Dimers on surface graphs and spin structures. II
DIMERS ON SURFACE GRAPHS AND SPIN STRUCTURES. II
DAVID CIMASONI AND NICOLAI RESHETIKHIN
Abstract. In a previous paper [3], we showed how certain orientations of the
edges of a graph Γ embedded in a closed oriented surface Σ can be understood
as discrete spin structures on Σ. We then used this correspondence to give a
geometric proof of the Pfaffian formula for the partition function of the dimer
model on Γ. In the present article, we generalize these results to the case of
compact oriented surfaces with boundary. We also show how the operations
of cutting and gluing act on discrete spin structures and how they change the
partition function. These operations allow to reformulate the dimer model as
a quantum field theory on surface graphs.
Contents
Introduction 2
Acknowledgements 3
1. The dimer model on graphs with boundary 3
1.1. Dimers on graphs with boundary 3
1.2. Dimers on surface graphs with boundary 4
2. Kasteleyn orientations on surface graphs with boundary 5
2.1. Kasteleyn orientations 5
2.2. Discrete spin structures 5
2.3. The Pfaffian formula for the partition function 6
3. Cutting and gluing 8
3.1. Cutting and gluing graphs with boundary 8
3.2. Cutting and gluing surface graphs with boundary 9
3.3. Cutting and gluing discrete spin structures 10
3.4. Cutting Pfaffians 12
4. Quantum field theory for dimers 13
4.1. Quantum field theory on graphs 13
4.2. Quantum field theory for dimers on graphs 13
4.3. The dimer model as the theory of free Fermions 14
5. Dimers on bipartite graphs and height functions 16
5.1. Composition cycles on bipartite graphs 16
5.2. Height functions for planar bipartite graphs 16
5.3. Height functions for bipartite surface graphs 19
5.4. The dimer quantum field theory on bipartite surface graphs 22
References 23
Date: October 22, 2018.
1991 Mathematics Subject Classification. Primary: 82B20; Secondary: 57R15.
http://arxiv.org/abs/0704.0273v1
2 DAVID CIMASONI AND NICOLAI RESHETIKHIN
Introduction
A dimer configuration on a graph Γ is a choice of a family of edges of Γ, called
dimers, such that each vertex of Γ is adjacent to exactly one dimer. Assigning
weights to the edges of Γ allows to define a probability measure on the set of dimer
configurations. The study of this measure is called the dimer model on Γ. Dimer
models on graphs have a long history in statistical mechanics [6, 12], but also show
interesting aspects involving combinatorics, probability theory [10, 4], real algebraic
geometry [9, 8], etc...
A remarkable fact about dimer models was discovered by P.W. Kasteleyn in
the 60’s: the partition function of the dimer model can be written as a linear
combination of 22g Pfaffians of N ×N matrices, where N is the number of vertices
in the graph and g the genus of a closed oriented surface Σ where the graph can be
embedded. The matrices are signed-adjacency matrices, the sign being determined
by an orientation of the edges of Γ called a Kasteleyn orientation. If the graph
is embedded in a surface of genus g, there are exactly 22g equivalence classes of
Kasteleyn orientations, defining the 22g matrices. This Pfaffian formula for the
partition function was proved by Kasteleyn in [6] for the cases g = 0, 1, and only
stated for the general case [7]. A combinatorial proof of this fact and the exact
description of coefficients for all oriented surfaces first appeared much later [11, 14].
The number of equivalence classes of Kasteleyn orientations on a graph Γ em-
bedded in Σ is also equal to the number of equivalence classes of spin structures on
Σ. An explicit construction relating a spin structure on a surface with a Kasteleyn
orientation on a graph with dimer configuration was suggested in [10]. In [3], we in-
vestigated further the relation between Kasteleyn orientations and spin structures.
This allows to understand Kasteleyn orientations on a graph embedded in Σ as
discrete spin structures on Σ. We also used this relation to give a geometric proof
of the Pfaffian formula for closed surfaces. Our final formula can be expressed as
follows: given a graph Γ embedded in a closed oriented surface Σ of genus g, the
partition function of the dimer model on Γ is given by
Z(Γ) =
ξ∈S(Σ)
Arf(ξ)Pf(Aξ(Γ)),
where S(Σ) denotes the set of equivalence classes of spin structures on Σ, Arf(ξ) =
±1 is the Arf invariant of the spin structure ξ, and Aξ(Γ) is the matrix given by
the Kasteleyn orientation corresponding to ξ.
The first part of the present paper is devoted to the extension of the results
obtained in [3] to dimer models on graphs embedded in surfaces with boundary
(Sections 1 and 2). We then show how the operations of cutting and gluing act
on discrete spin structures and how they change the partition function (Section 3).
These operations define the structure of a functorial quantum field theory in the
spirit of [2, 13], as detailed in Section 4. We then give two equivalent reformulations
of the dimer quantum field theory: the “Fermionic” version, which describes the
partition function of the dimer model as a Grassman integral, and the “Bosonic”
version, the equivalent description of dimer models on bipartite surface graphs in
terms of height functions. This special case of bipartite graphs is the subject of
Section 5.
DIMERS ON SURFACE GRAPHS AND SPIN STRUCTURES. II 3
Throughout this paper, Σ is a compact surface, possibly disconnected and possi-
bly with boundary, endowed with the counter-clockwise orientation. All results can
be extended to the case of non-orientable surfaces, which will be done in a separate
publication. We refer to [14] for a combinatorial treatment of dimer models on
non-orientable surface graphs.
Acknowledgements. We are grateful to J. Andersen, M. Baillif, P. Teichner and
A. Vershik for inspiring discussions. We also thankfully acknowledge the hospitality
of the Department of Mathematics at the University of Aarhus. The work of D.C.
was supported by the Swiss National Science Foundation. This work of N.R. was
partially supported by the NSF grant DMS–0307599, by the CRDF grant RUM1–
2622, by the Humboldt foundation and by the Niels Bohr research grant.
1. The dimer model on graphs with boundary
1.1. Dimers on graphs with boundary. In this paper, a graph with boundary
is a finite graph Γ together with a set ∂Γ of one valent vertices called boundary
vertices . A dimer configuration D on a graph with boundary (Γ, ∂Γ) is a choice of
edges of Γ, called dimers , such that each vertex that is not a boundary vertex is
adjacent to exactly one dimer. Note that some of the boundary vertices may be
adjacent to a dimer of D, and some may not. We shall denote by ∂D this partition
of boundary vertices into matched and non-matched. Such a partition will be called
a boundary condition for dimer configurations on Γ.
A weight system on Γ is a positive real valued function w on the set of edges of
Γ. It defines edge weights on the set D(Γ, ∂Γ) of dimer configurations on (Γ, ∂Γ)
w(D) =
w(e),
where the product is taken over all edges occupied by dimers of D.
Fix a boundary condition ∂D0. Then, the Gibbs measure for the dimer model
on (Γ, ∂Γ) with weight system w and boundary condition ∂D0 is given by
Prob(D | ∂D0) =
Z(Γ;w | ∂D0)
where
Z(Γ;w | ∂D0) =
D:∂D=∂D0
w(D),
the sum being on all D ∈ D(Γ, ∂Γ) such that ∂D = ∂D0.
Let V (Γ) denote the set of vertices of Γ. The group
G(Γ) = {s : V (Γ) → R>0}
acts on the set of weight systems on Γ as follows: (sw)(e) = s(e+)w(e)s(e−), where
e+ and e− are the two vertices adjacent to the edge e. Note that (sw)(D) =
v s(v)w(D) and Z(Γ; sw | ∂D0) =
v s(v)Z(Γ;w | ∂D0), both products being on
the set of vertices of Γ matched by D0. Therefore, the Gibbs measure is invariant
under the action of the group G(Γ).
Note that the dimer model on (Γ, ∂Γ) with boundary condition ∂D0 is equivalent
to the dimer model on the graph obtained from Γ by removing all edges adjacent
to non-matched boundary vertices.
4 DAVID CIMASONI AND NICOLAI RESHETIKHIN
Given two dimer configurations D and D′ on a graph with boundary (Γ, ∂Γ),
let us define the (D,D′)-composition cycles as the connected components of the
symmetric difference C(D,D′) = (D ∪D′)\(D ∩D′). If ∂D = ∂D′, then C(D,D′)
is a 1-cycle in Γ with Z2-coefficients. In general, it is only a 1-cycle (rel ∂Γ).
1.2. Dimers on surface graphs with boundary. Let Σ be an oriented compact
surface, not necessarily connected, with boundary ∂Σ. A surface graph with bound-
ary Γ ⊂ Σ is a graph with boundary (Γ, ∂Γ) embedded in Σ, so that Γ ∩ ∂Σ = ∂Γ
and the complement of Γ \ ∂Γ in Σ \ ∂Σ consists of open 2-cells. These conditions
imply that the graph Γ := Γ ∪ ∂Σ is the 1-skeleton of a cellular decomposition of
Note that any graph with boundary can be realized as a surface graph with
boundary. One way is to embed the graph in a closed surface of minimal genus,
and then to remove one small open disc from this surface near each boundary vertex
of the graph.
A dimer configuration on a surface graph with boundary Γ ⊂ Σ is simply a
dimer configuration on the underlying graph with boundary (Γ, ∂Γ). Given two
dimer configurations D and D′ on a surface graph Γ ⊂ Σ, let ∆(D,D′) denote
the homology class of C(D,D′) in H1(Σ, ∂Σ;Z2). We shall say that two dimer
configurations D and D′ are equivalent if ∆(D,D′) = 0 ∈ H1(Σ, ∂Σ;Z2). Note
that given any three dimer configurations D,D′, and D′′ on Γ ⊂ Σ, we have the
identity
(1) ∆(D,D′) + ∆(D′, D′′) = ∆(D,D′′)
in H1(Σ, ∂Σ;Z2).
Fix a homology class β ∈ H1(Σ, ∂Σ;Z2), a dimer configuration D1 ∈ D(Γ, ∂Γ)
and a boundary condition ∂D0. The associated partial partition function is defined
Zβ,D1(Γ;w | ∂D0) =
D:∂D=∂D0
∆(D,D1)=β
w(D),
where the sum is taken over all D ∈ D(Γ, ∂Γ) such that ∂D = ∂D0 and ∆(D,D1) =
The equality (1) implies that
Zβ,D1(Γ;w | ∂D0) = Zβ+∆(D0,D1),D0(Γ;w | ∂D0).
Furthermore, the relative homology class β′ = β + ∆(D0, D1) lies in the image of
the canonical homomorphism j : H1(Σ,Z2) → H1(Σ, ∂Σ;Z2). Hence,
Zβ′,D0(Γ;w | ∂D0) =
α:j(α)=β′
Zα(Γ, w | ∂D0),
where the sum is taken over all α ∈ H1(Σ,Z2) such that j(α) = β
′, and
Zα(Γ;w | ∂D0) =
D:∂D=∂D0
∆(D,D0)=α
w(D).
Therefore the computation of the partition function Zβ,D1(Γ;w | ∂D0) boils down
to the computation of Zα(Γ;w | ∂D0) with α ∈ H1(Σ;Z2). We shall give a Pfaffian
formula for this latter partition function in the next section (see Theorem 2.4).
DIMERS ON SURFACE GRAPHS AND SPIN STRUCTURES. II 5
2. Kasteleyn orientations on surface graphs with boundary
2.1. Kasteleyn orientations. Let K be an orientation of the edges of a graph Γ,
and let C be an oriented closed curve in Γ. We shall denote by nK(C) the number
of times that, traveling once along C following its orientation, one runs along an
edge in the direction opposite to the one given by K.
A Kasteleyn orientation on a surface graph with boundary Γ ⊂ Σ is an orien-
tation K of the edges of Γ = Γ ∪ ∂Σ which satisfies the following condition: for
each face f of Σ, nK(∂f) is odd. Here ∂f is oriented as the boundary of f , which
inherits the orientation of Σ.
Using the proof of [3, Theorem 3.1], one easily checks that if ∂Σ is non-empty,
then there always exists a Kasteleyn orientation on Γ ⊂ Σ. More precisely, we have
the following:
Proposition 2.1. Let Γ ⊂ Σ be a connected surface graph, possibly with boundary,
and let C1, . . . , Cµ be the boundary components of Σ with the induced orientation.
Finally, let n1, . . . , nµ be 0’s and 1’s. Then, there exists a Kasteleyn orientation on
Γ ⊂ Σ such that 1 + nK(−Ci) ≡ ni (mod 2) for all i if and only if
n1 + · · ·+ nµ ≡ V (mod 2),
where V is the number of vertices of Γ.
Proof. First, let us assume that there is a Kasteleyn orientation K on Γ ⊂ Σ such
that 1 + nK(−Ci) ≡ ni for all i. Let Σ
′ be the closed surface obtained from Σ by
pasting a 2-disc Di along each boundary component Ci. Let Γ
′ ⊂ Σ′ be the surface
graph obtained from Γ as follows: for each i such that ni = 1, add one vertex in the
interior of Di and one edge (arbitrarily oriented) between this vertex and a vertex
of Ci. The result is a Kasteleyn orientation on Γ
′ ⊂ Σ′, with Σ′ closed. By [3,
Theorem 3.1], the number V ′ of vertices of Γ′ is even. Hence,
0 ≡ V ′ ≡ V + n1 + · · ·+ nµ (mod 2).
Conversely, assume Γ ⊂ Σ is a surface graph with n1+ · · ·+nµ ≡ V (mod 2). Paste
2-discs along the boundary components of Σ as before. This gives a surface graph
Γ′ ⊂ Σ′ with Σ′ closed and V ′ even. By [3, Theorem 3.1], there exists a Kasteleyn
orientation K ′ on Γ′ ⊂ Σ′. It restricts to a Kasteleyn orientation K on Γ ⊂ Σ with
1 + nK(−Ci) ≡ ni for all i. �
Recall that two Kasteleyn orientations are called equivalent if one can be ob-
tained from the other by a sequence of moves reversing orientations of all edges
adjacent to a vertex. The proof of [3, Theorem 3.2] goes through verbatim: if
non-empty, the set of equivalence classes of Kasteleyn orientations on Γ ⊂ Σ is an
affine H1(Σ;Z2)-space. In particular, there are exactly 2
b1(Σ) equivalence classes of
Kasteleyn orientations on Γ ⊂ Σ.
2.2. Discrete spin structures. As in the closed case, any dimer configuration
D on a graph Γ allows to identify equivalence classes of Kasteleyn orientations on
Γ ⊂ Σ with spin structures on Σ. Indeed, [3, Theorem 4.1] generalizes as follows.
Given an oriented simple closed curve C in Γ, let ℓD(C) denote the number of
vertices v in C whose adjacent dimer of D sticks out to the left of C in Σ. Also, let
V∂D(C) be the number of boundary vertices v in C not matched by D, and such
that the interior of Σ lies to the right of C at v.
6 DAVID CIMASONI AND NICOLAI RESHETIKHIN
Theorem 2.2. Fix a dimer configuration D on a surface graph with boundary
Γ ⊂ Σ. Given a class α ∈ H1(Σ;Z2), represent it by oriented simple closed curves
C1, . . . , Cm in Γ. If K is a Kasteleyn orientation on Γ ⊂ Σ, then the function
qKD : H1(Σ;Z2) → Z2 given by
qKD (α) =
Ci · Cj +
(1 + nK(Ci) + ℓD(Ci) + V∂D(Ci)) (mod 2)
is a well-defined quadratic form on H1(Σ;Z2).
Proof. Fix a dimer configuration D on (Γ, ∂Γ) and a Kasteleyn orientation K on
Γ ⊂ Σ. Let Σ′ be the surface (homeomorphic to Σ) obtained from Σ by adding a
small closed collar to its boundary. For every vertex v of ∂Γ that is not matched
by a dimer of D, add a vertex v′ near v in the interior of the collar and an edge
between v and v′. Let us denote by Γ′ the resulting graph in Σ′. Putting a
dimer on each of these additional edges, and orienting them arbitrarily, we obtain
a perfect matching D′ and an orientation K ′ on Γ′. Although Γ′ ⊂ Σ′ is not
strictly speaking a surface graph, all the methods of [3, Section 4] apply. Indeed,
Kuperberg’s vector field defined near Γ′ clearly extends continuously to the collar.
As in the closed case, it also extends to the faces with even index singularities. Using
the perfect matching D′ on Γ′, we obtain a vector field f(K ′, D′) with even index
singularities, which determines a spin structure ξf(K′,D′) on Σ
′. Johnson’s theorem
[5] holds for surfaces with boundary, so this spin structure defines a quadratic
form q on H1(Σ
′;Z2) = H1(Σ;Z2). If C is a simple close curve in Γ
′ ⊂ Σ′, then
q([C]) + 1 = nK
(C) + ℓD′(C) as in the closed case. The proof is completed using
the equalities nK
(C) = nK(C) and ℓD′(C) = ℓD(C) + V∂D(C). �
Since Johnson’s theorem holds true for surfaces with boundary and [3, Proposi-
tion 4.2] easily extends, we have the following corollary.
Corollary 2.3. Let Γ ⊂ Σ be a surface graph, non-necessarily connected, and pos-
sibly with boundary. Any dimer configuration D on Γ ⊂ Σ induces an isomorphism
of affine H1(Σ;Z2)-spaces
ψD : K(Γ ⊂ Σ) −→ S(Σ)
from the set of equivalence classes of Kasteleyn orientations on Γ ⊂ Σ onto the set
of spin structures on Σ. Furthermore, ψD − ψD′ is equal to the Poincaré dual of
∆(D,D′). In particular, ψD = ψD′ if and only if D and D
′ are equivalent dimer
configurations. �
2.3. The Pfaffian formula for the partition function. Let Γ be a graph, not
necessarily connected, and possibly with boundary, endowed with a weight system
w. Realize Γ as a surface graph Γ ⊂ Σ, and fix a Kasteleyn orientation K on it.
The Kasteleyn coefficient associated to an ordered pair (v, v′) of distinct vertices
of Γ is the number
aKvv′ =
εKvv′(e)w(e),
where the sum is on all edges e in Γ between the vertices v and v′, and
εKvv′(e) =
1 if e is oriented by K from v to v′;
−1 otherwise.
DIMERS ON SURFACE GRAPHS AND SPIN STRUCTURES. II 7
One also sets aKvv = 0. Let us fix a boundary condition ∂D0 and enumerate the
matched vertices of Γ by 1, 2, . . . , 2n. Then, the corresponding coefficients form a
2n× 2n skew-symmetric matrix AK(Γ;w | ∂D0) = A
K called the Kasteleyn matrix .
Let D be a dimer configuration on (Γ, ∂Γ) with ∂D = ∂D0, given by edges
e1, . . . , en matching vertices iℓ and jℓ for ℓ = 1, . . . , n. Let σ be the permutation
(1, . . . , 2n) 7→ (i1, j1, . . . , in, jn), and set
εK(D) = (−1)σ
εKiℓjℓ(eℓ),
where (−1)σ denotes the sign of σ. Note that εK(D) does not depend on the choice
of σ, but only on the dimer configuration D.
Finally, recall that the Arf invariant of a (possibly degenerate) quadratic form
q on H := H1(Σ;Z2) is defined by
Arf(q) =
(−1)q(α).
If there is a component γ of ∂Σ such that q(γ) 6= 0, then one easily checks that
Arf(q) = 0. On the other hand, if q(γ) = 0 for all boundary components γ of Σ,
then Arf(q) takes the values +1 or −1.
Theorem 2.4. Let Γ ⊂ Σ be a surface graph, not necessarily connected, and possi-
bly with boundary. Let b1(Σ) denote the dimension of H1(Σ;Z2), and let g denote
the genus of Σ. Then,
Zα(Γ;w | ∂D0) =
2b1(Σ)
(−1)q
εK(D0)Pf(A
for any α ∈ H1(Σ;Z2), and
Z(Γ;w | ∂D0) =
Arf(qKD0)ε
K(D0)Pf(A
where both sums are over the 2b1(Σ) equivalence classes of Kasteleyn orientations
on Γ ⊂ Σ. Furthermore, Arf(qKD0)ε
K(D0) does not depend on D0.
Proof. First note that if the theorem holds for two surface graphs, then it holds
for their disjoint union. Therefore, it may be assumed that Σ is connected. The
first formula follows from Theorem 2.2: the proof of Theorem 4 and the first half
of the proof of Theorem 5 of [3] generalize verbatim to the case with (possible)
boundary. The second formula can be obtained from the first one by summing
over all α ∈ H1(Σ;Z2). However, this requires some cumbersome computations,
so let us give another proof of this equality. As mentioned in Section 1, the dimer
model on (Γ, ∂Γ) with boundary condition ∂D0 is equivalent to the dimer model
on the graph Γ′ = Γ′(∂D0) obtained from Γ by removing all edges adjacent to non-
matched boundary vertices. Let w′ denote the restriction of w to Γ′. If Γ ⊂ Σ is a
surface graph with boundary, then Γ′ ⊂ Σ′ is a surface graph, where Σ′ is the closed
oriented surface obtained from Σ by gluing discs along all boundary components.
By [3, Theorem 5.3],
Z(Γ;w | ∂D0) = Z(Γ
′;w′) =
Arf(qK
(D0)Pf(A
K′(Γ′;w′)),
8 DAVID CIMASONI AND NICOLAI RESHETIKHIN
the sum being on all equivalence classes of Kasteleyn orientations on Γ′ ⊂ Σ′.
Such a Kasteleyn orientation K ′ extends uniquely to a Kasteleyn orientation K on
Γ ⊂ Σ such that qKD0(γ) = 0 for all boundary component γ of Σ. Furthermore,
(D0) = ε
K(D0) and A
K′(Γ′;w′) = AK(Γ;w | ∂D0). Since Arf(q
) = 0 for all
other Kasteleyn orientations, the theorem follows. �
3. Cutting and gluing
3.1. Cutting and gluing graphs with boundary. Let (Γ, ∂Γ) be a graph with
boundary, and let us fix an edge e of Γ. Let (Γ{e}, ∂Γ{e}) denote the graph with
boundary obtained from (Γ, ∂Γ) as follows: cut the edge e in two, and set ∂Γ{e} =
∂Γ ∪ {v′, v′′}, where v′ and v′′ are the new one valent vertices. Iterating this
procedure for some set of edges E leads to a graph with boundary (ΓE, ∂ΓE), which
is said to be obtained by cutting (Γ, ∂Γ) along E.
Note that a dimer configuration D ∈ D(Γ, ∂Γ) induces an obvious dimer config-
uration DE ∈ D(ΓE, ∂ΓE): cut in two the dimers of D that belong to E.
A weight system w on Γ induces a family of weight systems (wt
)t on ΓE indexed
by t : E → R>0, as follows: if e is an edge of Γ which does not belong to E, set
(e) = w(e); if e ∈ E is cut into two edges e′, e′′ of ΓE, set w
(e′) = t(e)w(e)1/2 and
(e′′) = t(e)−1w(e)1/2. Note that this family of weight systems is an orbit under
the action of the subgroup of G(ΓE) consisting of elements s such that s(v) = 1 for
all v ∈ V (Γ) and s(v′) = s(v′′) whenever v′, v′′ ∈ ∂ΓE come from the same edge of
Let us now formulate how the cutting affects the partition function. The proof
is straightforward.
Proposition 3.1. Fix a boundary condition ∂D0 on (Γ, ∂Γ) and a set E of edges
of Γ. Then, given any parameter t : E → R>0,
Z(Γ;w | ∂D0) =
Z(ΓE;w
| ∂DI0),
where the sum is taken over all subsets I of E and ∂DI0 is the boundary condition
on (ΓE, ∂ΓE) induced by ∂D0 and I: a vertex of ∂ΓE is matched in ∂D
0 if and only
if it is matched in ∂D0 or it comes from an edge in I. �
The operation opposite to cutting is called gluing: pick a pair of boundary
vertices of Γ, and glue the adjacent edges e′, e′′ along these vertices into a single
edge e. In order for the result to be a graph, it should be assumed that e′ and e′′
are different edges of Γ. We shall denote by (Γϕ, ∂Γϕ) the graph obtained by gluing
(Γ, ∂Γ) according to a pairing ϕ of several vertices of ∂Γ.
Note that a dimer configuration D ∈ D(Γ, ∂Γ) induces a dimer configuration
Dϕ ∈ D(Γϕ, ∂Γϕ) if and only if the boundary condition ∂D on ∂Γ is compatible
with ϕ, i.e: ϕ relates matched vertices with matched vertices. Obviously, a dimer
configuration DE is compatible with the pairing ϕ which glues back the edges of E,
and (DE)ϕ = D on ((ΓE)ϕ, (∂ΓE)ϕ) = (Γ, ∂Γ).
An edge weight system w on Γ induces an edge weight system wϕ on Γϕ as
follows:
wϕ(e) =
w(e) if e is an edge of Γ;
w(e′)w(e′′) if e is obtained by gluing the edges e′ and e′′ of Γ.
DIMERS ON SURFACE GRAPHS AND SPIN STRUCTURES. II 9
If E is a set of edges of Γ and ϕ is the pairing which glues back these edges, then
)ϕ = w for any t : E → R>0.
The effect of gluing on the partition function is best understood in the language
of quantum field theory. We therefore postpone its study to Section 4.
3.2. Cutting and gluing surface graphs with boundary. Let Γ ⊂ Σ be a
surface graph with boundary. Let C be a simple curve in Σ which is “in general
position” with respect to Γ, in the following sense:
(i) it is disjoint from the set of vertices of Γ;
(ii) it intersects the edges of Γ transversally;
(iii) its intersection with any given face of Σ is connected.
Let ΣC be the surface with boundary obtained by cutting Σ open along C. Also,
let ΓC := ΓE(C) be the graph with boundary obtained by cutting (Γ, ∂Γ) along the
set E(C) of edges of Γ which intersect C, as illustrated in Figure 1.
Γ ⊂ Σ
ΓC ⊂ ΣC
Figure 1. Cutting a surface graph Γ ⊂ Σ along a curve C.
Obviously, ΓC ⊂ ΣC is a surface graph with boundary. We will say that it is
obtained by cutting Γ ⊂ Σ along C. Abusing notation, we shall write wtC for the
weight system wt
E(C) on ΓC .
A class β ∈ H1(Σ, ∂Σ;Z2) induces βC ∈ H1(ΣC , ∂ΣC ;Z2) via
H1(Σ, ∂Σ;Z2) → H1(Σ, ∂Σ ∪N(C);Z2) ≃ H1(ΣC , ∂ΣC ;Z2).
Here N(C) denotes a neighborhood of C in Σ, the first homomorphism is induced
by inclusion, and the second one is the excision isomorphism. Note that given
any two dimers configurations D and D′ on Γ ⊂ Σ, ∆(DC , D
C) = ∆(D,D
′)C in
H1(ΣC , ∂ΣC ;Z2).
This easily leads to the following refinement of Proposition 3.1.
Proposition 3.2. Fix β ∈ H1(Σ, ∂Σ;Z2), D
′ ∈ D(Γ, ∂Γ), and a boundary condi-
tion ∂D0 on (Γ, ∂Γ). Then, given any parameter t : E(C) → R>0,
Zβ,D′(Γ;w | ∂D0) =
I⊂E(C)
ZβC ,D′C (ΓC ;w
C | ∂D
where the sum is taken over all subsets I of E(C) and ∂DI0 is the boundary condition
on (ΓC , ∂ΓC) induced by ∂D0 and I. �
Let us now define the operation opposite to cutting a surface graph with bound-
ary. Pick two closed connected subsets M1,M2 of ∂Σ, which are not points, and
satisfy the following properties:
(i) M1 ∩M2 ⊂ ∂M1 ∪ ∂M2 and ∂M1 ∪ ∂M2 is disjoint from ∂Γ;
10 DAVID CIMASONI AND NICOLAI RESHETIKHIN
(ii) the intersection of each given face of Σ with M1 ∪M2 is connected;
(iii) there exists an orientation-reversing homeomorphism ϕ : M1 → M2 which
induces a bijection M1 ∩ ∂Γ → M2 ∩ ∂Γ such that for all v in M1 ∩ ∂Γ, v
and ϕ(v) are not adjacent to the same edge of Γ.
Let Γϕ ⊂ Σϕ be obtained from the surface graph Γ ⊂ Σ by identifying M1 and M2
via ϕ and removing the corresponding vertices of Γ. This is illustrated in Figure 2.
By the conditions above, the pair Γϕ ⊂ Σϕ remains a surface graph. It is said to
be obtained by gluing Γ ⊂ Σ along ϕ.
Γ ⊂ Σ Γϕ ⊂ Σϕ
−→ M2
Figure 2. Gluing a surface graph Γ ⊂ Σ along ϕ : M1 → M2.
Note that any surface graph ΓC ⊂ ΣC obtained by cutting Γ ⊂ Σ along some
curve C in general position with respect to Γ satisfies the conditions listed above.
Furthermore, (ΓC)ϕ ⊂ (ΣC)ϕ = Γ ⊂ Σ, where ϕ is the obvious homeomorphism
identifying the two closed subsets of ∂ΣC coming from C. Conversely, if C denotes
the curve in Σϕ given by the identification ofM1 andM2 via ϕ, then it is in general
position with respect to Γϕ, and (Γϕ)C ⊂ (Σϕ)C .
3.3. Cutting and gluing discrete spin structures. Let Γ ⊂ Σ be a surface
graph with boundary, and let C be a simple curve in Σ in general position with
respect to Γ. As noted above, any dimer configurationD on (Γ, ∂Γ) induces a dimer
configuration DC on (ΓC , ∂ΓC). If two dimer configurations D,D
′ ∈ D(Γ, ∂Γ) are
equivalent, then DC , D
C ∈ D(ΓC , ∂ΓC) are equivalent as well:
∆(DC , D
C) = ∆(D,D
′)C = 0 ∈ H1(ΣC , ∂ΣC ;Z2).
A Kasteleyn orientation K on Γ ⊂ Σ induces a Kasteleyn orientation KC on
ΓC ⊂ ΣC as follows. Let KC be equal to K on all edges of ΓC coming from edges
of Γ. For all the new edges of ΓC , there is a unique orientation which satisfies the
Kasteleyn condition, since each face of Σ is crossed at most once by C. One easily
checks that if K and K ′ are equivalent Kasteleyn orientations, then KC and K
are also equivalent. Hence, there is a well-defined operation of cutting discrete spin
structures on a surface with boundary.
This is not a surprise. Indeed, the inclusion ΣC ⊂ ΣC ∪ N(C) = Σ induces
a homomorphism i∗ : H1(ΣC ;Z2) → H1(Σ;Z2). The assignment q 7→ qC = q ◦ i∗
defines a map from the quadratic forms on H1(Σ;Z2) to the quadratic forms on
H1(ΣC ;Z2), which is affine over the restriction homomorphism i
∗ : H1(Σ;Z2) →
H1(ΣC ;Z2). By Johnson’s theorem, it induces an affine map between the sets of
DIMERS ON SURFACE GRAPHS AND SPIN STRUCTURES. II 11
spin structures S(Σ) → S(ΣC). By Corollary 2.3, there is a unique map K(Γ ⊂
Σ) → K(ΓC ⊂ ΣC) which makes the following diagram commute:
(2) K(Γ ⊂ Σ)
∼= ψD
// K(ΓC ⊂ ΣC)
∼= ψDC
S(Σ) // S(ΣC).
This map is nothing but [K] 7→ [KC ].
Now, let K be a Kasteleyn orientation on a surface graph Γ ⊂ Σ, and let
ϕ : M1 →M2 be an orientation-reversing homeomorphism between two closed con-
nected subsets in ∂Σ, as described above. We shall say that a Kasteleyn orientation
K on Γ ⊂ Σ is compatible with ϕ if the following conditions hold:
(i) whenever two edges e′, e′′ of Γ are glued into a single edge e of Γϕ, the
orientation K agrees on e′ and e′′, giving an orientation Kϕ on e;
(ii) the induced orientation Kϕ is a Kasteleyn orientation on Γϕ ⊂ Σϕ.
The Kasteleyn orientationKϕ on Γϕ ⊂ Σϕ is said to be obtained by gluing K along
Given any Kasteleyn orientation K on Γ ⊂ Σ, the induced orientation KC on
ΓC ⊂ ΣC is compatible with the map ϕ such that (ΣC)ϕ = Σ; furthermore, (KC)ϕ
is equal to K. Conversely, if K is a Kasteleyn orientation on Γ ⊂ Σ which is
compatible with ϕ, and C denotes the curve in Σϕ given by the identification of
M1 and M2 via ϕ, then (Kϕ)C is equal to K. With these notations, any dimer
configuration D on Γ which is compatible with ϕ satisfies (Dϕ)C = D. Therefore,
diagram (2) gives
K(Γϕ ⊂ Σϕ)
∼= ψDϕ
// K(Γ ⊂ Σ)
∼= ψD
S(Σϕ) // S(Σ),
where both horizontal maps are affine over i∗ : H1(Σϕ;Z2) → H
1(Σ;Z2). Under-
standing the gluing of Kasteleyn orientations (up to equivalence) now amounts to
understanding the restriction homomorphism i∗. Using the exact sequence of the
pair (Σϕ,Σ), one easily checks the following results:
– The restriction homomorphism i∗ is injective, unlessM1 andM2 are disjoint
and belong to the same connected component of Σ. In this case, the kernel
of i∗ has dimension 1.
– The homomorphism i∗ is onto unless M1 ∪ M2 is a 1-cycle and the cor-
responding connected component of Σϕ is not closed. In this case, the
cokernel of i∗ has dimension 1.
This leads to the four following cases. Fix a Kasteleyn orientation K on Γ ⊂ Σ.
(1) If i∗ is an isomorphism, then there exist a Kasteleyn orientation K ′ equiv-
alent to K which is compatible with ϕ. Furthermore, the assignment
[K] 7→ [K ′ϕ] gives a well-defined map between K(Γ ⊂ Σ) and K(Γϕ ⊂ Σϕ).
(2) If i∗ is onto but not injective, then there exist K ′,K ′′ ∼ K which are
compatible with ϕ, inducing two distinct well-defined maps [K] 7→ [K ′ϕ]
and [K] 7→ [K ′′ϕ] between K(Γ ⊂ Σ) and K(Γϕ ⊂ Σϕ).
12 DAVID CIMASONI AND NICOLAI RESHETIKHIN
(3) If i∗ is injective but not onto, then M1 ∪M2 is a 1-cycle, oriented as part
of the boundary of Σ. There exist K ′ ∼ K which is compatible with ϕ if
and only if the following condition holds:
nK(M1) + n
K(M2) ≡ 0 (mod 2) if M1 and M2 are disjoint;
nK(M1 ∪M2) ≡ 1 (mod 2) otherwise.
(Note that this condition only depends on the equivalence class of K.) In
this case, it induces a well-defined class [K ′ϕ] in K(Γϕ ⊂ Σϕ).
(4) Finally, assume i∗ is neither onto nor injective. If K satisfies the condition
above, then there exist K ′,K ′′ ∼ K which are compatible with ϕ, inducing
two well-defined maps [K] 7→ [K ′ϕ] and [K] 7→ [K
ϕ]. On the other hand,
if K does not satisfy the condition above, then it does not contain any
representative which is compatible with ϕ.
3.4. Cutting Pfaffians. Let us conclude this section with one last observation.
Let Γ ⊂ Σ be a surface graph with boundary, and let C be a simple curve in Σ.
The equality
Z(Γ;w | ∂D0) =
I⊂E(C)
Z(ΓC ;w
C | ∂D
of Proposition 3.1 can be understood as the Taylor series expansion of the function
Z(Γ;w | ∂D0) in the variables (w(e))e∈E(C). Clearly, if E(C) = {ei1 , . . . , eik}, then
w(eiℓ)
∂kZ(Γ;w | ∂D0)
∂w(ei1) · · · ∂w(eik)
(0) = Z(ΓC ;w
C | ∂D
By Theorem 2.4, the partition function Z(Γ;w | ∂D0) can be expressed as a lin-
ear combination of Pfaffians of matrices AK(Γ;w | ∂D0) depending on Kasteleyn
orientations K of Γ ⊂ Σ such that qKD0(γ) = 0 for all boundary component γ of
Σ. Recall that any such orientation K extends to a Kasteleyn orientation KC on
ΓC ⊂ ΣC . Furthermore, all equivalence classes of Kasteleyn orientations such that
(D0)C
(γ) = 0 for all boundary component γ of ΣC are obtained in this way. (This
follows from the fact that the map [K] 7→ [KC ] is affine over the restriction homo-
morphism.) Finally, the partition function Z(ΓC ;w
C | ∂D0) can also be expressed
as a linear combination of Pfaffians of matrices AKC (ΓC ;w
C | ∂D0) via Theorem
Gathering all these equations, we obtain a relation between the Pfaffian of the
matrix AK(Γ;w | ∂D0) and the Pfaffian of the matrix A
KC (ΓC ;w
C | ∂D0). This
relation turns out to be exactly the equation below, a well-known property of Pfaf-
fians.
Proposition 3.3. Let A = (aij) be a skew-symmetric matrix of size 2n. Given
an ordered subset I of the ordered set α = (1, . . . , 2n), let AI denote the matrix
obtained from A by removing the ith row and the ith column for all i ∈ I. Then,
for any ordered set of indices I = (i1, j1, . . . , ik, jk),
∂kPf(A)
∂ai1j1 · · · ∂aikjk
= (−1)σ(I)Pf(AI),
where (−1)σ(I) denote the signature of the permutation which sends α to the ordered
set I(α\I). �
DIMERS ON SURFACE GRAPHS AND SPIN STRUCTURES. II 13
4. Quantum field theory for dimers
4.1. Quantum field theory on graphs. Let (Γ, ∂Γ) be a graph with boundary,
and let us assume that each vertex v in ∂Γ is oriented, that is, endowed with some
sign εv. In the spirit of the Atiyah-Segal axioms for a (0 + 1)-topological quantum
field theory [2, 13], let us define a quantum field theory on graphs as the following
assignment:
(1) Fix a finite dimensional complex vector space V .
(2) To the oriented boundary ∂Γ, assign the vector space
Z(∂Γ) =
εv=+1
εv=−1
where V ∗ denotes the vector space dual to V .
(3) To a finite graph Γ with oriented boundary ∂Γ and weight system w, assign
some vector Z(Γ;w) ∈ Z(∂Γ), with Z(∅;w) = 1 ∈ C = Z(∅).
Note that any orientation preserving bijection f : ∂Γ → ∂Γ′ induces an isomor-
phism Z(f) : Z(∂Γ) → Z(∂Γ′) given by permutation of the factors. This assign-
ment is functorial: if g : ∂Γ′ → ∂Γ′′ is another orientation preserving bijection,
then Z(g ◦ f) = Z(g) ◦Z(f). Finally, if f : ∂Γ → ∂Γ′ extends to a homeomorphism
F : Γ → Γ′, then Z(f) maps Z(Γ) to Z(Γ′). Note also that Z(−∂Γ) = Z(∂Γ)∗, and
that Z(∂Γ ⊔ ∂Γ′) = Z(∂Γ)⊗ Z(∂Γ′).
The main point is that we require the following gluing axiom. Let Γ be a graph
with oriented boundary ∂Γ, such that there exists two disjoint subsets X1, X2 of ∂Γ
and an orientation reversing bijection ϕ : X1 → X2 (i.e. εϕ(v) = −εv for all v ∈ X1).
Obviously, ϕ induces a linear isomorphism Z(ϕ) : Z(X1) → Z(X2)
∗. Let Γϕ denote
the graph with boundary ∂Γϕ = ∂Γ \ (X1 ∪X2) obtained by gluing Γ according to
ϕ, and let wϕ be the corresponding weight system on Γϕ (recall Section 3.1). Let
Bϕ denote the composition
Z(∂Γ) = Z(∂Γϕ)⊗ Z(X1)⊗ Z(X2) → Z(∂Γϕ)⊗ Z(X2)
∗ ⊗ Z(X2) → Z(∂Γϕ),
where the first homomorphism is given by id⊗Z(ϕ)⊗ id, and the second is induced
by the natural pairing Z(X2)
∗ ⊗ Z(X2) → C. We require that
Bϕ(Z(Γ;w)) = Z(Γϕ;wϕ).
Remark. In the same spirit, one can define a quantum field theory on surface graphs .
Here, the vector Z(Γ ⊂ Σ;w) ∈ Z(∂Γ) might depend on the realization of Γ as a
surface graph Γ ⊂ Σ, and the gluing axiom concerns gluing of surface graphs, as
defined in Section 3.2.
4.2. Quantum field theory for dimers on graphs. Let us now explain how
the dimer model on weighted graphs with boundary defines a quantum field theory.
As vector space V , choose the 2-dimensional complex vector space with fixed basis
a0, a1. Let α0, α1 denote the dual basis in V
∗. To a finite graph Γ with oriented
boundary ∂Γ and weight system w, assign
Z(Γ;w) =
Z(Γ;w | ∂D) a(∂D) ∈ Z(∂Γ),
where the sum is on all possible boundary conditions ∂D on ∂Γ, and
a(∂D) =
εv=+1
aiv(∂D) ⊗
εv=−1
αiv(∂D) ∈ Z(∂Γ).
14 DAVID CIMASONI AND NICOLAI RESHETIKHIN
Here, iv(∂D) = 1 if the vertex v is matched by ∂D, and iv(∂D) = 0 otherwise.
Let us check the gluing axiom. First note that Bϕ(a(∂D)) = 0 unless ∂D is
compatible with ϕ (i.e: unless ϕ(v) is matched in ∂D if and only if v is matched in
∂D). In such a case, B(a(∂D)) = a(∂D|∂Γϕ), where ∂D|∂Γϕ denotes the restriction
of the boundary condition ∂D to ∂Γϕ ⊂ ∂Γ. All the possible boundary conditions
∂Dϕ on ∂Γϕ are given by such restrictions. Therefore,
Bϕ(Z(Γ;w)) =
∂D⊃∂Dϕ
Z(Γ;w | ∂D)
a(∂Dϕ),
the interior sum being on all boundary conditions ∂D on ∂Γ that are compatible
with ϕ, and such that ∂D|∂Γϕ = ∂Dϕ. By definition,
∂D⊃∂Dϕ
Z(Γ;w | ∂D) =
D:∂D⊃∂Dϕ
w(D) = Z(Γϕ;wϕ | ∂Dϕ).
Therefore, the gluing axiom is satisfied.
4.3. The dimer model as the theory of free Fermions. Let W be an n-
dimensional vector space. The choice of an ordered basis in W induces an isomor-
phism between its exterior algebra
W = ⊕nk=0
W and the algebra generated
by elements φ1, . . . , φn with defining relations φiφj = −φjφi. This space is known
as the Grassman algebra generated by φ1, . . . , φn. The choice of an ordered basis
in W also defines a basis in the top exterior power of W . The integral over the
Grassman algebra of W of an element a ∈
W is the coordinate of a in the top
exterior power of W with respect to this basis. It is denoted by
a dφ.
There is a scalar product on the Grassman algebra generated by φ1, . . . , φn; it
is given by the Grassman integral
(3) < F,G >=
F (φ)G(ψ)dφdψ.
Note that the monomial basis is orthonormal with respect to this scalar product.
One easily shows (see e.g. the Appendix to [3]) that the Pfaffian of a skew symmetric
matrix A = (aij) can be written as
Pf(A) =
i,j=1
φiaijφj
Let us now use this to reformulate the quantum field theory of dimers in terms of
Grassman integrals. Let Γ ⊂ Σ be a (possibly disconnected) surface graph, possibly
with boundary. Let us fix a numbering of the vertices of Γ, a boundary condition
∂D0 on ∂Γ and a Kasteleyn orientation K on Γ ⊂ Σ. Let a
ij be the Kasteleyn
coefficient associated to K and the vertices i, j of Γ (recall Section 2). By Theorem
2.4 and the identity above,
Z(Γ;w | ∂D0) =
Arf(qKD0)ε
K(D0)
i,j∈V (D0)
dφ∂D0 ,
where the sum is over all 2b1(Σ) equivalence classes of Kasteleyn orientations on
Γ ⊂ Σ, V (D0) denotes the set of vertices of Γ that are matched by D0, and dφ∂D0 =
DIMERS ON SURFACE GRAPHS AND SPIN STRUCTURES. II 15
∧i∈V (D0)dφi. This leads to the formula
Z(Γ;w | ∂D0) =
i,j∈V (D0)
DK∂D0φ,
where DK∂D0φ = Arf(q
)εK(D0) d∂D0φ. Let us point out that this measure does
not depend on the choice of D0, but only on the induced boundary condition ∂D0.
Now, the numbering of the vertices of Γ gives a numbering of the vertices of ∂Γ.
This induces a linear isomorphism between Z(∂Γ) and the Grassman algebra
generated by (φi)i∈∂Γ. The image of the partition function under this isomorphism
is the following element of the Grassman algebra of boundary vertices:
Z(Γ;w) =
Z(Γ;w | ∂D0)
i∈V (∂D0)
(∂Γ),
where V (∂D0) = V (D0) ∩ ∂Γ. This leads to
Z(Γ;w) =
i,j∈V (D0)
DK∂D0φ
i∈V (∂D0)
i,j∈V (Γ)
where DKφ = Arf(qKD0)ε
K(D0) ∧i/∈∂Γ dφi. This measure depends only on K, but
not on D0.
We can now formulate the dimer model as the theory of free (Gaussian) Fermions:
(1) To the boundary of Γ ⊂ Σ, we assign
(∂Γ), the Grassman algebra gener-
ated by the ordered set ∂Γ;
(2) To a surface graph Γ ⊂ Σ with ordered set of vertices V (Γ) and weight
system w, we assign the element Z(Γ ⊂ Σ;w) of
(∂Γ) given by
Z(Γ ⊂ Σ;w) =
i,j∈V (Γ)
where the sum is over all 2b1(Σ) equivalence classes of Kasteleyn orientations
on Γ ⊂ Σ, and DKφ = Arf(qKD0)ε
K(D0) ∧i/∈∂Γ dφi.
The gluing axiom now takes the following form. Let Γϕ ⊂ Σϕ denote the surface
graph with boundary obtained by gluing Γ ⊂ Σ along some orientation-reversing
homeomorphism ϕ : M1 → M2 (see Section 3.2). Recall that ϕ induces a bijection
between the two disjoint sets X1 = ∂Γ ∩M1 and X2 = ∂Γ ∩M2. Therefore, it
induces an isomorphism Z(ϕ) :
(X1) →
(X2). Consider the map Bϕ given by
the composition
(∂Γ) =
(∂Γϕ)⊗
(X1)⊗
(X2) →
(∂Γϕ)⊗
(X2) →
(∂Γϕ).
Here, the first homomorphism is given by id⊗ (h ◦Z(ϕ))⊗ id, where h :
(X2) →
∗ is the isomorphism induced by the scalar product (3). Then, we require
Bϕ(Z(Γ ⊂ Σ;w)) = Z(Γϕ ⊂ Σϕ;wϕ).
We already know that this equality holds. Indeed, Z(Γ ⊂ Σ;w) just depends on
(Γ, w), and the formula above is nothing but the gluing axiom for Z(Γ;w) translated
in the formalism of Grassman algebras. However, it can also be proved from scratch
using the results of Section 3.3 together with well-known properties of Pfaffians.
16 DAVID CIMASONI AND NICOLAI RESHETIKHIN
5. Dimers on bipartite graphs and height functions
5.1. Composition cycles on bipartite graphs. Recall that a bipartite structure
on a graph Γ is a partition of its set of vertices into two groups, say blacks and
whites, such that no edge of Γ joins two vertices of the same group. Equivalently,
a bipartite structure can be regarded as a 0-chain
v black
v white
v ∈ C0(Γ;Z).
A bipartite structure induces an orientation on the edges of Γ, called the bipartite
orientation: simply orient all the edges from the white vertices to the black ones.
Using this orientation, a dimer configuration D ∈ D(Γ, ∂Γ) can now be regarded
as a 1-chain with Z-coefficients
e ∈ C1(Γ;Z)
such that ∂D = β in C0(Γ, ∂Γ;Z) = C0(Γ;Z)/C0(∂Γ;Z). Therefore, given two
dimer configurations D,D′ on Γ, their difference D −D′ is a 1-cycle (rel ∂Γ) with
Z-coefficients, denoted by C(D,D′). Its connected components are called (D,D′)-
composition cycles . In short, a bipartite structure on a graph allows to orient the
composition cycles.
5.2. Height functions for planar bipartite graphs. Let us now assume that
the bipartite graph Γ is planar without boundary, i.e. that it can be realized
as a surface graph Γ ⊂ S2. Let X denote the induced cellular decomposition
of the 2-sphere, which we endow with the counter-clockwise orientation. Since
H1(X ;Z) = H1(S
2;Z) = 0, the 1-cycle C(D,D′) is a 1-boundary, so there exists
σD,D′ ∈ C2(X ;Z) such that ∂σD,D′ = C(D,D
′). Let hD,D′ ∈ C
2(X ;Z) be given
by the equality
σD,D′ =
f∈F (X)
hD,D′(f) f ∈ C2(X ;Z),
where the sum is over all faces of X . The cellular 2-cochain hD,D′ is called a height
function associated to D,D′. Since H2(X ;Z) = H2(S
2;Z) = Z, the 2-chain σD,D′
is uniquely defined by D,D′ up to a constant, and the same holds for hD,D′ . Hence,
one can normalize all height functions by setting hD,D′(f0) = 0 for some fixed face
f0. This is illustrated in Figure 3.
Alternatively, hD,D′ can be defined as the only h ∈ C
2(X ;Z) such that h(f0) = 0
and h increases by 1 when a (D,D′)-composition cycle is crossed in the positive
direction (left to right as we cross). It follows that for any height function h and
any two 2-cells f1 and f2,
|h(f1)− h(f2)| ≤ d(f1, f2),
where d(f1, f2) is the distance between f1 and f2 in the dual graph, i.e. the minimal
number of edges crossed by a path connecting an point inside f1 with a point inside
f2. This can be regarded as a Lipschitz property of height functions. Note also
that for any three dimer configurations D, D′ and D′′ on Γ, the following cocycle
equality holds:
hD,D′ + hD′,D′′ = hD,D′′ .
DIMERS ON SURFACE GRAPHS AND SPIN STRUCTURES. II 17
Figure 3. An example of a bipartite planar graph with two dimer
configurations D (solid) and D′ (traced lines). The corresponding
height function hD,D′ (where f0 is the outer face) and (D,D
composition cycles are pictured on the right hand side.
The Lipschitz condition stated above leads to the following definition. Given a
fixed 2-cell f0 of the cellular decomposition X induced by Γ ⊂ S
2, set
H(X, f0) = {h ∈ C
2(X ;Z) |h(f0) = 0 and |h(f1)− h(f2)| ≤ d(f1, f2) ∀f1, f2}.
Given h ∈ H(X, f0), let C(h) denote the oriented closed curves formed by the
set of oriented edges e of Γ such that h increases its value by 1 when crossing e in
the positive direction. (In other words, C(h) = ∂σ, where σ ∈ C2(X ;Z) is dual to
h ∈ C2(X ;Z).) Obviously, there is a well-defined map
D(Γ)×D(Γ) → H(X, f0), (D,D
′) 7→ hD,D′
with C(hD,D′) = C(D,D
′). However, this map is neither injective nor surjective
in general. Indeed, the number of preimages of a given h is equal to the number of
dimer configurations on the graph obtained from Γ by removing the star of C(h).
Depending on Γ ⊂ S2, this number can be zero, or arbitrarily large.
To obtain a bijection, we proceed as follows. Fix a dimer configuration D0 on Γ.
Let C(D0) denote the set of all C ⊂ Γ consisting of disjoint oriented simple 1-cycles,
such that the following condition holds: for all e ∈ D0, either e is contained in C
or e is disjoint from C. Finally, set
HD0 (X, f0) = {h ∈ H(X, f0) |C(h) ∈ C(D0)}.
Proposition 5.1. Given any h ∈ HD0(X, f0), there is unique dimer configuration
D ∈ D(Γ) such that hD,D0 = h. Furthermore, given any two dimer configurations
D0, D1 on Γ, we have a canonical bijection
HD0(X, f0) → HD1(X, f0)
given by h 7→ h+ hD0,D1 .
Proof. One easily checks that the assignment D 7→ C(D,D0) defines a bijection
D(Γ) → C(D0). Furthermore, there is an obvious bijection HD0(X, f0) → C(D0)
given by h 7→ C(h). This induces a bijection D(Γ) → HD0(X, f0) and proves the
first part of the proposition. The second part follows from the first one via the
cocycle identity hD,D0 + hD0,D1 = hD,D1 . �
18 DAVID CIMASONI AND NICOLAI RESHETIKHIN
Let us now consider an edge weight system w on the bipartite planar graph Γ.
Recall that the Gibbs measure of D ∈ D(Γ) is given by
Prob(D) =
Z(Γ;w)
where w(D) =
e∈D w(D) and Z(Γ;w) =
D∈D(Γ)w(D). Let us now fix a dimer
configurationD0 and a face f0 ofX , and use the bijectionD(Γ) → HD0(X, f0) given
by D 7→ hD,D0 to translate this measure into a probability measure on HD0(X, f0).
To do so, we shall need the following notations: given an oriented edge e of Γ,
wβ(e) =
w(e) if the orientation on e agrees with the bipartite orientation;
w(e)−1 otherwise.
This defines a group homomorphism wβ : C1(X ;Z) → R>0. Finally, given any
f ∈ F (X), set
qf = wβ(∂f),
where ∂f is oriented as the boundary of the counter-clockwise oriented face f . This
number qf is called the volume weight of the face f .
Proposition 5.2. The Gibbs measure on D(Γ) given by the edge weight system w
translates into the following probability measure on HD0(X, f0):
ProbD0(h) =
ZD0,f0(X, q)
where
q(h) =
f∈F (X)
f and ZD0,f0(X ; q) =
h∈HD0(X,f0)
q(h).
Furthermore, this measure is independant of the choice of f0. Finally, the bijection
HD0(X, f0) → HD1(X, f0) given by h 7→ h+hD0,D1 is invariant with respect to the
measures ProbD0 and ProbD1 .
Proof. For any D ∈ D(Γ), we have
w(D)w(D0)
w(e)−1 = wβ(C(D,D0))
= wβ(∂σD,D0) = wβ
f∈F (X)
hD,D0(f)∂f
f∈F (X)
wβ(∂f)
hD,D0(f) =
f∈F (X)
hD,D0 (f)
f = q(hD,D0).
The proposition follows easily from this equality. �
Let V (Γ) (resp. E(Γ)) denote the set of vertices (resp. of edges) of Γ. Recall
that the group
G(Γ) = {s : V (Γ) → R>0}
acts on the set of weight systems on Γ by (sw)(e) = s(e+)w(e)s(e−), where e+ and
e− are the two vertices adjacent to the edge e. As observed in Section 1.1, the
Gibbs measure on D(Γ) is invariant under the action of the group G(Γ).
Note also that this action is free unless Γ is bipartite. In this later case, the
1-parameter family of elements sλ ∈ G(Γ) given by sλ(v) = λ if v is black and
DIMERS ON SURFACE GRAPHS AND SPIN STRUCTURES. II 19
sλ(v) = λ
−1 if v is white act as the identity on the set of weight systems. Hence, if
Γ is bipartite, the number of “essential” parameters is equal to |E(Γ)|− |V (Γ)|+1.
If this bipartite graph is planar, then
|E(Γ)| − |V (Γ)|+ 1 = |F (X)| − χ(S2) + 1 = |F (X)| − 1.
The |F (X)| volume weights qf are invariant with respect to the action of G(Γ). They
can be normalized in such a way that
f∈F (X) qf = 1, giving exactly |F (X)| − 1
parameters. Thus, in the height function formulation of the Gibbs measure, only
essential parameters appear.
5.3. Height functions for bipartite surface graphs. Let us now address the
general case of a bipartite surface graph Γ ⊂ Σ, possibly disconnected, and possibly
with boundary ∂Γ ⊂ ∂Σ. Fix a family γ = {γi}
i=1 of oriented simple curves in Γ
representing a basis in H1(Σ, ∂Σ;Z). Note that such a family of curves exists since
Γ is the 1-squeletton of a cellular decomposition X of Σ.
Given any D,D′ ∈ D(Γ, ∂Γ), the homology class of C(D,D′) = D −D′ can be
written in a unique way
[C(D,D′)] =
D,D′(i)[γi] ∈ H1(Σ, ∂Σ;Z),
with a
D,D′(i) ∈ Z. Hence, C(D,D
i=1 a
D,D′(i)γi is a 1-boundary (rel ∂X),
that is, there exists σ
D,D′ ∈ C2(X, ∂X ;Z) = C2(X ;Z) such that
(4) C(D,D′)− ∂σ
D,D′ −
D,D′(i)γi ∈ C1(∂X ;Z).
The 2-cochain h
D,D′ ∈ C
2(X ;Z) dual to σ
D,D′ is called a height function associated
to D,D′ with respect to γ. Since Z2(X, ∂X ;Z) = H2(X, ∂X ;Z) = H2(Σ, ∂Σ;Z) ∼=
H0(Σ;Z), the 2-chain σ
D,D′ is uniquely determined byD,D
′ and γ up to an element
of H0(Σ;Z), and the same holds for h
D,D′ . In other words, the set of height
functions associated toD,D′ with respect to γ is an affineH0(Σ;Z)-space: it admits
a freely transitive action of the abelian group H0(Σ;Z). One can normalize the
height functions by choosing some family F0 of faces of X , one for each connected
component of X , and by setting h
D,D′(f0) = 0 for all f0 ∈ F0.
Given h ∈ C2(X ;Z), set C(h) = ∂σ ∈ C1(X ;Z), where σ ∈ C2(X ;Z) is dual to
h ∈ C2(X ;Z). Given a fixed D0 ∈ D(Γ, ∂Γ), let C(D0) denote the set of all C ⊂ Γ
consisting of disjoint oriented 1-cycles (rel ∂Γ) such that the following condition
holds: for all e ∈ D0, either e is contained in C or e is disjoint from C.
Finally, let H
(X,F0) denote the set of pairs (h, a) ∈ C
2(X ;Z) × Zb1 which
satisfy the following properties:
– h(f0) = 0 for all f0 in F0;
– there exists C ∈ C(D0) such that C − C(h)−
i=1 a(i)γi ∈ C1(∂X ;Z).
We obtain the following generalization of Proposition 5.1. The proof is left to the
reader.
Proposition 5.3. Given any (h, a) ∈ H
(X,F0), there is a unique dimer config-
uration D ∈ D(Γ, ∂Γ) such that h
= h and a
= a. Furthermore, given any
20 DAVID CIMASONI AND NICOLAI RESHETIKHIN
two dimer configuration D0, D1 ∈ D(Γ, ∂Γ), there is a canonical bijection
(X,F0) → H
(X,F0)
given by (h, a) 7→ (h+ h
D0,D1
, a+ a
D0,D1
Recall that the boundary conditions on dimer configurations induce a partition
D(Γ, ∂Γ) =
D(Γ, ∂Γ | ∂D′0),
where D(Γ, ∂Γ | ∂D′0) = {D ∈ D(Γ, ∂Γ) | ∂D = ∂D
0}. This partition translates
into a partition of H
(X,F0) via the bijection D(Γ, ∂Γ) → H
(X,F0) given by
D 7→ (h
). Indeed, let F∂(X) denote the set of boundary faces of X , that
is, the set of faces of X that are adjacent to ∂Σ. The choice of a boundary condition
∂D′0 (together with F0) determines h
(f) for all D such that ∂D = ∂D′0 and
all f ∈ F∂(X). The actual possible values of h
on the boundary faces depend
on γ, D0 and F0; they can be determined explicitely. We shall denote by ∂h such
a value of a height function on boundary faces, and call it a boundary condition for
height functions. In short, we obtain a partition
(X,F0) =
(X,F0 | ∂h
indexed by all possible boundary conditions on height functions h
. Each bound-
ary condition on dimer configurations corresponds to one boundary condition on
height functions via D 7→ h
Let us now consider an edge weight system w on the bipartite graph Γ, and a
fixed boundary condition ∂D′0. Recall that the Gibbs measure for the dimer model
on (Γ, ∂Γ) with weight system w and boundary condition ∂D′0 is given by
Prob(D | ∂D′0) =
Z(Γ;w | ∂D′0)
where
Z(Γ;w | ∂D′0) =
D∈D(Γ,∂Γ | ∂D′0)
w(D).
Let us realize Γ as a surface graph Γ ⊂ Σ, fix a dimer configuration D0 ∈
D(Γ, ∂Γ), a family γ = {γi} of oriented simple curves in Γ representing a basis
in H1(Σ, ∂Σ;Z), and a collection F0 of faces of the induced cellular decomposition
X of Σ, one face for each connected component of X . We can use the bijection
D(Γ, ∂Γ | ∂D′0) → H
(X,F0 | ∂h
0) given by D 7→ (h
) to translate the
Gibbs measure into a probability measure on H
(X,F0 | ∂h
To do so, let us first extend the weight system w to all edges of X by setting
w(e) = 1 for all boundary edges of X . As in the planar case, define wβ : C1(X ;Z) →
R>0 as the group homomorphism such that, for any oriented edge e of X ,
wβ(e) =
w(e) if the orientation on e agrees with the bipartite orientation;
w(e)−1 otherwise.
DIMERS ON SURFACE GRAPHS AND SPIN STRUCTURES. II 21
Note that this makes sense even for boundary edges where there is no bipartite
orientation, as w(e) = 1 for such edges. Consider the parameters
qf = wβ(∂f) for all f ∈ F (X) \ F∂(X);
qi = wβ(γi) for all 1 ≤ i ≤ b1.
We obtain the following generalization of Proposition 5.2:
Proposition 5.4. Given an element (h, a) ∈ H
(X,F0), set
q(h, a) =
f∈F (X)\F∂(X)
1≤i≤b1
Then, the Gibbs measure for the dimer model on (Γ, ∂Γ) with weight system w
and boundary condition ∂D′0 translates into the following probability measure on
(X,F0 | ∂h
ProbD0(h, a | ∂h
q(h, a)
D0,F0
(X ; q | ∂h′0)
where
D0,F0
(X ; q | ∂h′0) =
(h,a)∈H
(X,F0 | ∂h
q(h, a).
Furthermore, the measure is independant of the choice of F0. Finally, the bijection
(X,F0 | ∂h
0) → H
(X,F0 | ∂h
1) given by (h, a) 7→ (h+ h
D0,D1
, a+ a
D0,D1
invariant with respect to the measures ProbD0 and ProbD1 .
Proof. For any D ∈ D(Γ, ∂Γ | ∂D′0), equation (4) leads to
C(D,D0)− ∂σ
i=1 a
(i)γi
Computing the first term, we get
wβ(C(D,D0)) =
w(e)−1 = w(D)w(D0)
As for the second one,
wβ(∂σ
) = wβ
f∈F (X)
hD,D0(f)∂f
f∈F (X)
hD,D0 (f)
Since wβ(γi) = qi, these equations lead to
w(D) = w(D0)
f∈F (X)
1≤i≤b1
i = λ · q(h
where λ = w(D0)
f∈F∂(X)
f depends only on D0 and ∂D
0. The proposi-
tion follows easily from this equality. �
Let us count the number of essential parameters in the dimer model on (Γ, ∂Γ)
with some boundary condition partitioning ∂Γ into (∂Γ)nm ⊔ (∂Γ)m, matched and
non-matched vertices. We have |E(Γ)|− |(∂Γ)nm| edge weights, with an action of a
(|V (Γ)|−|(∂Γ)nm|)-parameter group. Since Γ is bipartite, there is a b0(Γ)-parameter
22 DAVID CIMASONI AND NICOLAI RESHETIKHIN
subgroup acting as the identity. Therefore, the number of essential parameters is
equal to
|E(Γ)| − |V (Γ)|+ b0(Γ) = |E(X)| − |∂Γ| − |V (X)|+ b0(X)
= |F (X)| − |∂Γ| − χ(X) + b0(X)
= |F (X) \ F∂(X)|+ b1(Σ)− b2(Σ).
The numbers |F (X) \ F∂(X)| and b1(Σ) correspond to the parameters qf and qi.
Furthermore, the parameters qf can be normalized by
f qf = 1, the product being
on all faces of a given closed component of Σ. Therefore, we obtain exactly the
right number of parameters in this height function formulation of the dimer model.
Remark. Note that all the results of the first part of the present section can be
adapted to the general case of a non-necessarily bipartite surface graph: one simply
needs to work with Z2-coefficients. However, the height function formulation of the
dimer model using volume weights does require a bipartite structure. It is unknown
whether a reformulation of the dimer model with the right number of parameters
is possible in the general case.
5.4. The dimer quantum field theory on bipartite surface graphs. Let us
now use these results to reformulate the dimer quantum field theory on bipartite
graphs. Let Γ ⊂ Σ be a bipartite surface graph, and let X denote the induced
cellular decomposition of Σ. Fix a dimer configuration D0 ∈ D(Γ, ∂Γ), a family
γ = {γi} of oriented simple curves in Γ representing a basis in H1(Σ, ∂Σ;Z), and a
choice F0 of one face in each connected component of X .
(1) To ∂X , assign
Z(∂X) =
f∈F∂(X)
where W is the complex vector space with basis {αn}n∈Z, and F∂(X) de-
notes the set of faces of X adjacent to the boundary.
(2) To X with weight system q = {qf}f∈F (X) ∪ {qi}1≤i≤b1(Σ), assign
D0,F0
(X ; q) =
D0,F0
(X ; q | ∂h)α(∂h) ∈ Z(∂X),
where
D0,F0
(X ; q | ∂h) =
(h,a)∈H
(X,F0 | ∂h)
f∈F (X)\F∂(X)
1≤i≤b1(Σ)
and α(∂h) =
f∈F∂(X)
f αh(f).
Recall the notation a(∂D) ∈ Z(∂Γ) of Section 4.2. The bijection D(Γ, ∂Γ) →
(X,F0) induces an inclusion j : Z(∂Γ) →֒ Z(∂X) such that
j(a(∂D)) =
f∈F∂(X)
DIMERS ON SURFACE GRAPHS AND SPIN STRUCTURES. II 23
Therefore, using the proof of Proposition 5.4,
j(Z(Γ;w)) =
Z(Γ;w | ∂D) j(a(∂D))
D0,F0
(X ; q | ∂h)w(D0)
f∈F∂(X)
f∈F∂(X)
αh(f)
= w(D0)Z
D0,F0
(X ; q),
where the weight system q is obtained from w by qf = wβ(∂f) and qi = wβ(γi).
In this setting, the gluing axiom makes sense only when the data β, D0 and γ
are compatible with the gluing map ϕ. In such a case case, it holds by the equality
above and the results of Section 4.2.
The equivalence between the quantum field theories formulated in Section 4.3
and in the present section should be regarded as a discrete version of the boson-
fermion correspondence on compact Riemann surfaces (see [1]).
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(1988) 175–186.
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Department of Mathematics, UC Berkeley, 970 Evans Hall, Berkeley, CA 94720, USA
E-mail address: cimasoni@math.berkeley.edu
E-mail address: reshetik@math.berkeley.edu
	Introduction
	Acknowledgements
	1. The dimer model on graphs with boundary
	1.1. Dimers on graphs with boundary
	1.2. Dimers on surface graphs with boundary
	2. Kasteleyn orientations on surface graphs with boundary
	2.1. Kasteleyn orientations
	2.2. Discrete spin structures
	2.3. The Pfaffian formula for the partition function
	3. Cutting and gluing
	3.1. Cutting and gluing graphs with boundary
	3.2. Cutting and gluing surface graphs with boundary
	3.3. Cutting and gluing discrete spin structures
	3.4. Cutting Pfaffians
	4. Quantum field theory for dimers
	4.1. Quantum field theory on graphs
	4.2. Quantum field theory for dimers on graphs
	4.3. The dimer model as the theory of free Fermions
	5. Dimers on bipartite graphs and height functions
	5.1. Composition cycles on bipartite graphs
	5.2. Height functions for planar bipartite graphs
	5.3. Height functions for bipartite surface graphs
	5.4. The dimer quantum field theory on bipartite surface graphs
	References
