[theorem] [lemma] [proposition] [corollary]

# Spectral transform for the Ising model

Terrence George  Department of Mathematics, University of California, Los
Angeles, CA 90095, USA<EMAIL_ADDRESS>
(Date: August 28, 2024)

###### Abstract.

We prove a correspondence between Ising models in a torus and the algebro-
geometric data of a Harnack curve with a certain symmetry and a point in the
real part of its Prym variety, extending the correspondence between dimer
models and Harnack curves and their Jacobians due to Kenyon and Okounkov.

##  1\. Introduction

$J_{1}$$J_{2}$(a)$\longrightarrow$$1$$s_{e}$$c_{e}$$s_{e}$$c_{e}$$1$(b)$e$$s_{2}$$c_{2}$$s_{2}$$c_{2}$$s_{1}$$c_{1}$$c_{1}$$s_{1}$(c)
Figure 1. (a) An Ising model $(G,J)$ in a torus $\mathbb{T}$, (b) the mapping
from $(G,J)$ to $(G^{\square},[\mathrm{wt}^{\square}])$ and (c) the dimer
model $(G^{\square},[\mathrm{wt}^{\square}])$.

An Ising model in a torus $\mathbb{T}$ is a pair $(G,J)$ where $G=(V,E,F)$ is
a graph embedded in $\mathbb{T}$ such that every face of $G$ is a topological
disk and $J:E(G)\rightarrow\mathbb{R}_{>0}$ is a function called the coupling
constant. A dimer model in $\mathbb{T}$ is a pair $(\Gamma,[\mathrm{wt}])$
where $\Gamma=(B\sqcup W,E,F)$ is a bipartite graph in $\mathbb{T}$ and
$[\mathrm{wt}]$ is a function
$\mathrm{wt}:E(\Gamma)\rightarrow\mathbb{R}_{>0}$ called edge weight defined
modulo a certain gauge equivalence.

Following [FW70, Dub11, BdT14], we can associate to an Ising model $(G,J)$ a
dimer model $(G^{\square},[\mathrm{wt}^{\square}])$ as follows. We define two
functions $s,c:E\rightarrow(0,1)$ by

$s_{e}:=\operatorname{sech}(2J_{e})\text{ and
}c_{e}:=\operatorname{tanh}(2J_{e})$

for every edge $e\in E$ and replace $e$ with the bipartite graph shown in
Figure 1(b).

$1$$\frac{d}{ac+bd}$$\frac{c}{ac+bd}$$\frac{b}{ac+bd}$$\frac{a}{ac+bd}$$1$$1$$a$$d$$c$$b$$1$$\longleftrightarrow$(a)(b)$b$$a$$d$$c$$\longleftrightarrow$$c$$d$$b$$a$$1$$1$
Figure 2. (a) The square move and (b) the contraction-uncontraction move.
Using gauge equivalence, we can assume that the original weight is as shown on
the right. Then the new weight is as shown on the left.

There are two local moves on bipartite torus graphs (Figure 2). If we apply
square moves at all the square faces of $G^{\square}$ corresponding to edges
of $G$, the resulting graph $\overline{G^{\square}}$ is the same as
$G^{\square}$ but with all vertices having opposite colors. Let
$\mu([\mathrm{wt}])$ denote the resulting weight on $\overline{G^{\square}}$.
Let $[\overline{\mathrm{wt}}]$ denote the weight where the weight of every
edge is the same as that of $G^{\square}$.

Our first result is the following characterization of the subset of dimer
models that corresponds to Ising models.

######  Theorem 1.1.

A dimer model $(G^{\square},[\mathrm{wt}])$ arises from an Ising model if and
only if $[\overline{\mathrm{wt}}]=\mu([\mathrm{wt}])$.

One direction is easy. If $\mathrm{wt}^{\square}$ is as in Figure 1(b) then
$a=c=s_{e}$ and $b=d=c_{e}$ in Figure 2(a). Since
$ac+bd=s_{e}^{2}+c_{e}^{2}=1$, we get
$[\overline{\mathrm{wt}^{\square}}]=\mu([\mathrm{wt}^{\square}])$. To prove
the other direction, we have to study the conditions imposed by setting
$[\overline{\mathrm{wt}}]=\mu([\mathrm{wt}])$ on the gauge invariant $X$
coordinates on the set of dimer models. This is done in Section 5.

Kenyon and Okounkov [KO06] showed that for every (minimal) bipartite graph
$\Gamma$ in $\mathbb{T}$ there is a bijection called the spectral transform
between the following sets of data.

  1. Data A.

    1. (a)

$(\Gamma,[\mathrm{wt}])$ a dimer model in $\mathbb{T}$.

    2. (b)

$[\kappa]$ a Kasteleyn sign on $\Gamma$ (equivalent to a choice of one of the
four elements of $H^{1}(\mathbb{T},\\{\pm 1\\})$).

  2. Data B.

    1. (a)

$C\subset(\mathbb{C}^{\times})^{2}$ a spectral curve which is a real algebraic
curve of a special type called a Harnack curve.

    2. (b)

$D$ a divisor on $C$ with one point on each compact oval of $C$ (i.e.,
connected component of the real locus $C(\mathbb{R})$) called a standard
divisor. Such divisors form a component of the real part of the Jacobian
variety $\operatorname{Jac}(C)$ of $C$.

    3. (c)

An ordering of the points at toric infinity of $C$.

Our second result is the following proved in Section 6.

######  Theorem 1.2.

For every (minimal) graph $G$ in $\mathbb{T}$, the spectral transform
restricts to a bijection between the following sets of data.

  1. Data A.

    1. (a)

$(G,J)$ an Ising model in $\mathbb{T}$.

    2. (b)

$[\kappa]$ a Kasteleyn sign on $G^{\square}$.

  2. Data B.

    1. (a)

$C$ a Harnack curve that is invariant under the involution $\sigma$ on
$(\mathbb{C}^{\times})^{2}$ defined by $(z,w)\mapsto(z^{-1},w^{-1})$. Whenever
a curve $C$ carries an involution, it defines a linear subvariety of
$\operatorname{Jac}(C)$ called the Prym variety of $C$.

    2. (b)

$D$ a standard divisor in the Prym variety of $C$.

    3. (c)

An ordering of the points at toric infinity of $C$ that is invariant under
$\sigma$.

This correspondence is interesting for several reasons. Firstly, it
generalizes the $Z$-invariant Ising models of Baxter [Bax78, Bax86, Bax89],
which correspond to genus-one spectral curves, to arbitrary genus. These Ising
models and their genus-zero degenerations (critical or isoradial Ising models)
have been extensively studied [CS11, BdT11, CS12, Cim12, CDC13, Li14, BdTR19,
Gal22].

Secondly, it completes the following table of correspondences between
statistical-mechanical models and algebro-geometric data.

|  disk |  torus
---|---|---
dimer models |  positive Grassmannian [Pos06] |  Harnack curves and standard divisors [KO06]
electrical networks |  positive Lagrangian Grassmannian [Lam18, BGKT21, CGS21] |  symmetric Harnack curves with a node and standard divisors in Prym varieties [Geo24]
Ising models |  positive orthogonal Grassmannian [HWX14, GP20] |  symmetric Harnack curves and standard divisors in Prym varieties (this paper)

Thirdly, combining results of Fock [Foc15] with compatibility of the Ising
Y-$\Delta$ move with the local moves on bipartite graphs [KP16], we get that
the discrete dynamical systems on Ising models arising from the Ising
Y-$\Delta$ move are linearized on Prym varieties of spectral curves; in this
sense, Ising models in $\mathbb{T}$ give rise to integrable systems extending
the cluster integrable systems of [GK13] constructed from dimer models in
$\mathbb{T}$.

Finally, the spectral transform for the dimer model has been recently used in
statistical mechanics to understand limit shapes [BD23, BB23]. We expect that
the spectral transform for the Ising model will have similar applications in
statistical mechanics.

### Acknowledgements.

I first started thinking of the spectral transform of the Ising model in 2019
when I was a graduate student after discussions with Richard Kenyon. I am
grateful to Cédric Boutillier, Béatrice de Tilière and Pavel Galashin for
discussions.

##  2\. Background on the Ising model

In this section, we collect some background on the Ising model. For further
background, see [Che18, CCK17]. Let $G=(V,E,F)$ be a graph in $\mathbb{T}$ and
let $\mathcal{I}_{G}(\mathbb{R}_{>0}):=\\{J:E(G)\rightarrow\mathbb{R}_{>0}\\}$
be the set of Ising models with graph $G$.

###  2.1. Zig-zag paths and the Newton polygon

A zig-zag path in $G$ is an oriented path in $G$ that alternately turns
maximally left or right at each vertex. Zig-zag paths in $G$ come in pairs
with opposite orientations; we denote by $\overline{\alpha}$ the zig-zag path
opposite to the zig-zag path $\alpha$. Let $Z_{G}$ denote the set of zig-zag
paths in $G$.

Let $\pi:\mathbb{R}^{2}\rightarrow\mathbb{T}$ denote the universal cover of
$\mathbb{T}$. We say that $G$ is minimal if the lift of any zig-zag path to
$\mathbb{R}^{2}$ does not have a self-intersection and the lifts of two zig-
zag paths to $\mathbb{R}^{2}$ share at most one edge. Hereafter, we assume
that our graphs are minimal.

Choose a fundamental rectangle $R$ for $\mathbb{T}$ and let
$\gamma_{z},\gamma_{w}$ be loops in $\mathbb{T}$ along the sides of $R$ as
shown in Figure 3(a). Then $\\{[\gamma_{z}],[\gamma_{w}]\\}$ is a basis for
$H_{1}(\mathbb{T},\mathbb{Z})$ identifying $H_{1}(\mathbb{T},\mathbb{Z})$ with
$\mathbb{Z}^{2}$. A nonzero vector $v\in\mathbb{Z}^{2}$ is called primitive if
$v$ is not a multiple of another vector in $\mathbb{Z}^{2}$, i.e., if
$v=\lambda w$ for $w\in\mathbb{Z}^{2}$ and $\lambda\in\mathbb{Z}_{>0}$ then
$v=w$ and $\lambda=1$. A convex polygon $N\subset\mathbb{R}^{2}$ is said to be
integral if all of its vertices are contained in $\mathbb{Z}^{2}$. Let
$\sigma$ denote the involution $v\mapsto-v$ of $\mathbb{R}^{2}$. A convex
integral polygon $N$ is called centrally symmetric if $N$ is invariant under
$\sigma$.

Associated to a minimal $G$ is a centrally-symmetric convex integral polygon
$N_{G}\subset\mathbb{R}^{2}$ as follows: each zig-zag path $\alpha\in Z_{G}$
defines a primitive vector given by the homology class $[\alpha]\in
H_{1}(\mathbb{T},\mathbb{Z})\cong\mathbb{Z}^{2}$. There is a unique (modulo
translation) convex integral polygon in $\mathbb{R}^{2}$ whose
counterclockwise-oriented boundary consists of the vectors
$\\{[\alpha]\in\mathbb{Z}^{2}:\alpha\in Z_{G}\\}$. The translation is fixed by
centering at $(0,0)$.

######  Example 2.1.

Let $(G,J)$ denote the Ising model in Figure 1(a). There are four zig-zag
paths in $G$. Let $\alpha$ (resp., $\beta$) denote the red (resp., blue) zig-
zag path in Figure 3(a). The other two zig-zag paths are $\overline{\alpha}$
and $\overline{\beta}$. The homology classes are

$[\alpha]=(1,1),\quad[\beta]=(-1,1),\quad[\overline{\alpha}]=(-1,-1),\quad[\overline{\beta}]=(1,-1),$

so the Newton polygon is the convex integral polygon shown in Figure 3(b).

$\gamma_{z}$$\gamma_{w}$(a)(b) Figure 3. (a) Two of the zig-zag paths of the
Ising model $(G,J)$ in Figure 1(a) and (b) its Newton polygon.

###  2.2. Kramers–Wannier duality

Let $(G,J)$ be an Ising model in $\mathbb{T}$. Let $G^{*}$ denote the dual
graph of $G$ in $\mathbb{T}$. For $e\in E(G)$, let $e^{*}\in E(G^{*})$ denote
the dual edge. Define $J^{*}:E(G^{*})\rightarrow\mathbb{R}_{>0}$ by the
condition

$\operatorname{sinh}(2J_{e^{*}}^{*})=\frac{1}{\operatorname{sinh}(2J_{e})}.$

Then $(G^{*},J^{*})$ is called the dual Ising model. Duality defines a
bijection

$\mathcal{I}_{G}(\mathbb{R}_{>0})\rightarrow\mathcal{I}_{G^{*}}(\mathbb{R}_{>0}).$

It is convenient to introduce the following coordinates. Let
$x:E(G)\rightarrow\mathbb{R}_{>0}$ be defined as $x_{e}:=\exp(2J_{e})$. Then
$x^{*}_{e^{*}}:=\exp(2J^{*}_{e^{*}})$ is given by the unique positive solution
to

$x_{e}+x^{*}_{e^{*}}+x_{e}x^{*}_{e^{*}}=1.$

###  2.3. Y-$\Delta$ move

$a$$c$$b$$\longleftrightarrow$$B$$A$$C$ Figure 4. The Ising Y-$\Delta$ move.

Ising models have a local transformation called the Y-$\Delta$ or star-
triangle move which replaces a portion of a graph $G$ that looks like one side
of Figure 4 with a portion that looks like the other side to get a graph
$G^{\prime}$ and modifies the $x$ weights by

$\displaystyle A=\sqrt{\frac{(abc+1)(a+bc)}{(b+ac)(c+ab)}},\quad
B=\sqrt{\frac{(abc+1)(b+ac)}{(a+bc)(c+ab)}},\quad
C=\sqrt{\frac{(abc+1)(c+ab)}{(a+bc)(b+ac)}}.$

The Y-$\Delta$ move gives rise to a bijection
$\mathcal{I}_{G}(\mathbb{R}_{>0})\rightarrow\mathcal{I}_{G^{\prime}}(\mathbb{R}_{>0})$.
Moreover, the Y-$\Delta$ move and duality are compatible, i.e., the following
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We say that two graphs $G$ and $G^{\prime}$ are move-equivalent if they are
related a sequence of Y-$\Delta$ moves and duality.

######  Theorem 2.2 ([GK13, Theorem 5.4]).

For every centrally symmetric convex integral polygon $N$, there is a family
of minimal graphs with Newton polygon $N$. Any two members of a family are
move equivalent. In other words, the set of move equivalence classes of
minimal graphs is in bijection with the set of centrally symmetric convex
integral polygons.

##  3\. Background on the dimer model

In this section, we give a brief background on the dimer model and its
spectral transform, mostly following [Ken09, GK13].

###  3.1. Dimer models in $\mathbb{T}$

A dimer model in $\mathbb{T}$ is a pair $(\Gamma,[\mathrm{wt}])$ where

  1. (1)

$\Gamma=(B\sqcup W,E,F)$ is a bipartite graph embedded in $\mathbb{T}$ such
that the faces of $\Gamma$ are topological disks,

  2. (2)

$[\mathrm{wt}]$ is the gauge-equivalence class of
$\mathrm{wt}:E\rightarrow\mathbb{C}^{\times}$ which is a function assigning to
each edge its edge weight, and two edge weights $\mathrm{wt}_{1}$ and
$\mathrm{wt}_{2}$ are gauge equivalent if there is a function $f:B\sqcup
W\rightarrow\mathbb{C}^{\times}$ such that for every edge
$e=\\{\mathrm{b},\mathrm{w}\\}\in E$ (where $\mathrm{b}\in B,\mathrm{w}\in
W$),

$\mathrm{wt}^{\prime}(e)=f(\mathrm{b})^{-1}\mathrm{wt}(e)f(\mathrm{w}).$

Let $\mathcal{X}_{\Gamma}$ denote the space of edge weights modulo gauge
equivalence. It will be convenient to rephrase the above in the language of
algebraic topology. In particular, doing so will help us identify coordinates
on $\mathcal{X}_{\Gamma}$ that are invariant under gauge equivalence.

We consider $\Gamma$ as a cell complex where the vertices are the $0$-cells
and the edges are the $1$-cells, where we orient each edge
$e=\\{\mathrm{b},\mathrm{w}\\}$ from $\mathrm{b}$ to $\mathrm{w}$. The
cellular chain complex is

$0\rightarrow
C_{1}(\Gamma,\mathbb{Z})\xrightarrow[]{\partial}C_{0}(\Gamma,\mathbb{Z})\rightarrow
0,$

where $C_{0}(\Gamma,\mathbb{Z})=\mathbb{Z}B\oplus\mathbb{Z}W$,
$C_{1}(\Gamma,\mathbb{Z})=\mathbb{Z}E$ and $\partial e=\mathrm{w}-\mathrm{b}$.
Therefore,

$H_{1}(\Gamma,\mathbb{Z})=\ker(C_{1}(\Gamma,\mathbb{Z})\xrightarrow[]{\partial}C_{0}(\Gamma,\mathbb{Z})),$

so $1$-homology classes in $\Gamma$ are the same thing as $1$-cycles in
$\Gamma$.

The cellular cochain complex is

$1\rightarrow
C^{0}(\Gamma,\mathbb{C}^{\times})\xrightarrow[]{d}C^{1}(\Gamma,\mathbb{C}^{\times})\rightarrow
1,$

where

  1. (1)

$C^{0}(\Gamma,\mathbb{C}^{\times})=\text{Hom}_{\mathbb{Z}}(C_{0}(\Gamma,\mathbb{Z}),\mathbb{C}^{\times})\cong\\{f:B\sqcup
W\rightarrow\mathbb{C}^{\times}\\}$,

  2. (2)

$C^{1}(\Gamma,\mathbb{C}^{\times})=\text{Hom}_{\mathbb{Z}}(C_{1}(\Gamma,\mathbb{Z}),\mathbb{C}^{\times})\cong\\{\mathrm{wt}:E\rightarrow\mathbb{C}^{\times}\\}$,

  3. (3)

$(df)(e)={f(\mathrm{b})}^{-1}{f(\mathrm{w})}$,

where $\text{Hom}_{\mathbb{Z}}(\cdot,\mathbb{C}^{\times})$ denotes the space
of abelian group homomorphisms from $\cdot$ to $\mathbb{C}^{\times}$.
Therefore, a $1$-cocycle is the same thing as an edge weight and two
$1$-cocycles differ by a $1$-coboundary if they are gauge equivalent, so

$\mathcal{X}_{\Gamma}=H^{1}(\Gamma,\mathbb{C}^{\times}).$

With this identification, we can describe coordinates on
$\mathcal{X}_{\Gamma}$. We describe this abstractly first and then explain
what it means concretely.

Since
$H^{1}(\Gamma,\mathbb{C}^{\times})=\mathrm{Hom}_{\mathbb{Z}}(H_{1}(\Gamma,\mathbb{Z}),\mathbb{C}^{\times})$
is the algebraic torus with lattice of characters $H_{1}(\Gamma,\mathbb{Z})$,
its coordinate ring is

$\mathbb{C}[H_{1}(\Gamma,\mathbb{Z})]=\mathbb{C}\left[\sum_{\gamma\in
H_{1}(\Gamma,\mathbb{Z})}a_{\gamma}X_{\gamma}:a_{\gamma}=0\text{ for all but
finitely many }\gamma\right]$

with multiplication given by $X_{\gamma_{1}}\cdot
X_{\gamma_{2}}=X_{\gamma_{1}+\gamma_{2}}$, where
$X_{\gamma}:\mathcal{X}_{\Gamma}\rightarrow\mathbb{C}^{\times}$ is the
character given by $X_{\gamma}([\mathrm{wt}])=[\mathrm{wt}](\gamma)$, i.e., by
evaluating the cohomology class $[\mathrm{wt}]$ on the cycle $\gamma$.
Explicitly, if $\gamma$ is the cycle ${\rm b}_{1}\xrightarrow[]{e_{1}}{\rm
w}_{1}\xrightarrow[]{e_{2}}{\rm b}_{2}\xrightarrow[]{e_{3}}{\rm
w}_{2}\xrightarrow[]{e_{4}}\cdots\xrightarrow[]{e_{2n-2}}{\rm
b}_{n}\xrightarrow[]{e_{2n-1}}{\rm w}_{n}\xrightarrow[]{e_{2n}}{\rm b}_{1}$ ,
then

$[\mathrm{wt}](\gamma)=\frac{\mathrm{wt}({e_{1}})\cdots\mathrm{wt}{(e_{2n-1}})}{\mathrm{wt}{(e_{2})}\cdots\mathrm{wt}{(e_{2n})}}.$

In down to earth terms, this means that the $X_{\gamma}$’s are a set
coordinates on $\mathcal{X}_{\Gamma}$ but they are not independent and have to
satisfy some relations. If we choose a basis for $H_{1}(\Gamma,\mathbb{Z})$,
then we obtain a basis for the set of coordinates. There is no canonical
choice for a basis but the following choice is convenient for computations. We
identify faces of $\Gamma$ with the cycles given by their counterclockwise-
oriented boundaries. Then each face $f$ determines a regular function $X_{f}$
on $\mathcal{X}_{\Gamma}$. Since $\sum_{f\in F}f=0$ in
$H_{1}(\Gamma,\mathbb{Z})$, we have $\prod_{f\in F}X_{f}=1$. Additionally if
we choose two cycles $a,b$ in $\Gamma$ generating
$H_{1}(\mathbb{T},\mathbb{Z})$, then $\\{\text{all faces except one},a,b\\}$
is a basis for $H_{1}(\Gamma,\mathbb{Z})$ and so

(3.1) $\\{X_{f}\\}_{\text{all faces $f$ except one}}\sqcup\\{X_{a},X_{b}\\}$

is a basis for the coordinate ring.

Let $\mathcal{X}_{\Gamma}(\mathbb{R}_{>0})$ denote the positive real valued
points of $\mathcal{X}_{\Gamma}$, i.e., the set of points $[\mathrm{wt}]$
where $X_{\gamma}([\mathrm{wt}])\in\mathbb{R}_{>0}$ for all $\gamma\in
H_{1}(\Gamma,\mathbb{Z})$.

###  3.2. Zig-zag paths

A zig-zag path in $\Gamma$ is a cycle that turns maximally right at black
vertices and maximally left at white vertices. We denote the set of zig-zag
paths of $\Gamma$ by $Z_{\Gamma}$. The unique (modulo translation) convex
integral polygon $N_{\Gamma}\subset\mathbb{R}^{2}$ whose counterclockwise-
oriented boundary consists of $\\{[\alpha]\in
H_{1}(\mathbb{T},\mathbb{Z})\cong\mathbb{Z}^{2}:\alpha\in Z_{\Gamma}\\}$ is
called the Newton polygon of $\Gamma$. For a side $S$ of $N$, let
$Z_{\Gamma,S}$ denote the subset of zig-zag paths whose homology classes form
the side $S$.

Recall that $\pi:\mathbb{R}^{2}\rightarrow\mathbb{T}$ is the universal cover.
We say that $\Gamma$ is minimal if there are no zig-zag paths with zero
homology, no lift of a zig-zag path has a self-intersection, and lifts of two
zig-zag paths do not form parallel bigons, i.e., they do no pass through two
edges $e_{1}\neq e_{2}$ of the biperiodic graph
$\pi^{-1}(\Gamma)\subset\mathbb{R}^{2}$ with both lifts oriented from $e_{1}$
to $e_{2}$. Hereafter we assume that all our bipartite torus graphs are
minimal.

###  3.3. Moves

${\rm v}$$\delta$$\gamma$ Figure 5. The local pairing at a trivalent vertex
$\rm v$ (which may be black or white) is defined by $\epsilon_{\rm
v}(\gamma,\delta)=\frac{1}{2}$ and bilinearity and antisymmetry.
(a)(b)(c)(d)(e)$f_{1}$$f_{2}$$f_{3}$$f_{4}$$f_{1}^{\prime}+f^{\prime}$$f_{2}^{\prime}$$f_{3}^{\prime}+f^{\prime}$$f_{4}^{\prime}$$\mathrm{w}_{2}$$\mathrm{w}_{1}$$\mathrm{w}_{4}$$\mathrm{w}_{3}$$\mathrm{b}_{1}^{\prime}$$\mathrm{b}_{2}^{\prime}$$f^{\prime}$$f_{2}^{\prime}$$f_{1}^{\prime}$$f_{4}^{\prime}$$f_{3}^{\prime}$$\mathrm{w}_{1}$$\mathrm{w}_{4}$$\mathrm{w}_{3}$$\mathrm{w}_{2}$$\mathrm{b}_{1}$$\mathrm{b}_{2}$$f$$f_{1}$$f_{4}$$f_{3}$$f_{2}$
Figure 6. (a) Labeling of the graph near a square face $f$ and (b)–(e) the
correspondence $s_{*}$ between cycles in $\Gamma$ and $\Gamma^{\prime}$.

There are two local moves on bipartite torus graphs shown in Figure 2. Each
move $\Gamma\stackrel{{\scriptstyle s}}{{\rightsquigarrow}}\Gamma^{\prime}$
induces a pair of maps:

  1. (1)

An isomorphism $s_{*}:H_{1}(\Gamma,\mathbb{Z})\rightarrow
H_{1}(\Gamma^{\prime},\mathbb{Z})$ such that $s_{*}$ restricts to a bijection
between $Z_{\Gamma}$ and $Z_{\Gamma^{\prime}}$ and such that
$[s_{*}(\gamma)]=[\gamma]\in H_{1}(\mathbb{T},\mathbb{Z})$,

  2. (2)

A bijection
$\mu_{s}:\mathcal{X}_{\Gamma}(\mathbb{R}_{>0})\rightarrow\mathcal{X}_{\Gamma^{\prime}}(\mathbb{R}_{>0})$.

We discuss the case of the square move at a face $f$ in more detail (see
[GK13, GLSBS22]). Let $\Gamma_{f}$ and $\Gamma_{f}^{\prime}$ denote the
portions of the graphs shown in Figure 6. Suppose the vertices and faces near
$f$ are labeled as in Figure 6 (left). Let $\mathbf{M}$ denote the set
$\\{\mathrm{w}_{1},\mathrm{w}_{2},\mathrm{w}_{3},\mathrm{w}_{4}\\}$. Then each
cycle in $\Gamma$ (resp., $\Gamma^{\prime}$) restricts to a relative cycle in
$H_{1}(\Gamma_{f},\mathbf{M},\mathbb{Z})$ (resp.,
$H_{1}(\Gamma_{f}^{\prime},\mathbf{M},\mathbb{Z})$) and
$f_{1},f_{2},f_{3},f_{4}$ (resp.,
$f_{1}^{\prime},f_{2}^{\prime},f_{3}^{\prime},f_{4}^{\prime}$) is a basis for
$H_{1}(\Gamma_{f},\mathbf{M},\mathbb{Z})$ (resp.,
$H_{1}(\Gamma_{f}^{\prime},\mathbf{M},\mathbb{Z})$). Note that
$f=-f_{1}-f_{2}-f_{3}-f_{4}$ and similarly for $f^{\prime}$.

In these bases, $s_{*}$ is given by the formula

(3.2) $s_{*}(f_{i})=f_{i}^{\prime}+\text{max}(0,\langle
f_{i},f\rangle)f^{\prime},$

where $\langle\cdot,\cdot\rangle:H_{1}(\Gamma,\mathbb{Z})\times
H_{1}(\Gamma,\mathbb{Z})$ is an antisymmetric bilinear form on cycles defined
by the intersection pairing on the conjugate surface (see [GK13, Section
1.1.1]). It can be computed by

$\langle\gamma,\delta\rangle=\sum_{{\rm b}\in
B}\epsilon_{\mathrm{b}}(\gamma,\delta)-\sum_{{\rm w}\in
W}\epsilon_{\mathrm{w}}(\gamma,\delta),$

where $\epsilon_{\rm v}(\gamma,\delta)$ is the local pairing shown in Figure
5. Since all the vertices in $G^{\square}$ are trivalent, we do not give the
general rule (see [GK13, Appendix]).

For example, we have

$\langle
f_{1},f\rangle=\epsilon_{\mathrm{b}_{1}}(f_{1},f)-\epsilon_{\mathrm{w}_{4}}(f_{1},f)=\frac{1}{2}-\left(-\frac{1}{2}\right)=1,$

so $s_{*}(f_{1})=f_{1}^{\prime}+f^{\prime}$. The cycles
$s_{*}(f_{i}),i\in\\{1,2,3,4\\}$, are shown in Figure 6(b)–(e). Note that

$s_{*}(f)=s_{*}(-f_{1}-f_{2}-f_{3}-f_{4})=-(f_{1}^{\prime}+f^{\prime})-f_{2}^{\prime}-(f_{3}^{\prime}+f^{\prime})-f_{4}^{\prime}=-f^{\prime}.$

The bijection
$\mu_{s}:\mathcal{X}_{\Gamma}(\mathbb{R}_{>0})\rightarrow\mathcal{X}_{\Gamma^{\prime}}(\mathbb{R}_{>0})$
is given by

(3.3)
$X_{s_{*}(\gamma)}(\mu_{s}([\mathrm{wt}]))=X_{\gamma}([\mathrm{wt}])(1+X_{f}([\mathrm{wt}]))^{-\langle\gamma,f\rangle},$

or more succinctly,
$\mu^{*}_{s}X_{s_{*}(\gamma)}=X_{\gamma}(1+X_{f})^{-\langle\gamma,f\rangle}$,
which is the formula for mutation of $X$ cluster variables. For example,

$\displaystyle\mu^{*}_{s}X_{f^{\prime}}=\mu^{*}_{s}X_{s_{*}(-f)}=X_{-f}(1+X_{f})^{-\langle-f,f\rangle}=X_{f}^{-1},$

and

$\mu^{*}_{s}X_{f_{1}^{\prime}}=\mu^{*}_{s}X_{s_{*}(f_{1}+f)}=X_{f_{1}+f}(1+X_{f})^{-1}=X_{f_{1}}(1+X_{f}^{-1})^{-1}.$

###  3.4. The mapping from Ising models to dimer models

(a)(b) Figure 7. (a) Two of the zig-zag paths of the graph $G^{\square}$ from
Figure 1(b) and (b) its Newton polygon.

Recall the mapping from Ising models to dimer models from Figure 1(b). There
is a natural bijection between $Z_{G}$ and $Z_{G^{\square}}$ that preserves
homology classes (compare Figure 3(a) and Figure 7(a)). Therefore,
$N_{G^{\square}}=N_{G}$.

A Y-$\Delta$ move $G_{1}\rightarrow G_{2}$ can be realized as a sequence of
moves $G_{1}^{\square}\rightarrow G_{2}^{\square}$ [KP16, Figure 6] (see also
[AGPR24, Section 4.3]), i.e., we have a commuting diagram

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and the dual Ising model gives the dimer model
$(\overline{G^{\square}},[\overline{\mathrm{wt}^{\square}}])$. Therefore, the
mapping is compatible with respect to moves on both models.

###  3.5. The spectral transform

Let $(\Gamma,[\mathrm{wt}])$ be a dimer model in $\mathbb{T}$. We assign to
each edge $e=\\{\mathrm{b},\mathrm{w}\\}$ of $\Gamma$ a monomial
$\phi(e):=z^{i}w^{j}$ where $i$ (resp., $j$) records the number of signed
intersections of $e$ with $\gamma_{w}$ (resp., $\gamma_{z}$) when we orient
$e$ from $\mathrm{b}$ to $\mathrm{w}$.

A Kasteleyn sign on $\Gamma$ is a $1$-cohomology class $[\kappa]\in
H^{1}(\Gamma,\\{\pm 1\\})$ such that for any face $f\in F$,

$[\kappa](f)=(-1)^{\frac{\\#\partial f}{2}+1},$

where $e\in f$ means $e$ is incident to $f$. Here $\\{\pm 1\\}$ is the
multiplicative group of square roots of unity. There are four choices of
$[\kappa]$ corresponding to the four elements of $H^{1}(\mathbb{T},\\{\pm
1\\})$. If we use the basis (3.1), a choice of $[\kappa]$ is equivalent to a
choice of $(X_{a},X_{b})\in\\{\pm 1\\}^{2}$.

Let $\mathrm{wt}$ be a $1$-cocycle representing the cohomology class
$[\mathrm{wt}]$ and let $\kappa:E\rightarrow\\{\pm 1\\}$ be a $1$-cocycle
representing $[\kappa]$. The Kasteleyn matrix
$K_{(\Gamma,\mathrm{wt},\kappa)}(z,w)$ is the Laurent-polynomial-valued matrix
with rows and columns indexed by white and black vertices respectively defined
by

$K_{(\Gamma,\mathrm{wt},\kappa)}(z,w)_{\mathrm{w}\mathrm{b}}:=\sum_{e=\\{\mathrm{b},\mathrm{w}\\}}\mathrm{wt}(e)\kappa(e)\phi(e),$

where the sum is over all edges with endpoints $\mathrm{b}$ and $\mathrm{w}$.
The determinant

$P_{(\Gamma,\mathrm{wt},\kappa)}(z,w):=\det
K_{(\Gamma,\mathrm{wt},\kappa)}(z,w)$

is a Laurent polynomial called the characteristic polynomial. The Newton
polygon of $P_{(\Gamma,\mathrm{wt},\kappa)}(z,w)$ is defined as

$\text{convex-hull}\\{(i,j)\in\mathbb{R}^{2}:\text{coefficient of $z^{i}w^{j}$
is nonzero in $P_{(\Gamma,\mathrm{wt},\kappa)}(z,w)$}\\}.$

For minimal $\Gamma$, the Newton polygon of
$P_{(\Gamma,\mathrm{wt},\kappa)}(z,w)$ coincides with the Newton polygon of
$\Gamma$, whence the name. The vanishing locus
$C^{\circ}:=\\{(z,w)\in(\mathbb{C}^{\times})^{2}:P_{(\Gamma,\mathrm{wt},\kappa)}(z,w)=0\\}$
is called the open spectral curve.

The amoeba $\mathbb{A}(C^{\circ})$ is defined to be the image of $C^{\circ}$
under the map
$\operatorname{Log}:(\mathbb{C}^{\times})^{2}\rightarrow\mathbb{R}^{2}$
defined by $(z,w)\mapsto(\log|z|,\log|w|)$. A curve $C^{\circ}$ defined by a
real Laurent polynomial $P(z,w)$ is said to be a Harnack curve if the
restriction

$\operatorname*{Log}:C^{\circ}\rightarrow\mathbb{A}(C^{\circ})$

is $2:1$ over the interior of $\mathbb{A}(C^{\circ})$. For Harnack curves, the
boundary of $\mathbb{A}(C^{\circ})$ is the image of the real points
$C^{\circ}(\mathbb{R})$.

######  Theorem 3.1 ([KS04, KOS06]).

The open spectral curve $C^{\circ}$ of a dimer model is a Harnack curve.

The Newton polygon $N$ defines a compactification

$\mathcal{N}:=\overline{\\{[z^{i}w^{j}]_{(i,j)\in
N\cap\mathbb{Z}^{2}}:(z,w)\in(\mathbb{C}^{\times})^{2}\\}}\subset\mathbb{C}\mathbb{P}^{\\#N\cap\mathbb{Z}^{2}-1}$

of $(\mathbb{C}^{\times})^{2}$ called a toric surface. The boundary

$\mathcal{N}-(\mathbb{C}^{\times})^{2}=\bigcup_{S\text{ side of }N}D_{S}$

where each $D_{S}\cong\mathbb{C}\mathbb{P}^{1}$ is called a line at infinity.
Taking closure, we get a compactification

$C:=\overline{C^{\circ}}\subset\mathcal{N}$

called the spectral curve. The points $C-C^{\circ}$ are called points at
infinity; we call the points in $C\cap D_{S}$ the points at infinity
corresponding to the side $S$. The number of such points is equal to the
number of primitive vectors in $S$.

Points at infinity corresponding to the side $S$ are in bijection with
tentacles of the amoeba in the direction of the outward-pointing normal to
$S$. Each tentacle is asymptotic to a line $ix+jy+c=0$ where $(i,j)$ is the
primitive vector along the side $S$ of $N$. There is a bijection

$\nu_{\Gamma}:Z_{\Gamma}\rightarrow\bigsqcup_{S\text{ side of }N}C\cap D_{S}$

which can be described as follows: The tentacle corresponding to a zig-zag
path $\alpha$ with $[\alpha]$ along $S$ is the one asymptotic to the line

(3.4) $ix+jy+\log|X_{\alpha}([\mathrm{wt}])|=0.$

Recall that $Z_{\Gamma,S}$ is the subset of zig-zag paths corresponding to
$S$. Note that by definition, this bijection has the property that
$\nu_{\Gamma}(Z_{\Gamma,S})=C\cap D_{S}$.

The real locus of a Harnack curve has $g+1$ components called ovals, one of
which corresponds to the boundary of $N$ and is called the outer oval, and $g$
interior ovals that are in bijection with the interior lattice points in $N$.

By definition of the spectral curve, the Kasteleyn matrix
$K_{(\Gamma,\mathrm{wt},\kappa)}(z,w)$ has nonzero kernel and cokernel over
$C^{\circ}$. If $(z,w)$ is a smooth point of $C^{\circ}$ then the kernel and
cokernel of $K_{(\Gamma,\mathrm{wt},\kappa)}(z,w)$ are $1$-dimensional. Hence,
if $C^{\circ}$ is smooth then the kernel and cokernel are line bundles.

For a white vertex $\mathrm{w}$, the image of the function
$\delta_{\mathrm{w}}\in\mathbb{C}^{W}$ in
$\operatorname{coker}K_{(\Gamma,\mathrm{wt},\kappa)}(z,w)$ defines a section
of the cokernel. This section vanishes on a set of $g$ points
$\\{(p_{i},q_{i})\\}_{i=1}^{g}$ in $\mathbb{C}^{\circ}$, where $g$ is the
arithmetic genus of $C$ and is equal to the number of interior points in
$N\cap\mathbb{Z}^{2}$. We call

(3.5) $D_{\mathrm{w}}:=\sum_{i=1}^{g}(p_{i},q_{i})$

the divisor of $\mathrm{w}$. By the same construction applied to
$K_{(\Gamma,\mathrm{wt},\kappa)}(z,w)^{T}$, we define degree-$g$ divisors
$D_{\mathrm{b}}$ for black vertices $\mathrm{b}\in B$. These divisors can be
described more explicitly as follows. Let
$Q_{(\Gamma,\mathrm{wt},\kappa)}(z,w)$ denote the cofactor matrix of
$K_{(\Gamma,\mathrm{wt},\kappa)}(z,w)$. The divisor $D_{\mathrm{b}}$ (resp.,
$D_{\mathrm{w}}$) is given by the vanishing of the $\mathrm{b}$-row (resp.,
$\mathrm{w}$-column) of $Q_{(\Gamma,\mathrm{wt},\kappa)}(z,w)$.

A degree-$g$ divisor $D$ is called a standard divisor if $D$ contains one
point in each interior oval. If $C$ is smooth, the set of standard divisors is
a component of the real locus of the Jacobian of $C$ and is a $g$-dimensional
real torus.

######  Theorem 3.2 ([KO06]).

Let $(\Gamma,[\mathrm{wt}])$ be a dimer model in $\mathbb{T}$ and let
$\mathrm{v}$ be a vertex of $\Gamma$. The spectral curve $C$ is a Harnack
curve and the divisor $D_{\mathrm{v}}$ is a standard divisor.

######  Remark 3.3.

The only possible singularities of Harnack curves occur when some ovals shrink
to zero size forming isolated real nodes or when tentacles in the same
direction merge [MRr01]. Suppose $C$ is singular with $b$ nodes. Let
$\tilde{C}$ denote its desingularization which has genus $\tilde{g}=g-b$. If
$\pi:\tilde{C}\rightarrow C$ is the canonical map gluing pairs of points into
nodes, then $\pi^{-1}(D_{\rm v})$ has $g+b$ points: each smooth point lifts to
one point and each node to two points. Let $\tilde{D}_{\rm v}$ denote the
$\tilde{g}$ points of $\pi^{-1}(D_{\rm v})$ corresponding to the smooth
points. All the constructions work for singular $C$ as for smooth $C$ if we
replace $C$ with $\tilde{C}$ and $D_{\rm v}$ with $\tilde{D}_{\rm v}$, and we
do this without further mention below.

$\mathrm{b}$$\mathrm{w}$$s_{2}$$c_{2}$$s_{2}$$-c_{2}$$-s_{1}$$c_{1}$$c_{1}$$s_{1}$$w$$\frac{1}{z}$$\frac{z}{w}$(a)(b)
Figure 8. (a) A Kasteleyn sign and $\phi$ for the dimer model in Figure 1(b)
and (b) the amoeba of $C$ and the standard divisor $D_{\mathrm{w}}$ for
$c_{1}=\frac{1}{\sqrt{2}}$ and $c_{2}=\frac{\sqrt{3}}{2}$.

######  Example 3.4.

Let $(G^{\square},[\mathrm{wt}^{\square}])$ be the dimer model from Figure
1(c). Consider the Kasteleyn sign $\kappa$ and $\phi$ shown in Figure 8. The
Kasteleyn matrix is (with vertices ordered from left to right)

$K_{(G^{\square},\mathrm{wt}^{\square},\kappa)}(z,w)=\begin{bmatrix}c_{2}&s_{2}&0&\frac{z}{w}\\\
s_{2}&-c_{2}&1&0\\\ 0&w&-s_{1}&c_{1}\\\
\frac{1}{z}&0&c_{1}&s_{1}\end{bmatrix},$

and the characteristic polynomial is

$\displaystyle
P_{(G^{\square},\mathrm{wt}^{\square},\kappa)}(z,w)=1+c_{1}^{2}c_{2}^{2}+c_{2}^{2}s_{1}^{2}+c_{1}^{2}s_{2}^{2}+s_{1}^{2}s_{2}^{2}-c_{2}s_{1}w-c_{1}s_{2}z-\frac{c_{2}s_{1}}{w}-\frac{c_{1}s_{2}}{z}.$

The adjugate matrix $Q_{(G^{\square},\mathrm{wt}^{\square},\kappa)}(z,w)$ is

$\begin{bmatrix}c_{1}^{2}c_{2}+c_{2}s_{1}^{2}-s_{1}w&c_{1}^{2}s_{2}-c_{1}z+s_{1}^{2}s_{2}&s_{1}s_{2}-\frac{c_{1}c_{2}z}{w}&-c_{1}s_{2}-\frac{c_{2}s_{1}z}{w}+z\\\
c_{1}^{2}s_{2}-\frac{c_{1}}{z}+s_{1}^{2}s_{2}&-c_{1}^{2}c_{2}-c_{2}s_{1}^{2}+\frac{s_{1}}{w}&-\frac{c_{1}s_{2}z}{w}-c_{2}s_{1}+\frac{1}{w}&c_{1}c_{2}-\frac{s_{1}s_{2}z}{w}\\\
s_{1}s_{2}w-\frac{c_{1}c_{2}}{z}&-\frac{c_{1}s_{2}}{z}-c_{2}s_{1}w+1&-c_{2}^{2}s_{1}+\frac{c_{2}}{w}-s_{1}s_{2}^{2}&c_{1}c_{2}^{2}+c_{1}s_{2}^{2}-s_{2}z\\\
-c_{1}s_{2}w-\frac{c_{2}s_{1}}{z}+\frac{w}{z}&c_{1}c_{2}w-\frac{s_{1}s_{2}}{z}&c_{1}c_{2}^{2}+c_{1}s_{2}^{2}-\frac{s_{2}}{z}&c_{2}^{2}s_{1}-c_{2}w+s_{1}s_{2}^{2}\end{bmatrix}.$

Setting the $\mathrm{w}$-column and $\mathrm{b}$-row respectively equal to $0$
(with $\mathrm{w}$ and $\mathrm{b}$ as shown in Figure 8), we get

$D_{\mathrm{w}}=\left(\frac{c_{1}}{s_{2}(c_{1}^{2}+s_{1}^{2})},\frac{s_{1}}{c_{2}(c_{1}^{2}+s_{1}^{2})}\right)\text{
and
}D_{\mathrm{b}}=\left(\frac{s_{2}}{c_{1}(c_{2}^{2}+s_{2}^{2})},\frac{c_{2}}{s_{1}(c_{2}^{2}+s_{2}^{2})}\right).$

Note that $P_{(G^{\square},\mathrm{wt}^{\square},\kappa)}(z,w)$ is invariant
under the involution $\sigma:(z,w)\mapsto(z^{-1},w^{-1})$. Using
$c_{1}^{2}+s_{1}^{2}=c_{2}^{2}+s_{2}^{2}=1$, we also see that
$D_{\mathrm{b}}=\sigma(D_{\mathrm{w}})$. Both of these properties are true for
all spectral curves and standard divisors that arise from the Ising model and
characterize them as we will prove in Theorem 6.1.

A spectral data associated to $\Gamma$ is a triple $(C,D,\nu_{\Gamma})$ where

  1. (1)

$C$ is a Harnack curve with Newton polygon $N_{\Gamma}$,

  2. (2)

$D$ is a standard divisor on $C$,

  3. (3)

$\nu_{\Gamma}:Z_{\Gamma}\rightarrow\bigsqcup_{S\text{ side of }N}C\cap D_{S}$
is a bijection between zig-zag paths and points at infinity such that for
every side $S$ of $N_{\Gamma}$, $\nu_{\Gamma}(Z_{\Gamma,S})=C\cap D_{S}$.

Let $\mathcal{S}_{\Gamma}$ denote the set of spectral data associated to
$\Gamma$.

The spectral transform is the map

$\lambda_{(\Gamma,\mathrm{v})}:\mathcal{X}_{\Gamma}(\mathbb{R}_{>0})\times
H^{1}(\mathbb{T},\\{\pm 1\\})\rightarrow\mathcal{S}_{\Gamma}$

sending $([\mathrm{wt}],[\kappa])$ to the spectral data
$(C,D_{\mathrm{v}},\nu_{\Gamma})$ where

  1. (1)

$C$ is the spectral curve,

  2. (2)

$D_{\mathrm{v}}$ is the divisor as in (3.5),

  3. (3)

$\nu_{\Gamma}$ is the bijection in (3.4).

Although the Kasteleyn matrix depends on the choice of cochains $\mathrm{wt}$
and $\kappa$ representing $[\mathrm{wt}]$ and $[\kappa]$, the spectral data
only depends on the cohomology classes $[\mathrm{wt}]$ and $[\kappa]$.

######  Theorem 3.5 ([KO06]).

The spectral transform is a bijection.

$1$$\kappa_{2}$$\kappa_{1}$$\kappa_{4}$$\kappa_{3}$$1$$1$$\kappa_{1}$$\kappa_{4}$$\kappa_{3}$$\kappa_{2}$$1$$\longleftrightarrow$$\longleftrightarrow$$1$$-1$
Figure 9. Transformation of $\kappa$ under moves.

Suppose $\Gamma\stackrel{{\scriptstyle s}}{{\rightsquigarrow}}\Gamma^{\prime}$
is a move. By a gauge transformation, we can assume that $\mathrm{wt}$ and
$\kappa$ are as shown on one side of Figure 9. Then $\mu_{s}([\kappa])$ is as
shown on the other side.

Finally, we need a construction of Fock [Foc15] called the discrete Abel map.
Recall that $\pi:\mathbb{R}^{2}\rightarrow\mathbb{T}$ is the universal cover.
Let $\tilde{\Gamma}:=\pi^{-1}(\Gamma)$ denote the bi-periodic graph in the
plane associated to $\Gamma$. The discrete Abel map
$\mathbf{d}_{\tilde{\Gamma}}$ associates to each vertex of $\tilde{\Gamma}$ a
divisor at infinity of $C$ and is defined as follows:

  1. (1)

Let $\mathbf{d}_{\tilde{\Gamma}}({\mathrm{w}})=0$ where $\mathrm{w}$ is a
fixed white vertex (this is a normalization).

  2. (2)

For any edge $\\{\mathrm{b},\mathrm{w}\\}$ contained in the fundamental
rectangle with zig-zag paths $\alpha,\beta$ passing through it,

$\mathbf{d}_{\tilde{\Gamma}}(\mathrm{b})-\mathbf{d}_{\tilde{\Gamma}}(\mathrm{w})=\nu_{\Gamma}(\alpha)+\nu_{\Gamma}(\beta).$

Then
$\mathbf{d}_{\tilde{\Gamma}}({\mathrm{v}}+(i,j))=\mathbf{d}_{\tilde{\Gamma}}({\mathrm{v}})+\operatorname{div}z^{i}w^{j}$
so $\mathbf{d}_{\tilde{\Gamma}}$ is not well-defined on $\Gamma$ (different
lifts of a vertex have different values). However, since $z^{i}w^{j}$ is a
rational function, the linear equivalence class of
$\mathbf{d}_{\tilde{\Gamma}}(\mathrm{v})$ is well-defined for every vertex
$\mathrm{v}$ of $\Gamma$ independent of the choice of lift of $\mathrm{v}$ to
$\tilde{\Gamma}$; we denote this linear equivalence class by
$\mathbf{d}_{\Gamma}({\mathrm{v}})$.

A move $\Gamma\stackrel{{\scriptstyle s}}{{\rightsquigarrow}}\Gamma^{\prime}$
induces a discrete Abel map $\mathbf{d}_{\Gamma^{\prime}}$ as follows: There
is a unique way to define $\mathbf{d}_{\Gamma^{\prime}}$ on $\Gamma^{\prime}$
so that
$\mathbf{d}_{\Gamma^{\prime}}(\mathrm{v})=\mathbf{d}_{\Gamma}(\mathrm{v})$ for
vertices $\mathrm{v}$ that are in both $\Gamma$ and $\Gamma^{\prime}$;
essentially we want the two discrete Abel maps to be consistently normalized.

The following theorem summarizes how the spectral data changes upon doing
moves and changing the vertex $\rm v$.

######  Theorem 3.6.

Suppose $\Gamma\stackrel{{\scriptstyle s}}{{\rightsquigarrow}}\Gamma^{\prime}$
is a sequence of moves. Then the following diagram commutes

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}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},$

where the map $\mathcal{S}_{\Gamma}\rightarrow\mathcal{S}_{\Gamma^{\prime}}$
is given by $(C,D_{\rm v},\nu_{\Gamma})\mapsto(C,D_{{\rm
v}^{\prime}},\nu_{\Gamma^{\prime}})$ where

  1. (1)

$D_{\mathrm{v}^{\prime}}$ is the unique degree $g$ effective divisor
satisfying (where $\sim$ denotes linear equivalence of divisors)

    1. (a)

$D_{\rm v}+\mathbf{d}_{\Gamma}(\rm v)\sim D_{{\rm
v}^{\prime}}+\mathbf{d}_{\Gamma^{\prime}}({\rm v}^{\prime})$ if $\mathrm{v}$
and $\mathrm{v}^{\prime}$ have the same color.

    2. (b)

$D_{\rm v}+D_{\mathrm{v}^{\prime}}\sim
K_{C}+\mathbf{d}_{\Gamma}({\mathrm{v}})-\mathbf{d}_{\Gamma^{\prime}}(\mathrm{v}^{\prime})$
if $\mathrm{v}$ is black and $\mathrm{v}^{\prime}$ is white, where $K_{C}$ is
the canonical divisor of $C$.

  2. (2)

$\nu_{\Gamma^{\prime}}(s_{*}(\alpha)):=\nu_{\Gamma}(\alpha).$

Theorem 3.6(1)(a) is due to Fock [Foc15] and Theorem 3.6(1)(b) is well known;
see for example [GGK23, Corollary 6.2] or [BCdT23, Lemma 32].

##  4\. Color change

Bipartite torus graphs have a global transformation which will play an
important role in this paper. Let
$(\overline{\Gamma},[\overline{\mathrm{wt}}])$ be the weighted bipartite graph
obtained from $(\Gamma,[\mathrm{wt}])$ by changing the colors of all the
vertices and keeping the weights of all edges the same. For a vertex ${\rm v}$
in $\Gamma$ let $\overline{\rm v}$ denote the corresponding oppositely colored
vertex in $\overline{\Gamma}$. So if $\mathrm{wt}$ is an edge weight
representing $[\mathrm{wt}]$ and $e=\\{\mathrm{b},\mathrm{w}\\}$ is an edge of
$\Gamma$, then

$\overline{\mathrm{wt}}(\\{\overline{\mathrm{w}},\overline{\mathrm{b}}\\})=\mathrm{wt}(\\{\mathrm{b},\mathrm{w}\\})$

is an edge weight representing $[\overline{\mathrm{wt}}]$.

For a cycle $\gamma$ in $\Gamma$, let $\overline{\gamma}$ denote the
corresponding cycle in $\overline{\Gamma}$. Then
$\alpha\mapsto-\overline{\alpha}$ is a bijection between $Z_{\Gamma}$ and
$Z_{\overline{\Gamma}}$ so $N_{\overline{\Gamma}}=-N_{\Gamma}$.

If $\kappa$ is a Kasteleyn sign on $\Gamma$, then $\overline{\kappa}$ is a
Kasteleyn sign on $\overline{\Gamma}$. The Kasteleyn matrices of $\Gamma$ and
$\overline{\Gamma}$ are related by

(4.1)
$K_{(\overline{\Gamma},\overline{\mathrm{wt}},\overline{\kappa})}(z,w)_{\overline{\mathrm{b}}\overline{\mathrm{w}}}=K_{(\Gamma,\mathrm{wt},\kappa)}(z^{-1},w^{-1})_{\mathrm{w}\mathrm{b}}^{T},$

so we get

$P_{(\overline{\Gamma},\overline{\mathrm{wt}},\overline{\kappa})}(z,w)=P_{(\Gamma,\mathrm{wt},\kappa)}(z^{-1},w^{-1}).$

Let $\sigma:(\mathbb{C}^{\times})^{2}\rightarrow(\mathbb{C}^{\times})^{2}$
denote the involution $(z,w)\mapsto(z^{-1},w^{-1})$. Recall that
$Q_{(\Gamma,\mathrm{wt},\kappa)}(z,w)$ denotes the cofactor matrix of
$K_{(\Gamma,\mathrm{wt},\kappa)}(z,w)$ and that for a vertex $\rm v$ of
$\Gamma$, the divisor $D_{\rm v}$ is defined by the vanishing of the row or
column of $Q_{(\Gamma,\mathrm{wt},\kappa)}(z,w)$ corresponding to $\rm v$.
From (4.1), we get

$Q_{(\overline{\Gamma},\overline{\mathrm{wt}},\overline{\kappa})}(z,w)_{\overline{\mathrm{w}}\overline{\mathrm{b}}}=Q_{(\Gamma,\mathrm{wt},\kappa)}(z^{-1},w^{-1})_{\mathrm{b}\mathrm{w}}^{T},$

which implies that $D_{\overline{\rm v}}=\sigma(D_{\rm v})$.

Since
$X_{\overline{\alpha}}([\overline{\mathrm{wt}}])=X_{\alpha}([\mathrm{wt}])^{-1}$,
multiplying (3.4) by $-1$ we get

(4.2)
$\nu_{\overline{\Gamma}}(\overline{\alpha})=\sigma(\nu_{\Gamma}(\alpha)).$

We have shown:

######  Proposition 4.1.

The following diagram commutes:

${\mathcal{X}_{\Gamma}(\mathbb{R}_{>0})\times H^{1}(\mathbb{T},\\{\pm
1\\})}$${\mathcal{X}_{\overline{\Gamma}}(\mathbb{R}_{>0})\times
H^{1}(\mathbb{T},\\{\pm
1\\})}$${\mathcal{S}_{\Gamma}}$${\mathcal{S}_{\overline{\Gamma}}}$$\scriptstyle{\lambda_{(\Gamma,\mathrm{v})}}$$\scriptstyle{\lambda_{(\overline{\Gamma},\overline{\mathrm{v}})}}$

where the top map is
$([\mathrm{wt}],[\kappa])\mapsto([\overline{\mathrm{wt}}],[\overline{\kappa}])$
and the bottom map is
$(C,D,\nu_{\Gamma})\mapsto(\sigma(C),\sigma(D),\nu_{\overline{\Gamma}})$ with
$\nu_{\overline{\Gamma}}$ defined as in (4.2).

##  5\. A characterization of
$\mathcal{I}_{G}(\mathbb{R}_{>0})\subset\mathcal{X}_{G^{\square}}(\mathbb{R}_{>0})$

If we apply square moves at all the square faces of $G^{\square}$, then the
resulting graph is $\overline{G^{\square}}$. Let
$s_{*}:H_{1}(G^{\square},\mathbb{Z})\rightarrow
H_{1}(\overline{G^{\square}},\mathbb{Z})$ denote the induced map of cycles and
$\mu:\mathcal{X}_{G^{\square}}(\mathbb{R}_{>0})\rightarrow\mathcal{X}_{\overline{G^{\square}}}(\mathbb{R}_{>0})$
be induced map of weights. Therefore, starting with
$(G^{\square},[\mathrm{wt}])$, there are two ways to get weights on
$\overline{G^{\square}}$: $[\overline{\mathrm{wt}}]$ and $\mu([\mathrm{wt}])$.
The main result of this section is:

######  Theorem 5.1.

The subset
$\mathcal{I}_{G}(\mathbb{R}_{>0})\subset\mathcal{X}_{G^{\square}}(\mathbb{R}_{>0})$
is the set of weights such that $\mu([\mathrm{wt}])=[\overline{\mathrm{wt}}]$.

The proof of Theorem 5.1 relies on the following lemma.

######  Lemma 5.2.

Suppose $[\mathrm{wt}]\in\mathcal{X}_{G^{\square}}(\mathbb{R}_{>0})$ is such
that $\mu([\mathrm{wt}])=[\overline{\mathrm{wt}}].$ Then $[\mathrm{wt}]$ is
determined by $(X_{f}([\mathrm{wt}]))_{\text{square faces }f}$.

###### Proof.

Let $\gamma$ be a cycle in $G^{\square}$. Then we have
$X_{\overline{\gamma}}([\overline{\mathrm{wt}}])=X_{\gamma}([\mathrm{wt}])^{-1}$.
On the other hand, by (3.2), we have

(5.1) $\overline{\gamma}=s_{*}(\gamma)+\sum_{\text{square faces
}f}a_{f}s_{*}(f),$

for some $a_{f}\in\mathbb{Z}$. Using (3.3), we get

$X_{\overline{\gamma}}(\mu([\mathrm{wt}]))=X_{\gamma}([\mathrm{wt}])\prod_{\text{square
faces
}f}X_{f}([\mathrm{wt}])^{a_{f}}(1+X_{f}([\mathrm{wt}]))^{-\langle\gamma,f\rangle}.$

Since $\mu([\mathrm{wt}])=[\overline{\mathrm{wt}}]$, we have

(5.2) $X_{\gamma}([\mathrm{wt}])^{2}=\prod_{\text{square faces
}f}X_{f}([\mathrm{wt}])^{-a_{f}}(1+X_{f}([\mathrm{wt}]))^{\langle\gamma,f\rangle}.$

Therefore, every $X_{\gamma}^{2}$ is a function of $(X_{f})_{\text{square
faces }f}$. For positive weights, there is a natural square root, so
$X_{\gamma}$ is determined by $(X_{f})_{\text{square faces }f}$. ∎

$f_{2}$$f_{1}$$f_{3}$$f_{4}$$b$$a$(a)$\overline{f_{2}}$$\overline{f_{1}}$$\overline{f_{3}}$$\overline{f_{4}}$$s_{*}(b)$$s_{*}(a)$(b)
Figure 10. (a) A basis for $H_{1}(G^{\square},\mathbb{Z})$ and (b) the cycles
corresponding to $a$ and $b$ in $\overline{G^{\square}}$. From (b), we see
that $\overline{a}=s_{*}(a)-\overline{f_{2}}=s_{*}(a)+s_{*}(f_{2})$ and
similarly the equation for $\overline{b}$.

######  Example 5.3.

Consider the basis $\\{f_{1},f_{2},f_{3},a,b\\}$ for
$H_{1}(G^{\square},\mathbb{Z})$ shown in Figure 10(a). Then the equations
(5.1) are (see Figure 10(b))

$\displaystyle\overline{f_{1}}$ $\displaystyle=s_{*}(f_{1})-2s_{*}(f_{1}),$
$\displaystyle\overline{f_{2}}$ $\displaystyle=s_{*}(f_{2})-2s_{*}(f_{2}),$
$\displaystyle\overline{f_{3}}$
$\displaystyle=s_{*}(f_{3})+2s_{*}(f_{1})+2s_{*}(f_{2}),$
$\displaystyle\overline{a}$ $\displaystyle=s_{*}(a)+s_{*}(f_{2}),$
$\displaystyle\overline{b}$ $\displaystyle=s_{*}(b)-s_{*}(f_{1}),$

and the equations (5.2) are

$\displaystyle X_{f_{1}}^{2}$ $\displaystyle=X_{f_{1}}^{2},$ $\displaystyle
X_{f_{2}}^{2}$ $\displaystyle=X_{f_{2}}^{2},$ $\displaystyle X_{f_{3}}^{2}$
$\displaystyle=X_{f_{1}}^{-2}X_{f_{2}}^{-2}(1+X_{f_{1}})^{2}(1+X_{f_{2}})^{2},$
$\displaystyle X_{a}^{2}$
$\displaystyle=X_{f_{2}}(1+X_{f_{1}})^{-1}(1+X_{f_{2}})^{-1},$ (5.3)
$\displaystyle X_{b}^{2}$
$\displaystyle=X_{f_{1}}^{-1}(1+X_{f_{1}})(1+X_{f_{2}}).$

Theorem 5.1 says that the subset
$\mathcal{I}_{G}(\mathbb{R}_{>0})\subset\mathcal{X}_{G^{\square}}(\mathbb{R}_{>0})$
is defined by these five equations (two of which are trivial). Using the
weights in Figure 1(c), we compute

$\displaystyle X_{f_{1}}=\frac{s_{1}^{2}}{c_{1}^{2}},\quad
X_{f_{2}}=\frac{s_{2}^{2}}{c_{2}^{2}},\quad
X_{f_{3}}=\frac{1}{s_{1}^{2}s_{2}^{2}},\quad X_{a}=c_{1}s_{2},\quad
X_{b}=\frac{1}{c_{2}s_{1}}.$

Plugging into (5.3), we see that all the equations are satisfied.

###### Proof of Theorem 5.1.

Let $[\mathrm{wt}]$ be a weight such that
$\mu([\mathrm{wt}])=[\overline{\mathrm{wt}}]$. For an edge $e\in E(G)$, let
$f$ denote the corresponding square in $G^{\square}$. Consider the Ising
weight $\mathrm{wt}^{\square}$ on $G^{\square}$ with
$s_{e}:=\sqrt{\frac{X_{f}}{1+X_{f}}}$ and $c_{e}:=\sqrt{\frac{1}{1+X_{f}}}$.
Then $[\mathrm{wt}^{\square}]$ also satisfies

$\mu([\mathrm{wt}^{\square}])=[\overline{\mathrm{wt}^{\square}}].$

Since $X_{f}([\mathrm{wt}^{\square}])=X_{f}([\mathrm{wt}])$ for all square
faces $f$, $[\mathrm{wt}^{\square}]=[\mathrm{wt}]$ by Lemma 5.2.

∎

##  6\. Ising spectral data

The goal of this section is to describe the subset of spectral data that
corresponds to Ising models, i.e., the image of

$\mathcal{I}_{G}(\mathbb{R}_{>0})\hookrightarrow\mathcal{X}_{G^{\square}}(\mathbb{R}_{>0})\rightarrow\mathcal{S}_{G^{\square}}.$

######  Theorem 6.1.

Let $G$ be a minimal graph in $\mathbb{T}$, $G^{\square}$ the corresponding
bipartite graph, ${\mathrm{w}}$ a white vertex of $G^{\square}$ and $[\kappa]$
a Kasteleyn sign on $G^{\square}$. Suppose ${\mathrm{b}}$ denotes the black
vertex incident to $\mathrm{w}$ that is not a part of the square containing
$\mathrm{w}$ and let $\alpha,\overline{\alpha}$ denote the zig-zag paths that
contain the edge $\\{\mathrm{b},\mathrm{w}\\}$. Let
$[\mathrm{wt}]\in\mathcal{X}_{G^{\square}}(\mathbb{R}_{>0})$ and let
$(C,D_{{\mathrm{w}}},\nu_{G^{\square}})=\lambda_{(G^{\square},{\mathrm{w}})}([\mathrm{wt}],[\kappa])$.
Then $[\mathrm{wt}]\in\mathcal{I}_{G}(\mathbb{R}_{>0})$ if and only if

  1. (1)

$C$ is invariant under the involution
$\sigma:(z,w)\rightarrow(z^{-1},w^{-1})$.

  2. (2)

$D_{{\mathrm{w}}}$ is a standard divisor satisfying

$D_{{\mathrm{w}}}+\sigma(D_{{\mathrm{w}}})\sim
K_{C}+\nu_{G^{\square}}(\alpha)+\nu_{G^{\square}}(\overline{\alpha}).$

Such divisors are in the Prym variety of $C$ (see e.g., [vMM79, Section 3]).
If $C$ is singular, we replace it with its desingularization $\tilde{C}$ as in
Remark 3.3.

  3. (3)

For every zig-zag path $\alpha$ of $G^{\square}$ we have

$\nu_{G^{\square}}(\overline{\alpha})=\sigma(\nu_{G^{\square}}(\alpha)).$

###### Proof.

Since
$\mathbf{d}_{G^{\square}}(\mathrm{b})-\mathbf{d}_{G^{\square}}(\mathrm{w})=\nu_{G^{\square}}(\alpha)+\nu_{G^{\square}}(\overline{\alpha})$,
by Theorem 3.6(1)(b) the condition (2) is equivalent to

$(2^{\prime})\leavevmode\nobreak\ D_{{\mathrm{w}}}=\sigma(D_{{\mathrm{b}}}).$

Now we have

$\displaystyle[\mathrm{wt}]\in\mathcal{I}_{G}(\mathbb{R}_{>0})$
$\displaystyle\Longleftrightarrow[\overline{\mathrm{wt}}]=\mu([\mathrm{wt}])$
(Theorem 5.1)
$\displaystyle\Longleftrightarrow\lambda_{(\overline{\Gamma},\overline{{\mathrm{b}}})}([\overline{\mathrm{wt}}],[\overline{\kappa}])=\lambda_{(\overline{\Gamma},\overline{{\mathrm{b}}})}(\mu([\mathrm{wt}]),\mu([\kappa]))$
(Theorem 3.5 and $\mu([\kappa])=[\overline{\kappa}]$ (Figure 9))
$\displaystyle\Longleftrightarrow(\sigma(C),\sigma(D_{{\mathrm{b}}}),\sigma(\nu_{G^{\square}}(\overline{\cdot})))=(C,D_{{\mathrm{w}}},\nu_{G^{\square}}(\cdot))$
(Proposition 4.1 and Theorem 3.6) $\displaystyle\Longleftrightarrow\text{(1),
($2^{\prime}$) and (3)}.$

∎

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