives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7 Historical remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.8 The structure of the paper. . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Newton polygons and minimal bipartite graphs on surfaces
2.1 Zig-zag paths and minimal bipartite surface graphs. . . . . . . . . . . . .
2.2 From Newton polygons to collections of admissible graphs. . . . . . . . .
2.2.1 From admissible surface graphs to bipartite surface graphs. . . . .
2.2.2 Minimal triple point diagrams [Thur04]. . . . . . . . . . . . . . .
2.3 From minimal triple point diagrams to minimal bipartite surface graphs.
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Boundary intervals of the Newton polygon and zig-zag paths. . . . . . . .
Elementary transformations of bipartite surface graphs. . . . . . . . . . .
From minimal bipartite surface graphs to minimal triple point diagrams.
Graphs from polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Integrability of the dimer model
3.1 Edge weights and line bundles with connections on a graph
3.2 The partition function and the Hamiltonians . . . . . . . .
3.2.1 The weight of a dimer cover . . . . . . . . . . . . .
3.2.2 The homology class of a dimer cover . . . . . . . .
3.2.3 The α-deformed partition function . . . . . . . . .
3.2.4 The Hamiltonians . . . . . . . . . . . . . . . . . . .
3.2.5 A coordinate expression. . . . . . . . . . . . . . . .
3.3 The classical Hamiltonians commute . . . . . . . . . . . .
3.4 The matching polytope . . . . . . . . . . . . . . . . . . . .
3.5 The Newton polygon . . . . . . . . . . . . . . . . . . . . .
3.6 Counting triple crossings and faces . . . . . . . . . . . . .
3.7 The number of Hamiltonians . . . . . . . . . . . . . . . . .
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4 Dimers, cluster Poisson transformations, and quantum integrability
4.1 Dimer models on surfaces and cluster Poisson varieties. . . . . . . . . . .
4.1.1 Gluing the conjugated surfaces according to a spider move . . . .
4.1.2 Seeds, mutations, conjugated surfaces and spider moves . . . . . .
4.1.3 Quantum cluster transformations [FG03b] . . . . . . . . . . . . .
4.1.4 Cluster Poisson varieties [FG03b] . . . . . . . . . . . . . . . . . .
4.1.5 Gluing the moduli spaces LΓ according to a spider move . . . . .
4.1.6 Traditional cluster Poisson variety related to the dimer model . .
4.2 Spider moves and partition functions . . . . . . . . . . . . . . . . . . . .
4.3 Quantum integrable system . . . . . . . . . . . . . . . . . . . . . . . . .
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5 Resistor network model and its cluster nature
5.1 Resistor network model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1 Laplacians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.2 Dual graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.3 Zig-zag paths for a graph G on a torus. . . . . . . . . . . . . . . . .
5.1.4 Elementary transformations. . . . . . . . . . . . . . . . . . . . . . .
5.2 The associated dimer model on a torus . . . . . . . . . . . . . . . . . . . .
5.2.1 The bipartite graph assigned to a graph on a torus. . . . . . . . . .
5.2.2 The resistor network subvariety. . . . . . . . . . . . . . . . . . . . .
5.2.3 Determinant of the Laplacian and the dimer model partition function
5.2.4 Zig-zag paths and the Newton polygon . . . . . . . . . . . . . . . .
5.2.5 Equations defining the resistor network subvariety. . . . . . . . . .
5.2.6 The resistor network subvariety is Lagrangian. . . . . . . . . . . . .
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5.2.7 Quantized cluster monomial Lagrangian subvarieties .
5.3 Cluster nature of the resistor network space . . . . . . . . .
5.3.1 The basic commutative diagram for a graph G . . . .
5.3.2 Action of the Y-∆-move on coordinates. . . . . . . .
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6 Discrete cluster integrable systems
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6.1 Discrete cluster integrable systems on a square grid on a torus . . . . . . . 60
6.2 Cluster automorphisms for resistor networks on a torus. . . . . . . . . . . . 62
6.3 The cube recurrence discrete cluster integrable system . . . . . . . . . . . . 63
7 Newton polygons, toric surfaces and integrable
7.1 Newton polygons and toric surfaces . . . . . . .
7.2 The spectral data [KO03] . . . . . . . . . . . . .
7.3 The real positive points [KO03] . . . . . . . . .
7.4 Independence of the Hamiltonians . . . . . . . .
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8 Appendix: Calculating the Poisson structure on LΓ
8.1 The local pairing. . . . . . . . . . . . . . . . . . . . .
8.2 The global pairing . . . . . . . . . . . . . . . . . . .
8.3 Amalgamation and the Poisson structure on LΓ . . . .
8.3.1 Bipartite ribbon graphs . . . . . . . . . . . .
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Introduction
A line bundle V on a graph Γ is given by assigning a 1-dimensional complex vector space
Vv to each vertex of Γ. A connection on V is a choice of isomorphism φvv′ : Vv → Vv′
whenever v, v ′ are adjacent, satisfying φv′ v = φ−1
vv′ .
We construct a Poisson structure on the space of line bundles with connections on
a bipartite graph embedded on a surface. When the graph is embedded on a torus T
we construct an algebraic integrable system, with commuting Hamiltonians which can be
written as Laurent polynomials in natural coordinates on LΓ , the moduli space of line
bundles with connections on Γ. The Hamiltonians are sums of dimer covers of Γ.
A dimer cover of a graph Γ is a set of edges with the property that every vertex is
the endpoint of a unique edge in the cover. Probability measures on dimer covers have
been the subject of much recent work in statistical mechanics. In this paper we study
some non-probabilistic applications of the dimer model, in particular to the construction
of cluster integrable systems.
Below we introduce the key non-technical definitions which we use throughout the
paper, and discuss the main results.
4
1.1
Dimer models and Poisson geometry.
A surface graph Γ is a graph embedded on a compact oriented surface S whose faces, i.e.
the connected components of S − Γ, are contractible. It is homotopy equivalent to an
open surface S0 ⊂ S, obtained by putting a puncture in every face. A bipartite graph is a
graph with vertices of two types, black and white, such that each edge has one black and
one white vertex.
1.1.1
Conjugate surface graph and Poisson structure
Let LΓ be the moduli space of line bundles with connections on a graph Γ. Any oriented
loop L on Γ gives rise to a function WL on LΓ given by the monodromy of a line bundle
with connection along L. One has W−L = WL−1 , where −L is the loop L with the opposite
orientation. The monodromies around the loops provide an isomorphism
LΓ ∼
= Hom(H1 (Γ, Z), C∗ ) = H 1 (Γ, C∗ ).
(1)
For any surface graph Γ, the moduli space LΓ has the standard Poisson structure.
Namely, since Γ is homotopy equivalent to the surface S0 , we have LΓ = H 1 (S0 , C∗ ). This
latter space has a Poisson structure provided by the intersection pairing on H1 (S0 , Z).
Now let Γ be a bipartite surface graph. Then we introduce a new Poisson structure
on LΓ related to the standard one by a “twist”.
A surface graph is the same thing as a ribbon graph: a ribbon graph is a graph with an
additional structure given by, for each vertex, a cyclic order of the edges at that vertex.
Given a ribbon graph, we can replace its edges by ribbons, getting an oriented surface.
bw = Γ,
b the conjugate
A bipartite surface graph Γ gives rise to a new ribbon graph Γ
1
graph, obtained by reversing the cyclic orders at all white vertices . The topological
b w is called the conjugated
surface with boundary corresponding to the ribbon graph Γ
b
surface Sw .
b is the same thing as a line
Evidently a line bundle with connection on the graph Γ
bundle with connection on the original graph Γ. So there is a canonical isomorphism
LΓ = LΓb .
(2)
The standard Poisson structure on the moduli space LΓb combined with the isomorphism
(2) provides the Poisson structure on LΓ we need. Precisely, consider the intersection
pairing on Sbw :
(3)
εSbw : H1 (Sbw , Z) ∧ H1 (Sbw , Z) −→ Z.
Given two loops L1 , L2 on Γ, we define the Poisson bracket {WL1 , WL2 } by setting
{WL1 , WL2 } = εSbw (L1 , L2 )WL1 WL2 .
(4)
and extending it via the Leibniz rule to a Poisson bracket on the algebra generated by the
functions WL , which coincides with the algebra of regular functions on the variety LΓ.
b b , which differs from Γ
b w by
Reversing the cyclic order at the black vertices we get a ribbon graph Γ
the orientation.
1
5
1.1.2
Natural coordinates on LΓ .
The orientation of the surface S induces orientations of the faces of Γ. The group H1 (Γ, Z)
is generated by the oriented boundaries ∂(F ) of the faces F of Γ, whose orientation is
induced by the orientation of F , and loops generating H1 (S, Z). The only relation is that
the sum of boundaries of all faces is zero. There is an exact sequence, where the map π
is induced by the embedding Γ ֒→ S:
π
0 −→ Ker π −→ H1 (Γ; Z) −→ H1 (S; Z) −→ 0.
The subgroup Ker π is generated by the oriented boundaries ∂(F ) of faces F .
Summarising, the monodromies of connections around all but one oriented faces F of
Γ, augmented by the monodromies around loops generating H1 (S, Z), provide a coordinate
system {Xi } on the moduli space LΓ . The Poisson structure in these coordinates has a
standard quadratic form
{Xi , Xj } = εij Xi Xj ,
εij ∈ Z,
(5)
where ε is the intersection pairing on Ŝw .
1.1.3
Zig-zag loops and the center of the Poisson algebra of functions on LΓ .
A zig-zag path (see [K02, Post06]) on a ribbon graph graph Γ is an oriented path on Γ
which turns maximally left at white vertices and turns maximally left at black vertices;
see Figure 1. Zig-zag paths necessarily close up to form zig-zag loops. There is a bijection
{zig-zag loops on Γ} ↔ {boundaries of the holes on Sbw }.
(6)
The orientations of zig-zag loops match the orientations of the boundary loops on Sbw
induced by the orientation of Sbw .
Figure 1: A Zig-zag path on a ribbon graph.
Lemma 1.1. Let Z be an (oriented) zig-zag loop on a bipartite oriented surface graph Γ.
Then as Z runs over zig-zag loops, the functions WZ generate the center of the Poisson
algebra O(LΓ ) of functions on LΓ. The product of all of them is 1. This is the only
relation between them.
We call the functions WZ the Casimirs.
6
Proof. It follows from the canonical bijection (6). Evidently the boundary loops on Sbw
generate the kernel of the intersection pairing on H1 (Sbw , Z). Clearly the sum of them is
zero, and this is the only relation between them.
There is a similar Poisson moduli space LΓ for a bipartite graph Γ on an oriented
surface S with boundary.
1.1.4
Gluing the Poisson phase space
The set of homology classes of dimer coverings of Γ (as defined below) defines a convex
polyhedron N ⊂ H1 (S, R). It is the Newton polyhedron of the dimer partition function.
Two bipartite surface graphs are equivalent if their Newton polyhedra coincide up to
translation. In each equivalence class there are minimal bipartite surface graphs, which
have the minimal number of faces. We prove that every two minimal bipartite surface
graphs are related by certain elementary transformations, consisting of “spider” moves
and “shrinking of a 2-valent vertex” moves. Each elementary transformation Γ1 → Γ2
gives rise to a cluster Poisson birational isomorphism
∼
iΓ1 ,Γ2 : LΓ1 −→ LΓ2
(7)
We define a Poisson variety XN , the phase space, by gluing the moduli spaces LΓ via
transformations (7). The variety XN depends only on N.
Since the sets of zig-zag loops on equivalent bipartite surface graphs are naturally
identified, the gluing maps (7) preserve the Casimirs. Therefore the center of the Poisson
algebra of regular functions on the phase space XN is also described by the Casimirs.
1.2
1.2.1
Cluster integrable systems for dimer models on a torus.
Classical integrable system
Let us restrict to the case when Γ is a bipartite graph on a torus T. In this case the
Newton polyhedron N is a convex polygon in the plane H1 (T, R) with vertices at the
lattice H1 (T, Z). Zig-zag loops are parametrized by the intervals on the boundary ∂N
between two consecutive integral points, called primitive boundary intervals.
We show that the partition function of the dimer model on Γ gives rise to a regular
function P on XN , the modified partition function, well defined up to a multiplication
by a monomial of Casimirs. It has a natural decomposition into a sum of components
Ha , called the Hamiltonians, parametrized by the internal integral points of the Newton
polygon N. These Hamiltonians are Laurent polynomials in the Xi ’s. The Hamiltonian
flows are well defined.
Theorem 1.2. The Hamiltonian flows of the Ha commute, providing an integrable system
on XN . Precisely, we get integrable systems on the generic symplectic leaves of XN , given
by the level sets of the Casimirs.
7
In particular,
dimXN = 2i(N) + e(N) − 1,
where i(N) (respectively e(N)) is the number of integral points inside (respectively on
the boundary) of the polygon N.
1.2.2
Quantum integrable system
The algebra of regular functions on any cluster Poisson variety X has a natural qdeformation to a ∗-algebra Oq (X ), defined in [FG03b, Section 3] It can be realized, in
many different ways, as a subalgebra in the quantum torus algebra. The latter is generated by the elements Xi±1 satisfying the following relations, where q is a formal variable
commuting with Xi ’s:
q −εij Xi Xj = q −εji Xj Xi ,
εij ∈ Z.
(8)
It has an involutive antiautomorphism ∗, acting on the generators by
∗(q) = q −1 .
∗(Xi ) = Xi ,
So in our case we get a ∗-algebra Oq (XN ). The εij in (8) is the same as the Poisson
tensor in (5). The quantum analog of Theorem 1.2 is
Theorem 1.3. The Hamiltonians Ha give rise to commuting selfadjoint elements Ha in
the ∗-algebra Oq (XN ), which we call quantum Hamiltonians. The Casimirs give rise to
the generators of the center of the algebra.
In any of the coordinate systems provided by bipartite graphs, the quantum Hamiltonians are given by Laurent polynomials with positive integral coefficients.
Let us assume now that |q| = 1. Then, according to [FG07], for any cluster Poisson
variety X , there is a family of Hilbert spaces Hχ , parametrized by the characters χ of the
center of the algebra Oq (X ), and a ∗-representation of the ∗-algebra Oq (X ) in (a certain
Schwartz space Sχ , dense in) Hχ .
So in our case we get a family of Hilbert spaces HN,χ , parametrized by a collection of
real numbers χ, the eigenvalues of the Casimirs. The quantum Hamiltonians are realized
by commuting unbounded selfadjoint operators in HN,χ .
Problem 1. Find the generalized eigenfunctions of the quantum Hamiltonians, and the
spectral decomposition.
We call the emerging object a cluster integrable system. A key point is that a cluster
integrable system admits a quantum version, as described above. We believe that many
interesting integrable systems are cluster integrable systems.
8
1.2.3
Discrete cluster integrable systems.
In the case of the dimer model on a square grid on a torus2 our quantum integrable system
has a remarkable automorphism A, called the octahedron recurrence or, in other language,
Hirota’s bilinear difference equation (HBDE) [Hir81]. We show that it commutes with the
quantum Casimirs and quantum Hamiltonians, and is given by an element of the cluster
modular group [FG03b] of the cluster variety X – see Section 6.
There are similar discrete cluster integrable systems related to hexagonal graphs G
on a torus, where the discrete evolution operator is described by the cube recurrence, see
Sections 6.2-6.3. Its phase space is the phase space of the dimer model related to the
resistor network model on G, see Section 1.3.
We stress that discrete cluster integrable systems are discrete quantum integrable
systems. Indeed, the ring of regular functions on any cluster Poisson variety admits a
canonical non-commutative q-deformation, and the discrete evolution operator is given
by an element of the cluster modular group, which acts by an automorphism of the qdeformed space [FG03b]. Finally, the classical Hamiltonains, in all examples we know,
are given by functions which in any of the coordinate systems are Laurent polynomials
with positive integral coefficients. Therefore they admit a natural upgrade to their noncommutative counterparts.
1.3
1.3.1
Cluster nature of a resistor network model on a torus
Dimer model arising from a graph on a torus
A special case of the dimer model is one arising from a planar resistor network G via
the generalized Temperley’s trick [KPW00]. Namely, an arbitrary graph G on a torus
gives rise to a bipartite graph Γ = ΓG , obtained by taking a barycentric subdivision of
the graph G and declaring the centers of the edges of G to be the white vertices and the
centers of the faces of G together with the vertices of G to be the black vertices. The
graph Γ contains the dual graph G′ to the graph of G, see Figure 2, where G is shown by
a solid lines, and G′ by dotted lines.
e embedded on the
Given a surface graph G embedded on a surface S there is a graph G
e
universal cover S, which is the natural lift of G. A surface graph G embedded on surface
S is called minimal if zig-zag paths (defined as unoriented paths which turn alternately
e do not have selfcrossings
maximally left and maximally right at consecutive vertices) on G
and different zig-zag paths do not intersect in more than one edge.
The Newton polygon N(G) of the bipartite graph ΓG is a centrally-symmetric integral
polygon in H1 (T, Z) centered at the origin. It coincides with the Newton polygon of the
spectral curve of the Laplacian related to a generic resistor network on G, see Section 5.
Any such polygon arises as a Newton polygon N(G) for a non-empty finite collection
of minimal graphs G on a torus. Any two minimal graphs G1 and G2 on a torus with
2
The square grid dimer models on a torus are parametrized by sublattices of the isomorphism group
Z of the standard square grid bipartite graph in R2 .
2
9
Figure 2: The bipartite graph ΓG (dashed and solid edges) assigned to a graph G (solid
edges).
the same Newton polygon are related by a sequence of elementary transformations, called
Y-∆ moves, see Figure 24.
A Y-∆ move G1 → G2 induces a transformation Γ1 → Γ2 of the corresponding bipartite graphs. We decompose it into a composition of four spider moves (Lemma 5.11),
which provides us with a cluster birational isomorphism
µY −∆ : LΓ1 −→ LΓ2 .
(9)
Gluing the tori LΓ assigned to the bipartite graphs Γ related to the equivalent minimal
graphs G, we get a cluster Poisson space X . 3
1.3.2
Resistor network Lagrangian subvariety
The moduli space LΓG of line bundles with connections on the graph ΓG has a subvariety
RG defined by the condition that the monodromies over any loop α in G and any loop α′
in G′ in the same homology class on T are equal. These equations are given by equating
certain monomials in the Xi ’s to 1.
We prove that RG is a Lagrangian subvariety of the Poisson variety LΓG , i.e. its
intersections with the generic symplectic leaves are Lagrangian subvarieties. We show
that the space RG parametrizes resistor networks on G, justifying the name resistor
network Lagrangian subvariety.
Equivalently, the conjugated surface SbΓG is canonically realized as the boundary of a
certain handlebody.
We show that the Y-∆-cluster transformation (9) provides a birational isomorphism
of the Lagrangian subvarieties
µR
Y −∆ : RG1 −→ RG2 .
3
Gluing this space we use only the graphs Γ which are equivalent by the Y-∆-moves. We may have
other bipartite graphs which are still equivalent as bipartite graphs.
10
Gluing the subvarieties RG via these birational isomorphisms, we get a Lagrangian subspace R of the corresponding cluster Poisson space X . It gives rise to a uniquely defined
up to a constant functional on the Schwartz space Sχ , see Section 5.2.7.
1.4
Toric surfaces, spectral data and the Beauville integrable
system
Let T := TT ∼
= (C∗ )2 be the algebraic torus related to the torus T. Recall that a bipartite
graph Γ on T gives rise to a Newton polygon N in H1 (T, R) with vertices at H1 (T, Z).
Such a Newton polygon N, considered up to translation, provides us a toric surface N
together with a line bundle L. The algebraic torus T acts on the pair (N , L). The points
of N where the action is not free form the divisor at infinity N∞ of N . It is a collection
of P1 ’s which is combinatorially isomorphic to the polygon N.
1.4.1
The Kasteleyn operator
A Kasteleyn line bundle with connection K on a bipartite surface graph Γ is a line bundle
with connection with monodromy (−1)ℓ/2+1 around faces having ℓ edges, and ±1 around
topologically nontrivial loops. There are 22g non-isomorphic choices of K when S has
genus g. Given a line bundle with connection V on a bipartite surface graph Γ, we
construct a new one V ⊗ K, and set
M
VB :=
(V ⊗ K)v ,
(10)
v∈{black vertices of Γ}
VW :=
M
(V ⊗ K)v .
(11)
v∈{white vertices of Γ}
A connection on the line bundle V ⊗ K is encoded by an operator
K : VB −→ VW ,
the Kasteleyn operator, whose matrix elements are given by parallel transports for the
connection along the edges of Γ. We let
L = detVB∗ ⊗ detVW
(12)
be the determinant line (here det V denotes the highest exterior power of a vector space
V ). The determinant det K is an element of the determinant line.
1.4.2
The spectral data.
Let
∼
T := Hom(H1 (T, Z), C∗ ) = H1 (T, C∗ ) = (C∗ )2 ;
The group T is identified with the group of complex line bundles with flat connections on
the torus T. The embedding i of the graph Γ into the torus T provides a free action of
11
the group T on the moduli space LΓ , given by ∗ 7−→ ∗ ⊗ i∗ L, where L is a flat line bundle
on the torus T. This action preserves the Poisson structure on LΓ .
The partition function can be calculated as the determinant of the Kasteleyn operator.
It is a section of the determinant line bundle over LΓ . Let us restrict it to a T-orbit on
LΓ . Then it extends to a section of the line bundle L ⊗ L over the toric surface N which
compactifies the orbit. The divisor of zeros of this section is called the spectral curve C.
The spectral curve C intersects the divisor at infinity N∞ at a divisor C∞ . A choice of
a point of N0 := N − N∞ provides a coordinate on every component of the divisor N∞ .
The divisors C∞ satisfy a single condition:
The product of the coordinates of all points of C∞ is equal to 1.
(13)
We call such divisors admissible divisors at infinity. There is a bijection
∼
ν : {zig-zag paths on Γ} −→ divisor at infinity C∞ .
(14)
It is uniquely determined by the condition that the Casimir functions on the phase space
given by the monodromies along zig-zag paths α are proportional to the Casimir functions
given by the coordinates of the boundary points ν(α) on the corresponding components
of N∞ .
Pick a black vertex b of Γ. A non-zero vector in the fiber Vb provides a section sb of
the sheaf CokerK. The latter sheaf is a line bundle for a smooth C. The restriction of its
divisor of zeros to N0 is a degree g effective divisor S, where g is the genus of C.
The triple (C, S, ν) is called the spectral data. It was studied in [KO03].
1.4.3
A cover of the Beauville integrable system.
Let S be the moduli space parametrizing the spectral data. Let B be the moduli space
of pairs (C, ν). Forgetting S, we get a projection
π : S −→ B,
(C, S, ν) 7−→ (C, ν).
(15)
We claim that it is an integrable system, which covers the Beauville integrable system –
the latter does not take into account parametrizations (13).
Namely, let Sχ ⊂ S be the subspace parametrizing the spectral data (C, S, ν) with
a given divisor C∞ . It carries a rational symplectic form ΩS defined as follows. The
canonical T-invariant symplectic form ω on N0 gives rise to the Beauville symplectic form
on Symg N0 :
g
X
s∗i ω.
ΩB :=
i=1
g
Here si : N −→ N is the projection to the i-th factor. There is a projection
p : Sχ −→ Symg N0 ,
12
(C, S, ν) 7−→ S.
It is a Galois cover, whose Galois group is a product of symmetric groups acting by
permuting the points of C∞ at each boundary component, and hence altering the bijection
(14). Indeed, one shows that, given a generic admissible divisor at infinity C∞ ⊂ N∞ and
a generic degree g divisor S ⊂ N0 , there is a unique spectral curve C passing through
S and intersecting N∞ at the divisor C∞ . So the pull back Ω := p∗ ΩB is a rational
symplectic form on Sχ . Let Bχ ⊂ B be the subspace given by the condition that C ∩ N∞
is a given divisor. There is a fibration
πχ : Sχ −→ Bχ .
(16)
Its fibers are Lagrangian. The fiber over a point (C, ν) is identified with an open part
of Picg (C), which is a principal homogeneous space over the Jacobian of C. So (16) and
therefore (15) are algebraic integrable systems.
1.4.4
Connection with the dimer integrable system.
A proof of Theorem 1.4 below will be given in [GK]. Combined with the results of this
paper it allows to quantize the integrable system (15).
Theorem 1.4. For any Newton polygon N, the spectral data provides a birational isomorphism S over the base B:
S
X
−→
ց
S
ւ
(17)
B
It identifies the dimer integrable system with the integrable system (15), matching the
symplectic leaves of X and the symplectic varieties (Sχ , Ω).
The map S intertwines the action of the torus T on X with the action on S provided
by the action of T on the surface N .
In Section 7 we prove a weaker statement: S is a finite cover over the generic part of
S. It implies the independence of the Hamiltonians.
1.5
Analogies between dimers, Teichmüller theory and cluster
varieties.
Cluster Poisson varieties provide a framework for study of both (classical and higher)
Teichmüller theory [FG03a] and theory of dimers. Here is a dictionary relating key objects
in these three theories.
13
Dimer Theory
Convex integral polygon N
Minimal bipartite graphs
on a torus
spider moves of graphs
Face weights
Moduli space of spectral
data on toric surface N
simple Harnack curve + divisor
Moduli space of
simple Harnack curves + divisors
Tropical Harnak curve
with divisor
Moduli space of tropical
Harnack curve + divisors
Dimer integrable system
Hamiltonians
Teichmüller Theory
Oriented surface S
with n > 0 punctures
ideal triangulations of S
Cluster varieties
flips of triangulations
cross-ratio coordinates
seed mutations
Poisson cluster
coordinates
cluster Poisson variety
Moduli space of framed
PGL(2) local systems
complex structure on S
Teichmüller space of S
seeds
positive real points of
cluster Poisson variety
Measured lamination
space of measured
real tropical points of
laminations on S
cluster Poisson variety
Integrable system related to
pants decomposition of S
Monodromies around loops
of a pants decomposition
What distinguishes these two examples – the dimer theory and the Teichmüller theory
– from the general theory of cluster Poisson varieties is that in each of them the set of
real / tropical real points of the relevant cluster variety has a meaningful and non-trivial
interpretation as the moduli space of some geometric objects.
Here by moduli space of certain objects related to the toric surface N we mean the
space parametrizing the orbits of the torus T acting on the objects. For example, the
moduli space of spectral data means the space S/T. So combining results of [GK] with
Theorem 4.6 would imply that the latter corresponds to the hypersurface XN0 in the cluster
Poisson variety XN′ .
Discrete cluster integrable systems in Teichmüller theory. Consider the moduli
space XPGL2 ,S of framed PGL2 -local systems on S [FG03a]. If S has no punctures, it is just
the moduli space of PGL2 -local systems on S, which has the classical symplectic structure.
If S is a surface with punctures, the moduli space XPGL2 ,S has a cluster Poisson variety
structure (loc. cit.). Take a collection of loops {αi } providing a pants decomposition of S.
The monodromies of local systems over the loops αi commute under the Poisson bracket
and, together with the Casimirs given by the monodromies around the punctures, provide
an integrable system. When S is a surface with punctures it is a cluster integrable system.
Furthermore, the Dehn twists along the loops αi generate a commutative subgroup
3g−3
Z
in the modular group Γ(S) of the surface S, which commutes with the Hamiltonians
14
and acts by cluster Poisson automorphisms. So we get a discrete cluster integrable system.
We continue to develop these analogies in [GK]. Yet parts of this dictionary are still
missing. In particular, it would be very interesting to construct a canonical basis – in the
spirit of the Duality Conjectures from [FG03b] – in the space of regular functions on the
cluster variety XN .
1.6
Further perspectives.
We assigned to a Newton polygon N a cluster integrable system. Here is another way to
assign to N an integrable system:
1. Newton polygons N, considered up to translation, parametrize smooth toric CalabiYau threefolds. Namely, the canonical bundle KN of the toric surface N assigned to N is
a toric Calabi-Yau threefold: the multiplicative group Gm acts on the fibers, so the torus
T × Gm acts on KN . Alternatively, it can be described as follows. Put a Newton polygon
N in the z = 1 plane in the lattice Z3 , and make a cone over it. The corresponding
toric threefold is a Calabi-Yau. Every smooth quasiprojective toric Calabi-Yau threefold
is obtained this way. See [IU09] for the connections between dimer models and toric
Calabi-Yau threefolds.
2. Any Calabi-Yau threefold Y gives rise to an N = 2 supersymmetric Yang-Mills
(N = 2 SUYM) theory on R4 . Namely, consider a IIB type string theory in 10-dimensional
space Y (C) × R4 and compactify it on Y (C).
3. An N = 2 SUYM theory on R4 gives rise to an algebraic integrable system – see
[NS] and references there.
Combining these three steps we conclude that any Newton polygon N gives rise to
an algebraic integrable system. It is natural to conjecture that it is the dimer cluster
integrable system assigned to N. It would be interesting to prove this, and understand
the role of the cluster structure, and in particular of quantum integrability.
1.7
Historical remarks
Several of the concepts we use are present with different citations in the literature on the
dimer model and the literature on cluster algebras/cluster varieties.
Zig-zag paths were defined in the dimer model in [K02] but are implicitly present in
work of Baxter on the Yang-Baxter equation in the 6-vertex model, see e.g. [Bax]. They
also occur in a more general form, for “bicolored” planar graphs, in [Post06].
Elementary transformations for dimers and in particular the spider move were first
defined by Kuperberg in the 90s (using the term “urban renewal” which was coined by
Propp), see [Ciu98, KPW00]. In the cluster algebra literature these are sometimes referred
to as Postnikov moves. The analogous elementary transformations for electrical networks
(the star-triangle move, also known as Y-Delta move) appeared in Kennelly [Kenn1899].
The notion of minimal network appeared for resistor networks in [CdV94] and [CIM98],
and for dimers/clusters in [Thur04] and [Post06].
15
Cluster algebras were introduced in [FZ01]. A family of Poisson structures on cluster
algebras was introduced in [GSV02].
Cluster Poisson varieties were introduced in [FG03b]. They are in duality with cluster
algebras. The field of rational functions on a cluster Poisson variety has a natural noncommutative q-deformation, see Section 3 of loc. cit.. The cluster modular group, defined
there as well, acts by automorphisms of the cluster Poisson variety, its q-deformation, and
the cluster algebra. In general it does not, however, preserve the Poisson structures introduced in [GSV02]. Non-commutative q-deformations of the cluster algebra corresponding
to these Poisson structures were defined in [BZ04] and called quantum cluster algebras.
Cluster Poisson varieties are related to cluster algebras as follows. The subvariety of a
cluster Poisson variety obtained by equating all monomial Casimirs to 1 is a distinguished
symplectic leaf. The restrictions of the cluster Poisson coordinates to that leaf can be
realized as ratios of cluster monomials in the corresponding cluster algebra, considered
already in [GSV02]. The obtained elements generate a subfield in the field of fractions
of the cluster algebra. Its transcendentality degree is smaller than the one of the field of
fractions of the cluster algebra by the number of independent Casimirs, which coincides
with the codimension of the generic symplectic leaf in the cluster Poisson variety.
In [GSV09] Gekhtman, Shapiro and Vainshtein study a related model of paths in
directed networks on an annulus, originally defined by Postnikov [Post06] for planar networks. The relation with our model is not completely straightforward, but in the cases
of interest we believe the models are essentially equivalent. The results of [GSV09] have
some overlap with ours: in particular they construct the same Poisson structure, but do
not discuss integrability (the annular system is not integrable). It would be worthwhile
to study more closely the connection between these models.
The octahedron recurrence, although known as Hirota’s bilinear difference equation,
goes back to Hirota [Hir81]. Thecube recurrence, also known as the discrete BKP equation
or Hirota-Miwa equation, goes back to the work of Miwa [Miwa82], and was rediscovered
and studied several times afterwords, see [FZL01], [CS04].
1.8
The structure of the paper.
In Section 2 we study the set of minimal bipartite graphs Γ with a given Newton polygon.
In Section 3 we define a modified partition function, and prove that it is a sum of
commuting Hamiltonians, providing a quantum integrable system.
In Section 4 we show that the spider move gives rise to a cluster Poisson transformation.
We show that these transformations preserve the partition functions. We also briefly
recall, for the convenience of the reader, basic definitions of quantum cluster varieties.
In Section 5 we study the dimer model of special kind corresponding to an arbitrary
torus graph G, describing a resistor network on G.
In Section 6 we construct discrete cluster integrable systems related to the dimer
models on square grids and dimer models related to hexagonal resistor networks.
16
In Section 7 we turn to the algebraic-geometric perspective of the story. As a biproduct
we prove the independence of the Hamiltonians.
In Section 8 we give a local calculation of the Poisson structure on the space LΓ , and
generalize the story to bipartite graphs on open surfaces.
Acknowledgments. We are very grateful to Vladimir Fock, Francois Labourie, Grisha
Mikhalkin, Michael Shapiro, Andy Neitzke, Andrei Okounkov, Dylan Thurston, Masahito
Yamazaki for fruitful discussions. We thank the referees for useful comments. A.G. was
supported by NSF grant DMS-1059129. He is grateful to Aspen Center of Physics and
IHES for hospitality; R.K. is supported by the NSF.
2
2.1
Newton polygons and minimal bipartite graphs on
surfaces
Zig-zag paths and minimal bipartite surface graphs.
Let E = v1 v2 be an oriented edge of an oriented ribbon graph. There is a notion of the left
and right edges adjacent to E: the left (respectively the right) edge to E is the previous
(respectively the next) for the cyclic order at v2 .
Let Γ be a bipartite graph on a surface S. The medial graph of Γ is the 4-valent graph
with a vertex for each edge of Γ and an edge whenever two edges of Γ share a vertex and
are consecutive in the cyclic order around that vertex. A strand or zig-zag strand is a
path in the medial graph passing consecutively through the edges of a zig-zag path of Γ
as in Figure 3; equivalently it is a path in the medial graph which goes “straight” at each
(degree-4) vertex. For every edge E of Γ there are exactly two strands passing through
E, see Figure 3. For every m-valent vertex v of Γ there is a unique m-gon formed by the
parts of the strands intersecting the consecutive pairs of the edges incident to v, see the
right picture on Figure 3. We call it the arc m-gon surrounding v.
Figure 3: Zig-zag strands and surrounding 4-gon.
Definition 2.1. A bipartite surface graph Γ is minimal if, in the universal cover Γ̃ ⊂ S̃,
strands have no self-intersections, and there are no parallel bigons, i.e. pairs of strands
intersecting at two points and oriented the same way – see the left pictures on Figure 4.
17
Figure 4: Minimal bipartite surface graphs have no self-intersections and parallel bigons
(left); antiparallel bigons (right) are allowed.
2.2
From Newton polygons to collections of admissible graphs.
Let N be a convex polygon in R2 with vertices in Z2 , considered up to a translation by a
vector in Z2 , called below a Newton polygon. It is described by a collection {ei } of integral
primitive vectors with the zero sum, obtained as follows. Take the set of all integral points
on the boundary of N. It is cyclically ordered by the counterclockwise orientation of the
boundary. The vectors ei are obtained by connecting each of these points with the next
one. So the number of integral primitive vectors on the boundary coincides with the
number of integral points on the boundary.
Consider the torus T = R2 /Z2 . Each vector ei determines a homology class [ei ] ∈
H1 (T, Z). There is a unique up to translation geodesic representing this class – the
projection of the vector ei to the torus.
Take a family of oriented loops {αi } on the torus T in generic position such that the
isotopy class of the loop αi coincides with the isotopy class of the geodesic representing
[ei ]. The union
G = ∪i αi
is an oriented graph on the torus T.
Definition 2.2. A graph G given by a collection of oriented loops on an oriented surface
S is an admissible minimal graph if the following two conditions hold:
1. Going along any of the loops αi , the directions of the loops intersecting it alternate.
2. The total number of intersection points is minimal.
The graph G provides a decomposition of the surface S into a union of polygons Pj .
The sides of the polygons inherit orientations from the loops. The admissibility condition
1) is equivalent to the following:
1′ . The sides of any of the polygons Pi are either oriented the same way, clockwise or
counterclockwise, or their directions alternate.
2.2.1
From admissible surface graphs to bipartite surface graphs.
The admissibility condition 1′ ) for a surface graph implies that there are three types of
domains Pi :
18
1. Black domains: the sides are oriented counterclockwise.
2. White domains: the sides are oriented clockwise.
3. Faces: the directions of the sides alternate.
Therefore an admissible graph G on S gives rise to a bipartite graph Γ on S: the black
(white) vertices of Γ are the black (white) domains in S − G. Edges connecting black
and white vertices correspond to the vertices of the graph G. Indeed, every vertex of G
is 4-valent (since the loops αi are in generic position), and locally looks as an intersection
of two arrows. So it determines the black and the white domains.
2.2.2
Minimal triple point diagrams [Thur04].
A triple point diagram4 is a collection of oriented arcs in the disc with the ends at the
boundary of the disc considered modulo isotopies, such that
1. Three arcs cross at each intersection point.
2. The endpoints of the arcs are distinct points on the boundary of the disc.
3. The orientation of the arcs induce consistent orientations on the complementary
regions.
Observe that each complimentary region to a triple point diagram is a disc.
The orientation condition 3) is equivalent to the following:
3′ . Orientations of the arcs alternate “in” and “out” around each intersection point.
Let n be the number of arcs. Then each of the 2n endpoints is oriented “in” or “out”,
and these orientations alternate as we move around the boundary. Furthermore, given a
collection of 2n alternatively oriented points on the boundary, a triple crossing diagram
inducing such a collection provides a matching between the “in” and “out” points.
A triple point diagram is minimal if the number of its intersection points is not bigger
than for any other diagram inducing the same matching on the boundary.
Theorem 2.3 ([Thur04]). 1) In a disc with 2n endpoints on the boundary, all n! matchings of “in” endpoints with “out” endpoints are achieved by minimal triple point diagrams.
2) Any two minimal triple point diagrams with the same matching on the endpoints
can be related by a sequence of 2 ↔ 2 moves, shown on Figure 5.
4
This is called in [Thur04] a connected triple point diagram.
19
~
Figure 5: The 2 - 2 move.
2.3
From minimal triple point diagrams to minimal bipartite
surface graphs.
Each crossing point of a triple point diagram has a canonical resolution, obtained by
moving a little bit one of the strands passing through the point so that the emerging little
triangle is oriented by the strands the same way as the surface – the counterclockwise
resolution shown on Figure 6. We apply this procedure to every intersection point of a
triple point diagram.
The complimentary domains for the obtained configuration of arcs are of three possible types: black domains, white domains and faces, according to definitions (1)-(3) in
section 2.2.1. Namely, the black and white domains are the ones consistently oriented—
counterclockwise and clockwise respectively—while the directions of the sides of the remaining domains, the faces, alternate. So we arrive at a bipartite graph associated to
a minimal triple point diagram, see Figure 6. Its black vertices correspond to the triple
crossing points, and thus are 3-valent. The white vertices can have valencies two, three
or higher.
Let us shrink all edges incident to white 2-valent vertices (and remove the white vertex,
gluing its two neighbors into a single vertex). The bipartite graph obtained can have both
black and white vertices of valencies four or higher.
Figure 6: The bipartite graph obtained by counterclockwise resolution of a triple crossing
diagram.
Let us call two bipartite graphs similar if the graphs obtained by shrinking their 2valent white vertices coincide. There is a canonical bijection between the faces as well as
between the perfect matchings for similar bipartite graphs. So similar bipartite graphs
have the same partition function, discussed in Section 3.2.
Remark. There is another option: resolving all triple crossings clockwise. If a triple
crossing diagram was associated to a Newton polygon N, the counterclockwise resolution
is the same as the clockwise resolution of the diagram associated to the polygon N rotated
20
by 180 degrees. Indeed, the rotation changes the orientations of all strands of the triple
crossing diagram. Then resolving clockwise is the same as resolving counterclockwise the
original triple crossing diagram.
2.4
Boundary intervals of the Newton polygon and zig-zag paths.
Recall that a zig-zag path has a natural orientation such that it turns right at the black
vertices. Therefore the homology class of a zig-zag path is well defined. It is a natural invariant of a zig-zag path. On a minimal bipartite graph Γ zig-zag path has no
selfintersections, and so it is naturally a simple loop.
Lemma 2.4. Let N be a Newton polygon, and Γ a minimal bipartite graph on the torus
assigned to N. The homology class of a zig-zag loop is a primitive vector on a side of N,
whose orientation is induced by the orientation of N. The cyclically ordered set of zig-zag
paths with a given homology class is identified with the cyclically ordered set of boundary
intervals on the corresponding side of N. So the number of zig-zag paths on Γ equals to
the number of integral boundary points on N.
Proof. It is clear from Figure 7 that zig-zag paths on the bipartite graph arising from a
triple crossing diagram correspond to the strands of the triple crossing diagram. On the
other hand by construction the latter correspond to the primitive vectors of the boundary
of the Newton polygon.
Figure 7: Zig-zag paths correspond to strands of the triple crossing diagram.
2.5
Elementary transformations of bipartite surface graphs.
Figure 8 shows an elementary transformation Γ1 → Γ2 of bipartite graphs, called a spider
move. (There are two versions of the move depending on the color of the two interior
vertices.) We will see below that for appropriate choice of weights the spider move does
not change the partition function for the dimer model.
Figure 8 shows a pair of elementary transformations of bipartite graphs: splitting a
vertex of valency k > 3 and collapsing a 2-valent vertex. We will see below that it does
not change the partition function.
2.6
From minimal bipartite surface graphs to minimal triple
point diagrams.
A minimal bipartite graph can be transformed back into a triple point diagram in two
steps.
21
Figure 8: A spider move Γ1 → Γ2 of bipartite graphs. The white vertices are attached to
the rest of the graph; the black vertices are only attached to white vertices as shown.
Step 1. By a sequence of moves inverse to the shrinking of a 2-valent white vertex, we
replace an arbitrary bipartite graph by a bipartite graph with black vertices of valency 3,
see Figure 9.
Figure 9: Replacing a 4-valent black vertex by two 3-valent black vertices. A similar
vertex splitting splits any k-valent black vertex into k − 2 3-valent ones.
Step 2. Take a minimal bipartite graph Γ on S with the black vertices of valency 3.
Draw zig-zag strands on S assigned for all edges of Γ – two per edge. So each vertex
of Γ is surrounded by an oriented arc m-gon. Then there is a unique way to shrink the
counterclockwise triangles into points, getting a triple point diagram. This procedure is
evidently inverse to the one we used to get a bipartite graph from a triple point diagram.
Figure 10: From bipartite graphs (solid) to triple point diagrams (dashed).
2.7
Graphs from polygons
Here is the main result of this section.
22
Theorem 2.5. i) For any Newton polygon N there exists a minimal admissible graph G
on a torus associated with N. It produces a minimal bipartite torus graph Γ associated
with N.
ii) Any two minimal bipartite graphs on a torus associated with N are related by a
sequence of spider moves and shrinking / expanding of 2-valent vertices.
This result is illustrated in an example in Figure 11.
→
→
→
Figure 11: Procedure for going from a convex integer polygon to a bipartite torus graph:
1. A convex integer polygon. 2. The corresponding geodesics on a torus. 3. Isotoping
the geodesics to a triple-crossing diagram having specified order of intersection with the
boundary of the square. 4. The resulting bipartite graph.
Proof. Let us glue a torus T from a rectangle R by identifying opposite edges. Consider
a finite collection of oriented simple loops αi on T, whose homology classes correspond
to the primitive edge vectors of N, read counterclockwise around ∂N. We choose the
loops so that the total number of intersections of any loop with an edge of R is minimal:
this can be accomplished by starting with Euclidean geodesics (but we will isotope these
around preserving the number of intersections with R).P
Since ∂N is a closed path the total homology class i [αi ] represented by these loops
is zero. So the total intersection number of these loops with any edge of the rectangle
R is zero. Therefore we can isotope the loops, by moving them one through the other,
so that the intersection points of the loops with an edge of R alternate in orientation:
outgoing loops interlaced with ingoing loops. During this isotopy we do not add any more
crossings of loops with ∂R, and furthermore we do not exchange two outgoing or two
ingoing loops, that is, the order of outgoing paths and the order of ingoing paths does not
change. We can do this simultaneously for all edges of R, so that the crossings of the loops
with ∂R alternate around the entire boundary of R. Note that we have not introduced
any new crossings of ”outgoing” paths or crossings of ”ingoing” paths, only crossings of
one outgoing and one ingoing path. Then we can apply the part 1) of Theorem 2.3, and
isotope the paths αi into a minimal triple point diagram while fixing the boundary points
on ∂R.
Note that each path αi intersects the boundary with a consistent orientation (consistent with being a straight geodesic). Thus no path αi can self-intersect: a ”monogon” on
T is homotopically trivial and so must either lie entirely in the interior of R (disallowed
23
by minimality of the triple point diagram) or intersect one of the sides of R with both
orientations (disallowed by construction). Furthermore in T2 there are no parallel bigons:
such a bigon cannot lie in the interior of R by minimality of the triple point diagram,
and if it crosses the boundary, its vertices correspond to crossings of two outgoing or two
ingoing paths which are not present in the original geodesic path diagram (in other words,
along any isotopy back to a straight geodesic diagram there must occur at some point an
uncrossing of two outgoing paths, a contradiction).
a
b
c b
c
a
Figure 12: Permuting the boundary intersections.
The components of the intersection of a loop with the interior of R will be called
strands. Thanks to the part 2) of Theorem 2.3, any two minimal triple point diagrams
whose strands intersect ∂R in the same order are related by 2 ↔ 2 moves. We claim that
by isotoping the loops on the torus T any two minimal triple point diagrams for the same
polygon N are related by 2 ↔ 2 moves. Consider the set of strands intersecting an edge E
of R. Some of these strands may be part of the same loop, or part of isotopic loops. The
relative order of intersection of these subsets of strands with ∂R is fixed. However the
strands which are parts of different non-isotopic geodesics do not have a unique relative
order along ∂R. We need to show that by isotopy and 2 ↔ 2 moves we can permute the
order of intersection of these sets of strands with E.
Let a, c be consecutive “ingoing” strands on ∂R which by the alternating property
necessarily are separated by a single “outgoing” strand b, so that a, b, c are three consecutive intersections of strands with ∂R. Suppose that the strands ending at a, c cross inside
the rectangle R. We show that using 2 ↔ 2 moves and isotopy on T we can rearrange
these strands so that they intersect ∂R consecutively in the reverse order c, b, a without
changing the order of the other intersections. In other words, we can arbitrarily permute
the “ingoing” strands by doing a sequence of these transpositions {a, c} → {c, a} as long
as they intersect in R, and similarly for the “outgoing” strands. To accomplish this transposition, [Thur04] shows that we can use 2 ↔ 2 moves so that the triple point diagram
inside R has the three strands at a, b, c meeting just adjacent to the boundary (that is,
before meeting any other strands). Then one can simply isotope this triple crossing across
the boundary which has the effect of switching the intersections order of the strands to
c, b, a as desired. See Figure 12.
Now given a triple point diagram one can construct the minimal bipartite graph as
above. Observe that there are two different 2 ↔ 2 move diagrams: the one is obtained
from the other by reversing the strand orientations. Resolving their triple intersection
24
points in the standard (counterclockwise) way, we get two elementary moves of bipartite
graphs: the spider move on Figure 13 and the move on Figure 14 which amounts to
shrinking the edge with a 2-valent (white) vertex, and expanding it the other way.
Figure 13: This 2 ↔ 2 move amounts to a spider move of bipartite graphs.
Figure 14: This 2 ↔ 2 move amounts to collapsing the edges incident to a white 2-valent
vertex and expanding the other way.
3
3.1
Integrability of the dimer model
Edge weights and line bundles with connections on a graph
A vector bundle with connection on a graph Γ is the following data: we assign to each
vertex v of Γ a vector space Vv , and to each oriented edge E from a vertex v1 to a vertex
v2 an isomorphism
iE : Vv1 → Vv2
called parallel transport, so that i−E = i−1
E , where −E is the edge E with the opposite
orientation. The monodromy of the connection along a loop is the composition of the
parallel transports around the loop. Two vector bundle with connections {V, iE }, {V ′ , i′E }
are isomorphic (gauge equivalent) if there are isomorphisms ψv : Vv → Vv′ such that
i′E = ψv2 ◦ iE ◦ ψv−1
for all edges E = v1 v2 .
1
25
The moduli space LΓ is the space of line bundles with connections on a graph Γ
modulo isomorphisms. Here is another description of the moduli space LΓ . Choose a
non-zero vector ev in each space Vv . Then a line bundle with connection on Γ amounts to
a collection of non-zero numbers, the edge weights aE , assigned to the oriented edges E
of the graph, so that a−E = a−1
E :
E = v1 → v2 .
iE ev1 = aE ev2 ,
Another choice of base vectors e′v = bv ev gives rise to a different set of edge weights a′E ,
related to the original one by a gauge transformation: a′E = bv aE b−1
v′ . So
LΓ =
edge weights
.
gauge transformations
Given a trivializations {ev } of a line bundle with connection {V, iE }, we can encode
the connection by the Kasteleyn operator K, see Section 1.4.1, which in this case is just a
matrix. However the determinant det(K) is not gauge invariant, and so is not a function
on the moduli space LΓ .
3.2
3.2.1
The partition function and the Hamiltonians
The weight of a dimer cover
A dimer cover of a bipartite graph Γ, also known as a perfect matching, is a collection of
edges of Γ such each vertex is the endpoint of a unique edge in the dimer cover.
Recall the determinant line L, see (12). Given a connection, a dimer cover M on Γ
provides an L-valued function W (M) on LΓ , called the weight of M. For an edge E = bw,
let iE be the parallel transport along E of L. Then iE ∈ Vb∗ ⊗ Vw . The product of these
vectors over the edges of M is the weight:
W (M) ∈ O(LΓ ) ⊗ L.
Precisely, choose an order of the edges E1 , ..., EN of the matching. Write iE = a∗E ⊗ bE .
Then W (M) := a∗E1 ∧... ∧a∗EN ⊗bE1 ∧· · ·∧bEN . It does not change when we alter the order
of edges. The following result, due to Kasteleyn (in different language), is well known.
Theorem 3.1 ([Kast63]). We have
X
det K =
sgn(M)W (M) ∈ O(LΓ ) ⊗ L
(18)
M
where the sum is over dimer covers and the sign sgn(M) only depends on the choice of the
Kasteleyn line bundle with connection (see Section 1.4.1) and the homology class modulo
2 of M (defined in Section 3.2.2).
Definition 3.2. The partition function is PΓ = det K.
The partition function depends on our choice of κ, which we fixed once and for all.
26
3.2.2
The homology class of a dimer cover
A dimer cover M on Γ provides a function
ωM : {Edges of Γ} → {0, 1}.
Its value on an edge is 1 if and only if the edge belongs to M. We view it as a 1-chain
X
[M] := −
ωM (E)[E],
E
where the sum is over all edges, and [E] is the edge E oriented from black to white. The
condition that M is a dimer cover means that
X
sgn(v)[v]
d[M] =
v
where the sum is over all vertices v of Γ, and sgn(v) = 1 for the black vertex v, and
sgn(v) = −1 for the white.
If M1 , M2 are dimer covers, [M1 ] − [M2 ] is a 1-cycle, and thus each dimer cover defines
an “affine” homology class. It is more convenient to define an absolute homology class
for each dimer cover by fixing a chain Φ such that [M] − Φ is a cycle (for all M). We do
this in Section 3.2.3.
Remark. A dimer cover M2 trivialises the determinant line L by providing isomorphisms Vb → Vw for each edge E = bw. In Section 3.2.3 we consider “fractional” dimer
covers provided by weighted zig-zag paths, and use them in a similar way to trivialize L.
3.2.3
The α-deformed partition function
For every edge of an oriented ribbon graph there are exactly two zig-zag paths containing
the edge (one in which the edge is traversed in either orientation). Let E be an edge of Γ.
Denote by z r (respectively z l ) the zig-zag path containing E in which the white vertex of
E precedes the black (respectively follows the black). Let us consider a map
α : {zig-zag paths on Γ} −→ R/Z.
(19)
It provides a function
ϕα = ϕ : {Edges of Γ} −→ R,
ϕ(E) := αr − αl .
(20)
where αr (respectively αl ) is the evaluation of the map α on the zig-zag path z r (respectively z l ) crossing the edge E, see Figure 15, and αr − αl denotes the length of the
counterclockwise arc on S 1 = R/Z from αl to αr .
The strands, and thus the oriented zig-zag paths, are in bijection with the set {e1 , . . . , en }
of primitive edges of the boundary ∂N of N, and so inherit a circular order from ∂N.
27
zl
zr
Figure 15: The weight is ϕ(E) = αr − αl .
Theorem 3.3. Let X be the set of circular-order-preserving maps
{e1 , . . . , en } → R/Z.
For any map α ∈ X, the function ϕα satisfies
X
dϕα =
sgn(v)[v].
(21)
v
The result is more generally true for arbitrary triple crossing diagrams in a disk.
Proof. We need to show that dϕα = 1, respectively −1, at a black, respectively white,
vertex. For this it suffices to show that at each vertex, the set of adjacent strands (the
strands intersecting edges emanating from the vertex) has the same circular order as the
circular order of the edges containing the vertex. This follows from Lemma 3.4 below.
Lemma 3.4. Given a triple-crossing diagram in a disk, at any triple-crossing the three
intersecting strands occur in the same circular order as their ingoing endpoints on the
boundary of the disk.
Proof. The proof is by induction on the number of strands. Order the strands according to
the counterclockwise circular order of their ingoing endpoints. We recall the construction
of a triple crossing diagram of [Thur04]: take a strand with the property that on one
side, no other strand has both its endpoints. One can draw this strand following the
boundary as illustrated in Figure 16, crossing the intermediate strands in pairs. Each of
these crossings is positive. The remaining region again has the property that the strands
alternate orientation along the boundary and so can be completed to a triple crossing
diagram with one fewer strands by induction. Moreover the ingoing strands of the new
region retain their original circular order.
By taking limits in which all points α(ei ) collide (but preserve their circular order)
there are circular-order-preserving maps α for which ϕα takes integer values ϕα ∈ {0, 1}.
Given a map α provided by Theorem 3.3, there is a 1-chain
X
X
Φα =
ϕα (E)[E] with dΦα =
sgn(v)[v].
v
E
It is an integral 1-chain if φα is integer-valued. Thus [M] − Φα is an integral cycle. It
provides a monomial function on LΓ
ωα (M) : LΓ −→ C∗ ,
28
(22)
Figure 16:
given by the monodromy around the cycle [M] − Φα . It is the complex character of the
group LΓ corresponding to the homology class [M] − Φα via the isomorphism (1).
Definition 3.5. Let Γ be a bipartite surface graph. Let α be a map satisfying (21). The
α-deformed partition function Pα of the dimer model on Γ is a function on the space LΓ
given as the sum of monomials ωα (M) over all dimer covers M of Γ, with the same signs
as in (18):
X
Pα :=
sgn(M)ωα (M).
(23)
M
3.2.4
The Hamiltonians
Let [M]α ∈ H1 (Γ, Z) be the homology class of the cycle [M] − Φα . Recall the projection
π : H1 (Γ, Z) → H1 (S, Z). Given a homology class a ∈ H1 (S, Z), take the part of the sum
(23) over the matchings M such that the projection of [M]α to H1 (S, Z) is the class a:
X
Hα;a :=
sgn(M)ωα (M).
M :π([M ]α )=a
By Theorem 3.1 the sign sgn(M) depends only on a, and the partition function is a sum
X
Pα =
sgn(a)Hα;a .
(24)
a∈H1 (S,Z)
Definition 3.6. The functions Hα;a in (24) are the Hamiltonians of the dimer system.
Remark. A different map α′ leads to another collection of Hamiltonians {Hα′ ;a } which
differ from {Hα;a } by a common monomial factor, which lies in the center of the Poisson
algebra O(LΓ ). Indeed, the monodromy along any zig-zag loop is a Casimir element.
Therefore the Hamiltonian flows do not depend on the choice of α, and on each symplectic
leaf any two collections of Hamiltonians differ by a common factor.
29
3.2.5
A coordinate expression.
P
Let z1 , ..., z2g be loops on Γ whose homology classes generate H1 (S, Z). Let ni [zi ] be the
homology class of the cycle [M] − Φα . Then there is a function ψM : {Faces of Γ} → R,
which we interpret as a 2-chain, such that
X
dψM = [M] − Φα −
ni [zi ].
So one has
Pα :=
X
M
±
Y
ψ
WF M
F
(F )
Y
zini .
i
Each Hamiltonian is, up to a sign, a sum of monomials with coefficients +1 in variables
WF , zi with a given value of (n1 , ..., n2g ). If S is the torus, Pα is a polynomial in z1 , z2 .
3.3
The classical Hamiltonians commute
A minimal bipartite graph on a torus provides an algebraic integrable system:
Theorem 3.7. Let Γ be a minimal bipartite graph on a torus T. Then
i) The Hamiltonians Hα;a commute under the Poisson bracket on LΓ .
ii) The Hamiltonians are independent and their number is the half of the dimension
of the generic symplectic leaf.
The part i) of Theorem 3.7 is proved in Section 3.3. The number of Hamiltonians is
counted in Section 3.7, based on the results of Section 3.6. Finally, the independence of
Hamiltonians is proved in Section 7.4.
The quantum version of our integrable system is established in Section 4.3. The key
point is that the classical Hamiltonians are Laurent polynomials in any coordinate system
related to a minimal bipartite graph on the torus, and thus admit a natural upgrade to
elements of the corresponding quantum tori. The claim that they commute follows then
immediately from the combinatorial proof of commutativity of the classical Hamiltonians
given below. We also need the independence of the classical Hamiltonians, proved in
Section 7.4.
Section 5 is totaly independent: it neither relies on nor used in the proof of integrability.
Section 6 is not used in the proof of integrability. It discusses more specific discrete cluster
integrable systems, assuming Theorem 3.7.
Proof of part i) Take a pair of matchings (M1 , M2 ) on Γ. Let us assign to them another
f1 , M
f2 ) on Γ. Observe that [M1 ] − [M2 ] is a 1-cycle. It is a disjoint
pair of matchings (M
union of
1. homologically trivial loops,
2. homologically non-trivial loops,
3. edges shared by both matchings.
30
For every edge E of each homologically trivial loop we switch the label of E: if E belongs
f2
to a matching M1 (respectively M2 ), we declare that it will belong to a matching M
f1 ). For all other edges we keep their labels intact. Denote by µi and µ
(respectively M
ei
f
f
fi ] −Φα . Clearly (M
f1 , M
f2 ) = (M1 , M2 ).
the homology classes of the cycles [Mi ] −Φα and [M
We use a shorthand ε(∗, ∗) for ε(∗, ∗)ŜΓ .
Lemma 3.8. One has
ε(µ1 , µ2 ) + ε(e
µ1 , µ
e2 ) = 0.
(25)
Proof. We use a local calculation of the Poisson bracket on LΓ given in Section 8. Let us
write (25) as a sum of the local contributions corresponding to the vertices v of Γ, and
break it in three pieces as follows:
X X X
) δv (µ1 , µ2 ) + δv (e
µ1 , µ
e2 ) .
+
+
ε(µ1 , µ2) + ε(e
µ1 , µ
e2 ) = (
v∈E1
v∈E2
v∈E3
Here E1 (respectively E2 and E3 ) is the set of vertices which belongs to the homologically
trivial loops (respectively homologically non-trivial loops, double edges).
Then for any v ∈ E1 we have δv (µ1 , µ2 ) + δv (e
µ1 , µ
e2 ) = 0 since the elements lv (µ1 ) ∈ Av
provided by µi by the very construction coincides with the element lv (e
µ2 ), and similarly
lv (µ2 ) = lv (e
µ1 ). So the first sum is zero since it is a sum of zeros.
The third sum is also a sum of zeros. Indeed, the contribution of the matchings M1
and M2 at every vertex v ∈ E3 is the same, so the local pairing vanishes by the skew
symmetry.
It remains to prove that
X
δv (µ1 , µ2 ) + δv (e
µ1 , µ
e2 ) = 0.
v∈E2
Let γ be an oriented loop on Γ. We define the bending bv (γ; ϕ) of the function ϕ at a
vertex v of γ as
X
X
ϕ(E) ∈ R.
ϕ(E) −
bv (γ; ϕ) =
E∈Lv
E∈Rv
Here Rv (respectively Lv ) is the set of all edges sharing the vertex v which are on the
right (respectively left) of the path γ, following the direction of the path γ. Lemma 3.8
now follows from Lemma 3.9 below.
Recall the set X of circular-order-preserving maps from the zig-zag paths to R/Z.
Lemma 3.9. There is an α ∈ X such that the corresponding function ϕα satisfies, for
any simple topologically nontrivial loop γ,
X
bv (γ; ϕα ) = 0.
v∈γ
31
Proof. On a Euclidean torus, a simple closed polygonal curve which is not null-homotopic
has the property that its net curvature is zero, where the net curvature is the total amount
of left turning minus right turning. This fact can be seen by taking an isotopy of the curve
to a straight geodesic through a family of simple polygonal curves; the total curvature is
an integer multiple of 2π but changes continuously under the isotopy, and for a geodesic
it is zero.
A map α ∈ X allows us to draw our graph Γ on a Euclidean torus in such a way that
bv (γ, ϕα ) is (with a small proviso) the net curvature of γ at vertex v. This is accomplished
as follows. To each edge E, recall that αr , αl are the angles of the strands crossing E.
We define a 1-cochain f by f (E) = eiαl + eiαr . We have df = 0, since strands cross two
edges of every face, once with each orientation. Thus there is a multivalued function F on
vertices of Γ, which is locally well defined but has additive periods ω1 , ω2 for the generators
of H1 (S). Extending F linearly over the edges gives a map from Γ to a Euclidean torus
C/hω1, ω2 i. Locally, to each edge we are associating a Euclidean rhombus with vertices
z, z + eiαl , z + eiαl + eiαr , z + eiαr , where z is determined by F ; the edge E runs from z to
z + eiαl + eiαr , and the vertices z + eiαl , z + eiαr are the centers of the faces adjacent to
E. These rhombi are glued edge to edge to form a Euclidean torus (possibly with conical
singularities).
If for each E the angle ϕ(E) = αr − αl is less than π, then the rhombi are positively
oriented and, by Lemma 3.3, the total angle at each vertex is 2π and our Euclidean surface
is smooth, that is, has no singularities. If a rhombus for edge E, connecting vertices v, v ′,
has angle larger than π, by Lemma 3.3 it will be the only rhombus at v or v ′ with this
property. Hence the singularities in our Euclidean structure are folds, as shown in Figure
17, and are separated from each other.
It remains to check that if γ contains a vertex v which has a singularity then it is still
true that bv (γ, ϕ) measures the net curvature of γ at v. If γ passes through v but not along
edge E this is trivially true. In the remaining case, γ contains edge E = vv ′: suppose γ
contains consecutive vertices v1 , v, v ′ , v2 in order. Then it is straightforward to check that
the sum of the curvature contributions at v and v ′ equals the change in Euclidean angle
of the edge v1 v to the edge v ′ v2 . In all cases therefore bv is the net curvature of γ, which
is zero.
Figure 17:
32
The part i) of Theorem 3.7 follows immediately from Lemma 3.8. Indeed, the sum
calculating the Poisson bracket of two Hamiltonians can be presented as a sum of contributions (25). The part i) of Theorem 3.7 is proved.
3.4
The matching polytope
Let Ω be the matching polytope of Γ. By definition this is the subset
Ω ⊂ [0, 1]{edges of Γ}
consisting of functions ω on edges of Γ with values in [0, 1] such that the sum of the values
at every vertex is 1. The subset Ω is a convex polyhedron.
Lemma 3.10 ([LP]). The vertices of Ω are the perfect matchings of Γ.
Proof. To make the paper self-contained, we give a proof. Let ω be a vertex of Ω. Let us
show that for every vertex v of Γ there is just one edge E incident to v with ω(E) 6= 0.
Indeed, otherwise there is a vertex of v1 incident to edges E1 , E2 with the weights in (0, 1).
Let v2 be another vertex of E2 . It is incident to another edge E3 with the weight in (0, 1)
– otherwise the sum of the weights at v2 will be less then 1. Let v3 be another edge of E3 ,
and so on. We get a closed path on Γ whose edges have the weights in (0, 1). Let us add
ε to the weight of each of these edges, where |ε| is very small, and the signs alternate.
The sign condition does not lead to a problem since Γ is a bipartite graph. Then we still
get an element ωε ∈ Ω. So ω is inside of a segment in Ω. Thus it is not a vertex of Ω.
It follows that the values of ω at the edges are either 1 or 0. The edges where the
value is 1 give a matching.
Elements of Ω are called fractional matchings.
Lemma 3.11. Let Γ be a minimal bipartite graph on a torus. Then Γ has a perfect
matching.
Proof. It suffices to show that Ω is nonempty. Indeed, by Lemma 3.10 any vertex of the
polyhedron Ω provides a perfect matching. Any minimal bipartite graph Γ is obtained
from some polygon N. The function ϕα from Theorem 3.3 is then a fractional matching
by (21).
Let ω0 ∈ Ω be a fixed fractional matching. Then for any ω ∈ Ω the difference
[ω] − [ω0 ] is a 1-cycle on Γ. Taking its homology class we get a map Ω → H1 (Γ, R). Its
image is a convex polyhedron [Ω] ⊂ H1 (Γ, R). Its projection under the canonical map
π : H1 (Γ, R) → H1 (T, R) is a convex polygon
π[Ω] ⊂ H1 (T, R) ∼
= R2 ,
which depends on ω0 only through a translation.
33
Recall the space X of order-preserving maps from {e1 , . . . , en } to R/Z. The image of
X under the map α → ϕ(α) is a subpolyhedron Ω0 ⊂ Ω. It is in fact a simplex: the
simplex of order-preserving maps from the strands to R/Z, modulo rotation.
Let αi ∈ X be the map which sends all Ei to 0 and assigns 1 to the arc between Ei
and Ei+1 and 0 to all other arcs. The matching ϕ(αi ) maps to a vertex of Ω0 , and every
vertex of Ω0 is of this type.
3.5
The Newton polygon
Theorem 3.12. Let Γ be a minimal bipartite graph with polygon N. Then the Newton
polygon of the partition function Pα (z1 , z2 ) for Γ is N (up to translation).
Proof. Set ω0 = Φα . The Newton polygon of Pα (z1 , z2 ) is a translate of π[Ω]. Indeed, by
definition the coefficient in P of z1i1 z2i2 is the sum of weights of dimer covers M of Γ for
which the homology class [M] − Φα satisfies [M] − Φα = i1 [z1 ] + i2 [z2 ].
We
Pcan define the intersection pairing of a chain with a zig-zag path Z asPfollows. If
C = E CE [E] is a chain, the intersection pairing of C with Z is hC, [Z]i = E∈Z ±CE
where the sign is alternating, and depends to whether the orientation of the edge of Z
agrees with the white-to-black orientation or disagrees with it. In other words, this is the
intersection pairing of C with the strand corresponding to Z.
We define another polygon NΓ ⊂ H1 (T, R) by a set of inequalities
l(Z)
NΓ := γ ∈ H1 (T, R) | hγ, [Z]i ≤
− h[ω0 ], [Z]i for every zig-zag path Z . (26)
2
We claim that N, NΓ and π[Ω] are translates of each other.
We first show that π[Ω] ⊂ NΓ . Let ω ∈ Ω. The sum of ω(E) over the edges E in
a zig-zag path Z is at most equal to l(Z)/2, since this is the number of white vertices
adjacent to Z (each white vertex adjacent to Z has total flow at most 1 across Z). Thus
h[ω − ω0 ], [Z]i = h[ω], [Z]i − h[ω0 ], [Z]i ≤
l(Z)
− h[ω0 ], [Z]i
2
which implies that π[Ω] ⊂ NΓ .
We now show that the edges of N and NΓ coincide. Take γi = [ω(αi )−ω0 ]. This satisfies
the inequalities (26) and two of these are equalities: those for Zi and Zi+1 , because in αi
every white vertex adjacent to Zi is matched to a black vertex on the other side of Zi ,
and similarly for Zi+1 . Note moreover that γi+1 = γi + [Zi ] so we see that [Zi ] = ei is an
edge of NΓ . Thus NΓ has sides consisting of the ei in order, which proves that N and NΓ
are translates. Note moreover that each corner of NΓ is in the image of π[Ω] (in fact in
the image of π[Ω0 ]). Thus π[Ω] = NΓ .
3.6
Counting triple crossings and faces
From Thurston [Thur04], the number of triple crossings in any triple crossing diagram in
a disk is a function solely of the strand order along the boundary, and can be obtained as
34
follows. Take a linear order of the strand endpoints which is consistent with the circular
order. Then we say two strands ac and bd, directed from a to c and b to d respectively,
have a parallel crossing if their endpoints are in the order a < b < c < d or d < c < b < a.
Lemma 3.13 ([Thur04]). In a minimal triple-crossing diagram in a disk, the number of
triple crossings is the number of pairs of strands with a parallel crossing.
Now take a triple crossing diagram on the unit square [0, 1]2 obtained from a convex
integral polygon N as in Section 2. Let e1 , . . . , en ∈ Z2 be the primitive sides of N.
Lemma 3.14. The number of triple crossings is twice the area of N.
Proof. A minimal triple crossing diagram can be obtained by starting with a configuration
of straight geodesics on the torus [0, 1]2 , one for each ei , and then isotoping them into
position.
We count the number of parallel crossings in a standard configuration of the geodesics
ei and then notice that this number is invariant when we isotope the strands to their final
position.
Let τ ∈ R2 be a vector with positive coordinates having irrational ratio, and let
vmin , vmax be respectively the unique vertices of N which minimize, respectively maximize
v · τ . Let NR be the set of sides of N between vmin and vmax in counterclockwise order
and NL be the set of sides of N between vmax and vmin in counterclockwise order. Then
after a circular re-indexing, say that e1 , . . . , ek ∈ NR and ek+1 , . . . , en ∈ NL .
Let p = ετ ∈ [0, 1)2 for some sufficiently small ε. Start by drawing all geodesics ei
in the torus through p. No geodesic contains the origin by irrationality of τ ; order the
boundary points on the square counterclockwise starting from the origin. Then ei and
ej have a parallel crossing at p if and only if ei and ej are either both in NR or both in
NL . Moreover if p is sufficiently close to the origin all other crossings of ei and ej will be
parallel or non-parallel according to the crossing at p.
Thus the number of parallel crossing strands in total is
X
X
|ei ∧ ej |.
|ei ∧ ej | +
1≤i<j≤k
k+1≤i<j≤n
However the area of the part of N below the line through vmin and vmax is 1/2 of
X
e1 ∧ e2 + (e1 + e2 ) ∧ e3 + · · · + (e1 + · · · + ek−1 ) ∧ ek =
|ei ∧ ej |,
1≤i<j≤k
as can be seen by triangulating this polygon using extra edges from vmin to each other v.
Similarly for the upper part of N.
Now as we isotope the geodesics ei , without letting a strand cross the origin, it is easy
to see that the number of parallel crossings does not change.
35
Let us denote by B and W the number of black and white faces for a triple point
diagram graph G on the torus arising from a Newton polygon N. Let V be the number
of vertices of G, that is the number of the triple crossings.
Proposition 3.15. One has
B = W = V = number of the triple crossings = 2(area of N ).
Proof. Euler’s formula for the triple point diagram G gives V − E + F = 0. Counting
flags (vertex, edge) we get 6V = 2E. Therefore F = 2V .
Lemma 3.11 implies that B = W . Indeed, the white (respectively black) domains of
a triple point diagram G became the white (respectively black) vertices of the bipartite
graph Γ. Thus F = B + W implies B = W = V . Lemma 3.14 implies the last statement.
3.7
The number of Hamiltonians
Let i(N) and e(N) be the number of interior and exterior points for the Newton polygon
N. Pick’s formula plus Proposition 3.15 gives
2i(N) + e(N) − 2 = 2(area of N ) = B.
(27)
Observe that i(N) is the number of Hamiltonians, and e(N) is the number of Casimirs
plus one, that is the dimension of center of the Poisson algebra O(LΓ) minus one. On the
other hand, B is the number of faces minus one, and rkH1 (S, Z) = 2. So B + 1 = dimLΓ .
So (27) implies that
2(number of Hamiltonians) = dimLΓ − dim Center O(LΓ ) .
4
4.1
4.1.1
Dimers, cluster Poisson transformations, and quantum integrability
Dimer models on surfaces and cluster Poisson varieties.
Gluing the conjugated surfaces according to a spider move
Given a bipartite ribbon graph Γ, let SbΓ be the corresponding conjugated surface. Since
b coincides with Γ as a graph, there are canonical isomorphisms:
the conjugated graph Γ
b Z) = H1 (Γ, Z).
H1 (SbΓ , Z) = H1 (Γ,
(28)
We denote by ΛΓ the lattice (28). It is equipped with the skew symmetric integral bilinear
form (∗, ∗)Γb given by the intersection pairing on SbΓ . The boundaries ∂F of the faces F
of Γ, whose orientations are induced by the orientation of S, give rise to a collection of
36
cycles {γF } on SbΓ . These cycles
P plus the generators α1 , ..., α2g of H1 (S, Z) generate ΛΓ ,
and satisfy the only relation F γF = 0.
A spider move is a transformation of a bipartite graph shown in dashed lines on Figure
18. We show on the same figure the transformation of the accompanying zig-zag paths
(solid), and the unique cyclic order (1, 3, 2, 4) of the zig-zag paths compatible with the
cyclic orders of the triples of the zig-zag paths at the black vertices. The white vertices
give rise to the opposite cyclic order. These cyclic orders are the same for both graphs.
4
4
1
4
1
1
2
3
2
2
3
3
Figure 18: A spider move and zig-zag paths.
Let F be the rectangular face on Γ which determines a spider move s : Γ −→ Γ′ .
Denote by F1 , F2 , F3 , F4 the faces of Γ sharing edges with F . We assume for simplicity of
the exposition that these are different faces. The spider move does not affect the graph
Γ outside of a little domain U containing the face F and the two extra edges emanating
from the two black vertices of F . We identify the conjugated surfaces SbΓ and SbΓ′ outside
of the U. We want to extend this identification to a homeomorphism of surfaces
sb : SbΓ −→ SbΓ′ .
(29)
Lemma 4.1. There is a unique up to isotopy homeomorphism (29) identifying the bound′
ary loops and identical outside of U-domains. Let γG
:= sb(γG ). Then one has
γF′ = −γF ,
γF′ 1 = γF1 + γF ,
γF′ 2 = γF2 ,
γF′ 3 = γF3 + γF ,
γF′ 4 = γF4 .
(30)
′
Otherwise γG
= γG .
b
Proof. Figure 19 (and Figure 20 in the untwisted version) shows the ribbon graphs Γ
′
b
and Γ inside of the U-domains. Each component of the oriented boundary of a ribbon
is a part of an oriented zig-zag path on Γ. Any map (29) preserving the boundaries and
identical outside of U-domains identifies the zig-zag paths on the original graphs. This
determines the gluing uniquely up to an isotopy. So we get the first claim of the Lemma.
On each surface the four zig-zag paths inside of U bound an annulus. A pair of parallel
zig-zag paths together with a pair of connecting them little segnents form a boundary
component of the annulus. Our gluing process amounts to gluing these two annuli.
Taking a loop going around the face on the left of Figure 19, and tracing its image on
the right surface, we get γF′ = −γF . Similarly we get the other identities.
37
Figure 19: Identifying the conjugated surfaces according to a spider move. Recall our
convention that a zig-zag path goes counterclockwise around black vertices and clockwise
around the white ones. When we turn the surface over, the notion of counterclockwise
and clockwise are interchanged. Since to get Γ̂ we turned the white vertices over, the new
orientations is counterclockwise for all vertices. (In the figure, we are seeing the underside
of the surface near the black vertices.)
4.1.2
Seeds, mutations, conjugated surfaces and spider moves
Definition 4.2. A seed s is a datum Λ, {ei }, (∗, ∗) , where
• Λ is a lattice, i.e. a free abelian group, {ei } is a collection of non-zero vectors in Λ;
• (∗, ∗) a skew-symmetric Z-valued bilinear form on Λ.
Definition 4.3. A mutation of a seed s in the direction of a basis vector ek is a new seed
es. The lattice and the form of the seed es are the same as of s. The collection of vectors
{e
ei } of e
s is defined by applying a half reflection at ek map to the old one:
ei + (ei , ek )+ ek if i 6= k
a+ := max(a, 0).
(31)
eei :=
−ek
if i = k.
Definitions 4.2 and 4.3 generalize slightly the standard definitions of seeds and seed
mutations, see Sections 1.2.1-1.2.4 of [FG03b]. Unlike loc. cit, we do not demand neither
that the vectors ei are linearly independent, nor that they span the lattice Λ.5
Here is our main example.
Definition 4.4. The seed sΓ assigned to a bipartite graph Γ on a genus g surface S is
sΓ := (ΛΓ , (∗, ∗)Γb , {γF }).
5
We simplify the discussion by considering only the “simply-laced” case, with the “multipliers” di = 1.
38
4
1
4
2
1
2
3
3
Figure 20: Identifying the conjugated surfaces according to a spider move, shown in the
untwisted version.
Lemma 4.5. The spider move Γ −→ Γ′ centered at a face F leads to a mutation of seeds
sΓ −→ sΓ′ in the direction of the vector γF .
Proof. The only non-zero values (γF , γG )Γb are
(γF , γF1 )Γb = 1,
(γF , γF2 )Γb = −1,
(γF , γF3 )Γb = 1,
(γF , γF4 )Γb = −1
(32)
Thus formalae (30) from Lemma 4.1 are just equivalent to the mutation formulae (31).
Therefore the geometric objects – the conjugated surfaces and the spider moves – are
encoded, respectively, by the corresponding seeds and their mutations.
4.1.3
Quantum cluster transformations [FG03b]
A lattice Λ with a skew-symmetric Z-valued bilinear form (∗, ∗) gives rise to a quantum
torus ∗-algebra TΛ . The algebra TΛ has a basis {Xv } over the ring Z[q, q −1 ] parametrized
by the vectors v of the lattice Λ. The multiplication is given by
q −(v1 ,v2 ) Xv1 Xv2 = Xv1 +v2 .
There is an involutive antiautomorphism
∗ : TΛ −→ TΛ ,
∗(Xv ) = Xv , ∗ (q) = q −1 .
Let us choose an order of the basis {vi } of Λ. Then
Xv = q
−
P
i<j
ai aj (vi ,vj )
n
Y
i=1
39
Xiai ,
v=
n
X
i=1
ai vi ,
(33)
Consider the following formal power series, called the q-exponential:
Ψq (x) :=
∞
Y
(1 + q 2a−1 x)−1 =
a=1
1
(1 + qx)(1 +
q 3 x)(1
+ q 5 x) . . .
.
(34)
It is characterized up to a constant by the difference relation
Ψq (q 2 x) = (1 + qx)Ψq (x).
(35)
Denote by TΛ the non-commutative fraction field of the algebra TΛ . The quantum
mutation map µqek in the direction of a vector ek is an automorphism of the skew field TΛ
given by the conjugation by Ψq (Xek ):
µqk := AdΨq (Xek ) : TΛ −→ TΛ .
(36)
Although Ψq (Xek ) is not a rational function, this is a rational map.
4.1.4
Cluster Poisson varieties [FG03b]
Cluster Poisson varieties, also known as cluster X -varieties [FG03b], are geometric objects
which are in duality to cluster algebras of Fomin-Zelevinsky [FZ01].
We assign to a seed s a complex algebraic torus XΛ := Hom(Λ, C∗ ), called the seed
torus. A vector v ∈ Λ gives rise to a function Xv on the torus obtained by evaluation
at v. A mutation in the direction of a vector ek gives rise to a birational isomorphism
µek : XΛ −→ XΛ , obtained by specilalising q = 1 in (36). It acts on the function Xv by
µ∗ek : Xv 7−→ Xv (1 + Xek )−(ei ,ek ) .
(37)
The Poisson bracket is the quasiclassical limit of the commutator:
{Xv1 , Xv2 } := lim[Xv1 , Xv2 ]/2(q − 1) = (v1 , v2 )Xv1 Xv2 .
q→1
Thus (37) is a Poisson map. An isomorphism of seeds leads to a Poisson isomorphism
of the seed tori. Compositions of seed mutations and seed isomorphisms are seed cluster
transformations. So a seed cluster transformation gives rise to a Poisson birational isomorphism of the seed tori. A seed cluster transformation s → s acting trivially on the
functions Xv is a trivial seed cluster transformation. Consider all seeds obtained from a
given seed s by a sequence of mutations. We assign to these seeds their seed torus, and
glue pairs of tori related by mutations into a space Xe by using birational isomorphisms
(37). Trivial seed cluster transformations give rise to automorphisms of Xe. Taking the
quotient of Xe by these automorphisms, we get a cluster Poisson variety X .6
Non-trivial cluster seed transformations s → s lead to automorphisms of the X , which
form the cluster modular group. We will see such examples in Section 6.
6
It could be a non-separated space rather then a variety.
40
A cluster Poisson variety is nothing else but a collection of seed tori XΛ glued via the
cluster transformations. The algebra of regular functions O(XΛ ) on a cluster torus is an
algebra of Laurent polynomials. So a cluster Poisson variety is described by a collection of
algebras of Laurent polynomials and birational automorphisms (37) providing the gluing
transformations. The algebra O(X ) of regular functions on a cluster Poisson variety X
can be identified with the subalgebra of the algebra O(XΛ ) consisting of the elements
which remains Laurent polynomials after any cluster transformation.
This suggests that one should think about the quantum torus algebra TΛ as of the
algebra of regular functions on a non-commutative quantum torus, and that these algebras
together with the mutation maps (36) should be thought of as a quantum cluster variety.
We define the algebra Oq (X ) of regular functions on a quantum cluster variety as the
subalgebra of the quantum torus algebra TΛ consisting of all elements which remain
Laurent polynomials after any quantum cluster transformation.
4.1.5
Gluing the moduli spaces LΓ according to a spider move
Instead of using the set of all seeds related to an initial seed s by seed cluster transformations, we can pick any subset of this set, and define a Poisson space by gluing the
corresponding cluster tori. The cluster Poisson spaces which we need in our paper are
defined this way: we use only finitely many seeds corresponding to minimal bipartite
graphs associated with a given Newton polygon.
4.1.6
Traditional cluster Poisson variety related to the dimer model
Traditional cluster Poisson varieties are the ones defined by using seeds where the vectors
{ei } form a basis of the lattice. Let
∼
TS := H1 (S, C∗ ) = (C∗ )2g .
It is the group of line bundles with flat connections on S. The embedding Γ ֒→ S provides
a free Poisson action of the group TS on the moduli space LΓ .
The monodromy of a line bundle on Γ around a face F is a function WF on LΓ . The
product of all WF ’s is 1. Define
XΓ′ = (C∗ ){faces of Γ} .
Let W : XΓ′ → C∗ be the product of all coordinates. Assigning to a line bundle on Γ its
monodromies around the faces we get a canonical isomorphism
∼
W
(LΓ /TS ) −→ Ker XΓ′ −→ C∗ .
The birational Poisson transformations (7) are TS -equivariant. So to describe them on
the quotient is just the same as to describe their action on the face weights.
41
Lemma 4.6. The algebraic tori {XΓ′ } parametrized by equivalent minimal bipartite graphs
Γ on S are glued by cluster Poisson transformations (7) into a cluster Poisson variety
XN′ . The maps (7) preserve the function W. So there is a hypersurface XN0 ⊂ XN′ given
by the equation W = 1. The group TS acts freely on XN . There is a Poisson isomorphism
∼
XN /TS −→ XN0 .
(38)
Proof. We just need to show that the product of the face weights is preserved by the
spider move cluster transformations. This is clear from formulas (39) below.
4.2
Spider moves and partition functions
We have seen that elementary transformations Γ → Γ′ give rise to rational maps between
the moduli spaces LΓ → LΓ′ . In this Section we show that they do not change the
partition functions for the dimer models related to the graphs Γ and Γ′ . Moreover, we
show that the rational maps are characterised by the condition that they preserve the
partition functions.
Theorem 4.7. Given a spider move, there is a unique rational transformation of the face
weights preserving the modified partition function Pα . This transformation is a cluster
Poisson transformation, see Figure 21:
W ′ = W −1 ,
Wi′ = Wi (1 + W ), i = 1, 3,
Wj′ = Wj (1 + W −1 )−1 , j = 2, 4.
(39)
It preserves the product of the face weights.
W1
W1'
W4
W'
W
W2
W4'
W3
W2'
W3'
Figure 21: A spider move and a mutation of the face weights.
Proof. Let us show first that formula (39) is a cluster Poisson transformation. Precisely:
Lemma 4.8. Formula (39) describes the cluster Poisson mutation centered at γF , where
F is the center of the spider move.
42
Proof. Let us compare formula (39) with the formula for the cluster transformation at γF .
For this purpose, let us write the action of the mutation at the vector ek on the functions
Xe′i in terms of the original functions Xei :
µ∗ek
: Xe′i 7−→
Xk−1
−sgn(ei ,ek ) −(ei ,ek )
Xi (1 + Xk
)
if
if
i = k,
i 6= k.
(40)
Evidently, thanks to (32), formula (39) is a specilaization of the general formula (40).
Proof of Theorem 4.7. We use the edge weights, where the notation are as on Figure
22. The equality of the modified partition functions reads as follows:
PΓ
PΓ′
= ′
Mα
Mα
(41)
where PΓ is the partition function of the dimer model on the bipartite graph Γ defined
using the edge weights, and
Y
weight(E)ϕ(E) .
Mα :=
edges E
Here ϕ(E) = αr − αl is determined by the function α, see (19), (20), and Figure 15.
Observe that the maps α for the graphs Γ and Γ′ are the same maps.
The face weights are calculated via the edge weights as follows, where Wi∗ are factors
common for both graphs Γ and Γ′ , and the edge weights are as on Figure 22:
W =
W′ =
b
a
e
f
dc
, W1 = W1∗ , W2 = W2∗ , W3 = W3∗ , W4 = W4∗ .
be
a
c
f
d
AC
F
B
E
D
, W1′ = W1∗ , W2′ = W2∗ , W3′ = W3∗ , W4′ = W4∗ .
BD
A
E
C
F
4
b
4
d
f
1 a
c
F
A
3
D
1
3
B
e
2
C
2
E
Figure 22: A spider move and a mutation of the edge weights.
Let us number the external vertices on the graphs shown on Figure 22 by 1, 2, 3, 4.
We present the graphs as unions of the inside and outside graphs, where the inside graphs
43
are the ones shown on Fig 22. Given a pair of vertices i, j ∈ {1, 2, 3, 4}, consider all
perfect matchings where the vertices i and j are matched to the outside vertices. Their
contribution to the partition function is the product of the total weight of the perfect
matchings for the outside graph by the weight pij of the perfect matchings for the inside
graph. Similarly p′ij for Γ′ . One should have
pij = Cp′ij ,
(42)
i.e. the points in CP5 with the coordinates (p12 : p13 : ... : p34 ) and (p′12 : p′13 : ... : p′34 )
coincide. One has
p12 = bf, p13 = be + dc, p14 = cf, p23 = ad, p24 = af, p34 = ae.
p′12 = CF, p′13 = EF, p′14 = DE, p′23 = BF, p′24 = AC + BD, p′34 = AE.
(43)
(44)
For example, there are exactly two matchings of the inside graph which do not use the
vertices 1, 3, and their total weight is p13 = be + dc.
Then for every pair (ij) and (kl) there is a condition
p′ij
pij
= ′ .
pkl
pkl
Let us investigate its consequences. We deduce that W ′ = W −1 by looking at
(be + dc)af
dc
p13 p24
=
=1+
= 1 + W;
p12 p34
(bf )(ae)
be
(EF )(AC + BD)
p′13 p′24
=
= 1 + (W ′ )−1 .
′
′
p12 p34
(CF )(AE)
The condition
p12
p24
=
p′12
p′24
leads to the following:
p12
b p′
AC
F
= , ′12 =
p24
a p24
A AC + BD
=> W1′ = W1 (1 + W ).
14
23
, pp34
, and pp24
.
The rest of the relations (39), see Figure 21 are obtained by looking at pp24
24
′
5
Conversely, transformation formulas (39) imply that the vectors (pij ) and (pij ) in CP
coincide.
Furthermore, we claim that (42) implies (41). The subtlety is that the modified
partition function involves the function α. Let us set, see the right picture on Figure 18,
q1 = p3 − p1 , q2 = p2 − p3 , q3 = p4 − p2 , q4 = p1 − p4 .
Then the denominators in (41) for the inner graphs are
Mαin = ap1 −p2 bp2 −p3 cp3 −p1 dp4 −p2 ep1 −p4 f p2 −p1 = (cf )q1 (bf )q2 (ad)q3 (ae)q4 ,
in
M ′ α = Ap1 −p4 B p4 −p2 C p2 −p3 D p3 −p1 E p3 −p4 F p4 −p3 = (DE)q1 (CF )q2 (BF )q3 (AE)q4 .
′ in
Therefore, using (42), Mαin = C q1 +q2 +q3 +q4 M ′ in
α = CM α since q1 + q2 + q3 + q4 = 1. So,
using (42) again, we get (41).
44
Figure 23: Shrinking a 2-valent vertex.
There is another transformation, see Figure 23, which obviously does not change the
face weights and the Poisson structure. Its inverse involves splitting of a high valency
vertices by adding valency two vertices.
4.3
Quantum integrable system
Recall the quantum torus algebra TΛ assigned to a lattice Λ with a skew-symmetric
integral form (∗, ∗), see Section 4.1.3. Each vector v ∈ Λ gives rise to an element Xv ∈ TΛ .
Each Hamiltonian Hα,a is a sum of different monomials with coefficients +1. So it
can be upgraded uniquely to an element of the quantum torus algebra TΛ assigned to a
minimal bipartite surface graph Γ:
X
Hα;a :=
Xv ∈ TΛ
(45)
v
where the sum is over the vectors v parametrising the monomials in the classical Hamiltonian Hα,a . The element (45) is the quantum Hamiltonian. Similarly one defines quantum
Casimirs corresponding to the classical Casimir monomials.
Proposition 4.9. Let Γ → Γ′ be a spider move. Then the corresponding quantum cluster
transformation preserves the quantum Hamiltonians and Casimirs.
Proof. The Casimirs lie in the center and thus commute with Ψq (Xv ).
For the quantum Hamiltonians we use a version of the strategy employed at Section 4.2.
First, let us do a calculation at q = 1. Let us denote by qij pij (respectively qij′ p′ij ) the part
of the modified partition function involving the dimer covers which connect the vertices i
and j to the outside vertices. Here pij (respectively p′ij ) is the factor calculated by using
the matchings involving only the six edges a, b, c, d, e, f (respectively, A, B, C, D, E, F ) on
Figure 22.
We write A ∼ B when A/B is independent of
W, W1 , W2 , W3 , W4 , and W ′ , W1′ , W2′ , W3′ , W4′ .
(46)
Since the product of all face weights is equal to one, we have
W W1 W2 W3 W4 ∼ 1,
W ′ W1′ W2′ W3′ W4′ ∼ 1.
Below we work modulo these relations. The 1-cycle assigned to any non trivial Laurent
polynomial in W, W1 , W2 , W3 , W4 has a non-trivial contribution of the edges a, b, c, d, e, f .
45
The qij are Laurent polynomials in the face weights for Γ, well defined modulo a
common Casimir used to correct the partition function. Therefore they do not depend on
the left face weights in (46). So we can calculate the contribution of the W ’s from (46)
to the ratios of p’s by looking at (43). For example, we get
be + dc
e
dc
p13
∼
= +
∼ W3 + W W3 .
p12
bf
f
bf
Indeed, the contribution of a, b, c, d, e, f on Γ to the 1-cycle assigned to W3 is [e] − [f ],
and the contribution to W W3 is [d] + [c] − [b] − [f ]. These Laurent polynomials in W are
uniquely defined modulo W W1 W2 W3 W4 ∼ 1.
The same argumentation in general delivers the following result:
p13
p14
p23
∼ W3 + W W3 ,
∼ W W3 W4 ,
∼ W W2 W3 ,
p12
p12
p12
p34
p24
∼ W1−1 ,
∼ W1−1 W3 .
p12
p12
Similarly, we get
p′13
p14
p′23
′
′
′
∼
W
,
∼ W2′ W3′ ,
∼
W
W
,
3
3
4
′
′
p12
p12
p12
p′34
p′24
′ −1
′ −1
∼
(W
)
+
W
(W
)
,
∼ (W1′ )−1 W3′ .
1
1
p′12
p′12
Now it is straitforward to chack that the mutation at W formula amounts to mapping the
five Laurent polynomilas in W ′ ’s to the similar Lauren polynomials in W ’s. Moreover,
the same is true for the quantum mutation maps, assuming that the Laurent polynomials
above are written with the correcting q-factors provided by the formula (33):
W3′ 7−→ W3 (1 + qW ). q −1 W3′ W4′ 7−→ W3 W W4 , . . .
It remains to notice that W W1 W2 W3 W4 7−→ W ′W1′ W2′ W3′ W4′ .
Theorem 4.10. Let Γ be a minimal bipartite graph on a torus T. Then the quantum
Hamiltonians (45) commute.
The quantum Casimirs generate the center. The quantum Hamiltonians Hα;a together
with the quantum Casimirs provide a quantum cluster integrable system in Oq (XN ).
Proof. One has
[Xv , Xw ] = (q (v,w) − q (w,v) )Xv+w .
Using the notation of Lemma 3.8, we have
[Xµ1 , Xµ2 ] + [Xµe1 , Xµe2 ] = (q (µ1 ,µ2 ) − q (µ2 ,µ1 ) )Xµ1 +µ2 + (q (eµ1 ,eµ2 ) − q (eµ2 ,eµ1 ) )Xµe1 +eµ2 .
By construction we have µ1 + µ2 = µ
e1 + µ
e2 . So the commutator is zero by Lemma 3.8.
Therefore the quantum Hamiltonians commute. The claim about Casimirs is obvious
since they are monomials. The last claim follows Proposition 4.9 and Theorem 3.7ii).
46
5
Resistor network model and its cluster nature
5.1
Resistor network model
Let G be a (not necessarily bipartite) surface graph on a torus T. Let c be a function
on the edges of G with values in C∗ , called the conductance function, considered up to a
multiplication by a non-zero scalar.7 A graph with a conductance function on its edges is
called a resistor network.
5.1.1
Laplacians
A line bundle with connection V on G gives rise to a vector space
V = ⊕v Vv ,
given by the sum of the fibers Vv of the line bundle over the vertices v of G, and a Laplace
operator ∆ : V → V defined by
X
c(v, v ′)(f (v) − iv′ v f (v ′))
∆(f )(v) :=
v′ ∼v
where the sum is over neighbors v ′ of v, and iv′ v f (v ′) is the parallel transport of the vector
f (v ′) to the fiber over v.
Suppose now that the bundle with connection V on G is flat, that is, the monodromies
around the faces of G are trivial. The connection still has non-trivial monodromies z1 , z2
around generators of H1 (T, Z). The determinant of ∆ is a Laurent polynomial det ∆ =
P (z1 , z2 ). It is symmetric, since ∆ is Hermitian when |z1 | = |z2 | = 1 and c is a positive
real function:
P (z1 , z2 ) = P (z1−1 , z2−1 ).
(47)
The Laplacian determinant is a weighted sum of combinatorial objects called cyclerooted spanning forests (CRSFs), see [K10]: these are maximal subsets of edges of G in
which each connected component contains a unique cycle, and this cycle has nontrivial
homology on the torus. The weight of a CRSF is the product over its edges of the
conductances, times the product over its cycles of 2 − w − 1/w where w is the monodromy
of the flat connection on that cycle.
Theorem 5.1 ([K10]). We have
X
det ∆ =
Y
CRSFs C e∈C
c(e)
Y
2 − w(γ) − 1/w(γ).
cycles γ
Two torus graphs G1 , G2 with conductances c1 , c2 are said to be electrically equivalent
if their associated Laplacian determinants P1 (z1 , z2 ) and P2 (z1 , z2 ) are proportional (that
is, their ratio is a monomial in z1 , z2 ). A weaker notion is topological equivalence: torus
graphs G1 , G2 are topologically equivalent if for (any choice of) positive conductances, the
Newton polygons of P1 (z1 , z2 ) and P2 (z1 , z2 ) are the same.
7
Usually in applications the conductance function is a positive real-valued function.
47
5.1.2
Dual graph
Given a resistor network on a surface, the dual graph G′ is also a resistor network, where
we assign conductance 1/c to an edge dual to an edge of conductance c. The Laplacian
determinants on G and G′ are related:
Lemma 5.2. For a graph G with dual G′ having the same flat line bundle, we have
!
Y
(48)
c(e) det ∆G′ = det ∆G .
edges
In particular a resistor network on a torus is electrically equivalent to its dual network.
Proof. Use Theorem 5.1 and note that, given a CRSF C on G, there is a dual CRSF C ′
on G′ consisting of the set of edges of G′ which are not crossed
Q by an edge of C. It has
the same homology class as C, and thus the same weight cycles 2 − w − 1/w, and the
Q
contribution from the edge weights is e∈E c(e)−1 times the product of the edge weights
of C.
5.1.3
Zig-zag paths for a graph G on a torus.
A zig-zag path on a surface graph G is a path which turns maximally left or right at
each vertex, and the left/right turns alternate. A zig-zag path on G is unoriented. The
collection {α} of zig-zag paths assigned to a graph G on a torus has the following property:
Lemma 5.3. Each zig-zag path intersects an even number of zig-zag paths (counted with
multiplicity).
Proof. Each vertex of a zig-zag path is either on the left or right of the corresponding
path in the medial graph, so a zig-zag path has an even number of steps. At each step it
crosses a single other zig-zag path.
Conversely, any collection of smooth unoriented loops {α} on a torus, in general
position, satisfying Lemma 5.3 and considered up to isotopy, arises as the set of medial
strands of a graph on the torus. Namely, one can color the complementary domains
black and white so that no domains of the same color share an edge. We declare the
black domains to be the vertices of G, and connect two vertices of G by an edge if the
corresponding domains are separated by an intersection point of two zig-zag paths. The
white domains are the faces of G.
5.1.4
Elementary transformations.
Elementary transformations are local rearrangements of a resistor network which preserve
electrical equivalence. There are four types: Y − ∆ transformations, parallel and series
reductions, and removal of dead branches. The Y − ∆ transformations appeared first in
1899 in the work of Kennelly [Kenn1899]
48
Recall that a Y-∆-transformation of a resistor network is given by replacing a triangle
∆ with conductances α, β, γ by the graph Y with conductances a, b, c, see Figure 24,
related as follows:
Β
Γ
a
Α
c
b
Figure 24: An example of a Y-∆-transformation.
bc
ac
ab
, β=
, γ=
.
(49)
a+b+c
a+b+c
a+b+c
αβ + βγ + γα
αβ + βγ + γα
αβ + βγ + γα
, b=
, c=
,
(50)
a=
α
β
γ
The graph transformation induced by moving a zig-zag strand across a crossing of two
other zig-zag strands is a Y-∆ move.
Two more elementary transformations of graphs with conductances, called parallel and
series reductions, are shown in Figure 25.
α=
c1
c2
c2
c1 c2
c1
c1 + c2
c1 + c2
Figure 25: Parallel and series reductions.
A last elementary transformation is removing a dead branch, which consist of contracting to a vertex any subgraph which is attached to the rest of the graph by a single
vertex.
A torus graph is said to be minimal if the lifts of any zig-zag path to the plane (the
universal cover of the torus) does not intersect itself, and the lift of two zig-zag paths to
the plane intersect at most once.
Theorem 5.4. Every torus graph G with conductances c can be reduced to a minimal
torus graph by a sequence of elementary transformations.
Two minimal torus graphs G1 , G2 are topologically equivalent if and only if G2 is related
to G1 or to its dual graph G′1 by Y-∆-moves.
We prove this at the end of Section 5.2.4. The corresponding statement for electrical
equivalence is false: there are minimal graphs which are electrically equivalent but not
related by Y − ∆ moves. In fact the electrical equivalence class is a subvariety of the
Hamiltonian dynamical system; see below.
49
5.2
5.2.1
The associated dimer model on a torus
The bipartite graph assigned to a graph on a torus.
Drawing the graph G and its dual graph G′ simultaneously one can make a bipartite
graph ΓG on the torus whose white vertices are the edges of G, black vertices are the
vertices and the faces of G, with an edge for each adjacency, see Figure 2. The graph ΓG
has the same number of white and black vertices since Euler’s formula on a torus reads
E = V + F . One can show [KPW00] that the graph ΓG admits dimer covers: in fact
these are in bijection with CRSFs in which both the cycles and dual cycles (the dual of
a CRSF is also a CRSF) have a chosen orientation.
5.2.2
The resistor network subvariety.
Let c be a conductance function on the edges of G, considered up to a non-zero scalar. It
gives rise to a line bundle with connection V (c) on ΓG , defined as follows. The fibers of
V (c) at the vertices of ΓG are identified with C. So the connection is defined via the edge
weights assigned to the oriented edges of ΓG . The weight of an edge incident to a vertex
corresponding to a face of G is 1. The weight of an edge going from a vertex of G to a
midpoint of an edge E of G is the conductance cE assigned to that edge.
The monodromies around a pair of generators of H1 (T, Z) provide an isomorphism of
the moduli space of line bundles with flat connections on ΓG with (C∗ )2 . Taking the tensor
product of V (c) with a line bundle with a flat connection on ΓG we get a line bundle with
connection on ΓG . We denote by RG the moduli space of line bundles with connections
on the graph ΓG constructed in this way, and call it the resistor network subvariety. It is
a closed subvariety
RG ⊂ LΓG ,
dimRG = EG + 1
where EG is the number of edges of G (note that multiplying all conductances on G by a
constant gives a gauge equivalent connection on ΓG ).
5.2.3
Determinant of the Laplacian and the dimer model partition function
Using the conductances on G to define the connection on ΓG , we have the following
relation between the dimer partition function and det ∆.
Lemma 5.5. The partition function for the dimer model on ΓG is equal to det∆G (up to
a monomial factor in z1 , z2 ).
Proof. Let G′ be the dual graph to G with conductance 1/c on an edge dual to an edge
of conductance c. Recall Lemma 48 relating the Laplacian determinants on G and G′ .
These two operators ∆G , ∆G′ acting on the black vertices of ΓG are related to the
Kasteleyn matrix K as follows:
∆G 0
= −K2∗ K1 ,
(51)
0 ∆G′
50
where K1 , K2 are two copies of the black-to-white Kasteleyn matrix with appropriate
gauge, defined as follows.
Embed G in a Euclidean torus with straight edges; this allows us to associate to each
directed edge b1 b2 a unit-modulus complex number ζ(b1 , b2 ) representing its direction.
Letting w be the vertex of ΓG in the center of this edge we then define K1 (b1 , w) =
c(b1 , b2 )ζ(b1 , b2 ) where c is the conductance of edge b1 b2 . If b3 is a black vertex of G′ in
a face of G adjacent to edge b1 b2 then K1 (b3 , w) is 1 times the unit modulus complex
number in the direction from b3 to w and perpendicular to ζ(b1 , b2 ). One can now check
that the monodromy around a face of ΓG is negative real (for positive conductances) and
thus K1 is a Kasteleyn matrix for ΓG . One defines K2 similarly but for the dual graph
G′ . Now (51) follows by inspection of the various cases: −K2∗ K1 (b, b) is the sum of the
conductances of edges emanating from b; −K2∗ K1 (b, b′ ) = 0 if b ∈ G and b′ ∈ G′ or vice
versa; −K2∗ K1 (b1 , b2 ) = −c if b1 , b2 are adjacent vertices of G and the edge connecting
them has conductance c; similarly for vertices adjacent in G′ .
Finally, det K2 , det K1 are both the dimer partition function (up to constants which
depend on our choice of gauge). Thus from (48) and (51) we have det ∆ = det K1 up to
multiplicative factor which is a monomial in z1 , z2 .
It follows from (47) that the Newton polygon of the dimer model on ΓG is centrally
symmetric with respect to the origin.
5.2.4
Zig-zag paths and the Newton polygon
Zig-zag paths are defined for both a graph G on a torus and the associated bipartite graph
ΓG ; zig-zag paths on G are unoriented. Zig-zag paths for ΓG are oriented. A zig-zag path
α on G corresponds to a pair of zig-zag paths on ΓG , isotopic to α and equipped with the
opposite orientations. In this way we get all zig-zag paths on ΓG .
The geometric Newton polygon N(G) is described as follows. Each zig-zag path α on
G gives rise to a pair of homology classes ±[α] ∈ H1 (T, Z), where [α] is the homology
class of the path α equipped with an orientation. There is a unique, centrally symmetric
with respect to the origin, convex integral polygon
N(G) ∈ H1 (T, Z)
with the sides given by the vectors ±[α] for all zig-zag paths α on G. Indeed, there is
evidently a unique up to a translation convex integral polygon in H1 (T, Z) with these
sides, which is centrally symmetric. Thanks to Lemma 5.3, it can be centered at the
origin.
Clearly the polygon N(G) coincides with the geometric Newton polygon N(ΓG ) assigned in a similar way, using the homology classes of zig-zag paths, to the bipartite graph
ΓG . Thus Lemma 5.5 implies
Lemma 5.6. Let G be a minimal graph on a torus. Then the geometric Newton polygon
N(G) coincides with the Newton polygon of det∆G .
51
Proof of Theorem 5.4. By Lemma 5.6, two equivalent minimal graphs G and G′ have
the same geometric Newton polygons, whose boundary sides determine the homology
classes of the zig-zag strands. Thus the collections of their zig-zag strands are isotopic.
The isotopy is realized by moving zig-zag strands through zig-zag strand crossings, and
thus the corresponding graphs are related by a sequence of Y-∆ moves.
To reduce a graph G to a minimal graph, consider the zig-zag strands for G, and
isotope each to a straight geodesic through a generic isotopy. Each type of singularity
which resolves corresponds to an elementary transformation: if it is a parallel, series, or
dead branch move, it may change the topology of the set of strands, but also strictly
reduces the number of crossings of strands. Eventually one arrives at a graph with a
minimal number of strand crossings and the remaining moves are Y − ∆ moves.
5.2.5
Equations defining the resistor network subvariety.
Denote by VG , EG , FG the number of vertices, edges and faces of the graph G, and
similarly VΓ , EΓ , FΓ those for ΓG . Then FΓ = 2EG . Indeed, every edge E of G is incident
to four faces of ΓG , and each face of ΓG is counted twice. So one has
dimLΓ = FΓ + 1 = 2EG + 1.
So the codimension of the resistor network subvariety RG is EG . On the other hand, a line
bundle with connection on ΓG which belongs to the subvariety RG satisfies the following
conditions:
1. The monodromies around the faces of the graph G and the graph G′ are equal to 1.
2. Given a loop α on the graph G and a loop α′ on the graph G′ whose homology
classes in H1 (T, Z) coincide, the monodromies around these loops also coincide.
Remarks. The first condition tells that for each black vertex b on ΓG , the product of
the face weights for the faces adjacent to b is equal to 1.
Secondly, the second condition implies the first: to see this, alter a loop α by adding
to it a loop going around a face of the graph G, and similarly for G′ .
Finally, given the first condition, the second condition imposes only two more equations
on the connection: one for each generator of H1 (T, Z), since the first condition already
implies that the monodromies around homologous paths, both in G or both in G′ , are
equal.
Since the product of the monodromies over all faces of G, as well as over all faces of
G , is 1, we get EG equations for the subvariety RG ⊂ LΓG :
′
(VG + FG − 2) + 2 = EG .
These conditions characterize the subvariety RG in LΓ :
Lemma 5.7. A line bundle with connection on ΓG arises from a resistor network on G
if and only if it satisfies the conditions 1 and 2 above.
52
Proof. We have seen already that the line bundle with connection on ΓG arising from a
resistor network satisfies the conditions 1 and 2.
Let us prove the converse. Notice that the condition 1 says that we have a flat line
bundle on the torus T assigned to the graph G, and another flat line bundle on the torus
assigned to the graph G′ . The condition 2 then says that these two flat line bundles are
isomorphic. So, untwisting by a flat line bundle on the torus, we can assume that our line
bundle has trivial monodromy around any loop on G and around any loop on G′ .
Let us trivialize the line bundle on ΓG on the dual graph G′ . Take an edge E0 of G and
assign it conductance 1. Recall that the medial graph of G is the graph with a vertex for
every edge of G and an edge whenever two edges of G share a vertex of G. The parallel
transports on ΓG determine a curl-free flow on the medial graph of G; the value of the
conductance of an edge E is then the integral of this flow along a path in the medial
graph from E0 to E. So we have reconstructed conductances up to a non-zero common
factor.
5.2.6
The resistor network subvariety is Lagrangian.
An irreducible subvariety of a Poisson variety is Lagrangian if the generic part of its intersection with the generic symplectic leaves are Lagrangian subvarieties of the symplectic
leaves.
Theorem 5.8. RG is a Lagrangian subvariety of the Poisson variety LΓG .
Proof. The following crucial lemma implies that the intersection of RG with a generic
symplectic leaf is coisotropic (that is, the symplectic form is zero on this intersection).
Lemma 5.9. The Poisson brackets between the defining equations 1-2 of the resistor
network subvariety RG are equal to zero.
Proof. Let us show first that the monodromies around the black vertices of the graph
ΓG Poisson commute. Let b1 , b2 be two black vertices of ΓG corresponding to the faces
F1 , F2 of G. Let Wb1 , Wb2 be the corresponding monodromies. If the faces F1 , F2 are not
adjacent, the Poisson bracket is obviously zero. If they are adjacent, they share an edge
E, which has, by construction, two black vertices at the ends and one white vertex at the
middle, see Figure 2. Thus {Wb1 , Wb2 } = 0. The argument in the dual case when b1 , b2
are two black vertices of ΓG corresponding to the vertices of G is the same. Finally, let b
be a black vertex assigned to a face F of G, and b′ a black vertex assigned to a vertex v of
G. Then the face F of G and the face F ′ of the dual graph G′ corresponding to v either
do not intersect, or intersect at two points. See Figure 2 where the boundary of F is a
black loop and the one for F ′ is a dashed loop. In the former case the Poisson bracket is
evidently zero. In the latter case, the contributions of the intersection points come with
the opposite signs, and thus cancel.
Continuation of proof of Theorem 5.8. Let 2m be the number of Casimirs for the Poisson
variety LΓG . It equals to the number of primitive boundary intervals of the Newton
53
polygon N(ΓG ) and thus is an even number, since the polygon is centrally symmetric.
Since the product of all Casimirs is 1, the dimension of the generic symplectic leaf on LΓG
is
2EG + 1 − (2m − 1) = 2(EG − m + 1).
The Casimirs are monodromies along the zig-zag paths, and the latter on the graph ΓG
′
come in pairs: {z1 , z1′ }, ..., {zm , zm
}, so that {zk , zk′ } have the opposite homology classes,
and thus the corresponding Casimirs Ck , Ck′ are inverse to each other, providing a relation
′
Ck Ck′ = 1. Since the product of all Casimirs is 1, the last relation Cm Cm
= 1 follows from
the others, while the first m − 1 are independent.
Since dimR equals to m − 1 plus half of the dimension of the generic symplectic leaf, it
is sufficient to show that the generic symplectic leaves intersected by R are parametrized
nm
= 1.
by (no less than) m−1 parameters. Let us assume that there is a relation C1n1 . . . Cm
Let us show that n1 = . . . = nm . Take a pair of Casimirs Ca and Cb corresponding to nonparallel sides of the Newton polygon. Then the corresponding zig-zag strands intersect.
So there is an edge E shared by them. Altering the conductance at the edge E we alter
just the Casimirs Ca and Cb . This implies that na = nb . Therefore n1 = . . . = nm .
5.2.7
Quantized cluster monomial Lagrangian subvarieties
The subvariety R is a cluster monomial Lagrangian subvariety – it is a Lagrangian subvariety of a cluster Poisson variety X , which is given in any cluster coordinate systems
by equating certain monomials to 1.
According to the correspondence principle of quantum mechanics, a symplectic manifold X should give rise to a Hilbert space HX – the quantization of X. Furthermore, a
Lagrangian subvariety L ⊂ X should provide a line in HX . Here is how the story goes in
the cluster situation.
The quantization of a cluster Poisson variety X [FG07] provides a collection of Gelfand
triples, parametrized by the eigenvalues χ of the Casimirs:
Sχ ⊂ Hχ ⊂ Sχ∗ .
Here Hχ is the Hilbert space discussed in Section 1.2.2. The modular double algebra, given
by the tensor product of quantum algebras at ~ and 1/~:
Oq (X ) ⊗ Oq∨ (X ),
q = eiπ~ , q ∨ = eiπ/~ ,
(52)
is realized by unbounded selfadjoint operators in the Hilbert space Hχ . The subspace
Sχ is the maximal subspace of Hχ where the algebra (52) acts, equipped with a natural
topology. The space Sχ∗ is its topological dual. One can show using the methods of loc.
cit. that any cluster monomial Lagrangian subvariety L ⊂ X gives rise to a functional
ψL ∈ Sχ∗ ,
defined uniquely up to a scalar by the condition that ψL is annihilated by the quantized
monomial equations in the algebra Oq (X ), and their counterparts in the algebra Oq∨ (X ).
54
Here is the basic example. Let H = L2 (R) be the Hilbert space of functions f (x).
Consider unbounded operators
Xf (x) := f (x + iπ~),
X ∨ f (x) := f (x + iπ),
Y f (x) := ex f (x),
Y ∨ f (x) := ex/~ f (x).
Then
XY = q 2 Y X,
2
X ∨Y ∨ = q∨ Y ∨X ∨.
So we get a representation of the modular double of the quantum torus algebra with two
generators. The subspace S ⊂ H consists of functions f (x) with exponential decay at
x → ±∞, whose Fourier transform has the same property. By the Payley-Wiener theorem
f (x) has an analytic continuation to C.
The equation X = 1 defines a Lagrangian subvariety L in the torus (C∗ )2 with coordinates (X, Y ). The quantized equations are (X − 1)f (x + iy) = 0, (X ∨ − 1)f (x + iy) = 0.
For irrational ~, they imply that f (x + iy) = f (x). This implies that the functional ψ is
a constant.
5.3
Cluster nature of the resistor network space
Recall that all our spaces are equipped with a free action of the two-dimensional torus T,
and relating them to cluster varieties we have to consider their quotients by this action,
or, equivalently, consider the action of all transformations on the face weights only.
In Section 5.3.1 we use ′ in the notation XG′ and R′G to indicate that, just like in
Section 4.1.6, we work with the quotients XG /T and RG /T. Moreover we do not define
′
the spaces AG and BG , only the “quotients” A′G and BG
.
5.3.1
The basic commutative diagram for a graph G
Given a graph G on a torus, let us define four split algebraic tori (that is, varieties (C∗ )k
′
for some k) A′G , BG
, R′G , XG′ , fitting into a commutative diagram
A′G
j
←֓
′
BG
p↓ ւs ↓q
XG′
i
←֓
(53)
R′G
′
The tori A′G , BG
, R′G , XG′ are defined as the tori with the following canonical coordinates:
• A′G : coordinates {Af } at the faces f of the graph ΓG .
′
• BG
: coordinates {Bb } at the black vertices b of the graph ΓG .
55
• R′G : coordinates {Cw } at the white vertices w of ΓG , defined up to a scalar.
• XG′ : coordinates {Xf } at the faces f of ΓG , whose product is 1.
The Cw -coordinates are nothing else but the conductances assigned to the edges of
the graph G, which are identified with the white vertices of ΓG .
We define the maps between these tori by their action on the coordinates. Below ∼
means an incidence relation. So b ∼ f means that a black vertex b is incident to a face f ,
etc.
First, the map p is the standard map [FG03b]
Y ε
p∗ Xf :=
Agf g ,
(54)
g
where the product is over faces g of ΓG , and εf g is the bilinear form on the faces defined
in Section 2.
Next,
Y
(55)
Bb = Bb Bb′ .
j ∗ Af :=
b∼f
Here b, b′ are the two black vertices of the face f of the graph ΓG .
b2
b1
f
b3
b4
Figure 26: The map s∗ Xf .
Then, we set
s∗ Xf :=
Bb1 Bb3
.
Bb2 Bb4
(56)
Here (b1 , b2 , b3 , b4 ) are the four black vertices which lie outside of the face and incident to
the two white vertices of the face, as shown on Figure 26.
Next,
Cw
.
(57)
i∗ Xf :=
Cw ′
Here w, w ′ are the two white vertices of the face f , and w, b, w ′ go counterclockwise around
the face, and the black vertex b is a vertex of f , Figure 27.
56
b1
w
w
b2
Figure 27: The map i∗ Xf := Cw /Cw′ .
Finally,
q ∗ Cw :=
Bc1 Bc3
.
Bc2 Bc4
(58)
where c1 , c2 , c3 , c4 are the four black vertices incident to the white vertex w as shown on
the Figure 28.
c1
c4
c2
c3
Figure 28: The map q ∗ Cw .
Define [FG03a] a 2-form on the space A′G by
X
εf g d log Af ∧ d log Ag .
ΩA :=
f,g
Proposition 5.10. i) The diagram (53) is commutative.
ii) The map j is an embedding. Its image is isotropic for the 2-form ΩA .
iii) The subvariety i(R′G ) is defined by monomial equations. It is Lagrangian.
′
iv) The subvariety s(BG
) is Lagrangian in the symplectic variety
UG′ := p(A′G ) ⊂ XG′ .
Proof. i) One checks this by looking at the Figure 26, 27, 28.
ii) Follows easily from the definitions.
iii) The subvariety i(R′G ) ⊂ XG′ is given by monomial equations
Y
Xf = 1.
f:
P
∂f =α−β
57
(59)
Here equations are assigned to pairs of loops α, β on G which are homologous on the torus.
The product is over a set of faces f of ΓG whose total boundary is α − β. In particular,
taking the empty loop β, we get equations
Y
Xf = 1 for each black vertex b of the graph ΓG .
(60)
f ∼b
Indeed, it is easy to see that i∗ (59) = 1. Recall that EG is the number of the edges of G.
It is also the number of the black as well as the white vertices of ΓG . One has
dim i(R′G ) = EG − 1.
Indeed, there are EG independent equations: EG − 2 are given by (60), and two more by
β’s generating H1 (T, Z). Since dim R′G = EG − 1, we proved the first claim. The second
is similar to the proof of Theorem 5.8.
iv) It is similar to the proof of Theorem 5.8.
B2
B4
B2
C2
c1
B6
c3 B0
B3
B6
B7
c2
B1
B4
C3
B3
C1
B5
B1
B5
Figure 29: Transformation law for B’s under a Y-∆-move.
5.3.2
Action of the Y-∆-move on coordinates.
A Y-∆-move acts on the conductances coordinates Ci as in (49)-(50). Referring to Figure
29, we have
c1 c2 + c2 c3 + c3 c1
B1 B7
= C1 =
=
B5 B6
c1
from which one obtains
B2 B3 B3 B1
B0 B4 B0 B5
+
B3 B1 B1 B2
B0 B5 B0 B6
B2 B3
B0 B4
+
B1 B2 B2 B3
B0 B6 B0 B4
B1 B4 + B2 B5 + B3 B6
.
(61)
B0
The same equation is obtained starting from the other two conductances C2 , C3 . This
equation is the so-called cube recurrence relation, see Section 6.3 below.
B7 =
58
Lemma 5.11. The Y − ∆ move G → G′ is a composition of four cluster mutations.
Precisely, there are commutative diagrams
j
A′G
XG′
′
←֓ BG
↓ µA
↓
↓ µX
j
A′G′
i
←֓ R′G
XG′ ′
′
←֓ BG
′
↓
i
←֓ R′G′
where the down arrows are birational maps induced by the Y − ∆ move action on the Band C-variables, given by (61) and (49)-(50), and the right ones are compositions of four
cluster A- and X -mutations.
Proof. i) See Figure 30. After this sequence of 4 cluster transformations, a little algebra
X8
X7
X9
X2
Μ1
X1
X6
X3
X2H2L
X2'
Μ2
X1'
X1H2L
X3H2L
X3'
X4
X5
X8H4L
X2H3L
Μ3
X2H4L
X7H4L
Μ1
X1H3L
X1H4L
X6H4L
X3H3L
X3H4L
X5H4L
Figure 30: The Y − ∆ move from spider moves.
59
X9H4L
X4H4L
using Theorem 4.7 shows that the new X variables are:
(4)
X1
(4)
X2
(4)
X3
(4)
X4
(4)
X5
(4)
X6
(4)
X7
(4)
X8
(4)
X9
1 + X1 + X1 X3
X3 (1 + X2 + X1 X2 )
1 + X2 + X1 X2
=
X1 (1 + X3 + X2 X3 )
1 + X3 + X2 X3
=
X2 (1 + X1 + X1 X3 )
=
= X4 (1 + X1 + X1 X3 )
X1 X3
= X5
1 + X1 + X1 X3
= X6 (1 + X3 + X2 X3 )
X2 X3
= X7
1 + X3 + X2 X3
= X8 (1 + X2 + X1 X2 )
X1 X2
= X9
.
1 + X2 + X1 X2
(4)
(4)
(4)
If X1 X2 X3 = 1, we will also have X1 X2 X3 = 1. Moreover if the conductances on
the edges of the triangle opposite faces 1, 2, 3 are C1 , C2 , C3 respectively then X1 = C2 /C3 ,
X2 = C3 /C1 , X3 = C1 /C2 . One checks that the new conductances are those satisfying
(50).
ii) It follows by comparing formulas (49)-(50).
It follows that the diagram (53) gives rise to a similar diagram of spaces, obtained
by gluing the tori assigned to minimal equivalent graphs G via the corresponding birational maps given either by the Y-∆-move or the corresponding sequence of four cluster
mutations.
6
6.1
Discrete cluster integrable systems
Discrete cluster integrable systems on a square grid on a
torus
Let us consider the dimer model on a square grid on a torus. Such examples are constructed as follows. Take the standard square grid bipartite graph in Z2 , and map it onto
torus by taking the quotient by a rank two subgroup of the (bipartite-coloring preserving)
translation group (that is, (1, 1)Z + (1, −1)Z), see Figure 33.
Let us construct an element A of the cluster modular group acting by a non-trivial
automorphism of the corresponding cluster Poisson variety XN .
There is a version of the spider move, see Figure 32, given by a composition of two
elementary moves on Figure 31.
60
Figure 31: Inserting two white 2-valent vertices, and performing the spider move in the
dotted region we get a version of the spider move on Figure 32.
W2
W2
W1
W3
W
W1
W
W
W3
W4
4
Figure 32: A version of the spider move.
The squares of the grid are of two types: one, which we refer to as the ∗-faces, have the
black bottom left vertex. Let us do the Figure 32 version of the spider move at all ∗-faces.
The result is illustrated at the right of Figure 34. Then we shrink the 2-valent vertices, as
shown on Figure 23. We get again a square grid, but now the colors of the vertices flipped,
see the bottom of Figure 34. So performing the same sequence of moves—the Figure 32
version of the spider moves at the ∗-faces and shrinking the 2-valent vertices—we get the
original grid. The resulting cluster transformation A is non-trivial. Thanks to Theorem
4.7 it commutes with the quantum Hamiltonians.
*
*
*
*
*
*
*
*
*
*
*
*
Figure 33: Discrete cluster integrable system on a square grid on a torus.
The transformation A has appeared in various guises in the literature. A version of
the transformation A first appeared in [Kor93]. It was called the octahedron recurrence
in [Spey04]; the relation of the octaherdon reccurence with the Hirota bilinear difference
61
equation [Hir81] is discussed there. A solution of the octaherdon reccurence on a simplex
was found in [FG03a]. See also [DFK10], [Naka10] for other examples which are special
cases.
The cluster transformation A together with the quantum Hamiltonians form a discrete
quantum integrable system—we think of A as of an evolution with a discrete time. In the
quantized picture [FG07] the operator A gives rise to a unitary operator in the Hilbert
space of the system commuting with the quantum Hamiltonians. It is a composition of
the quantum dilogarithm intertwiners parametrized by the faces.
*
spider moves
at the
*
* faces
shrink 2−valent
vertices
same sequence of
moves at the faces
*
*
*
Figure 34: Constructing an element A of the cluster modular group.
6.2
Cluster automorphisms for resistor networks on a torus.
One also finds cluster automorphisms for cluster varieties associated to resistor networks
on a torus. The pair (X , R) assigned to an equivalence class of minimal graphs G on a
torus has a large group A of cluster automorphisms, isomorphic to (a quotient of) Zs−2 ,
where 2s is the number of sides of the Newton polygon N(G). It is constructed as follows.
(This material is also found in [HS07].)
For each pair E, E ′ of centrally symmetric sides of the Newton polygon N(G) of a
minimal graph G on a torus there is a collection α1 , ..., αk of isotopic zig-zag strands
on G which do not intersect due to minimality of G: their (well defined up to sign)
homology class in H1 (T, Z) is a primitive vector on E. We assume that they are written
in their natural cyclic order provided by the embedding to the torus. Isotope them
cyclically (α1 , ..., αk ) → (α2 , ..., αk , α1 ), leaving the other zig-zag paths intact. Present
this as a sequence of Y-∆-moves corresponding to the zig-zag strands moving through
simple crossings of other zig-zag strands, which transforms the graph G to itself. Take
62
the product of the cluster transformations assigned by Lemma 5.11 to the Y-∆-moves.
We arrive at a cluster transformation, realized by a birational isomorphism of the pair
(X , R). It is an element of the cluster modular group, as defined in Section 2.4 of [FG03b].
The automorphisms assigned to different sides commute, and their product is 1.
6.3
The cube recurrence discrete cluster integrable system
Let us apply the construction of Section 6.2 to the honeycomb graph on a torus. In this
case there are three families of parallel zig-zag loops. Moving cyclically one of the families
we get a cluster automorphism. We show that its action on the B-variables is given by
the so-called cube recurrence [CS04], also known as the discrete BKP equation or
Hirota-Miwa equation, which goes back to the work of Miwa [Miwa82], see also [BS].
A function a : Z3 → C satisfies the cube recurrence if
ax+1,y+1,z+1ax,y,z = ax+1,y,z ax,y+1,z+1 + ax,y+1,z ax+1,y,z+1 + ax,y,z+1ax+1,y+1,z .
(62)
Note that if a is defined and nonzero for 0 ≤ x + y + z ≤ 2 then equation (62) defines it
for x + y + z = 3, also for x + y + z = −1, and by iteration for all (x, y, z) ∈ Z3 .
H1,-1,1L
H0,-1,2L
H-1,-1,2L
H0,-1,1L
H0,0,2L
H1,-1,0L
H1,0,1L
H0,0,1L
H-1,0,1L
H1,0,0L
H0,0,0L
H0,1,1L
H2,0,0L
H1,0,-1L
H1,1,0L
H0,1,0L
H-1,1,0L
H2,1,-1L
H1,1,-1L
H0,1,-1L
H1,1,-2L
H2,2,-2L
H1,2,-1L
H0,2,0L
Figure 35: Vertex and face labeling for the cube recurrence.
Let G be the honeycomb graph, thought of as a resistor network. Vertices of G are the
orthogonal projection to the plane {x + y + z = 0} of the set {(x, y, z) ∈ Z3 | x + y + z =
0 or 2}; edges connect vertices whose coordinate sum is 0 to those nearest neighbors whose
coordinate sum is 2. We can likewise index faces of G by (x, y, z) satisfying x + y + z = 1.
See Figure 35. Let Vk be the set of vertices or faces with x + y + z = k.
Assign variables Bx,y,z to the vertices and faces of G.
Perform a Y − ∆ move at each vertex vx,y,z where x + y + z = 0. The graph becomes
the regular triangulation with vertices at V2 , faces at V1 and new faces at V3 . Using (61),
the B-value at a new face is obtained from (62). See Figure 36.
63
Next, perform ∆ − Y moves on all faces in V1 . The resulting graph is again a honeycomb, with vertices V2 , V4 and faces at V3 . Iterating this procedure we produce variables
Bx,y,z for all (x, y, z) ∈ Z3 and these variables satisfy the cube recurrence.
Equivalently, we can look at zig-zag strands and move cyclically one of the families of
parallel zig-zag strands, say the vertical ones on Figure 35. Then each strand will cross
two different sets of crossings of other two familes, which can be thought of as the vertices
of the grid, resulting into Y − ∆ and ∆ − Y moves described above.
By Lemma 5.11, the composition B of these Y −∆ moves and ∆−Y moves is a cluster
automorphism, i.e. an element of the cluster modular group, of the dimer model phase
space assigned to the graph G. Therefore it gives rise to a discrete cluster integrable
system, with the discrete time evolution given by B. Thanks to (61), the cube recurrence
is obtained by restricting the cluster automorphism B to the isotropic subvariety B.
Figure 36: Steps of the cube recurrence. The black dots are the vertices vx,y,z with
x + y + z = 0.
7
7.1
Newton polygons, toric surfaces and integrable systems
Newton polygons and toric surfaces
Let us briefly recall this well known connection. See [Fult] for more details. Let T = (C∗ )2 .
Denote by X ∗ (T) the group of its characters, isomorphic to Z2 .
A Newton polygon N is a convex polygon in the real plane X∗ (T)R := X ∗ (T)⊗R = R2
with vertices at the lattice X ∗ (T). It gives rise to a polarized projective toric surface N ,
possibly singular. The polarization is given by a T-equivariant line bundle L on N .
Shifting the Newton polygon by a lattice vector l ∈ X ∗ (T) amounts to changing the
equivariant structure of the line bundle L by multiplying it by the trivial line bundle with
the torus T action given by the character χl assigned to l.
Each edge E of N gives rise to a ray ρE in the dual space X∗ (T)R , consisting of the
normals to the edge E looking inside of N. These rays form a fan ∆N . It subdivides the
plane X∗ (T)R into a disjoint union of cones. Each cone σ provides the dual cone σ ∨ , and
hence a semigroup Rσ := σ ∨ ∩ X ∗ (T) and an affine surface Uσ := Spec Z[Rσ ]. Gluing the
surfaces Uσ so that Uσ1 ∩σ2 = Uσ1 ∩ Uσ2 we get the surface N .
64
The torus T acts on N . There is a unique 2-dimensional orbit, U0 , where 0 is
the origin, denoted below by N0 . Its complement is the divisor at infinity N∞ , whose
shape is described by the polygon N: the irreducible components DE are projective lines
parametrized by the edges E of N, and their intersections match the vertices of N. The
kernel of the action of the torus T on the component DE is the cocharacter χ∨E : C∗ → T(C)
assigned to the conormal line to the edge E.
There is a canonical isomorphism, serving as a definition of the line bundle L:
X
∼
iN : L −→ O(D),
D=
aE DE .
E
where the coefficient aE ∈ Z is the distance from the origin to the edge E.8 The torus
T action on L comes from the natural T-action on O(D), provided by the action on N
preserving the divisor D. Shifting the polygon N by a vector l ∈ X ∗ (T) amounts to
multiplication of the isomorphism iN by the character χl , i.e. iN +l = iN χl . Using the
isomorphism iN , we have
∼
∼
H 0 (N , L) −→ H 0 (N , O(D)) ⊂ C(T) = C(z1 , z2 ),
(63)
where (z1 , z2 ) is a basis in the group of characters of T. It is identified with the space of
all Laurent polynomials with the Newton polygon N.
So given a pair (N , L), the polygon N is recovered as the convex hull of the set of
characters of the action of the torus T on H 0 (N , L).
Denote by ω the canonical up to a sign invariant 2-form on N − N∞ . Given a basis
(z1 , z2 ) of characters of the torus T, it can be written as
ω = d log z1 ∧ d log z2 .
So it is a 2-form on N with logarithmic singularities at N∞ .
7.2
The spectral data [KO03]
A parametrization of a degree d divisor C =
Pk
i=1
ni ci by a set S is a map
ν : S −→ {c1 , ..., ck } such that |ν −1 (ci )| = ni .
Let |L| be a linear system of curves on N given by the zeros of sections of L.
Definition 7.1. A spectral data is a triple (C, S, ν) where C is a genus g curve on the
toric surface N from the linear system |L|, S is a degree g effective divisor on C0 = C ∩N0 ,
and ν = {νE } are parametrizations of the divisors DE ∩ C.
The moduli space SN = S parametrizes the spectral data related to N.
8
The latter is the value at the origin of the unique surjective affine linear function αE : X∗ (T) → Z
which vanishes on E and is non-negative on N .
65
Let C be a curve on N from the linear system |L|. Its intersection with N0 is defined
by a Laurent polynomial P (z1 , z2 ) with the Newton polygon N. The intersection of
C ∩ DE is determined by the sum of monomials of P supported on the edge E. The
ratio of monomials assigned to two integral points on the sides of N are Casimirs. So the
Casimirs are naturally assigned to the primitive boundary edges of N, and their product
is equal to 1.
Proposition 7.2. Given a black vertex v of a minimal bipartite graph Γ assigned to N,
there is a natural rational map
κΓ,v : X −→ S.
Proof. A point of X parametrizes a line bundle with connection on Γ, encoded by the
Kasteleyn operator K. Its determinant is a section of the line bundle L ⊗ L. The divisor
of zeros of the section is the spectral curve C on N . The black point b defines a section
of a line bundle on C. Its degree is the genus g of C, and its divisor is the divisor S.
Homology classes [αE ] of zig-zag loops on Γ are in bijection with the sides of the Newton
polygon N. There is a parametrization
∼
γE : {Zig-zag loops on Γ in the homology class [αE ]} −→ DE ∩ C,
(64)
uniquely determined by the following condition. The function given by the monodromy
around a zig-zag loop α coincides with the function given by the coordinate of the boundary point γE (α).
Denote by Sχ the subvariety of S given by the condition that C ∩ N∞ is a given
divisor C∞ . Using the parametrization (64), the divisor C∞ provides a subvariety Xχ
in X defined by the condition that the monodromies around zig-zag paths are given by
the coordinates of the corresponding boundary points. Since this condition uses only the
boundary divisor of the spectral curve, there is a similar subvariety Bχ in B. The map
κΓ,v is evidently fibered over Bχ , providing a commutative diagram
κΓ,v
−→
Xχ
ց
Sχ
ւ
Bχ
Theorem 7.3. The image of the map κΓ,v is open. The map κΓ,v is a finite map over the
generic point.9
See below for the proof. Since both spaces have the same dimension, the first claim
implies the second.
9
In the sequel to this paper we prove that this map is a birational isomorphism.
66
7.3
The real positive points [KO03]
Since X is a union of cluster coordinate tori, and the transition functions between different
cluster coordinate systems are subtraction free rational functions, there is a well defined
set X (R>0 ) of its real positive points. It parametrizes connections on a minimal bipartite
graph Γ associated with N whose face weights are positive numbers, and does not depend
on the choice of Γ.
Simple Harnack curves were defined by G. Mikhalkin in [Mikh00] as real curves
P (z1 , z2 ) = 0 whose real components satisfy certain topological constraints. They are
a very special case of the curves which appear in the Harnack constructions [Har1876] of
several series of real curves of degree d with the maximal number of connected components
(M-curves). Several characterizations of simple Harnack curves were given in [Mikh00].
They include a definition via 2 − 1 covering of its amoeba. Recall that the amoeba is the
image of the curve under the map
C∗ × C∗ → R2 ,
(z1 , z2 ) 7→ (log |z1 |, log |z2 |).
Another characterization of simple Harnack curves, due to [MR00], is that they are the
curves which maximize the area of their amoeba among all curves with given Newton
polygon.
The results of Kenyon-Okounkov [KO03, Theorems 1,3] are as follows:
• The real positive part B(R>0 ) := π(X (R>0)) parametrizes simple Harnack curves
on N which lie in the linear system |L|. Such a genus g simple Harnack curve has
g ovals Oi matching the integral points inside of the Newton polygon N.
• The space X (R>0) parametrizes the data {A simple Harnack curve C on N from |L|,
a choice of a point on each oval Oi , a choice of parametrization of the intersection
of C with each divisor at infinity}.
Therefore the fibers of the fibration
π+ : X (R>0 ) −→ B(R>0 )
are real g-dimensional tori. They are the Liouville tori of the integrable system restricted
to X (R>0 ).
7.4
Independence of the Hamiltonians
Corollary 7.4. Forgetting the parametrization, the space B projects to an open part of
|L|.
Proof. Thanks to Section 7.3 the image of B is an algebraic subvariety of |L|, and its real
part contains an open subset of the real locus of |L|.
Corollary 7.5. The Hamiltonians are independent on X .
67
Proof. Denote by |L|χ the moduli space of curves from the linear system |L| with a given
intersection C∞ with N∞ . Thanks to Corollary 7.4, Bχ is a finite cover over on open part
of |L|χ . We claim that the Hamiltonians are local coordinate functions on |L|χ . Therefore
they are independent on X .
Let us prove the claim. Identify the sections of L with Laurent polynomials in (z1 , z2 )
with the Newton polygon N. Let PN0 be the subspace of Laurent polynomials in (z1 , z2 )
whose Newton polygon is the interior of N. Adding such a polynomial to a section of L,
we do not change the intersection with N∞ of its divisor of zeros. So we get an action
of the vector space PN0 on |L|χ . This action is transitive. The Hamiltonians are given by
the coefficients of the monomials in PN0 . So their differentials at a point C ∈ |L|χ form a
basis of TC∗ |L|χ .
Proof of Theorem 7.3. The restriction of the map κΓ,v to Xχ (R>0 ) is an isomorphism.
Thus its image is open in Sχ .
8
Appendix: Calculating the Poisson structure on LΓ
Let us give a local description of the intersection pairing (3) as a sum of local contributions
assigned to the vertices of Γ.
8.1
The local pairing.
The star of a vertex v in a ribbon graph, Figure 37, is a vertex v together with the
cyclically ordered edges incident to v.
Figure 37: The star of a vertex on an oriented surface.
The star of a vertex v gives rise to an abelian group Av , given by Z-linear combinations
of the oriented edges Ei at v of total degree zero:
X →
X
Av = {
ni E i |
ni = 0, ni ∈ Z},
→
where E i is the edge Ei oriented out of v. Clearly Av is a torsion free abelian group of
→
→
rank m − 1, where m is the valency of v. It is generated by the “oriented paths” E i − E j ,
→
where − E j is the edge Ej oriented towards v.
Lemma 8.1. There exists a unique skew symmetric bilinear form
1
δv : Av ∧ Av −→ Z
2
68
γ
E2
l
=
E0
E1
γr
Figure 38: One has δv (γr , γl ) = 21 .
such that for any triple E0 , E1 , E2 of edges as in Figure 38 one has
1
δv (γr ∧ γl ) = ,
2
→
→
γr :=E1 − E0 ,
→
→
γl :=E2 − E0 .
(65)
Proof. Let {x1 , ..., xm } be the cyclically ordered set of the endpoints of the edges sharing
the vertex v. Let γi be the oriented
P path xi vxi+1 . Then the elements γ1 , ..., γm generate
Av and satisfy the only relation i γi = 0. We define the local pairing by setting
1
δv (γi ∧ γi+1 ) = .
2
P
Then i γi is in the kernel, so the local pairing is well defined. It is easy to see that it
satisfies formula (65).
8.2
The global pairing
Let L1 , L2 be two oriented paths on Γ. Let v be a vertex shared by them. Each path Li
gives rise to an oriented path at the star of v, providing an element li ∈ Av . Let εv (L1 , L2 )
be the value of the local pairing δv on l1 ∧ l2 .
Definition 8.2. Let L1 , L2 be two oriented loops on a bipartite surface graph Γ. Then
X
ε(L1 , L2 ) :=
sgn(v)εv (L1 , L2 ),
v
where the sum is over all vertices v shared by L1 and L2 , and sgn(v) = 1 for the white
vertex v, and −1 for the black vertex.
γ
2
γ
4
γ
1
γ
3
Figure 39: One has ε(γ1, γ2 ) = −1, ε(γ3 , γ4 ) = 0.
69
γ
γ
2
γ
1
γ
γ
4
γ
3
1
2
Figure 40: One has ε(γ1, γ2 ) = 1, ε(γ3, γ4 ) = 0.
Proposition 8.3. The bilinear form ε(L1 , L2 ) from Definition 8.2 coincides with the
b in Section 1.1.1
bilinear form defined using the conjugated graph Γ
Proof. It is sufficient to show that in the examples on Figures 39 and 40 the sum of
the local contributions coincides with the local intersection number of the corresponding
paths on the conjugated graph. The latter clearly gives the results presented on Figures
39 and 40. So one needs to show that the former gives the same result. For example, to
check that ε(γ1 , γ2) = 1 for the left picture on Figure 40, notice that on Figure 41 we have
1
δv (γ1 , γ3 ) = − ,
2
γ
1
δv (γ1 , γ2 + γ3 ) = .
2
2
γ
γ1
γ
3
γ
γ
2
γ
1
γ
1
γ1
1
γ +γ
2
3
γ1
3
Figure 41: Proving ε(γ1 , γ2 ) = 1.
Example. The face orientation provides the counterclockwise orientation of the
boundary of the face. Given two different faces F1 , F2 , and an edge E, we calculate
that
if E 6∈ ∂(F1 ) ∩ ∂(F2 ),
0
+1 if the “black to white” orientation of E
is the same as the orientation induced
δE;F1 ,F2 :=
(66)
by
the
orientation
of
the
face
F
,
1
−1 otherwise.
Taking now the sum over all edges E, we get
ε(F1 , F2 ) := ε(∂(F1 ), ∂(F2 )) =
X
E
70
δE;F1,F2 .
Example. Given a face F and an oriented loop L on Γ, let us calculate ε(F, L).
Denote by OrE the “black to white” orientation of the edge E. The orientation of L
induces an orientation OrE,L of any edge E of L. Set
δE,L = −OrE,L /OrE ∈ {±1}.
(67)
So if the orientation of L is opposite to the canonical orientation of E, then δE,L = 1.
Otherwise δE,L = −1. Then it follows from Figure 39 that we have
X
ε(F, L) =
δE,L .
(68)
E∈L∩∂(F )
Here the sum is over all edges E shared by the face F and the loop L.
Changing the orientation of the surface amounts to changing the sign of the Poisson
bracket. Interchanging black and white we change the sign of the Poisson bracket.
8.3
Amalgamation and the Poisson structure on LΓ .
We extend the story to the case Γ is a bipartite graph on an oriented surface S with
boundary. Pairs (Γ, S) are encoded by bipartite ribbon graphs.
8.3.1
Bipartite ribbon graphs
Take a bipartite graph embedded into an oriented surface S with boundary, so that some
vertices of the graph may lie at the boundary of S. Let us add to every boundary vertex
of the graph a few edges sticking out of the surface, called external legs. The graph Γ is
then a bipartite ribbon graph. Every bipartite ribbon graph is obtained this way.
Definition 8.4. Given a bipartite ribbon graph Γ, the moduli space LΓ parametrizes line
bundles with connection over the graph Γ, equipped with a trivialization of the restriction
of the bundle to every external edge.
Let Γ1 and Γ2 be bipartite ribbon graphs. Let us assume that we are given a way to
glue some pairs of their external legs into a bipartite ribbon graph Γ. So each external leg
of each graph is glued to no more than one external leg of the other graph, and the result
is a bipartite ribbon graph. Then there is a map of moduli spaces, the amalgamation
map:
(69)
α : LΓ1 × LΓ2 −→ LΓ .
We use trivializations at the external legs of the bundles on Γ1 and Γ2 to glue a new
bundle on Γ. It inherits a connection. The following is obvious.
Lemma 8.5. The gluing map (69) is a principal fibration over LΓ with the fiber (C∗ )k ,
where k is the number of pairs of glued boundary legs.
71
Figure 42: White and black hedgehogs with six external legs.
A bipartite ribbon graph with single internal vertex is called a hedgehog, see Figure
42. It is a white / black hedgehog according to the color of its internal vertex. Let Hv be
a hedgehog with the internal vertex v. Then one has
(70)
LHv = Hom(Av , C∗ ) = Ker µ : (C∗ )m −→ C∗ .
Here m is the number of legs and µ is the multiplication map. The first isomorphism
assigns to an oriented path connecting two external legs the monodromy of the connection
along this path. Since the line bundle is trivialized at the endpoints, it is just a non zero
complex number. The second is obvious.
A ribbon graph Γ is glued from the hedgehogs provided by the stars of its internal
vertices. Amalgamating the hedgehog moduli spaces we get the moduli space LΓ .
Figure 43: Gluing bipartite ribbon graphs from hedgehogs.
Thanks to identification (70), the skew symmetric local pairing on the abelian group
Av provided by the color of the internal vertex v of the hedgehog gives rise to a Poisson
structure on the moduli space LHv . The following proposition follows easily from the
definitions.
Lemma 8.6. There is a unique Poisson structure on the space LΓ such that the Poisson structure on the space LHv assigned to a hedgehog Hv is the standard one, and the
amalgamation maps (69) are Poisson.
When Γ is a bipartite graph on a surface without boundary we recover its Poisson
structure.
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