Zhegalkin Zebra Motives Digital Recordings of Mirror Symmetry

Zhegalkin Zebra Motives
Digital Recordings of Mirror Symmetry††This paper is a contribution to the
Special Issue on Modular Forms and String Theory in honor of Noriko Yui. The
full collection is available at http://www.emis.de/journals/SIGMA/modular-
forms.html

Jan STIENSTRA

J. Stienstra

Department of Mathematics, Utrecht University,
P.O. Box 80010, 3508 TA Utrecht, The Netherlands

<EMAIL_ADDRESS>
Received May 10, 2018, in final form October 02, 2018; Published online
October 13, 2018

Zhegalkin zebra motives are tilings of the plane by black and white polygons
representing certain ${\mathbb{F}}_{2}$-valued functions on
${\mathbb{R}}^{2}$. They exhibit a rich geometric structure and provide easy
to draw insightful visualizations of many topics in the physics and
mathematics literature. The present paper gives some pieces of a general
theory and a few explicit examples. Many more examples will be shown in the
forthcoming article “Zhegalkin zebra motives: algebra and geometry in black
and white”.

Zhegalkin polynomials; motives; dimer models; mirror symmetry 52C20; 82B20;
14M25

###### Contents

  1. 1 Introduction
    1. 1.1 Zhegalkin zebra functions
    2. 1.2 The superpotential and weight functions
    3. 1.3 Dessins d’enfants
  2. 2 The algebraic geometry of weights
    1. 2.1 The (co)homology of $\Gamma_{\\!\\!\Lambda}$ and $\Gamma_{\\!\\!\Lambda}^{\vee}$
    2. 2.2 The geometry of ${\mathsf{Proj}(\mathbb{Z}[\mathcal{W}_{\Lambda}])}$
    3. 2.3 The geometry of ${\mathsf{Spec}(\mathbb{Z}[\mathcal{W}_{\Lambda}])}$
    4. 2.4 The geometry of a weight realization
  3. 3 The Jacobi algebra
    1. 3.1 General theory of the Jacobi algebra and master space
    2. 3.2 Weight realizations and Jacobi algebra
  4. 4 Symptoms of mirror symmetry
  5. 5 Practical matters
    1. 5.1 How to draw the picture of the tiling of $\mathcal{F}$
    2. 5.2 How to compute the superpotential $[\mathcal{F}]_{\Lambda}$ and the realization $\omega_{\mathcal{F}}$
    3. 5.3 How to compute the lattice and the quadrangles of a weight realization
  6. References

##  1 Introduction

###  1.1 Zhegalkin zebra functions

The constructions of motives in algebraic geometry heavily depend on the
intersection theory of algebraic cycles and, hence, on the fairly delicate
choice of an adequate equivalence relation on algebraic cycles. Chow motives,
for instance, are based on rational equivalence, which is the finest
equivalence relation on algebraic cycles yielding a good intersection theory
[15].

On the contrary, the “motives” in the present paper are built with the usual
set theoretical operations from simple subsets of the plane, which we call
zebras. In 1927 Zhegalkin pointed out that functions with values in the field
$\mathbb{F}_{2}=\mathbb{Z}/2\mathbb{Z}$ with the usual addition and
multiplication can replace the standard Boolean formalism. The zebra with
frequency $\mathfrak{v}\in\mathbb{R}^{2}$, $\mathfrak{v}\neq\mathsf{0}$, is
the function on $\mathbb{R}^{2}$ given by

$\displaystyle\mathsf{Z}^{\mathfrak{v}}(\mathbf{x})=\lfloor
2\mathbf{x}\raisebox{0.86108pt}{$\centerdot$}\mathfrak{v}\rfloor\bmod
2\qquad\textrm{for}\quad\mathbf{x}\in\mathbb{R}^{2}.$ (1.1)

Here $\centerdot$ is the dot product on $\mathbb{R}^{2}$ and for a real number
$r$ the integer $\lfloor r\rfloor$ is such that $0\leq r-\lfloor r\rfloor<1$.
It is sometimes convenient to identify the Euclidean plane $\mathbb{R}^{2}$
and the complex plane $\mathbb{C}$. In this paper we only use zebras for which
the frequencies are positive integer multiples of the complex numbers

$\displaystyle\mathfrak{v}_{1}=\sqrt{3}\varepsilon^{5},\qquad\mathfrak{v}_{2}=\varepsilon^{4},\qquad\mathfrak{v}_{3}=\sqrt{3}\mathbf{i},\qquad\mathfrak{v}_{4}=\varepsilon^{2},\qquad\mathfrak{v}_{5}=\sqrt{3}\varepsilon,\qquad\mathfrak{v}_{6}=1,$
(1.2)

with $\mathbf{i}=\sqrt{-1}$ and $\varepsilon=e^{\pi\mathbf{i}/6}$; see Fig. 1.

Figure 1: Left: (1.2). Right: Zebra $\mathsf{Z}^{\mathfrak{v}}$. All bands are
$\perp\mathfrak{v}$ and have width $\frac{1}{2|\mathfrak{v}|}$.

######  Definition 1.1.

We denote the zebra with frequency $k\mathfrak{v}_{j}$ by $\mathsf{Z}^{jk}$.
The elements of the ring of $\mathbb{F}_{2}$-valued functions on
$\mathbb{R}^{2}$ generated by the zebras $\mathsf{Z}^{jk}$ are called
Zhegalkin zebra functions. Such a function $\mathcal{F}$ gives a tiling of the
plane by white ($\mathcal{F}=0$) and black ($\mathcal{F}=1$) polygons. A
Zhegalkin zebra function is _convex_ (czzf) if all polygons in the tiling are
bounded and convex. These tilings are the Zhegalkin zebra motives in the
title.

######  Goal.

Understand the deformation theory of these tilings.

Every Zhegalkin zebra function can be written as a Zhegalkin zebra polynomial,
i.e., a polynomial in the variables $\mathsf{Z}^{jk}$ in which all monomials
have coefficients $1$ and the variables in each monomial have exponent $1$.
Section 5.1 describes an efficient way for evaluating a Zhegalkin zebra
polynomial and drawing the black-white picture. Convexity can easily be
checked by visual inspection. See Figs. 2, 3, 6 and 12 for examples.

A Zhegalkin zebra function $\mathcal{F}$ has an automorphism group consisting
of translations leaving the tiling invariant:

$\displaystyle\mathsf{Aut}(\mathcal{F})=\big{\\{}\tau\in\mathbb{R}^{2}\,|\,\mathcal{F}(\mathbf{x}+\tau)=\mathcal{F}(\mathbf{x}),\,\forall\,\mathbf{x}\in\mathbb{R}^{2}\big{\\}}.$

This is a lattice in $\mathbb{R}^{2}$ if $\mathcal{F}$ is convex. For every
sublattice $\Lambda\subset\mathsf{Aut}(\mathcal{F})$ the function
$\mathcal{F}$ descends to a function on the torus $\mathbb{R}^{2}/\Lambda$ and
gives a tiling of this torus by black and white polygons.

This brings us to the setting of dimer models (a.k.a. brane tilings), quivers
with superpotential and discrete differential geometry. There is an extensive
literature on these topics written from very different view-points, with very
different terminologies, for very different applications. Our view-point will
be that the pictures are realizations of an underlying combinatorial
structure. Our goal is: Understand the deformations of these realizations.

###  1.2 The superpotential and weight functions

The combinatorial structure consists of the set $\mathcal{E}$ of edges in the
picture, two permutations $\sigma_{0}$, $\sigma_{1}$ of $\mathcal{E}$ and an
injective homomorphism
$\mathfrak{p}\colon\mathbb{Z}^{2}\rightarrow\mathsf{Perm}(\mathcal{E})$ into
the permutation group of $\mathcal{E}$. The cycles (= orbits) of $\sigma_{0}$
and $\sigma_{1}$ correspond to the oriented boundaries of the white and black
polygons, while the cycles of the permutation
$\sigma_{2}=\sigma_{1}^{-1}\sigma_{0}$ correspond to the vertices in the
tiling. The orientation of the edges is such that the boundaries of the black
(resp. white) polygons are oriented clockwise (resp. counter-clockwise). The
homomorphism $\mathfrak{p}$ comprises the action of
$\mathsf{Aut}(\mathcal{F})$ and an isomorphism
$\mathsf{Aut}(\mathcal{F})\simeq\mathbb{Z}^{2}$. The permutations $\sigma_{0}$
and $\sigma_{1}$ commute with this action. Associated with a sublattice
$\Lambda$ of $\mathbb{Z}^{2}$ is then the finite set
${\mathcal{E}_{\Lambda}}=\mathcal{E}/\Lambda$ equipped with the permutations
$\sigma_{0}$, $\sigma_{1}$, $\sigma_{2}$ and an action of the finite group
$\mathbb{Z}^{2}/\Lambda$. Since the torus $\mathbb{R}^{2}/\Lambda$ has genus
$1$ the numbers of cycles of the permutations satisfy
$|\sigma_{0}|+|\sigma_{1}|+|\sigma_{2}|=|{\mathcal{E}_{\Lambda}}|$.

######  Definition 1.2.

We call
$[\mathcal{F}]_{\Lambda}=({\mathcal{E}_{\Lambda}},\sigma_{0},\sigma_{1})$ the
superpotential of the Zhegalkin zebra function $\mathcal{F}$ and the lattice
$\Lambda$.

######  Remark 1.3.

In the physics literature, e.g., [12], one writes the superpotential as a sum
of $|\sigma_{0}|+|\sigma_{1}|$ monomials in non-commuting variables
$\big{\\{}\widetilde{X}_{e}\big{\\}}_{e\in{\mathcal{E}_{\Lambda}}}$ such that
a cycle $(e_{1},\ldots,e_{r})$ of $\sigma_{c}$, $c=0,1$, contributes the
monomial $(-1)^{c}\widetilde{X}_{e_{1}}\cdots\widetilde{X}_{e_{r}}$. In this
formulation the cyclic structure is implicit.

$\begin{array}[]{l}\mathcal{F}_{2}=\mathsf{Z}^{21}+\mathsf{Z}^{41}\\\
\mathsf{Aut}(\mathcal{F}_{2})=\mathbb{Z}\mathfrak{v}_{6}\oplus\frac{1}{3}\mathbb{Z}\mathfrak{v}_{3}\\\
\sigma_{0}=(1,2,3,4)\\\ \sigma_{1}=(1,4,3,2)\end{array}$
$\begin{array}[]{l}\mathcal{F}_{4}=\mathsf{Z}^{21}+\mathsf{Z}^{31}+\mathsf{Z}^{41}+\mathsf{Z}^{61}\\\
\mathsf{Aut}(\mathcal{F}_{4})=\mathbb{Z}\mathfrak{v}_{6}\oplus\frac{1}{3}\mathbb{Z}\mathfrak{v}_{3}\\\
\sigma_{0}=(1,2,3)(4,5,6)(7,8,9)(10,11,12)\\\
\sigma_{1}=(1,11,9)(4,8,12)(7,2,6)(10,5,3)\end{array}$

Figure 2: Some classical patterns realized as czzf, with their automorphism
groups and corresponding superpotentials.

$\begin{array}[]{l}\mathcal{F}_{3}=\mathsf{Z}^{21}+\mathsf{Z}^{41}+\mathsf{Z}^{61}\\\
\mathsf{Aut}(\mathcal{F}_{3})=\mathbb{Z}\mathfrak{v}_{6}\oplus\frac{1}{3}\mathbb{Z}\mathfrak{v}_{5}\\\
\sigma_{0}=(1,2,3)\\\ \sigma_{1}=(1,3,2)\end{array}$
$\begin{array}[]{l}\mathcal{F}_{6}=\mathsf{Z}^{11}+\mathsf{Z}^{21}+\mathsf{Z}^{31}+\mathsf{Z}^{41}+\mathsf{Z}^{51}+\mathsf{Z}^{61}\\\
\mathsf{Aut}(\mathcal{F}_{6})=\frac{1}{3}\mathbb{Z}\mathfrak{v}_{1}\oplus\frac{1}{3}\mathbb{Z}\mathfrak{v}_{3}\\\
\sigma_{0}=(1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)\\\
\sigma_{1}=(1,8,18)(4,11,3)(7,14,6)(10,17,9)(13,2,12)(16,5,15)\end{array}$

Figure 3: Some classical patterns realized as czzf, with their automorphism
groups and corresponding superpotentials.

The superpotential
$[\mathcal{F}]_{\Lambda}=({\mathcal{E}_{\Lambda}},\sigma_{0},\sigma_{1})$
provides three abstract graphs

$\displaystyle\Gamma_{\\!\\!\Lambda}=\big{(}s,t\colon{\mathcal{E}_{\Lambda}}\rightrightarrows\mathsf{P}_{\\!\Lambda}^{\star}\big{)},\qquad\Gamma_{\\!\\!\Lambda}^{\vee}=\big{(}b,w\colon{\mathcal{E}_{\Lambda}}\rightrightarrows\mathsf{P}_{\\!\Lambda}^{\bullet}\cup\mathsf{P}_{\\!\Lambda}^{\circ}\big{)},$
$\displaystyle\mathcal{D}_{\mathcal{F},\Lambda}=\textsf{vertices}\colon\
\mathsf{P}_{\\!\Lambda}^{\star}\cup\mathsf{P}_{\\!\Lambda}^{\bullet}\cup\mathsf{P}_{\\!\Lambda}^{\circ},$
$\displaystyle\hphantom{\mathcal{D}_{\mathcal{F},\Lambda}=}{}\
\textsf{edges}\colon\
\\{(\mathbf{w},\mathbf{v})\in\mathsf{P}_{\\!\Lambda}^{\circ}\times\mathsf{P}_{\\!\Lambda}^{\star}\,|\,\exists\,e\in{\mathcal{E}_{\Lambda}}\colon\mathbf{w}=w(e),\,\mathbf{v}=t(e)\textrm{
or }s(e)\\}$
$\displaystyle\hphantom{\mathcal{D}_{\mathcal{F},\Lambda}=\textsf{edges}\colon}\
\cup\\{(\mathbf{b},\mathbf{v})\in\mathsf{P}_{\\!\Lambda}^{\bullet}\times\mathsf{P}_{\\!\Lambda}^{\star}\,|\,\exists\,e\in{\mathcal{E}_{\Lambda}}\colon\mathbf{b}=b(e),\,\mathbf{v}=t(e)\textrm{
or }s(e)\\}.$

Here $\mathsf{P}_{\\!\Lambda}^{\star}$, $\mathsf{P}_{\\!\Lambda}^{\bullet}$,
$\mathsf{P}_{\\!\Lambda}^{\circ}$ denote the respective sets of vertices,
black and white polygons in the tiling of the torus $\mathbb{R}^{2}/\Lambda$
and $s$, $t$, $b$, $w$ are the respective maps which assign to an edge its
source, target, adjacent black and white polygons.

In addition to the superpotential the actual pictures also contain a map
$\omega\colon\mathcal{E}\rightarrow\mathbb{R}^{2}\setminus\\{\mathsf{0}\\}$
which specifies for every edge the corresponding vector in $\mathbb{R}^{2}$.
For a sublattice $\Lambda\subset\mathbb{Z}^{2}$ we want this specification to
be $\Lambda$-invariant; i.e., it should be a map
$\omega\colon{\mathcal{E}_{\Lambda}}\rightarrow\mathbb{R}^{2}\setminus\\{\mathsf{0}\\}$.
We call such a map $\omega$ a realization of $[\mathcal{F}]_{\Lambda}$. It
also gives a realization $\Lambda_{\omega}$ of the lattice $\Lambda$ by
translations in the plane which leave the tiling specified by $\omega$
invariant. We denote the corresponding torus by

$\displaystyle\mathbb{T}_{\omega}=\mathbb{R}^{2}/\Lambda_{\omega}.$

The Zhegalkin zebra function $\mathcal{F}$ provides a tiling of
$\mathbb{R}^{2}$ with automorphism group $\mathsf{Aut}(\mathcal{F})$ and hence
a realization $\omega_{\mathcal{F}}$ of $[\mathcal{F}]_{\Lambda}$ for every
sublattice $\Lambda$ of $\mathbb{Z}^{2}$. It identifies $\Lambda$ with a
sublattice $\Lambda_{\omega_{\mathcal{F}}}$ of $\mathsf{Aut}(\mathcal{F})$.

A realization $\omega$ of the superpotential gives a tiling of
$\mathbb{R}^{2}$ which modulo $\Lambda_{\omega}$ gives an embedding of the
quiver (= graph with oriented edges) $\Gamma_{\\!\\!\Lambda}$ into the torus
$\mathbb{T}_{\omega}$ as the 0-cells and 1-cells in the tiling. One can
subsequently embed the graphs $\Gamma_{\\!\\!\Lambda}^{\vee}$ and
$\mathcal{D}_{\mathcal{F},\Lambda}$ into this torus by means of a function
$\theta\colon{\mathcal{E}_{\Lambda}}\rightarrow\mathbb{R}_{>0}$ for which the
sum over each cycle of $\sigma_{0}$ and each cycle of $\sigma_{1}$ is equal to
$1$. This function is used to mark in each black/white polygon a point by
taking a convex combination of the midpoints of its edges. This will be
discussed in detail in Section 5.3. In [10] such a function $\theta$ is called
a (positive) fractional matching. The existence of a fractional matching for
$[\mathcal{F}]_{\Lambda}$ implies
$|\mathsf{P}_{\\!\Lambda}^{\circ}|=|\mathsf{P}_{\\!\Lambda}^{\bullet}|$.

######  Definition 1.4.

An integer weight function for the superpotential $[\mathcal{F}]_{\Lambda}$ is
a map $\nu\colon{\mathcal{E}_{\Lambda}}\rightarrow\mathbb{Z}_{\geq 0}$ for
which the sum over each cycle of $\sigma_{0}$ and each cycle of $\sigma_{1}$
is equal to an integer $\deg\nu$ (the degree of $\nu$). The integer weight
functions with the operation $+$ form a graded semi-group
$\mathcal{W}_{\Lambda}$. An integer weight function of degree $1$ is called a
perfect matching, dimer covering or dimer configuration. The set of perfect
matchings is denoted by $\mathcal{M}_{\Lambda}$.

An integer weight function $\nu$ is said to be positive if $\nu(e)>0$ for all
$e\in{\mathcal{E}_{\Lambda}}$.

Perfect matchings play a crucial role all over the literature on dimer models.
From the permutations $\sigma_{0}$ and $\sigma_{1}$ one can easily check
whether perfect matchings exist and determine them all. Subsequently one can
check whether the sum of all perfect matchings is a positive weight function,
which then divided by its degree $|\mathcal{M}_{\Lambda}|$ yields a positive
fractional matching.

######  Definition 1.5.

We say that the superpotential $[\mathcal{F}]_{\Lambda}$ is dimer complete if
the sum of all perfect matchings is a positive weight function.

If $[\mathcal{F}]_{\Lambda}$ is dimer complete, the semi-group
$\mathcal{W}_{\Lambda}$ is generated by the perfect matchings (see Proposition
2.2):

$\displaystyle\mathcal{W}_{\Lambda}=\mathbb{Z}_{\geq 0}\mathcal{M}_{\Lambda}.$
(1.3)

Rescaling the axes in the picture of the realization $\omega_{\mathcal{F}}$ by
means of the diagonal matrix
$\mathsf{diag}\bigl{(}\frac{1}{2},\frac{\sqrt{3}}{2}\bigr{)}$ yields a
realization of the superpotential $[\mathcal{F}]_{\Lambda}$ with edge vectors
$\omega_{\mathcal{F}}(e)\mathsf{diag}\bigl{(}\frac{1}{2},\frac{\sqrt{3}}{2}\bigr{)}$.
It follows from (1.1) (see also Section 5.2) that the vertices in the rescaled
tiling have coordinates in $\mathbb{Q}$. By further rescaling with a positive
integer factor one can clear the denominators and obtain a realization
$\omega$ of $[\mathcal{F}]_{\Lambda}$ with edge vectors in $\mathbb{Z}^{2}$,
say $\omega(e)=(\omega_{1}(e),\omega_{2}(e))$ with
$\omega_{1},\omega_{2}\colon{\mathcal{E}_{\Lambda}}\rightarrow\mathbb{Z}$.

Now assume that the superpotential $[\mathcal{F}]_{\Lambda}$ is dimer complete
and let $\nu$ be a positive integer weight function. Then, for a sufficiently
large integer $N$ the maps $\nu_{1}=\omega_{1}+N\nu$,
$\nu_{2}=\omega_{2}+N\nu$ and $\nu_{3}=N\nu$ are positive weight functions
with $\deg\nu_{1}=\deg\nu_{2}=\deg\nu_{3}$. Then
$\omega=(\nu_{1}-\nu_{3},\nu_{2}-\nu_{3})$ and
$\theta=\frac{1}{\deg\nu_{3}}\nu_{3}$ yield for every edge $\widetilde{e}$ in
the tiling of $\mathbb{R}^{2}$ four points $s(\widetilde{e})$,
$t(\widetilde{e})$, $b(\widetilde{e})$, $w(\widetilde{e})$, namely the
endpoints of that edge and the marked (by $\theta$) points in the polygons
adjacent to that edge; see Figs. 4, 5, 6 and 8.

These quadrangles (for $e\in{\mathcal{E}_{\Lambda}}$) constitute a tiling of
$\mathbb{R}^{2}$. Taken modulo $\Lambda_{\omega}$ the vertices and edges of
the induced quadrangle-tiling give an embedding of the graph
$\mathcal{D}_{\mathcal{F},\Lambda}$ into the torus $\mathbb{T}_{\omega}$. So
$\mathcal{D}_{\mathcal{F},\Lambda}$ is an S-quad-graph in the sense of [4,
Definitions 3.1 and 4.3].

######  Definition 1.6.

Assume that the superpotential $[\mathcal{F}]_{\Lambda}$ is dimer complete. A
_weight realization_ of $[\mathcal{F}]_{\Lambda}$ is a triple of positive
weight functions $(\nu_{1},\nu_{2},\nu_{3})$ such that
$\omega=(\nu_{1}-\nu_{3},\nu_{2}-\nu_{3})$ is a realization of
$[\mathcal{F}]_{\Lambda}$ in which all black and white polygons and all
quadrangles determined by $\big{(}\omega,\frac{1}{\deg\nu_{3}}\nu_{3}\big{)}$
are strictly convex.

Since it is easy to draw pictures (see Section 5.3) the conditions in
Definition 1.6 can easily be checked by visual inspection. In many examples
one can find weight realizations by staring at the picture of the tiling for
the Zhegalkin zebra function $\mathcal{F}$ drawn with the method of Section
5.1.

One can collect the maps
$s,t\colon{\mathcal{E}_{\Lambda}}\rightarrow\mathsf{P}_{\\!\Lambda}^{\star}$
and
$\nu_{1},\nu_{2},\nu_{3}\colon{\mathcal{E}_{\Lambda}}\rightarrow\mathbb{Z}_{>0}$
into a matrix
${\mathfrak{A}}^{\star\star}\big{(}u_{1}^{\nu_{1}}u_{2}^{\nu_{2}}u_{3}^{\nu_{3}}\big{)}$
as follows. The rows and columns of
${\mathfrak{A}}^{\star\star}\big{(}u_{1}^{\nu_{1}}u_{2}^{\nu_{2}}u_{3}^{\nu_{3}}\big{)}$
correspond with the elements of $\mathsf{P}_{\\!\Lambda}^{\star}$ and its
entries lie in the polynomial ring $\mathbb{Z}[u_{1},u_{2},u_{3}]$; the entry
in row $\mathbf{s}$ and column $\mathbf{t}$ is

$\displaystyle{\mathfrak{A}}^{\star\star}\big{(}u_{1}^{\nu_{1}}u_{2}^{\nu_{2}}u_{3}^{\nu_{3}}\big{)}_{\mathbf{s},\mathbf{t}}=\sum_{e\in{\mathcal{E}_{\Lambda}}:s(e)=\mathbf{s},t(e)=\mathbf{t}}u_{1}^{\nu_{1}(e)}u_{2}^{\nu_{2}(e)}u_{3}^{\nu_{3}(e)}.$
(1.4)

The matrix
${\mathfrak{A}}^{\star\star}\big{(}u_{1}^{\nu_{1}}u_{2}^{\nu_{2}}u_{3}^{\nu_{3}}\big{)}$
can be written uniquely as a sum

$\displaystyle{\mathfrak{A}}^{\star\star}\big{(}u_{1}^{\nu_{1}}u_{2}^{\nu_{2}}u_{3}^{\nu_{3}}\big{)}=\sum_{e\in{\mathcal{E}_{\Lambda}}}\Phi_{\nu_{1},\nu_{2},\nu_{3}}(e)$

of matrices each of which has only one non-zero entry and this entry is a
monomial; see Section 3.2. The algebra generated by the matrices
$\Phi_{\nu_{1},\nu_{2},\nu_{3}}(e)$ is (isomorphic to) the Jacobi algebra
$\mathsf{Jac}([\mathcal{F}]_{\Lambda})$; see Theorem 3.7.

Figure 4: Quadrangles for the matrices
${\mathfrak{A}}^{\star\star}\big{(}u_{1}^{\nu_{1}}u_{2}^{\nu_{2}}u_{3}^{\nu_{3}}\big{)}$
in Example 1.7.

######  Example 1.7.

For $\mathcal{F}_{2}$ and $\mathcal{F}_{3}$ as in Figs. 2 and 3 one has the
weight realizations

$\displaystyle\left(\begin{matrix}0&u_{1}^{3}u_{2}u_{3}^{2}+u_{1}u_{2}^{3}u_{3}^{2}\\\
u_{1}u_{2}u_{3}^{2}+u_{1}^{3}u_{2}^{3}u_{3}^{2}&0\end{matrix}\right)\qquad\text{resp.}\quad
u_{1}^{3}u_{2}u_{3}^{3}+u_{1}^{4}u_{2}^{4}u_{3}^{3}+u_{1}^{2}u_{2}^{4}u_{3}^{3}$

The $2$-cells in these realizations are squares, resp. triangles with angles
$\frac{\pi}{4}$, $\frac{\pi}{4}$, $\frac{\pi}{2}$. The marked points in the
$2$-cells are their barycenters. The quadrangles for these weight realizations
are shown in Fig. 4.

###  1.3 Dessins d’enfants

The quadrangles in a weight realization constitute a tiling of the plane
$\mathbb{R}^{2}$. When taken modulo $\Lambda_{\omega}$ the
$s(\widetilde{e})t(\widetilde{e})$-diagonals and the
$w(\widetilde{e})b(\widetilde{e})$-diagonals show the graphs
$\Gamma_{\\!\\!\Lambda}$ and $\Gamma_{\\!\\!\Lambda}^{\vee}$ embedded in the
torus $\mathbb{T}_{\omega}$ and the duality between them.

######  Example 1.8.

The two ways of putting diagonals in the left-hand picture in Fig. 4 yield the
two pictures in Fig. 2. Both ways of putting diagonals in the right-hand
picture in Fig. 4 lead to triangulations equivalent with the left-hand picture
in Fig. 3.

######  Example 1.9.

The Zhegalkin zebra function shown on the left in Fig. 6 has no fractional
matchings because
$|\mathsf{P}_{\\!\Lambda}^{\circ}|\neq|\mathsf{P}_{\\!\Lambda}^{\bullet}|$.
Nonetheless if one takes the barycentres of the polygons, one finds the tiling
by quadrangles as shown on the right in Fig. 6. The two ways of putting
diagonals lead to respectively the left-hand picture in Fig. 6 and the right-
hand picture in Fig. 3.

Figure 5: Quadrangle for edge $\widetilde{e}$.

$\mathcal{F}=\mathsf{Z}^{21}+\mathsf{Z}^{41}+\mathsf{Z}^{61}+\mathsf{Z}^{62}$,
$\sigma_{0}=(1,4,5,3,2,6)$, $\sigma_{1}=(1,2,5)(3,4,6)$.

Figure 6: A Zhegalkin zebra function without fractional matchings.

Each of its two diagonals divides a quadrangle into two triangles which we
color black/white as indicated in Fig. 5. When the quadrangles are put
together to make a tiling of the plane the colored triangles for the
$s(\widetilde{e})t(\widetilde{e})$-diagonals fuse so as to form the black and
white polygons in a tiling which we want to think of as the deformation,
determined by $(\nu_{1},\nu_{2},\nu_{3})$, of the tiling given by the
Zhegalkin zebra function $\mathcal{F}$.

The weight realization $(\nu_{1},\nu_{2},\nu_{3})$ itself can be deformed by

$\displaystyle(\nu_{1},\nu_{2},\nu_{3})\rightsquigarrow\big{(}N\nu_{1}+\nu^{\prime}_{1}-\nu^{\prime\prime}_{1},N\nu_{2}+\nu^{\prime}_{2}-\nu^{\prime\prime}_{2},N\nu_{3}+\nu^{\prime}_{3}-\nu^{\prime\prime}_{3}\big{)},$
(1.5)

where
$\nu^{\prime}_{1},\nu^{\prime}_{2},\nu^{\prime}_{3},\nu^{\prime\prime}_{1},\nu^{\prime\prime}_{2},\nu^{\prime\prime}_{3}\in\mathcal{W}_{\Lambda}$
are such that $\deg\nu^{\prime}_{j}=\deg\nu^{\prime\prime}_{j}$ for $j=1,2,3$
and $N\in\mathbb{Z}_{\geq 0}$ is so large that the positivity and strict
convexity conditions are satisfied for the deformed triple.

The colored triangles for the $w(\widetilde{e})b(\widetilde{e})$-diagonals, on
the other hand, make up a tiling of the plane $\mathbb{R}^{2}$ by black and
white triangles such that each triangle has one $\star$-vertex, one
$\bullet$-vertex and one $\circ$-vertex. It is a well-known [14] that from
such a triangulation one can construct a branched covering
$\mathcal{B}\colon\mathbb{T}_{\omega}\rightarrow\mathbb{C}\mathbb{P}^{1}$ with
precisely three branch points $0$, $1$, $\infty$:

$\displaystyle\mathcal{B}\colon\
\mathbb{T}_{\omega}\longrightarrow\mathbb{C}\mathbb{P}^{1},\qquad\mathcal{B}(\mathsf{P}_{\\!\Lambda}^{\circ})=0,\qquad\mathcal{B}(\mathsf{P}_{\\!\Lambda}^{\bullet})=1,\qquad\mathcal{B}(\mathsf{P}_{\\!\Lambda}^{\star})=\infty.$

This is where Zhegalkin zebra motives meet dessins d’enfants. In the works on
dessins d’enfants on Riemann surfaces of genus $1$ one wants to find on the
torus a structure of elliptic curve over a number field such that the branched
covering map is a morphism of varieties, called a Belyi map. We will not
elaborate on dessins d’enfants, but refer instead to [14, 16, 17].

The map $\mathcal{B}$ induces unramified coverings of
$\mathbb{C}\setminus\\{0,1\\}=\mathbb{C}\mathbb{P}^{1}\setminus\\{0,1,\infty\\}$:

$\displaystyle\mathbb{T}_{\omega}\setminus\bigl{(}\mathsf{P}_{\\!\Lambda}^{\circ},\mathsf{P}_{\\!\Lambda}^{\bullet},\mathsf{P}_{\\!\Lambda}^{\star}\bigr{)}\stackrel{{\scriptstyle\mathcal{B}}}{{\longrightarrow}}\mathbb{C}\setminus\\{0,1\\}\stackrel{{\scriptstyle\widetilde{\mathcal{B}}}}{{\longleftarrow}}\mathbb{R}^{2}\setminus\bigl{(}{\mathsf{P}}^{\circ},{\mathsf{P}}^{\bullet},{\mathsf{P}}^{\star}\bigr{)}.$

One can normalize the formulas describing $\widetilde{\mathcal{B}}$ such that
the $s(\widetilde{e})t(\widetilde{e})$-diagonals of the quadrangles are mapped
to the line $\Re z=\textstyle{\frac{1}{2}}\displaystyle$ in $\mathbb{C}$ while
the midpoints of these diagonals are mapped to the point
$\textstyle{\frac{1}{2}}\displaystyle$. Every path in
$\mathbb{C}\setminus\\{0,1\\}$ starting at the point
$\textstyle{\frac{1}{2}}\displaystyle$ can be lifted uniquely to a collection
of paths in
$\mathbb{R}^{2}\setminus\bigl{(}{\mathsf{P}}^{\circ},{\mathsf{P}}^{\bullet},{\mathsf{P}}^{\star}\bigr{)}$
starting at the midpoints of the $s(\widetilde{e})t(\widetilde{e})$-diagonals.

The fiber $\mathcal{B}^{-1}\big{(}\textstyle{\frac{1}{2}}\displaystyle\big{)}$
can be identified with the set ${\mathcal{E}_{\Lambda}}$. The monodromy action
of the fundamental group
$\pi_{1}\bigl{(}\mathbb{C}\setminus\\{0,1\\},\textstyle{\frac{1}{2}}\displaystyle\bigr{)}$
on ${\mathcal{E}_{\Lambda}}$ is then exactly the permutation action described
by the superpotential
$[\mathcal{F}]_{\Lambda}=({\mathcal{E}_{\Lambda}},\sigma_{0},\sigma_{1})$.
This is illustrated in Fig. 7.

Lifting the figure-$\infty$-loop which starts at
$\textstyle{\frac{1}{2}}\displaystyle$ in direction NW yields a collection of
paths known as zigzags. It is evident from Fig. 7 that these correspond to the
orbits of the permutation $\sigma_{1}\sigma_{0}$.

Figure 7:
$\mathbb{P}^{1}\setminus\\{0,1,\infty\\}=\mathbb{C}\setminus\\{0,1\\}$.

Zigzag paths are visible in the quadrangle tiling as paths which enter and
leave the quadrangles at opposite sides; see Figs. 4 and 6. Through every
quadrangle pass two zigzag paths which one can let over/under cross as
indicated in the left-hand picture in Fig. 8. The zigzag paths form an
alternating strand diagram in the sense of [5, Definition 1.10].

Zigzag paths play an important role in the literature on dimer models and are
used to formulate consistency conditions [5, 6, 7, 11, 12]. Because not all
dimer models which come from Zhegalkin zebra functions do satisfy these
consistency conditions we will not say more about zigzag paths.

Instead we focus on the lifts of the arrows
$0\leftarrow\textstyle{\frac{1}{2}}\displaystyle$ and
$\textstyle{\frac{1}{2}}\displaystyle\rightarrow 1$ and
$\textstyle{\frac{1}{2}}\displaystyle\uparrow\infty$ shown in the right-hand
picture in Fig. 8 as the vectors $\mathsf{q}_{\mathbf{w}}(e)$,
$\mathsf{q}_{\mathbf{b}}(e)$ and $\mathsf{q}_{\mathbf{t}}(e)$, respectively.
Obviously, $\mathsf{q}_{\mathbf{t}}(e)=\frac{1}{2}\omega(e)$. Proposition 5.3
and formula (2.50) explicitly give the vectors $\mathsf{q}_{\mathbf{w}}(e)$
and $\mathsf{q}_{\mathbf{b}}(e)$.

Figure 8: Quadrangle for edge $e$.

Plan of the paper. In Sections 2 and 3 we give more details on, respectively,
the toric geometry and the non-commutative algebraic geometry behind Section
1.2. In Section 4 we attempt to put the results of Sections 2 and 3 into a
mirror symmetry perspective. Section 5 describes how one can solve some
practical matters (by computer). Formula (5.1) in Section 5.1 is basically
computer code for evaluating the Zhegalkin zebra function $\mathcal{F}$ and
drawing the picture of the tiling. In Section 5.2 I describe how one can
compute the superpotential $[\mathcal{F}]_{\Lambda}$ and the realization
$\omega_{\mathcal{F}}$. From the superpotential one can easily determine all
perfect matchings. It is described in the text between Definitions 1.5 and 1.6
how to obtain from this a weight realization, which subsequently can be
deformed with formula (1.5). In Section 5.3 I describe how one can construct
and draw the quadrangle tiling of $\mathbb{R}^{2}$ for a given weight
realization.

##  2 The algebraic geometry of weights

In this section $\mathcal{F}$ is a convex Zhegalkin zebra function and
$\Lambda$ is a sublattice of $\mathsf{Aut}(\mathcal{F})$, such that the
superpotential $[\mathcal{F}]_{\Lambda}$ is dimer complete and such that a
weight realization of the superpotential exists; see Definitions 1.1, 1.2, 1.5
and 1.6. So there is a tiling of $\mathbb{R}^{2}$ by strictly convex
quadrangles and the diagonals provide embeddings of the graphs
$\Gamma_{\\!\\!\Lambda}$ and $\Gamma_{\\!\\!\Lambda}^{\vee}$ into the torus
$\mathbb{T}=\mathbb{R}^{2}/\Lambda$.

###  2.1 The (co)homology of $\boldsymbol{\Gamma_{\\!\\!\Lambda}}$ and
$\boldsymbol{\Gamma_{\\!\\!\Lambda}^{\vee}}$

2.1.1.  Elements $\mathbf{v}\in\mathsf{P}_{\\!\Lambda}^{\star}$,
$\mathbf{b}\in\mathsf{P}_{\\!\Lambda}^{\bullet}$,
$\mathbf{w}\in\mathsf{P}_{\\!\Lambda}^{\circ}$ define maps
$\alpha_{\mathbf{v}},\beta_{\mathbf{b}},\beta_{\mathbf{w}}\colon{\mathcal{E}_{\Lambda}}\longrightarrow\mathbb{Z}$
such that

$\displaystyle\alpha_{\mathbf{v}}(e)=1\quad\text{if}\quad
t(e)=\mathbf{v},\qquad\alpha_{\mathbf{v}}(e)=-1\quad\text{if}\quad
s(e)=\mathbf{v},\qquad\alpha_{\mathbf{v}}(e)=0\quad\text{else},$ (2.1)
$\displaystyle\beta_{\mathbf{b}}(e)=1\quad\text{if}\quad
b(e)=\mathbf{b},\qquad\beta_{\mathbf{b}}(e)=0\quad\text{else},$ (2.2)
$\displaystyle\beta_{\mathbf{w}}(e)=1\quad\text{if}\quad
w(e)=\mathbf{w},\qquad\beta_{\mathbf{w}}(e)=0\quad\text{else}.$ (2.3)

There is only one linear relation between the maps $\alpha_{\mathbf{v}}$
($\mathbf{v}\in\mathsf{P}_{\\!\Lambda}^{\star}$) and there is only one linear
relation between the maps $\beta_{\mathbf{b}}$, $\beta_{\mathbf{w}}$
($\mathbf{b}\in\mathsf{P}_{\\!\Lambda}^{\bullet}$,
$\mathbf{w}\in\mathsf{P}_{\\!\Lambda}^{\circ}$), namely

$\displaystyle\sum_{\mathbf{v}\in\mathsf{P}_{\\!\Lambda}^{\star}}\alpha_{\mathbf{v}}=0,\qquad\sum_{\mathbf{b}\in\mathsf{P}_{\\!\Lambda}^{\bullet}}\beta_{\mathbf{b}}=\sum_{\mathbf{w}\in\mathsf{P}_{\\!\Lambda}^{\circ}}\beta_{\mathbf{w}}.$
(2.4)

2.1.2.  The first cohomology group
$\mathrm{H}^{1}(\Gamma_{\\!\\!\Lambda},\mathbb{Z})$ of the graph
$\Gamma_{\\!\\!\Lambda}$ is the subgroup of the group
$\mathbb{Z}^{{\mathcal{E}_{\Lambda}}}=\mathrm{Maps}({\mathcal{E}_{\Lambda}},\mathbb{Z})$
consisting of the maps
$\eta\colon{\mathcal{E}_{\Lambda}}\longrightarrow\mathbb{Z}$ which satisfy

$\displaystyle\sum_{e\in{\mathcal{E}_{\Lambda}}}\\!\eta(e)\alpha_{\mathbf{v}}(e)=0\qquad\textrm{for
all}\quad\mathbf{v}\in\mathsf{P}_{\\!\Lambda}^{\star}.$ (2.5)

The rank of this group is

$\displaystyle\operatorname{\mathsf{rank}}\mathrm{H}^{1}(\Gamma_{\\!\\!\Lambda},\mathbb{Z})=|{\mathcal{E}_{\Lambda}}|-|\mathsf{P}_{\\!\Lambda}^{\star}|+1=|\mathsf{P}_{\\!\Lambda}^{\bullet}|+|\mathsf{P}_{\\!\Lambda}^{\circ}|+1.$

The maps $\beta_{\mathbf{b}}$ and $\beta_{\mathbf{w}}$ for
$\mathbf{b}\in\mathsf{P}_{\\!\Lambda}^{\bullet}$,
$\mathbf{w}\in\mathsf{P}_{\\!\Lambda}^{\circ}$ are elements of
$\mathrm{H}^{1}(\Gamma_{\\!\\!\Lambda},\mathbb{Z})$. They generate a subgroup
of rank
$|\mathsf{P}_{\\!\Lambda}^{\bullet}|+|\mathsf{P}_{\\!\Lambda}^{\circ}|-1$; see
(2.2), (2.3) and (2.4).

The embedding $\Gamma_{\\!\\!\Lambda}\hookrightarrow\mathbb{T}$ induces a
homomorphism of homology groups

$\displaystyle\mathrm{H}_{1}(\Gamma_{\\!\\!\Lambda},\mathbb{Z})\longrightarrow\mathrm{H}_{1}(\mathbb{T},\mathbb{Z}).$

This homomorphism is surjective and its kernel is generated by the elements

$\displaystyle\check{\beta}_{\mathbf{b}}=\sum_{e\in{\mathcal{E}_{\Lambda}}}\beta_{\mathbf{b}}(e)e\qquad\textrm{and}\qquad\check{\beta}_{\mathbf{w}}=\sum_{e\in{\mathcal{E}_{\Lambda}}}\beta_{\mathbf{w}}(e)e$
(2.6)

for $\mathbf{b}\in\mathsf{P}_{\\!\Lambda}^{\bullet}$ and
$\mathbf{w}\in\mathsf{P}_{\\!\Lambda}^{\circ}$.

2.1.3.  A path of length $k$ on $\Gamma_{\\!\\!\Lambda}$ is a sequence
$\mathbf{p}=(e_{1},\ldots,e_{k})$ in ${\mathcal{E}_{\Lambda}}$ such that
$t(e_{i})=s(e_{i+1})$ for $i=1,\ldots,k-1$. We define
$s(\mathbf{p})=s(e_{1})$, $t(\mathbf{p})=t(e_{k})$. A path $\mathbf{p}$ on
$\Gamma_{\\!\\!\Lambda}$ is closed if $s(\mathbf{p})=t(\mathbf{p})$. In case
$k=0$ these are the constant paths supported on the vertices of
$\Gamma_{\\!\\!\Lambda}$.

The homology class of a closed path $\mathbf{p}=(e_{1},\ldots,e_{k})$ is

$\displaystyle[\mathbf{p}]=e_{1}+\dots+e_{k}.$ (2.7)

$\mathrm{H}_{1}(\Gamma_{\\!\\!\Lambda},\mathbb{Z})$ is generated by the
homology classes of closed paths on $\Gamma_{\\!\\!\Lambda}$. Special closed
paths on $\Gamma_{\\!\\!\Lambda}$ are given by the boundaries of the black and
white polygons in the tiling. Their homology classes are
$\check{\beta}_{\mathbf{b}}$ and $\check{\beta}_{\mathbf{w}}$ as in (2.6).
They generate a subgroup in
$\mathrm{H}_{1}(\Gamma_{\\!\\!\Lambda},\mathbb{Z})$ of rank
$|\mathsf{P}_{\\!\Lambda}^{\bullet}|+|\mathsf{P}_{\\!\Lambda}^{\circ}|-1$.

2.1.4.  The first cohomology group
$\mathrm{H}^{1}\big{(}\Gamma_{\\!\\!\Lambda}^{\vee},\mathbb{Z}\big{)}$ of the
graph $\Gamma_{\\!\\!\Lambda}^{\vee}$ is the subgroup of
$\mathbb{Z}^{{\mathcal{E}_{\Lambda}}}$ consisting of the maps
$\theta\colon{\mathcal{E}_{\Lambda}}\longrightarrow\mathbb{Z}$ which satisfy

$\displaystyle\forall\,\mathbf{w}\in\mathsf{P}_{\\!\Lambda}^{\circ},\
\forall\,\mathbf{b}\in\mathsf{P}_{\\!\Lambda}^{\bullet}\colon\quad\sum_{e\in{\mathcal{E}_{\Lambda}},w(e)=\mathbf{w}}\theta(e)=\sum_{e\in{\mathcal{E}_{\Lambda}},b(e)=\mathbf{b}}\theta(e)=0.$
(2.8)

Since there is only one linear relation between the equations in the system
(2.8) the rank of the cohomology group is

$\displaystyle\operatorname{\mathsf{rank}}\mathrm{H}^{1}\big{(}\Gamma_{\\!\\!\Lambda}^{\vee},\mathbb{Z}\big{)}=|{\mathcal{E}_{\Lambda}}|-|\mathsf{P}_{\\!\Lambda}^{\bullet}|-|\mathsf{P}_{\\!\Lambda}^{\circ}|+1=|\mathsf{P}_{\\!\Lambda}^{\star}|+1.$

The maps $\alpha_{\mathbf{v}}$ for
$\mathbf{v}\in\mathsf{P}_{\\!\Lambda}^{\star}$ are elements of
$\mathrm{H}^{1}\big{(}\Gamma_{\\!\\!\Lambda}^{\vee},\mathbb{Z}\big{)}$. They
generate a subgroup of rank $|\mathsf{P}_{\\!\Lambda}^{\star}|-1$; see (2.1)
and (2.4). The embedding
$\Gamma_{\\!\\!\Lambda}^{\vee}\hookrightarrow\mathbb{T}$ induces a
homomorphism of homology groups

$\displaystyle\mathrm{H}_{1}\big{(}\Gamma_{\\!\\!\Lambda}^{\vee},\mathbb{Z}\big{)}\longrightarrow\mathrm{H}_{1}(\mathbb{T},\mathbb{Z}).$
(2.9)

This homomorphism is surjective and its kernel is generated by the elements

$\displaystyle\check{\alpha}_{\mathbf{v}}=\sum_{e\in{\mathcal{E}_{\Lambda}}}\alpha_{\mathbf{v}}(e)e\qquad\textrm{for}\quad\mathbf{v}\in\mathsf{P}_{\\!\Lambda}^{\star}.$
(2.10)

###  2.2 The geometry of
$\boldsymbol{\mathsf{Proj}(\mathbb{Z}[\mathcal{W}_{\Lambda}])}$

In this section we investigate the geometry of the projective scheme
$\mathsf{Proj}(\mathbb{Z}[\mathcal{W}_{\Lambda}])$, which by general
constructions in algebraic geometry is associated with the graded semi-group
$\mathcal{W}_{\Lambda}$ of integer weight functions for
$[\mathcal{F}]_{\Lambda}$; see [13, Chapter II].

2.2.1.  By Definition 1.4 an integer weight function for the superpotential
$[\mathcal{F}]_{\Lambda}$ is a map
$\nu\colon{\mathcal{E}_{\Lambda}}\rightarrow\mathbb{Z}_{\geq 0}$ which
satisfies

$\displaystyle\forall\,\mathbf{w}\in\mathsf{P}_{\\!\Lambda}^{\circ},\
\forall\,\mathbf{b}\in\mathsf{P}_{\\!\Lambda}^{\bullet}\colon\quad\sum_{w(e)=\mathbf{w}}\nu(e)=\sum_{b(e)=\mathbf{b}}\nu(e)=\deg\nu.$
(2.11)

From (2.8) and (2.11) one sees that the difference $\nu-\nu^{\prime}$ of two
weight functions with the same degree is an element of
$\mathrm{H}^{1}\big{(}\Gamma_{\\!\\!\Lambda}^{\vee},\mathbb{Z}\big{)}$.

Conversely, if $\nu$ is a positive weight function (i.e., $\nu(e)>0$ for all
$e\in{\mathcal{E}_{\Lambda}}$) and $\theta$ is an element of
$\mathrm{H}^{1}\big{(}\Gamma_{\\!\\!\Lambda}^{\vee},\mathbb{Z}\big{)}$, then
for all sufficiently large integers $N$ the function $N\nu-\theta$ is a
positive weight function. Thus we find that

$\displaystyle\operatorname{\mathsf{rank}}\mathcal{W}_{\Lambda}=\operatorname{\mathsf{rank}}\mathrm{H}^{1}\big{(}\Gamma_{\\!\\!\Lambda}^{\vee},\mathbb{Z}\big{)}+1=|\mathsf{P}_{\\!\Lambda}^{\star}|+2.$
(2.12)

2.2.2.  On the semi-group $\mathcal{W}_{\Lambda}$ we define an equivalence
relation $\sim$ by

$\displaystyle\nu\sim\nu^{\prime}\Leftrightarrow\exists\,r\colon\
\mathsf{P}_{\\!\Lambda}^{\star}\rightarrow\mathbb{Z}\quad\textrm{s.t.}\quad\nu-\nu^{\prime}=\sum_{\mathbf{v}\in\mathsf{P}_{\\!\Lambda}^{\star}}r_{\mathbf{v}}\alpha_{\mathbf{v}}$
(2.13)

with $\alpha_{\mathbf{v}}$ as in (2.1). We denote the set of equivalence
classes by $\overline{\mathcal{W}}_{\Lambda}$:

$\displaystyle\overline{\mathcal{W}}_{\Lambda}=\mathcal{W}_{\Lambda}/{\sim}.$
(2.14)

This is a graded semi-group of rank $3$. The natural surjective homomorphism
of semi-groups
$\mathcal{W}_{\Lambda}\rightarrow\overline{\mathcal{W}}_{\Lambda}$ is the
analogue of the surjective homomorphism of groups
$\mathrm{H}_{1}\big{(}\Gamma_{\\!\\!\Lambda}^{\vee},\mathbb{Z}\big{)}\rightarrow\mathrm{H}_{1}(\mathbb{T},\mathbb{Z})$
induced by the embedding
$\Gamma_{\\!\\!\Lambda}^{\vee}\hookrightarrow\mathbb{T}$; cf. (2.9).

2.2.3.  Recall from Definition 1.4 that the integer weight functions of degree
$1$ are called perfect matchings and that $\mathcal{M}_{\Lambda}$ is the set
of perfect matchings. We denote the set of equivalence classes for the
relation $\sim$ on $\mathcal{M}_{\Lambda}$ by $\mathcal{A}_{\Lambda}$:

$\displaystyle\mathcal{A}_{\Lambda}=\mathcal{M}_{\Lambda}/{\sim}.$

######  Definition 2.1 (cf. [10, Sections 3.4 and 3.5]).

The convex hull $\mathsf{conv}(\mathcal{M}_{\Lambda})$ of
$\mathcal{M}_{\Lambda}$ in $\mathbb{R}^{{\mathcal{E}_{\Lambda}}}$ is called
the _matching polytope of $[\mathcal{F}]_{\Lambda}$_. The elements of
$\mathsf{conv}(\mathcal{M}_{\Lambda})$ are called _fractional matchings_. The
convex hull $\mathsf{conv}(\mathcal{A}_{\Lambda})$ of $\mathcal{A}_{\Lambda}$
is called the _Newton polygon of $[\mathcal{F}]_{\Lambda}$_.

######  Proposition 2.2 (cf. [10, Lemma 3.10]).

  1. $(i)$

The matching polytope satisfies

$\displaystyle\mathsf{conv}(\mathcal{M}_{\Lambda})=\left\\{\begin{array}[]{@{}l@{}}\text{maps
$\theta\colon{\mathcal{E}_{\Lambda}}\longrightarrow\mathbb{R}_{\geq 0}$ s.t.\
for all $\mathbf{w}\in\mathsf{P}_{\\!\Lambda}^{\circ}$ and}\\\ \textrm{all
}\mathbf{b}\in\mathsf{P}_{\\!\Lambda}^{\bullet}\colon\sum\limits_{e\in\mathbf{w}}\theta(e)=\sum\limits_{e\in\mathbf{b}}\theta(e)=1\end{array}\right\\}.$
(2.17)

The set of its vertices is precisely the set of perfect matchings
$\mathcal{M}_{\Lambda}$.

  2. $(ii)$

The semi-group $\mathcal{W}_{\Lambda}$ is generated by the perfect matchings
and the semi-group $\overline{\mathcal{W}}_{\Lambda}$ is generated by the set
$\mathcal{A}_{\Lambda}$:

$\displaystyle\mathcal{W}_{\Lambda}=\mathbb{Z}_{\geq
0}\mathcal{M}_{\Lambda},\qquad\overline{\mathcal{W}}_{\Lambda}=\mathbb{Z}_{\geq
0}\mathcal{A}_{\Lambda}.$ (2.18)

  3. $(iii)$

The matching polytope $\mathsf{conv}(\mathcal{M}_{\Lambda})$ has dimension
$|\mathsf{P}_{\\!\Lambda}^{\star}|+1$ and the Newton polygon
$\mathsf{conv}(\mathcal{A}_{\Lambda})$ has dimension $2$.

###### Proof.

$(i)$ This is Lemma 3.10 in [10].

$(ii)$ Let $\nu\in\mathcal{W}_{\Lambda}$, $\nu\neq\mathsf{0}$, be given. By
$(i)$ there are non-negative real numbers $r_{\nu,\mathsf{m}}$,
$\mathsf{m}\in\mathcal{M}_{\Lambda}$, such that

$\displaystyle\nu=\sum_{\mathsf{m}\in\mathcal{M}_{\Lambda}}r_{\nu,\mathsf{m}}\mathsf{m},\qquad\sum_{\mathsf{m}}r_{\nu,\mathsf{m}}=\deg\nu.$

Then $\nu(e)\geq r_{\nu,\mathsf{m}}\mathsf{m}(e)$ for all $\mathsf{m}$ and
$e$. Now take $\mathsf{m}$ such that $r_{\nu,\mathsf{m}}>0$. Then
$\nu(e)-\mathsf{m}(e)\geq 0$ for all $e\in{\mathcal{E}_{\Lambda}}$. This means
that $\nu-\mathsf{m}\in\mathcal{W}_{\Lambda}$. Note that
$\deg(\nu-\mathsf{m})=\deg\nu-1$. If $\nu-\mathsf{m}\neq\mathsf{0}$ we repeat
the preceding step with $\nu-\mathsf{m}$ instead of $\nu$. After finitely many
steps we arrive at the situation that $\nu$ minus some linear combination of
perfect matchings with positive integer coefficients is $0$. This result
passes well to $\sim$-equivalence classes.

$(iii)$ This follows from (2.12), (2.17) and (2.18). ∎

2.2.4.  General constructions in algebraic geometry associate with the graded
semi-groups $\mathcal{W}_{\Lambda}$ and $\overline{\mathcal{W}}_{\Lambda}$ the
projective schemes $\mathsf{Proj}(\mathbb{Z}[\mathcal{W}_{\Lambda}])$ and
$\mathsf{Proj}(\mathbb{Z}[\overline{\mathcal{W}}_{\Lambda}])$; see [13,
Chapter II]. The natural surjective homomorphism of semi-groups
$\mathcal{W}_{\Lambda}\rightarrow\overline{\mathcal{W}}_{\Lambda}$ becomes an
inclusion as closed subscheme

$\displaystyle\mathsf{Proj}\big{(}\mathbb{Z}[\overline{\mathcal{W}}_{\Lambda}]\big{)}\subset\mathsf{Proj}(\mathbb{Z}[\mathcal{W}_{\Lambda}]).$

2.2.5.  As a consequence of (2.18) we have a surjective homomorphism of rings

$\displaystyle\mathbb{Z}[X_{\mathsf{m}}\,|\,\mathsf{m}\in\mathcal{M}_{\Lambda}]\longrightarrow\mathbb{Z}[\mathcal{W}_{\Lambda}],\qquad
X_{\mathsf{m}}\mapsto X^{\mathsf{m}}$ (2.19)

from the polynomial ring in the variables $X_{\mathsf{m}}$,
$\mathsf{m}\in\mathcal{M}_{\Lambda}$, to the semi-group ring of
$\mathcal{W}_{\Lambda}$; here $X^{\mathsf{m}}$ denotes the element of
$\mathbb{Z}[\mathcal{W}_{\Lambda}]$ which corresponds to
$\mathsf{m}\in\mathcal{W}_{\Lambda}$. The kernel of the homomorphism (2.19) is
the ideal generated by the polynomials

$\displaystyle\prod_{\mathsf{m}\colon\mu(\mathsf{m})>0}X_{\mathsf{m}}^{\mu(\mathsf{m})}-\prod_{\mathsf{m}\colon\mu(\mathsf{m})<0}X_{\mathsf{m}}^{-\mu(\mathsf{m})}\qquad\textrm{for}\quad\mu\in\mathbb{M},$
(2.20)

where $\mathbb{M}$ denotes the lattice of $\mathbb{Z}$-linear relations
between the perfect matchings:

$\displaystyle\mathbb{M}=\bigg{\\{}\mu\in\mathbb{Z}^{\mathcal{M}_{\Lambda}}\,|\,\sum_{\mathsf{m}\in\mathcal{M}_{\Lambda}}\mu(\mathsf{m})\mathsf{m}=0\bigg{\\}}.$
(2.21)

It follows that $\mathsf{Proj}(\mathbb{Z}[\mathcal{W}_{\Lambda}])$ can be
identified with the closed subscheme of the projective space
$\mathbb{P}^{|\mathcal{M}_{\Lambda}|-1}=\mathsf{Proj}(\mathbb{Z}[X_{\mathsf{m}}\,|\,\mathsf{m}\in\mathcal{M}_{\Lambda}])$
given by the homogeneous equations

$\displaystyle\prod_{\mathsf{m}\colon\mu(\mathsf{m})>0}X_{\mathsf{m}}^{\mu(\mathsf{m})}=\prod_{\mathsf{m}\colon\mu(\mathsf{m})<0}X_{\mathsf{m}}^{-\mu(\mathsf{m})}\qquad\textrm{for}\quad\mu\in\mathbb{M}.$
(2.22)

2.2.6.  A perfect matching $\mathsf{m}_{0}$ defines an open subscheme of
$\mathsf{Proj}(\mathbb{Z}[\mathcal{W}_{\Lambda}])$, namely the affine scheme
$\mathsf{Spec}\big{(}\mathbb{Z}\big{[}\mathcal{W}_{\Lambda}^{0}[-\mathsf{m}_{0}]\big{]}\big{)}$
given by the semi-group

$\displaystyle\mathcal{W}_{\Lambda}^{0}[-\mathsf{m}_{0}]=\big{\\{}\nu-(\deg\nu)\mathsf{m}_{0}\in\mathbb{Z}^{{\mathcal{E}_{\Lambda}}}\,|\,\nu\in\mathcal{W}_{\Lambda}\big{\\}}$

(cf. [13, Chapter II, Proposition 2.5]). This is a sub-semi-group of
$\mathrm{H}_{1}\big{(}\Gamma_{\\!\\!\Lambda}^{\vee},\mathbb{Z}\big{)}$. The
schemes
$\mathsf{Spec}\big{(}\mathbb{Z}\big{[}\mathcal{W}_{\Lambda}^{0}[-\mathsf{m}_{0}]\big{]}\big{)}$
for $\mathsf{m}_{0}\in\mathcal{M}_{\Lambda}$ form a covering of
$\mathsf{Proj}(\mathbb{Z}[\mathcal{W}_{\Lambda}])$ by affine open subschemes.
Their intersection is
$\mathsf{Spec}\big{(}\mathbb{Z}\big{[}\mathrm{H}_{1}\big{(}\Gamma_{\\!\\!\Lambda}^{\vee},\mathbb{Z}\big{)}\big{]}\big{)}$.

######  Proposition 2.3.

The following diagram is commutative

$\displaystyle\begin{array}[]{@{}c@{\,\,}c@{\,\,}c@{}}\mathrm{H}^{1}(\mathbb{T},\mathbb{C}^{*})&\hookrightarrow&\mathsf{Proj}\big{(}\mathbb{Z}[\overline{\mathcal{W}}_{\Lambda}]\big{)}\\\
\downarrow&&\downarrow\\\
\mathrm{H}^{1}\big{(}\Gamma_{\\!\\!\Lambda}^{\vee},\mathbb{C}^{*}\big{)}&\hookrightarrow&\mathsf{Proj}(\mathbb{Z}[\mathcal{W}_{\Lambda}]).\end{array}$

The $(|\mathsf{P}_{\\!\Lambda}^{\star}|+1)$-dimensional complex torus
$\mathrm{H}^{1}\big{(}\Gamma_{\\!\\!\Lambda}^{\vee},\mathbb{C}^{*}\big{)}$ is
an open subset in the set of complex points of the
$(|\mathsf{P}_{\\!\Lambda}^{\star}|+1)$-dimensional scheme
$\mathsf{Proj}(\mathbb{Z}[\mathcal{W}_{\Lambda}])$.

The $2$-dimensional complex torus $\mathrm{H}^{1}(\mathbb{T},\mathbb{C}^{*})$
is an open subset in the set of complex points of $2$-dimensional scheme
$\mathsf{Proj}(\mathbb{Z}[\overline{\mathcal{W}}_{\Lambda}])$.

######  Remark 2.4.

The semi-group $\mathcal{W}_{\Lambda}^{0}[-\mathsf{m}_{0}]$ is generated by
the elements $\mathsf{m}_{1}-\mathsf{m}_{0}$ with $\mathsf{m}_{1}$ a perfect
matching $\neq\mathsf{m}_{0}$. By Proposition 2.2 these are precisely the
vectors along the edges of the matching polytope
$\mathsf{conv}(\mathcal{M}_{\Lambda})$ incident to the vertex
$\mathsf{m}_{0}$. Therefore the duals of the semi-groups
$\mathcal{W}_{\Lambda}^{0}[-\mathsf{m}_{0}]$ for
$\mathsf{m}_{0}\in\mathcal{M}_{\Lambda}$ give precisely the maximal cones in
the fan associated with the matching polytope
$\mathsf{conv}(\mathcal{M}_{\Lambda})$ by the construction in [9, p. 26]. Thus
$\mathsf{Proj}(\mathbb{Z}[\mathcal{W}_{\Lambda}])$ can also be obtained with
standard toric geometry constructions from the fan of outward pointing vectors
to the matching polytope $\mathsf{conv}(\mathcal{M}_{\Lambda})$.

2.2.7.  We recall from [10] the construction of the skew symmetric bilinear
form $\varepsilon$ on
$\mathrm{H}_{1}\big{(}\Gamma_{\\!\\!\Lambda}^{\vee},\mathbb{Z}\big{)}$ which
gives the Poisson structure on the group ring
$\mathbb{Z}\big{[}\mathrm{H}_{1}\big{(}\Gamma_{\\!\\!\Lambda}^{\vee},\mathbb{Z}\big{)}\big{]}$.
In order to facilitate the exposition we reproduce Fig. 38 and formula (65) of
[10] in our Fig. 9. The definition, in [10, Lemma 8.1 and Fig. 38], of the
local pairing at a white node $\mathbf{w}$ of the graph
$\Gamma_{\\!\\!\Lambda}^{\vee}$ can be phrased as follows. Let $e$,
$e^{\prime}$ and $e^{\prime\prime}$ in ${\mathcal{E}_{\Lambda}}$ be such that
$w(e)=w(e^{\prime})=w(e^{\prime\prime})=\mathbf{w}$. Let $\vec{E}$,
$\vec{E}^{\prime}$, $\vec{E}^{\prime\prime}$ be the edges of the graph
$\Gamma_{\\!\\!\Lambda}^{\vee}$ dual to $e$, $e^{\prime}$, $e^{\prime\prime}$,
respectively, and pointing away from the vertex $\mathbf{w}$. Write the cycle
of $\sigma_{0}$ which corresponds to $\mathbf{w}$ as $(e_{1},\ldots,e_{q})$
with $e_{1}=e$ and let $e^{\prime}=e_{j}$ and $e^{\prime\prime}=e_{h}$. Then
formula (65) in [10] can be stated as

$\displaystyle\delta_{\mathbf{w}}\bigl{(}\big{(}\vec{E}^{\prime}-\vec{E}\big{)}\wedge\big{(}\vec{E}^{\prime\prime}-\vec{E}\big{)}\bigr{)}=\textstyle{\frac{1}{2}}\displaystyle\mathrm{sign}(h-j).$
(2.23)

A similar formula holds for the local pairing at a black node $\mathbf{b}$ of
$\Gamma_{\\!\\!\Lambda}^{\vee}$, but since in our convention the boundaries of
the black polygons are oriented clockwise, there is an extra $-$-sign:

$\displaystyle\delta_{\mathbf{b}}\bigl{(}\big{(}\vec{E}^{\prime}-\vec{E}\big{)}\wedge\big{(}\vec{E}^{\prime\prime}-\vec{E}\big{)}\bigr{)}=-\textstyle{\frac{1}{2}}\displaystyle\mathrm{sign}(h-j).$
(2.24)

Definition 8.2 in [10] builds the skew symmetric bilinear form $\varepsilon$
on $\mathrm{H}_{1}\big{(}\Gamma_{\\!\\!\Lambda}^{\vee},\mathbb{Z}\big{)}$ from
these local pairings. For reasons that will become clear in (2.31) we denote
this form as $\varepsilon_{+}$. The defining formula in [10] can then be
stated as

$\displaystyle\varepsilon_{+}=\sum_{\mathbf{w}\in\mathsf{P}_{\\!\Lambda}^{\circ}}\delta_{\mathbf{w}}-\sum_{\mathbf{b}\in\mathsf{P}_{\\!\Lambda}^{\bullet}}\delta_{\mathbf{b}}.$
(2.25)

Figure 9: Fig. 38 + formula (65) in [10]: One has
$\delta_{v}(\gamma_{r},\gamma_{l})=\frac{1}{2}$.

By [10, Section 1.1] the Poisson bracket on
$\mathbb{Z}\big{[}\mathrm{H}_{1}\big{(}\Gamma_{\\!\\!\Lambda}^{\vee},\mathbb{Z}\big{)}\big{]}$
is then given by

$\displaystyle\bigl{\\{}X^{L_{1}},X^{L_{2}}\bigr{\\}}=\varepsilon_{+}(L_{1},L_{2})X^{L_{1}+L_{2}},$
(2.26)

where $X^{L}$ denotes the element of
$\mathbb{Z}\big{[}\mathrm{H}_{1}\big{(}\Gamma_{\\!\\!\Lambda}^{\vee},\mathbb{Z}\big{)}\big{]}$
which corresponds to the (homology class of) the loop $L$ on
$\Gamma_{\\!\\!\Lambda}^{\vee}$.

2.2.8.  We are now going to give a simple description of the form
$\varepsilon_{+}$ (2.25) in terms of the permutations $\sigma_{0}$ and
$\sigma_{1}$. Fix a perfect matching $\mathsf{m}$. Write $\sigma_{0}$ and
$\sigma_{1}$ as permutation matrices; i.e., matrices with rows and columns
indexed by the elements of ${\mathcal{E}_{\Lambda}}$ and in column $e$ only
one non-zero entry, namely $1$ in row $\sigma_{0}(e)$ (resp. $\sigma_{1}(e)$).
By multiplying for each $e\in{\mathcal{E}_{\Lambda}}$ the corresponding column
by $1-\mathsf{m}(e)$ we obtain two new matrices $\varsigma_{\mathsf{m},0}$ and
$\varsigma_{\mathsf{m},1}$, respectively. These are nilpotent matrices. We set

$\displaystyle\rho_{\mathsf{m},0}=(\mathbb{I}-\varsigma_{\mathsf{m},0})^{-1},\qquad\rho_{\mathsf{m},1}=(\mathbb{I}-\varsigma_{\mathsf{m},1})^{-1}.$
(2.27)

The meaning of these matrices is as follows. Write $\sigma_{0}$ as a product
of disjoint cycles such that the elements of $\mathsf{m}$ are in the first
position in their cycle. Then for $e\neq e^{\prime}\in{\mathcal{E}_{\Lambda}}$
the entry in column $e$ and row $e^{\prime}$ in the matrix
$\rho_{\mathsf{m},0}$ is $1$ if and only if $e$ and $e^{\prime}$ are in the
same cycle of $\sigma_{0}$ with $e$ to the left of $e^{\prime}$. And similarly
for $\sigma_{1}$ and $\rho_{\mathsf{m},1}$.

Let $\mathsf{m}^{\prime}$, $\mathsf{m}^{\prime\prime}$ be two perfect
matchings. Then
$\sum\limits_{e\in{\mathcal{E}_{\Lambda}}}(\mathsf{m}^{\prime}(e)-\mathsf{m}(e))e$
and
$\sum\limits_{e\in{\mathcal{E}_{\Lambda}}}(\mathsf{m}^{\prime\prime}(e)-\mathsf{m}(e))e$
are elements of
$\mathrm{H}_{1}\big{(}\Gamma_{\\!\\!\Lambda}^{\vee},\mathbb{Z}\big{)}$. It
follows from (2.23)–(2.25) that

$\displaystyle\varepsilon_{+}(\mathsf{m}^{\prime}-\mathsf{m},\mathsf{m}^{\prime\prime}-\mathsf{m})=\textstyle{\frac{1}{2}}\displaystyle{\mathsf{m}^{\prime}}^{t}\big{(}\rho_{\mathsf{m},0}-\rho_{\mathsf{m},0}^{t}+\rho_{\mathsf{m},1}-\rho_{\mathsf{m},1}^{t}\big{)}\mathsf{m}^{\prime\prime},$

where, for simplicity of notation, we have written on the left-hand side
$\mathsf{m}^{\prime}-\mathsf{m}$ and $\mathsf{m}^{\prime\prime}-\mathsf{m}$
for
$\sum\limits_{e\in{\mathcal{E}_{\Lambda}}}(\mathsf{m}^{\prime}(e)-\mathsf{m}(e))e$
and
$\sum\limits_{e\in{\mathcal{E}_{\Lambda}}}(\mathsf{m}^{\prime\prime}(e)-\mathsf{m}(e))e$,
respectively, while on the right-hand side we view $\mathsf{m}^{\prime}$ and
$\mathsf{m}^{\prime\prime}$ as column vectors.

By bilinearity this extends to all perfect matchings $\mathsf{m}_{1}$,
$\mathsf{m}_{2}$, $\mathsf{m}_{3}$, $\mathsf{m}_{4}$:

$\displaystyle\varepsilon_{+}(\mathsf{m}_{1}-\mathsf{m}_{2},\mathsf{m}_{3}-\mathsf{m}_{4})=\textstyle{\frac{1}{2}}\displaystyle(\mathsf{m}_{1}-\mathsf{m}_{2})^{t}\big{(}\rho_{\mathsf{m},0}-\rho_{\mathsf{m},0}^{t}+\rho_{\mathsf{m},1}-\rho_{\mathsf{m},1}^{t}\big{)}(\mathsf{m}_{3}-\mathsf{m}_{4}).$
(2.28)

In the matrix $\rho_{\mathsf{m},0}+\rho_{\mathsf{m},0}^{t}-\mathbb{I}$ the
$(e,e^{\prime})$-entry is $1$ if $e$ and $e^{\prime}$ sit in the same cycle of
$\sigma_{0}$ and is $0$ otherwise. Consequently,
$\bigl{(}\rho_{\mathsf{m},0}+\rho_{\rho,0}^{t}-\mathbb{I}\bigr{)}(\mathsf{m}_{3}-\mathsf{m}_{4})=0$
and

$\displaystyle\textstyle{\frac{1}{2}}\displaystyle\bigl{(}\rho_{\mathsf{m},0}-\rho_{\mathsf{m},0}^{t}\bigr{)}(\mathsf{m}_{3}-\mathsf{m}_{4})=\bigl{(}-\textstyle{\frac{1}{2}}\displaystyle\mathbb{I}+\rho_{\mathsf{m},0}\bigr{)}(\mathsf{m}_{3}-\mathsf{m}_{4}).$
(2.29)

Similarly

$\displaystyle\textstyle{\frac{1}{2}}\displaystyle\bigl{(}\rho_{\mathsf{m},1}-\rho_{\mathsf{m},1}^{t}\bigr{)}(\mathsf{m}_{3}-\mathsf{m}_{4})=\bigl{(}{-}\textstyle{\frac{1}{2}}\displaystyle\mathbb{I}+\rho_{\mathsf{m},1}\bigr{)}(\mathsf{m}_{3}-\mathsf{m}_{4}).$
(2.30)

Since $\mathrm{H}_{1}\big{(}\Gamma_{\\!\\!\Lambda}^{\vee},\mathbb{Z}\big{)}$
is the subgroup of $\mathbb{Z}^{{\mathcal{E}_{\Lambda}}}$ which is generated
by the differences of pairs of perfect matchings we conclude from
(2.28)–(2.30):

######  Proposition 2.5.

The bilinear form $\varepsilon_{+}$ on
$\mathrm{H}_{1}\big{(}\Gamma_{\\!\\!\Lambda}^{\vee},\mathbb{Z}\big{)}$ is the
restriction of the bilinear form on $\mathbb{Z}^{{\mathcal{E}_{\Lambda}}}$
associated with the matrix
$\rho_{\mathsf{m},0}+\rho_{\mathsf{m},1}-\mathbb{I}$:

$\displaystyle\forall\,\mathbf{h}_{1},\mathbf{h}_{2}\in\mathrm{H}_{1}\big{(}\Gamma_{\\!\\!\Lambda}^{\vee},\mathbb{Z}\big{)}\colon\quad\varepsilon_{+}(\mathbf{h}_{1},\mathbf{h}_{2})=\mathbf{h}_{1}^{t}\bigl{(}\rho_{\mathsf{m},0}+\rho_{\mathsf{m},1}-\mathbb{I}\bigr{)}\mathbf{h}_{2}.$
(2.31)

This holds for every perfect matching $\mathsf{m}$.

2.2.9.  The difference of the matrices
$-\textstyle{\frac{1}{2}}\displaystyle\mathbb{I}+\rho_{\mathsf{m},0}$ and
$-\textstyle{\frac{1}{2}}\displaystyle\mathbb{I}+\rho_{\mathsf{m},1}$ induces
another anti-symmetric bilinear form $\varepsilon_{-}$ on
$\mathrm{H}_{1}\big{(}\Gamma_{\\!\\!\Lambda}^{\vee},\mathbb{Z}\big{)}$:

$\displaystyle\forall\,\mathbf{h}_{1},\mathbf{h}_{2}\in\mathrm{H}_{1}\big{(}\Gamma_{\\!\\!\Lambda}^{\vee},\mathbb{Z}\big{)}\colon\quad\varepsilon_{-}(\mathbf{h}_{1},\mathbf{h}_{2})=\mathbf{h}_{1}^{t}\bigl{(}\rho_{\mathsf{m},0}-\rho_{\mathsf{m},1}\bigr{)}\mathbf{h}_{2}.$
(2.32)

The form $\varepsilon_{-}$ can also be defined with the method of Section 2.2,
i.e. (cf. (2.25))

$\displaystyle\varepsilon_{-}=\sum_{\mathbf{w}\in\mathsf{P}_{\\!\Lambda}^{\circ}}\delta_{\mathbf{w}}+\sum_{\mathbf{b}\in\mathsf{P}_{\\!\Lambda}^{\bullet}}\delta_{\mathbf{b}}.$

As a consequence the right-hand side of (2.32) is independent of the choice of
the perfect matching $\mathsf{m}$.

2.2.10.  The bilinear forms $\varepsilon_{+}$ and $\varepsilon_{-}$ in
(2.31)–(2.32) define two Poisson structures on
$\mathbb{Z}\big{[}\mathrm{H}_{1}\big{(}\Gamma_{\\!\\!\Lambda}^{\vee},\mathbb{Z}\big{)}\big{]}$;
see (2.26). So, they also define Poisson structures on
$\mathbb{Z}[\mathcal{W}_{\Lambda}^{0}[-\mathsf{m}_{0}]]$ for every
$\mathsf{m}_{0}\in\mathcal{M}_{\Lambda}$. Altogether this gives:

######  Theorem 2.6.

The bilinear forms $\varepsilon_{+}$, $\varepsilon_{-}$ in (2.31)–(2.32),
i.e., for
$\mathbf{h}_{1},\mathbf{h}_{2}\in\mathrm{H}_{1}\big{(}\Gamma_{\\!\\!\Lambda}^{\vee},\mathbb{Z}\big{)}$

$\displaystyle\varepsilon_{+}(\mathbf{h}_{1},\mathbf{h}_{2})=\mathbf{h}_{1}^{t}\bigl{(}\rho_{\mathsf{m},0}+\rho_{\mathsf{m},1}-\mathbb{I}\bigr{)}\mathbf{h}_{2},\qquad\varepsilon_{-}(\mathbf{h}_{1},\mathbf{h}_{2})=\mathbf{h}_{1}^{t}\bigl{(}\rho_{\mathsf{m},0}-\rho_{\mathsf{m},1}\bigr{)}\mathbf{h}_{2},$

define two Poisson structures on
$\mathsf{Proj}(\mathbb{Z}[\mathcal{W}_{\Lambda}])$ which extend the Poisson
structures on the torus
$\mathrm{H}^{1}\big{(}\Gamma_{\\!\\!\Lambda}^{\vee},\mathbb{C}^{*}\big{)}$
described in [10, Section 1.1].

2.2.11.  Note the equivalences of data

  * •

the pair of matrices $\rho_{\mathsf{m},0}+\rho_{\mathsf{m},1}-\mathbb{I}$ and
$\rho_{\mathsf{m},0}-\rho_{\mathsf{m},1}$,

  * •

the pair of matrices $-\mathbb{I}+2\rho_{\mathsf{m},0}$ and
$-\mathbb{I}+2\rho_{\mathsf{m},1}$,

  * •

the pair of permutations $\sigma_{0}$ and $\sigma_{1}$ plus the perfect
matching $\mathsf{m}$.

The four matrices define anti-symmetric bilinear forms on
$\mathrm{H}_{1}\big{(}\Gamma_{\\!\\!\Lambda}^{\vee},\mathbb{Z}\big{)}$:

$\displaystyle\varepsilon_{+}(\mathbf{h}_{1},\mathbf{h}_{2})=\mathbf{h}_{1}^{t}\cdot(\rho_{\mathsf{m},0}+\rho_{\mathsf{m},1}-\mathbb{I})\cdot\mathbf{h}_{2},$
(2.33)
$\displaystyle\varepsilon_{-}(\mathbf{h}_{1},\mathbf{h}_{2})=\mathbf{h}_{1}^{t}\cdot(\rho_{\mathsf{m},0}-\rho_{\mathsf{m},1})\cdot\mathbf{h}_{2},$
(2.34)
$\displaystyle\varepsilon_{\bullet}(\mathbf{h}_{1},\mathbf{h}_{2})=\mathbf{h}_{1}^{t}\cdot(-\mathbb{I}+2\rho_{\mathsf{m},1})\cdot\mathbf{h}_{2},$
(2.35)
$\displaystyle\varepsilon_{\circ}(\mathbf{h}_{1},\mathbf{h}_{2})=\mathbf{h}_{1}^{t}\cdot(-\mathbb{I}+2\rho_{\mathsf{m},0})\cdot\mathbf{h}_{2},$
(2.36)

for
$\mathbf{h}_{1},\mathbf{h}_{2}\in\mathrm{H}_{1}\big{(}\Gamma_{\\!\\!\Lambda}^{\vee},\mathbb{Z}\big{)}$;
cf. (2.31)–(2.32). One has the obvious relations

$\displaystyle\varepsilon_{+}=\textstyle{\frac{1}{2}}\displaystyle(\varepsilon_{\bullet}+\varepsilon_{\circ}),\qquad\varepsilon_{-}=\textstyle{\frac{1}{2}}\displaystyle(\varepsilon_{\circ}-\varepsilon_{\bullet}).$

Each of the forms $\varepsilon_{+}$, $\varepsilon_{-}$,
$\varepsilon_{\bullet}$, $\varepsilon_{\circ}$ defines a Poisson structure on
$\mathsf{Proj}(\mathbb{Z}[\mathcal{W}_{\Lambda}])$, independent of the choice
of the perfect matching $\mathsf{m}$.

The matrices $-\mathbb{I}+2\rho_{\mathsf{m},0}$ and
$-\mathbb{I}+2\rho_{\mathsf{m},1}$ are of the form
$\mathbb{I}+\mathrm{nilpotent}$ and are therefore invertible. Note that

$\displaystyle-\mathbb{I}+2\rho_{\mathsf{m},0}=(\mathbb{I}+\varsigma_{\mathsf{m},0})(\mathbb{I}-\varsigma_{\mathsf{m},0})^{-1},\qquad-\mathbb{I}+2\rho_{\mathsf{m},1}=(\mathbb{I}+\varsigma_{\mathsf{m},1})(\mathbb{I}-\varsigma_{\mathsf{m},1})^{-1}.$
(2.37)

The matrix $\rho_{\mathsf{m},0}-\rho_{\mathsf{m},1}$ is not invertible,
because for every $\mathbf{v}\in\mathsf{P}_{\\!\Lambda}^{\star}$:

$\displaystyle\bigl{(}\rho_{\mathsf{m},0}-\rho_{\mathsf{m},1}\bigr{)}\alpha_{\mathbf{v}}=0$
(2.38)

with $\alpha_{\mathbf{v}}$ as in (2.1), here viewed as a column vector. The
kernel of the homomorphism
$\mathrm{H}_{1}\big{(}\Gamma_{\\!\\!\Lambda}^{\vee},\mathbb{Z}\big{)}$
$\rightarrow\mathrm{H}_{1}(\mathbb{T},\mathbb{Z})=\Lambda$ induced by the
embedding $\Gamma_{\\!\\!\Lambda}^{\vee}\hookrightarrow\mathbb{T}$, is
generated by the elements $\check{\alpha}_{\mathbf{v}}$,
$\mathbf{v}\in\mathsf{P}_{\\!\Lambda}^{\star}$. So (2.38) means that the
bilinear form $\varepsilon_{-}$ in (2.32) is the pull-back of the intersection
form on $\mathrm{H}_{1}(\mathbb{T},\mathbb{Z})$; see also [10, Section 1.1].

######  Definition 2.7.

A _zigzag_ for the superpotential
$[\mathcal{F}]_{\Lambda}=({\mathcal{E}_{\Lambda}},\sigma_{0},\sigma_{1})$ is a
cycle of the permutation $\sigma_{1}\sigma_{0}$. The set of zigzags is denoted
by $\mathsf{P}_{\\!\Lambda}^{\mathsf{z}}$. Every zigzag $\mathbf{z}$ defines a
map
$\alpha_{\mathbf{z}}\colon{\mathcal{E}_{\Lambda}}\longrightarrow\mathbb{Z}$,

$\displaystyle\alpha_{\mathbf{z}}(e)=1\quad\text{if}\quad
e\in\mathbf{z},\qquad\alpha_{\mathbf{z}}(e)=-1\quad\textrm{if}\quad\sigma_{0}(e)\in\mathbf{z},\qquad\alpha_{\mathbf{z}}(e)=0\quad\textrm{else}.$

The matrix $\rho_{\mathsf{m},0}+\rho_{\mathsf{m},1}-\mathbb{I}$ is not
invertible either, because, as one easily checks,

$\displaystyle\bigl{(}\rho_{\mathsf{m},0}+\rho_{\mathsf{m},1}-\mathbb{I}\bigr{)}\alpha_{\mathbf{z}}=0$
(2.39)

for every zigzag $\mathbf{z}$. Compare formula (2.39) with [10, Lemma 1.1].

2.2.12.  The matrices $-\mathbb{I}+2\rho_{\mathsf{m},0}$ and
$-\mathbb{I}+2\rho_{\mathsf{m},1}$ have entries in $\mathbb{Z}_{\geq 0}$ and
are of the form $\mathbb{I}+\mathrm{nilpotent}$. So, they give injective (but
not surjective) homomorphisms of semi-groups

$\displaystyle\mathsf{T}_{\mathsf{m},0},\mathsf{T}_{\mathsf{m},1}\colon\
\mathcal{W}_{\Lambda}\longrightarrow\mathcal{W}_{\Lambda}^{\vee},$ (2.40)

which depend on the choice of the perfect matching $\mathsf{m}$. Here
$\mathcal{W}_{\Lambda}^{\vee}$ is the semi-group dual to
$\mathcal{W}_{\Lambda}$.

###  2.3 The geometry of
$\boldsymbol{\mathsf{Spec}(\mathbb{Z}[\mathcal{W}_{\Lambda}])}$

2.3.1.  By definition complex points of the scheme
$\mathsf{Spec}(\mathbb{Z}[\mathcal{W}_{\Lambda}])$ are ring homomorphisms
$\mathbb{Z}[\mathcal{W}_{\Lambda}]\rightarrow\mathbb{C}$, or equivalently,
homomorphism of semi-groups
$\xi\colon\mathcal{W}_{\Lambda}\rightarrow\mathbb{C}^{\times}$, where
$\mathbb{C}^{\times}$ denotes the set $\mathbb{C}$ with multiplication as
binary operation. Such a homomorphism $\xi$ is completely determined by the
complex numbers $\xi_{\mathsf{m}}=\xi(\mathsf{m})$,
$\mathsf{m}\in\mathcal{M}_{\Lambda}$, which must satisfy the equations (2.22).
So

$\displaystyle\mathbb{C}\mathsf{Spec}(\mathbb{Z}[\mathcal{W}_{\Lambda}])=\bigg{\\{}(\xi_{\mathsf{m}})\in\mathbb{C}^{\mathcal{M}_{\Lambda}}\,\big{|}\,\forall\,\mu\in\mathbb{M}\colon\prod_{\mu(\mathsf{m})>0}\xi_{\mathsf{m}}^{\mu(\mathsf{m})}=\prod_{\mu(\mathsf{m})<0}\xi_{\mathsf{m}}^{-\mu(\mathsf{m})}\bigg{\\}},$

where $\mathbb{M}$ denotes the lattice of $\mathbb{Z}$-linear relations
between the perfect matchings:

$\displaystyle\mathbb{M}=\bigg{\\{}\mu\in\mathbb{Z}^{\mathcal{M}_{\Lambda}}\,\big{|}\,\sum_{\mathsf{m}\in\mathcal{M}_{\Lambda}}\mu(\mathsf{m})\mathsf{m}=0\bigg{\\}}.$

A map $\psi\colon{\mathcal{E}_{\Lambda}}\rightarrow\mathbb{C}$ gives a point
in $\mathbb{C}\mathsf{Spec}(\mathbb{Z}[\mathcal{W}_{\Lambda}])$ through the
homomorphism

$\displaystyle E_{\psi}\colon\
\mathcal{W}_{\Lambda}\longrightarrow\mathbb{C}^{*}\subset\mathbb{C}^{\times},\qquad
E_{\psi}(\nu)=\exp\bigg{(}\sum_{e\in{\mathcal{E}_{\Lambda}}}\nu(e)\psi(e)\bigg{)}.$
(2.41)

The following commutative diagram helps to locate these points

$\displaystyle\begin{array}[]{@{}ccccccc}\mathsf{Spec}(\mathbb{Z}[\overline{\mathcal{W}}_{\Lambda}])&\rightarrow&\mathsf{Spec}(\mathbb{Z}[\mathcal{W}_{\Lambda}])&\rightarrow&\mathbb{A}^{|\mathcal{M}_{\Lambda}|}&\supset&\mathbb{C}^{|\mathcal{M}_{\Lambda}|}\\\\[8.61108pt]
\uparrow&&\uparrow&&\uparrow&&\uparrow\\\\[8.61108pt]
\mathsf{Spec}(\mathbb{Z}[\overline{\mathcal{W}}_{\Lambda}])\setminus\mathsf{0}&\rightarrow&\mathsf{Spec}(\mathbb{Z}[\mathcal{W}_{\Lambda}])\setminus\mathsf{0}&\rightarrow&\mathbb{A}^{|\mathcal{M}_{\Lambda}|}\setminus\mathsf{0}&\supset&(\mathbb{C}^{*})^{|\mathcal{M}_{\Lambda}|}\\\\[8.61108pt]
\downarrow&&\downarrow&&\downarrow&&\downarrow\\\\[8.61108pt]
\mathsf{Proj}(\mathbb{Z}[\overline{\mathcal{W}}_{\Lambda}])&\rightarrow&\mathsf{Proj}(\mathbb{Z}[\mathcal{W}_{\Lambda}])&\rightarrow&\mathbb{P}^{|\mathcal{M}_{\Lambda}|-1}&\supset&(\mathbb{C}^{*})^{|\mathcal{M}_{\Lambda}|}/\mathbb{C}^{*}\\\\[8.61108pt]
\uparrow&&\uparrow&&\uparrow&\nearrow_{\simeq}&\\\\[8.61108pt]
\mathrm{H}^{1}(\mathbb{T},\mathbb{C}^{*})&\rightarrow&\mathrm{H}^{1}(\Gamma_{\\!\\!\Lambda}^{\vee},\mathbb{C}^{*})&\rightarrow&(\mathbb{C}^{*})^{|\mathcal{M}_{\Lambda}|-1}&&\end{array}$
(2.49)

where on the second line $\setminus\mathsf{0}$ is shorthand for
$\cap\big{(}\mathbb{A}^{|\mathcal{M}_{\Lambda}|}\setminus\mathsf{0}\big{)}$.

The point given by (2.41) appears, for instance, in
$\mathbb{C}^{|\mathcal{M}_{\Lambda}|}$ as
$\bigl{(}\exp(E_{\psi}(\mathsf{m}))\bigr{)}_{\mathsf{m}\in\mathcal{M}_{\Lambda}}$.
Since its coordinates are $\neq 0$ it projects down into
$\mathrm{H}^{1}\big{(}\Gamma_{\\!\\!\Lambda}^{\vee},\mathbb{C}^{*}\big{)}\subset(\mathbb{C}^{*})^{|\mathcal{M}_{\Lambda}|-1}$.

Replacing $\psi$ by
$\psi+\sum\limits_{\mathbf{b}\in\mathsf{P}_{\\!\Lambda}^{\bullet}}c_{\mathbf{b}}\beta_{\mathbf{b}}+\sum\limits_{\mathbf{w}\in\mathsf{P}_{\\!\Lambda}^{\circ}}c_{\mathbf{w}}\beta_{\mathbf{w}}$
with $\beta_{\mathbf{b}}$, $\beta_{\mathbf{w}}$ as in (2.2), (2.3) and
$c_{\mathbf{b}},c_{\mathbf{w}}\in\mathbb{C}$ multiplies all coordinates of the
point (2.41) with
$\exp\Bigl{(}\sum\limits_{\mathbf{b}\in\mathsf{P}_{\\!\Lambda}^{\bullet}}c_{\mathbf{b}}+\sum_{\mathbf{w}\in\mathsf{P}_{\\!\Lambda}^{\circ}}c_{\mathbf{w}}\Bigr{)}$.
Such a replacement therefore shows no effect when we arrive downstairs in
projective space. Note that this observation agrees with (2.8) and
$\mathrm{H}^{1}\big{(}\Gamma_{\\!\\!\Lambda}^{\vee},\mathbb{C}^{*}\big{)}=\mathrm{H}^{1}\big{(}\Gamma_{\\!\\!\Lambda}^{\vee},\mathbb{Z}\big{)}\otimes_{\mathbb{Z}}\mathbb{C}^{*}$.

2.3.2.  The description (2.20) of the kernel of the homomorphism (2.19)
implies that $\mathsf{Spec}(\mathbb{Z}[\mathcal{W}_{\Lambda}])$ can be
identified with the closed subscheme of the affine space
$\mathbb{A}^{|\mathcal{M}_{\Lambda}|}=\mathsf{Spec}(\mathbb{Z}[X_{\mathsf{m}}\,|\,\mathsf{m}\in\mathcal{M}_{\Lambda}])$
given by the equations (2.22). The function
$\sum\limits_{\mathsf{m}\in\mathcal{M}_{\Lambda}}X_{\mathsf{m}}$ on
$\mathbb{A}^{|\mathcal{M}_{\Lambda}|}$ restricts to a function on
$\mathsf{Spec}(\mathbb{Z}[\mathcal{W}_{\Lambda}])$. At the point (2.41) this
function has the value

$\displaystyle\sum_{\mathsf{m}\in\mathcal{M}_{\Lambda}}\exp\bigg{(}\sum_{e\in{\mathcal{E}_{\Lambda}}}\mathsf{m}(e)\psi(e)\bigg{)}.$

###  2.4 The geometry of a weight realization

2.4.1.  Let $(\nu_{1},\nu_{2},\nu_{3})$ be a weight realization. Set
$\omega=(\nu_{1}-\nu_{3},\nu_{2}-\nu_{3})$ and
$\theta=\frac{1}{\deg\nu_{3}}\nu_{3}$. For $e\in{\mathcal{E}_{\Lambda}}$
consider the associated quadrangle. Proposition 5.3 gives values in
$\mathbb{R}^{2}$ for the vectors from $s(\widetilde{e})$ to
$b(\widetilde{e})$, to $w(\widetilde{e})$ and to $t(\widetilde{e})$. We
identify $\mathbb{R}^{2}$ with $\mathbb{C}$ and obtain three complex numbers
$q_{sb}(e)$, $q_{sw}(e)$, $\omega(e)$. We set

$\displaystyle\mathsf{q}_{\mathbf{b}}(e)=q_{sb}(e)-\textstyle{\frac{1}{2}}\displaystyle\omega(e),\qquad\mathsf{q}_{\mathbf{w}}(e)=q_{sw}(e)-\textstyle{\frac{1}{2}}\displaystyle\omega(e).$
(2.50)

The complex numbers $\mathsf{q}_{\mathbf{b}}(e)$ and
$\mathsf{q}_{\mathbf{w}}(e)$ are the vectors from the midpoint of the
$s(\widetilde{e})t(\widetilde{e})$-diagonal to $b(\widetilde{e})$ and
$w(\widetilde{e})$, respectively; see Fig. 10.

Figure 10: Quadrangle for edge $e$.

######  Proposition 2.8.

We have for every perfect matching $\mathsf{m}^{\prime}$

$\displaystyle\sum_{e\in{\mathcal{E}_{\Lambda}}}\mathsf{m}^{\prime}(e)\mathsf{q}_{\mathbf{b}}(e)=(\theta^{t}-\mathsf{m}^{\prime
t})\cdot\big{(}{-}\textstyle{\frac{1}{2}}\displaystyle\mathbb{I}+\rho_{\mathsf{m},1}\big{)}\cdot\omega,$
(2.51)
$\displaystyle\sum_{e\in{\mathcal{E}_{\Lambda}}}\mathsf{m}^{\prime}(e)\mathsf{q}_{\mathbf{w}}(e)=(\theta^{t}-\mathsf{m}^{\prime
t})\cdot\big{(}{-}\textstyle{\frac{1}{2}}\displaystyle\mathbb{I}+\rho_{\mathsf{m},0}\big{)}\cdot\omega.$
(2.52)

Here we view $\mathsf{m}^{\prime}$ as a column vector of $0$’s and $1$’s and
$\omega$ as a column vector of complex numbers.111The perfect matching
$\mathsf{m}$ appearing on the right-hand side has been fixed for the
computations in Section 5.3. Since the left-hand side does not involve
$\mathsf{m}$ the results do in fact not depend on the choice of $\mathsf{m}$.

###### Proof.

Equations (5.8) and (5.9) yield

$\displaystyle\mathsf{m}^{\prime
t}\cdot\mathbf{B}_{\mathsf{m},\theta}=\sum_{\mathbf{b}\in\mathsf{P}_{\\!\Lambda}^{\bullet}}\theta^{t}\cdot\mathsf{diag}(\beta_{\mathbf{b}})\cdot\big{(}{-}\textstyle{\frac{1}{2}}\displaystyle\mathbb{I}+\rho_{\mathsf{m},1}\big{)}=\theta^{t}\cdot\big{(}{-}\textstyle{\frac{1}{2}}\displaystyle\mathbb{I}+\rho_{\mathsf{m},1}\big{)},$
(2.53) $\displaystyle\mathsf{m}^{\prime
t}\cdot\mathbf{W}_{\mathsf{m},\theta}=\sum_{\mathbf{w}\in\mathsf{P}_{\\!\Lambda}^{\bullet}}\theta^{t}\cdot\mathsf{diag}(\beta_{\mathbf{w}})\cdot\big{(}{-}\textstyle{\frac{1}{2}}\displaystyle\mathbb{I}+\rho_{\mathsf{m},0}\big{)}=\theta^{t}\cdot\big{(}{-}\textstyle{\frac{1}{2}}\displaystyle\mathbb{I}+\rho_{\mathsf{m},0}\big{)}.$
(2.54)

The results (2.51) and (2.52) follow from (2.50), (2.53), (2.54) and
Proposition 5.3. ∎

2.4.2.  Subtracting (2.51) from (2.52) we get

$\displaystyle\sum_{e\in{\mathcal{E}_{\Lambda}}}\mathsf{m}^{\prime}(e)\bigl{(}\mathsf{q}_{\mathbf{w}}(e)-\mathsf{q}_{\mathbf{b}}(e)\bigr{)}=\big{(}\theta^{t}-\mathsf{m}^{\prime
t}\big{)}\cdot(\rho_{\mathsf{m},0}-\rho_{\mathsf{m},1})\cdot\omega.$

From (2.13) and (2.38) we see that if
$\mathsf{m}^{\prime\prime}\sim\mathsf{m}^{\prime}$, then

$\displaystyle(\theta^{t}-\mathsf{m}^{\prime\prime
t})\cdot(\rho_{\mathsf{m},0}-\rho_{\mathsf{m},1})\cdot\omega=\big{(}\theta^{t}-\mathsf{m}^{\prime
t}\big{)}\cdot(\rho_{\mathsf{m},0}-\rho_{\mathsf{m},1})\cdot\omega.$

This means that the map

$\displaystyle\mathcal{M}_{\Lambda}\longrightarrow\mathbb{R}^{2},\qquad\mathsf{m}^{\prime}\mapsto\sum_{e\in{\mathcal{E}_{\Lambda}}}\mathsf{m}^{\prime}(e)\bigl{(}\mathsf{q}_{\mathbf{w}}(e)-\mathsf{q}_{\mathbf{b}}(e)\bigr{)}=\big{(}\theta^{t}-\mathsf{m}^{\prime
t}\big{)}\cdot(\rho_{\mathsf{m},0}-\rho_{\mathsf{m},1})\cdot\omega$

induces an embedding of $\mathcal{A}_{\Lambda}=\mathcal{M}_{\Lambda}/\sim$ and
the Newton polygon $\mathsf{conv}(\mathcal{A}_{\Lambda})$ into
$\mathbb{R}^{2}$.

######  Example 2.9.

The above method yields for the superpotentials on the Zhegalkin zebra
functions $\mathcal{F}_{2}$, $\mathcal{F}_{3}$, $\mathcal{F}_{4}$,
$\mathcal{F}_{6}$ and $\mathcal{F}$ in Figs. 2, 3, 12 and Example 5.1 the
Newton polygons in Fig. 11, where we have also indicated the sizes of the
fibres of $\mathcal{M}_{\Lambda}\rightarrow\mathcal{A}_{\Lambda}$ if $>1$.

Figure 11: Newton polygons for the superpotentials on the Zhegalkin zebra
functions $\mathcal{F}_{2}$, $\mathcal{F}_{3}$, $\mathcal{F}_{4}$,
$\mathcal{F}_{6}$ and $\mathcal{F}$ in Figs. 2, 3, 12 and Example 5.1.

2.4.3.  For a map $\psi\colon{\mathcal{E}_{\Lambda}}\rightarrow\mathbb{C}$ and
$z\in\mathbb{C}$ we can construct the maps

$\displaystyle\psi+z\mathsf{q}_{\mathbf{b}},\qquad\psi+z\mathsf{q}_{\mathbf{w}}\colon\
{\mathcal{E}_{\Lambda}}\rightarrow\mathbb{C}.$

With the constructions in Section 2.3 we thus obtain two (parametrized) curves
in $\mathbb{C}\mathsf{Spec}(\mathbb{Z}[\mathcal{W}_{\Lambda}])$ through the
point given by $\psi$:

$\displaystyle{\mathcal{S}_{\bullet}}_{\psi},{\mathcal{S}_{\circ}}_{\psi}\colon\
\mathbb{C}\longrightarrow\mathbb{C}\mathsf{Spec}(\mathbb{Z}[\mathcal{W}_{\Lambda}]),$
(2.55)
$\displaystyle{\mathcal{S}_{\bullet}}_{\psi}(z)=\bigg{(}\exp\bigg{(}\sum_{e\in{\mathcal{E}_{\Lambda}}}\mathsf{m}^{\prime}(e)\bigl{(}\psi(e)+z\mathsf{q}_{\mathbf{b}}(e)\bigr{)}\bigg{)}\bigg{)}_{\mathsf{m}^{\prime}\in\mathcal{M}_{\Lambda}},$
(2.56)
$\displaystyle{\mathcal{S}_{\circ}}_{\psi}(z)=\bigg{(}\exp\bigg{(}\sum_{e\in{\mathcal{E}_{\Lambda}}}\mathsf{m}^{\prime}(e)\bigl{(}\psi(e)+z\mathsf{q}_{\mathbf{w}}(e)\bigr{)}\bigg{)}\bigg{)}_{\mathsf{m}^{\prime}\in\mathcal{M}_{\Lambda}}.$
(2.57)

It follows from (2.35), (2.36), (2.51) and (2.52) that for every
$\mathsf{m}^{\prime}\in\mathcal{M}_{\Lambda}$

$\displaystyle\frac{\partial}{\partial
z}\exp\bigg{(}\sum_{e\in{\mathcal{E}_{\Lambda}}}\mathsf{m}^{\prime}(e)\bigl{(}\psi(e)+z\mathsf{q}_{\mathbf{b}}(e)\bigr{)}\bigg{)}$
$\displaystyle\qquad{}=\textstyle{\frac{1}{2}}\displaystyle\varepsilon_{\bullet}\big{(}\theta^{t}-\mathsf{m}^{\prime
t},\omega\big{)}\exp\bigg{(}\sum_{e\in{\mathcal{E}_{\Lambda}}}\mathsf{m}^{\prime}(e)\bigl{(}\psi(e)+z\mathsf{q}_{\mathbf{b}}(e)\bigr{)}\bigg{)},$
(2.58) $\displaystyle\frac{\partial}{\partial
z}\exp\bigg{(}\sum_{e\in{\mathcal{E}_{\Lambda}}}\mathsf{m}^{\prime}(e)\bigl{(}\psi(e)+z\mathsf{q}_{\mathbf{w}}(e)\bigr{)}\bigg{)}$
$\displaystyle\qquad{}=\textstyle{\frac{1}{2}}\displaystyle\varepsilon_{\circ}\big{(}\theta^{t}-\mathsf{m}^{\prime
t},\omega\big{)}\exp\bigg{(}\sum_{e\in{\mathcal{E}_{\Lambda}}}\mathsf{m}^{\prime}(e)\bigl{(}\psi(e)+z\mathsf{q}_{\mathbf{w}}(e)\bigr{)}\bigg{)}.$
(2.59)

2.4.4.  Since the coordinates in (2.56) and (2.57) are non-zero, these
formulas also define two curves

$\displaystyle\overline{\mathcal{S}_{\bullet}}_{\psi},\overline{\mathcal{S}_{\circ}}_{\psi}\colon\
\mathbb{C}\longrightarrow\mathrm{H}^{1}\big{(}\Gamma_{\\!\\!\Lambda}^{\vee},\mathbb{C}^{*}\big{)}\subset\mathbb{C}\mathsf{Proj}(\mathbb{Z}[\mathcal{W}_{\Lambda}]).$

The symplectic forms $\varepsilon_{\bullet}$ and $\varepsilon_{\circ}$ on
$\mathrm{H}^{1}\big{(}\Gamma_{\\!\\!\Lambda}^{\vee},\mathbb{C}^{*}\big{)}$ and
the realization $\omega$ yield two vector fields
$\varepsilon_{\bullet}(\textrm{-},\omega)$ and
$\varepsilon_{\circ}(\textrm{-},\omega)$. The formulas (2.58), resp. (2.59),
show that the curves $\overline{\mathcal{S}_{\bullet}}_{\psi}$, resp.
$\overline{\mathcal{S}_{\circ}}_{\psi}$, are integral curves for these vector
fields.

##  3 The Jacobi algebra

Let $\mathcal{F}$ be a convex Zhegalkin zebra function and $\Lambda$ a
sublattice of $\mathsf{Aut}(\mathcal{F})$, such that the superpotential
$[\mathcal{F}]_{\Lambda}$ is dimer complete and such that a weight realization
of the superpotential exists; see Definitions 1.1, 1.2, 1.5 and 1.6. So there
is a tiling of the torus $\mathbb{R}^{2}/\Lambda$ by convex black and white
polygons. The vertices and edges of these polygons constitute a quiver (=
directed graph) $\Gamma_{\\!\\!\Lambda}$. In this section we study an algebra
which is naturally associated with the embedding of the quiver
$\Gamma_{\\!\\!\Lambda}$ into the torus $\mathbb{R}^{2}/\Lambda$.

###  3.1 General theory of the Jacobi algebra and master space

3.1.1.

######  Definition 3.1 (cf. [1, 5, 6, 8]).

  1. (i)

The path algebra $\mathbb{Z}[\mathsf{Path}(\Gamma_{\\!\\!\Lambda})]$ of the
quiver $\Gamma_{\\!\\!\Lambda}$ is the free abelian group on the set of all
paths on $\Gamma_{\\!\\!\Lambda}$. The product $\mathbf{p}\mathbf{p}^{\prime}$
is the concatenation of $\mathbf{p}$ and $\mathbf{p}^{\prime}$ if
$t(\mathbf{p})=s(\mathbf{p}^{\prime})$ and is $0$ otherwise. The constant
paths supported on the vertices of the quiver are idempotent elements in
$\mathbb{Z}[\mathsf{Path}(\Gamma_{\\!\\!\Lambda})]$.

  2. (ii)

The _Jacobi algebra_ of the superpotential $[\mathcal{F}]_{\Lambda}$ is the
algebra

$\displaystyle\mathsf{Jac}([\mathcal{F}]_{\Lambda})=\raisebox{2.15277pt}{$\mathbb{Z}[\mathsf{Path}(\Gamma_{\\!\\!\Lambda})]$}/\raisebox{-2.15277pt}{$\langle\mathsf{D}^{\circ}(e)\,|\,e\in{\mathcal{E}_{\Lambda}}\rangle$},$
(3.1)

where $\langle\mathsf{D}^{\circ}(e)\,|\,e\in{\mathcal{E}_{\Lambda}}\rangle$ is
the two sided ideal generated by the elements

$\displaystyle\mathsf{D}^{\circ}(e)=\prod_{e^{\prime}\neq e\colon
w(e^{\prime})=w(e)}^{\circlearrowleft}e^{\prime}-\prod_{e^{\prime}\neq e\colon
b(e^{\prime})=b(e)}^{\circlearrowright}e^{\prime}.$ (3.2)

  3. (iii)

The _master space_ 222In [1, 8] the master space is denoted as
$\mathcal{F}^{\flat}$. of $[\mathcal{F}]_{\Lambda}$ is the scheme
$\mathsf{Spec}(\mathsf{R}^{\flat}([\mathcal{F}]_{\Lambda}))$ with

$\displaystyle\mathsf{R}^{\flat}([\mathcal{F}]_{\Lambda})=\raisebox{2.15277pt}{$\mathbb{Z}[X_{e}\,|\,e\in{\mathcal{E}_{\Lambda}}]$}/\raisebox{-2.15277pt}{$\langle\mathsf{D}^{\flat}(e)|e\in{\mathcal{E}_{\Lambda}}\rangle$};$
(3.3)

here $\langle\mathsf{D}^{\flat}(e)\,|\,e\in{\mathcal{E}_{\Lambda}}\rangle$ is
the ideal generated by the elements

$\displaystyle\mathsf{D}^{\flat}(e)=\prod_{e^{\prime}\neq e\colon
w(e^{\prime})=w(e)}X_{e^{\prime}}-\prod_{e^{\prime}\neq e\colon
b(e^{\prime})=b(e)}X_{e^{\prime}}.$ (3.4)

######  Remark 3.2.

Note that $\mathsf{D}^{\flat}(e)=\frac{d\mathsf{F}}{dX_{e}}$ with

$\displaystyle\mathsf{F}=\sum_{\mathbf{w}\in\mathsf{P}_{\\!\Lambda}^{\circ}}\prod_{e\in{\mathcal{E}_{\Lambda}}\colon
w(e)=\mathbf{w}}X_{e}-\sum_{\mathbf{b}\in\mathsf{P}_{\\!\Lambda}^{\bullet}}\prod_{e\in{\mathcal{E}_{\Lambda}}\colon
b(e)=\mathbf{b}}X_{e}.$

So, $\langle\mathsf{D}^{\flat}(e)\,|\,e\in{\mathcal{E}_{\Lambda}}\rangle$ is
the Jacobi ideal of the polynomial $\mathsf{F}$.

The monomials in $\mathsf{F}$ correspond 1-1 with the cycles of the
permutations $\sigma_{0}$ and $\sigma_{1}$ with neglect of the cyclic
structure. Compare this with Remark 1.3 and the analogies (3.1)/(3.3) and
(3.2)/(3.4).

3.1.2.  The semi-group dual to $\mathcal{W}_{\Lambda}$ is
$\mathcal{W}_{\Lambda}^{\vee}=\mathrm{Hom}(\mathcal{W}_{\Lambda},\mathbb{Z}_{\geq
0})$. Evaluation of maps ${\mathcal{E}_{\Lambda}}\rightarrow\mathbb{Z}_{\geq
0}$ induces a map

$\displaystyle{\mathcal{E}_{\Lambda}}\longrightarrow\mathcal{W}_{\Lambda}^{\vee},\qquad
e\mapsto\bigl{(}\nu\mapsto\nu(e)\bigr{)}.$ (3.5)

Let ${\mathsf{Mat}}_{\mathsf{P}_{\\!\Lambda}^{\star}}$ denote the ring of
matrices with rows and columns indexed by the elements of
$\mathsf{P}_{\\!\Lambda}^{\star}$ and let
$\mathbb{Z}\big{[}\mathcal{W}_{\Lambda}^{\vee}\big{]}$ denote the semi-group
ring of $\mathcal{W}_{\Lambda}^{\vee}$. The map (3.5) can then be upgraded to
an algebra homomorphism

$\displaystyle\widetilde{\Phi}\colon\
\mathbb{Z}[\mathsf{Path}(\Gamma_{\\!\\!\Lambda})]\longrightarrow{\mathsf{Mat}}_{\mathsf{P}_{\\!\Lambda}^{\star}}\big{(}\mathbb{Z}\big{[}\mathcal{W}_{\Lambda}^{\vee}\big{]}\big{)},$

such that $\widetilde{\Phi}(\mathbf{p})$ is the matrix with all entries $0$
except for the $(s(\mathbf{p}),t(\mathbf{p}))$-entry, which is $\mathbf{p}$
viewed as an element of $\mathcal{W}_{\Lambda}^{\vee}$ through (3.5); i.e.,

$\displaystyle\mathbf{p}=(e_{1},\ldots,e_{k})\mapsto\bigg{(}\nu\mapsto\nu(\mathbf{p})=\sum_{j=1}^{k}\nu(e_{j})\bigg{)}.$

It is clear from (2.11) and (3.2) that $\widetilde{\Phi}$ induces an algebra
homomorphism

$\displaystyle\Phi\colon\
\mathsf{Jac}([\mathcal{F}]_{\Lambda})\longrightarrow{\mathsf{Mat}}_{\mathsf{P}_{\\!\Lambda}^{\star}}\big{(}\mathbb{Z}\big{[}\mathcal{W}_{\Lambda}^{\vee}\big{]}\big{)}.$
(3.6)

######  Definition 3.3.

We call the above homomorphism $\Phi$ the tautological representation of the
Jacobi algebra.

3.1.3.  It follows from (2.11) and (3.4) that the ring homomorphism

$\displaystyle\mathbb{Z}[X_{e}\,|\,e\in{\mathcal{E}_{\Lambda}}]\longrightarrow\mathbb{Z}\big{[}\mathcal{W}_{\Lambda}^{\vee}\big{]},\qquad
X_{e}\mapsto(\nu\mapsto\nu(e))$

induces a ring homomorphism

$\displaystyle\mathsf{R}^{\flat}([\mathcal{F}]_{\Lambda})\longrightarrow\mathbb{Z}\big{[}\mathcal{W}_{\Lambda}^{\vee}\big{]}$
(3.7)

and, hence, a morphism of schemes

$\displaystyle\mathsf{Spec}\big{(}\mathbb{Z}\big{[}\mathcal{W}_{\Lambda}^{\vee}\big{]}\big{)}\longrightarrow\mathsf{Spec}\big{(}\mathsf{R}^{\flat}([\mathcal{F}]_{\Lambda})\big{)}.$
(3.8)

######  Proposition 3.4.

The image of the morphism (3.8) is an irreducible closed subscheme of the
Master Space
$\mathsf{Spec}\big{(}\mathsf{R}^{\flat}([\mathcal{F}]_{\Lambda})\big{)}$.333In
[1, 8] the master space is denoted by $\mathcal{F}^{\flat}$ and the
irreducible component by ${}^{\textrm{Irr}}\\!\mathcal{F}^{\flat}$.

###### Proof.

Since $\mathcal{W}_{\Lambda}=\mathbb{Z}_{\geq 0}\mathcal{M}_{\Lambda}$ by
(2.18) an element $\alpha$ of $\mathcal{W}_{\Lambda}^{\vee}$ is completely
determined by its values $\alpha(\mathsf{m})$ for
$\mathsf{m}\in\mathcal{M}_{\Lambda}$. So there is an injective ring
homomorphism

$\displaystyle\mathbb{Z}\big{[}\mathcal{W}_{\Lambda}^{\vee}\big{]}\longrightarrow\mathbb{Z}[U_{\mathsf{m}}\,|\,\mathsf{m}\in\mathcal{M}_{\Lambda}],\qquad\alpha\mapsto\prod_{\mathsf{m}\in\mathcal{M}_{\Lambda}}U_{\mathsf{m}}^{\alpha(\mathsf{m})}.$
(3.9)

It follows that the ring
$\mathbb{Z}\big{[}\mathcal{W}_{\Lambda}^{\vee}\big{]}$ has no zero-divisors
and that the kernel of the ring homomorphism (3.7) is a prime ideal. ∎

3.1.4.  With $\mathbb{M}$ as in equation (2.21) we have

$\displaystyle\mathcal{W}_{\Lambda}^{\vee}=\bigg{\\{}\alpha\in\mathbb{Z}^{\mathcal{M}_{\Lambda}}\,|\,\forall\,\mu\in\mathbb{M}\colon\sum_{\mathsf{m}\in\mathcal{M}_{\Lambda}}\alpha(\mathsf{m})\mu(\mathsf{m})=0\bigg{\\}}.$

From this we see that (3.9) identifies
$\mathbb{C}\big{[}\mathcal{W}_{\Lambda}^{\vee}\big{]}$ with the sub-ring of
$\mathbb{C}[U_{\mathsf{m}}\,|\,\mathsf{m}\in\mathcal{M}_{\Lambda}]$ consisting
of those polynomials which are invariant under the (obvious) action of the
torus $\mathbb{M}\otimes\mathbb{C}^{*}$:

$\displaystyle\mathbb{C}\big{[}\mathcal{W}_{\Lambda}^{\vee}\big{]}=\mathbb{C}[U_{\mathsf{m}}\,|\,\mathsf{m}\in\mathcal{M}_{\Lambda}]^{\mathbb{M}\otimes\mathbb{C}^{*}}.$
(3.10)

The situation described by (3.10) is in an obvious sense dual to the situation
described in (2.19)–(2.22).

######  Remark 3.5.

The story in (3.3), (3.4), (2.21), (3.10) is well-known. It differs from the
discussion of the master space and its irreducible component in [1, 8] only in
terminology and style and in that we have highlighted the role of the weight
functions.

3.1.5.  By definition the center of the Jacobi algebra is

$\displaystyle\mathsf{Z}(\mathsf{Jac}([\mathcal{F}]_{\Lambda}))=\\{\pi\in\mathsf{Jac}([\mathcal{F}]_{\Lambda})\,|\,\forall\,e\in{\mathcal{E}_{\Lambda}}\colon\pi
e=e\pi\\}.$

Applying $\Phi$ (3.6) to an element $\pi$ in
$\mathsf{Z}(\mathsf{Jac}([\mathcal{F}]_{\Lambda}))$ yields the matrix
equations

$\displaystyle\forall\,e\in{\mathcal{E}_{\Lambda}}\colon\
\Phi(\pi)\Phi(e)=\Phi(e)\Phi(\pi).$

For $\nu\in\mathcal{W}_{\Lambda}$ “evaluation at $\nu$” defines a homomorphism
of semi-groups $\mathcal{W}_{\Lambda}^{\vee}\rightarrow\mathbb{Z}_{\geq 0}$
and, hence, a homomorphism of rings
$\mathbb{Z}\big{[}\mathcal{W}_{\Lambda}^{\vee}\big{]}\rightarrow\mathbb{Z}[u]$.
By combining this homomorphism with $\Phi$ we obtain an algebra homomorphism

$\displaystyle\Phi_{\nu}\colon\
\mathsf{Jac}([\mathcal{F}]_{\Lambda})\longrightarrow{\mathsf{Mat}}_{\mathsf{P}_{\\!\Lambda}^{\star}}(\mathbb{Z}[u]),$

such that for every $e\in{\mathcal{E}_{\Lambda}}$ the only non-zero entry of
the matrix $\Phi_{\nu}(e)$ is $u^{\nu(e)}$ in position $(s(e),t(e))$. The
matrix equations

$\displaystyle\forall\,e\in{\mathcal{E}_{\Lambda}}\colon\
\Phi_{\nu}(\pi)\Phi_{\nu}(e)=\Phi_{\nu}(e)\Phi_{\nu}(\pi)$

then imply that there is an element
$c_{\pi}\in\mathbb{Z}\big{[}\mathcal{W}_{\Lambda}^{\vee}\big{]}$ such that

$\displaystyle\Phi(\pi)=c_{\pi}\mathbb{I},\qquad\text{i.e.},\qquad\forall\,\nu\in\mathcal{W}_{\Lambda}\colon\
\Phi_{\nu}(\pi)=c_{\pi}(\nu)\mathbb{I}.$

Now let $\nu,\nu^{\prime}\in\mathcal{W}_{\Lambda}$ be such that
$\nu\sim\nu^{\prime}$ (see (2.13)), say

$\displaystyle\nu-\nu^{\prime}=\sum_{\mathbf{v}\in\mathsf{P}_{\\!\Lambda}^{\star}}r_{\mathbf{v}}\alpha_{\mathbf{v}}\qquad\textrm{with}\quad
r_{\mathbf{v}}\in\mathbb{Z},\quad\forall\,\mathbf{v}\in\mathsf{P}_{\\!\Lambda}^{\star}.$
(3.11)

Then we have for all $e\in{\mathcal{E}_{\Lambda}}$:

$\displaystyle\Phi_{\nu^{\prime}}(e)=\mathsf{diag}\bigl{(}\big{(}u^{r_{\mathbf{v}}}\big{)}_{\mathbf{v}\in\mathsf{P}_{\\!\Lambda}^{\star}}\bigr{)}\cdot\Phi_{\nu}(e)\cdot\mathsf{diag}\bigl{(}\big{(}u^{-r_{\mathbf{v}}}\big{)}_{\mathbf{v}\in\mathsf{P}_{\\!\Lambda}^{\star}}\bigr{)}.$
(3.12)

Consequently we have for $\pi$ in
$\mathsf{Z}(\mathsf{Jac}([\mathcal{F}]_{\Lambda}))$:

$\displaystyle\Phi_{\nu^{\prime}}(\pi)=\mathsf{diag}\bigl{(}\big{(}u^{r_{\mathbf{v}}}\big{)}_{\mathbf{v}\in\mathsf{P}_{\\!\Lambda}^{\star}}\bigr{)}\cdot\Phi_{\nu}(\pi)\cdot\mathsf{diag}\bigl{(}\big{(}u^{-r_{\mathbf{v}}}\big{)}_{\mathbf{v}\in\mathsf{P}_{\\!\Lambda}^{\star}}\bigr{)}$
(3.13)

and, hence, $c_{\pi}(\nu^{\prime})=c_{\pi}(\nu)$. This means that $c_{\pi}$ is
actually an element of the semi-group ring
$\mathbb{Z}\big{[}\overline{\mathcal{W}}_{\Lambda}^{\vee}\big{]}$ of the semi-
group $\overline{\mathcal{W}}_{\Lambda}^{\vee}$ dual to
$\overline{\mathcal{W}}_{\Lambda}$.

This proves:

######  Proposition 3.6.

There is an algebra homomorphism

$\displaystyle\mathsf{Z}(\mathsf{Jac}([\mathcal{F}]_{\Lambda}))\longrightarrow\mathbb{Z}\big{[}\overline{\mathcal{W}}_{\Lambda}^{\vee}\big{]}$
(3.14)

induced by the tautological representation $\Phi$ (3.6).

###  3.2 Weight realizations and Jacobi algebra

3.2.1.  For a weight realization $(\nu_{1},\nu_{2},\nu_{3})$ “evaluation” at
$(\nu_{1},\nu_{2},\nu_{3})$ defines a homomorphism of rings
$\mathbb{Z}\big{[}\mathcal{W}_{\Lambda}^{\vee}\big{]}\rightarrow\mathbb{Z}[u_{1},u_{2},u_{3}]$.
By combining this homomorphism with $\Phi$ we obtain an algebra homomorphism

$\displaystyle\Phi_{\nu_{1},\nu_{2},\nu_{3}}\colon\
\mathsf{Jac}([\mathcal{F}]_{\Lambda})\longrightarrow{\mathsf{Mat}}_{\mathsf{P}_{\\!\Lambda}^{\star}}(\mathbb{Z}[u_{1},u_{2},u_{3}]),$
(3.15)

such that for every $e\in{\mathcal{E}_{\Lambda}}$ the only non-zero entry of
the matrix $\Phi_{\nu_{1},\nu_{2},\nu_{3}}(e)$ is
$u_{1}^{\nu_{1}(e)}u_{2}^{\nu_{2}(e)}u_{3}^{\nu_{3}(e)}$ in position
$(s(e),t(e))$. So, in particular

$\displaystyle\Phi_{\nu_{1},\nu_{2},\nu_{3}}\bigg{(}\sum_{e\in{\mathcal{E}_{\Lambda}}}e\bigg{)}={\mathfrak{A}}^{\star\star}\big{(}u_{1}^{\nu_{1}}u_{2}^{\nu_{2}}u_{3}^{\nu_{3}}\big{)}$

with matrix
${\mathfrak{A}}^{\star\star}\big{(}u_{1}^{\nu_{1}}u_{2}^{\nu_{2}}u_{3}^{\nu_{3}}\big{)}$
as in (1.4).

3.2.2.  Let $(\nu_{1},\nu_{2},\nu_{3})$ and
$(\nu^{\prime}_{1},\nu^{\prime}_{2},\nu^{\prime}_{3})$ be weight realizations
such that $\nu_{1}\sim\nu^{\prime}_{1}$, $\nu_{2}\sim\nu^{\prime}_{2}$,
$\nu_{3}\sim\nu^{\prime}_{3}$; see (2.13). Then one can show as in
(3.11)–(3.13) that there is a diagonal matrix $D$ such that

$\displaystyle{\mathfrak{A}}^{\star\star}\big{(}u_{1}^{\nu^{\prime}_{1}}u_{2}^{\nu^{\prime}_{2}}u_{3}^{\nu^{\prime}_{3}}\big{)}=D\cdot{\mathfrak{A}}^{\star\star}\big{(}u_{1}^{\nu_{1}}u_{2}^{\nu_{2}}u_{3}^{\nu_{3}}\big{)}\cdot
D^{-1}.$

This means that the representations $\Phi_{\nu_{1},\nu_{2},\nu_{3}}$ and
$\Phi_{\nu^{\prime}_{1},\nu^{\prime}_{2},\nu^{\prime}_{3}}$ are isomorphic.

3.2.3. The matrix
${\mathfrak{A}}^{\star\star}\big{(}u_{1}^{\nu_{1}}u_{2}^{\nu_{2}}u_{3}^{\nu_{3}}\big{)}$
contains the complete information on the edge vectors with which one can draw
the quiver $\Gamma$ embedded in $\mathbb{R}^{2}$. More precisely, consider a
path $\mathbf{p}=(e_{1},\ldots,e_{k})$ and its subpaths
$\mathbf{p}_{j}=(e_{1},\ldots,e_{j})$ for $j=1,\ldots,k$. The path
$\mathbf{p}_{j}$ corresponds to the monomial

$\displaystyle\prod_{r=1}^{j}u_{1}^{\nu_{1}(e_{r})}u_{2}^{\nu_{2}(e_{r})}u_{3}^{\nu_{3}(e_{r})}$

in the $(s(e_{1}),t(e_{j}))$-entry of the matrix
$\bigl{(}{\mathfrak{A}}^{\star\star}\big{(}u_{1}^{\nu_{1}}u_{2}^{\nu_{2}}u_{3}^{\nu_{3}}\big{)}\bigr{)}^{j}$.
In this way one sees the actual path $\mathbf{p}$ as it runs through the end
points of the subpaths $\mathbf{p}_{j}$, $j=1,\ldots,k$. Thus one obtains from
${\mathfrak{A}}^{\star\star}\big{(}u_{1}^{\nu_{1}}u_{2}^{\nu_{2}}u_{3}^{\nu_{3}}\big{)}$
the paths on $\Gamma_{\\!\\!\Lambda}$, the boundary cycles of the polygons and
the period lattice. Since this is all one needs for (3.1)–(3.2) we conclude:

######  Theorem 3.7.

The algebra homomorphisms

$\displaystyle\Phi_{\nu_{1},\nu_{2},\nu_{3}}\colon\
\mathsf{Jac}([\mathcal{F}]_{\Lambda})$
$\displaystyle\longrightarrow{\mathsf{Mat}}_{\mathsf{P}_{\\!\Lambda}^{\star}}(\mathbb{Z}[u_{1},u_{2},u_{3}]),$
$\displaystyle\Phi\colon\ \mathsf{Jac}([\mathcal{F}]_{\Lambda})$
$\displaystyle\longrightarrow{\mathsf{Mat}}_{\mathsf{P}_{\\!\Lambda}^{\star}}\big{(}\mathbb{Z}\big{[}\mathcal{W}_{\Lambda}^{\vee}\big{]}\big{)},$
$\displaystyle\mathsf{Z}(\mathsf{Jac}([\mathcal{F}]_{\Lambda}))$
$\displaystyle\longrightarrow\mathbb{Z}\big{[}\overline{\mathcal{W}}_{\Lambda}^{\vee}\big{]}$

in (3.15), (3.6), (3.14) are injective.

######  Remark 3.8.

Theorem 3.7 is closely related to Theorem 3.17 and Definition 3.10 in [5]. So
it seems that the quivers with potential coming from Zhegalkin zebra functions
are cancellative in the sense of [5, Definition 3.10].

Theorem 3.20 in [5] states that $\mathsf{Jac}([\mathcal{F}]_{\Lambda})$ is a
non-commutative crepant resolution of the $3$-dimensional Gorenstein
singularity
$\mathsf{Spec}\bigl{(}\mathsf{Z}(\mathsf{Jac}([\mathcal{F}]_{\Lambda}))\bigr{)}$
if the quiver $\Gamma_{\\!\\!\Lambda}$ with superpotential
$[\mathcal{F}]_{\Lambda}$ is cancellative.

On the other hand, we do have examples of Zhegalkin zebra functions for which
the quiver with potential is not consistent in the sense of [5, Theorems 1.37
and 3.11].

######  Remark 3.9.

The above method of generating paths corresponds to the series expansion

$\displaystyle\bigl{(}\mathbb{I}-{\mathfrak{A}}^{\star\star}\big{(}u_{1}^{\nu_{1}}u_{2}^{\nu_{2}}u_{3}^{\nu_{3}}\big{)}\bigr{)}^{-1}=\sum_{j=0}^{\infty}\bigl{(}{\mathfrak{A}}^{\star\star}\big{(}u_{1}^{\nu_{1}}u_{2}^{\nu_{2}}u_{3}^{\nu_{3}}\big{)}\bigr{)}^{j}.$

Since every entry of the matrix
${\mathfrak{A}}^{\star\star}\big{(}u_{1}^{\nu_{1}}u_{2}^{\nu_{2}}u_{3}^{\nu_{3}}\big{)}$
is divisible by $u_{1}u_{2}u_{3}$ the series on the right-hand side converges
in the topology provided by the powers of the principal ideal
$u_{1}u_{2}u_{3}\mathbb{Z}[u_{1},u_{2},u_{3}]$.

##  4 Symptoms of mirror symmetry

In this section we put the results of Sections 2 and 3 into the perspective of
mirror symmetry. There are evidently two sides to the story with the graphs
$\Gamma_{\\!\\!\Lambda}^{\vee}$ and $\Gamma_{\\!\\!\Lambda}$ on different
sides and the S-quad-graph $\mathcal{D}_{\mathcal{F},\Lambda}$ providing a
“mirror correspondence”. The appearance of the semi-group ring
$\mathbb{Z}[\mathcal{W}_{\Lambda}]$ of $\mathcal{W}_{\Lambda}$ on one side and
the semi-group ring $\mathbb{Z}\big{[}\mathcal{W}_{\Lambda}^{\vee}\big{]}$ of
the dual semi-group
$\mathcal{W}_{\Lambda}^{\vee}=\mathrm{Hom}(\mathcal{W}_{\Lambda},\mathbb{Z}_{\geq
0})$ on the other side is reminiscent of mirror symmetry as in the work of
Batyrev and Borisov [2, 3].

The $\boldsymbol{\Gamma_{\\!\\!\Lambda}^{\vee}}$-side: The semi-group
$\mathcal{W}_{\Lambda}$ is to be put on the
$\Gamma_{\\!\\!\Lambda}^{\vee}$-side, because (see Section 2.2)

$\displaystyle\mathrm{H}^{1}\big{(}\Gamma_{\\!\\!\Lambda}^{\vee},\mathbb{Z}\big{)}=\bigl{\\{}\nu^{\prime}-\nu^{\prime\prime}\in\mathbb{Z}^{{\mathcal{E}_{\Lambda}}}\,|\,\nu^{\prime},\nu^{\prime\prime}\in\mathcal{W}_{\Lambda},\,\deg\nu^{\prime}=\deg\nu^{\prime\prime}\bigr{\\}}.$

$\Gamma_{\\!\\!\Lambda}$ induces an equivalence relation $\sim$ on
$\mathcal{W}_{\Lambda}$ (cf. Section 2.2):

$\displaystyle\nu\sim\nu^{\prime}\Longleftrightarrow\exists\,r\colon\mathsf{P}_{\\!\Lambda}^{\star}\rightarrow\mathbb{Z}\
\text{s.t.}\
\forall\,e\in{\mathcal{E}_{\Lambda}}\colon\nu(e)-\nu^{\prime}(e)=r(t(e))-r(s(e)).$

This corresponds to the equivalence relation on
$\mathrm{H}^{1}\big{(}\Gamma_{\\!\\!\Lambda}^{\vee},\mathbb{Z}\big{)}$ given
by the subgroup generated by the maps
$\alpha_{\mathbf{v}}\colon{\mathcal{E}_{\Lambda}}\rightarrow\mathbb{Z}$
defined in (2.1). Through equations (2.9)–(2.10) it can be traced back to the
inclusion
$\Gamma_{\\!\\!\Lambda}^{\vee}\hookrightarrow\mathbb{R}^{2}/\Lambda$.

The set of $\sim$-equivalence classes in $\mathcal{W}_{\Lambda}$ is denoted by
$\overline{\mathcal{W}}_{\Lambda}$; see (2.14). The diagram in equation (2.49)
shows the various schemes and their interrelations associated with the
(graded) semi-groups $\mathcal{W}_{\Lambda}$ and
$\overline{\mathcal{W}}_{\Lambda}$ and the group
$\mathrm{H}^{1}\big{(}\Gamma_{\\!\\!\Lambda}^{\vee},\mathbb{Z}\big{)}$.

######  Remark 4.1.

Notice the analogy between formula (1.5) for the deformations of weight
realizations and the action of $\mathbb{C}^{*}$ on
$\mathsf{Spec}(\mathbb{Z}[\mathcal{W}_{\Lambda}])$ and that of
$\mathrm{H}^{1}\big{(}\Gamma_{\\!\\!\Lambda}^{\vee},\mathbb{C}^{*}\big{)}$ on
$\mathsf{Proj}(\mathbb{Z}[\mathcal{W}_{\Lambda}])$.

The $\boldsymbol{\Gamma_{\\!\\!\Lambda}}$-side: The counterpart of
$\mathcal{W}_{\Lambda}$ on the $\Gamma_{\\!\\!\Lambda}$-side is the Jacobi
algebra $\mathsf{Jac}([\mathcal{F}]_{\Lambda})$ of the superpotential
$[\mathcal{F}]_{\Lambda}$. This is the quotient of the path algebra
$\mathbb{Z}[\mathsf{Path}(\Gamma_{\\!\\!\Lambda})]$ of the quiver
$\Gamma_{\\!\\!\Lambda}$ by a two-sided ideal provided by the permutations
$\sigma_{0}$ and $\sigma_{1}$; see (3.1) for a precise definition based on [5,
6]. The Jacobi algebra comes with an injective algebra homomorphism, the
tautological representation,

$\displaystyle\Phi\colon\
\mathsf{Jac}([\mathcal{F}]_{\Lambda})\longrightarrow{\mathsf{Mat}}_{\mathsf{P}_{\\!\Lambda}^{\star}}\big{(}\mathbb{Z}\big{[}\mathcal{W}_{\Lambda}^{\vee}\big{]}\big{)}$

into the algebra of
$|\mathsf{P}_{\\!\Lambda}^{\star}|\times|\mathsf{P}_{\\!\Lambda}^{\star}|$-matrices
over the semi-group ring
$\mathbb{Z}\big{[}\mathcal{W}_{\Lambda}^{\vee}\big{]}$ of the semi-group
$\mathcal{W}_{\Lambda}^{\vee}$ dual to $\mathcal{W}_{\Lambda}$. It restricts
to an injective algebra homomorphism

$\displaystyle\mathsf{Z}(\mathsf{Jac}([\mathcal{F}]_{\Lambda}))\longrightarrow\mathbb{Z}\big{[}\overline{\mathcal{W}}_{\Lambda}^{\vee}\big{]}$

from the center of the Jacobi algebra into the semi-group ring
$\mathbb{Z}\big{[}\overline{\mathcal{W}}_{\Lambda}^{\vee}\big{]}$; see Theorem
3.7.

The $\boldsymbol{\mathcal{D}_{\mathcal{F},\Lambda}}$-correspondence: A weight
realization $(\nu_{1},\nu_{2},\nu_{3})$ gives rise to a tiling of
$\mathbb{R}^{2}$ by quadrangles and, hence, for every
$e\in{\mathcal{E}_{\Lambda}}$ vectors $\mathsf{q}_{\mathbf{b}}(e)$ and
$\mathsf{q}_{\mathbf{w}}(e)$ as in Figs. 8 and 10. The vector
$\mathsf{q}_{\mathbf{b}}(e)-\mathsf{q}_{\mathbf{w}}(e)$ is the diagonal from
$w(e)$ to $b(e)$ in the quadrangle.

In this way the map
$\mathsf{q}_{\mathbf{b}}-\mathsf{q}_{\mathbf{w}}\colon{\mathcal{E}_{\Lambda}}\rightarrow\mathbb{R}^{2}$
realizes the duality between the graphs $\Gamma_{\\!\\!\Lambda}$ and
$\Gamma_{\\!\\!\Lambda}^{\vee}$.

For a perfect matching $\mathsf{m}$ we set, with the notations as in (2.37),

$\displaystyle\tau_{\mathsf{m},0}=(\mathbb{I}+\varsigma_{\mathsf{m},0})(\mathbb{I}-\varsigma_{\mathsf{m},0})^{-1},\qquad\tau_{\mathsf{m},1}=(\mathbb{I}+\varsigma_{\mathsf{m},1})(\mathbb{I}-\varsigma_{\mathsf{m},1})^{-1}.$

Then $\tau_{\mathsf{m},0}$ and $\tau_{\mathsf{m},1}$ are unipotent matrices of
size $|{\mathcal{E}_{\Lambda}}|\times|{\mathcal{E}_{\Lambda}}|$ with entries
in $\mathbb{Z}_{\geq 0}$. They define injective homomorphisms
$\mathsf{T}_{\mathsf{m},0}$ and $\mathsf{T}_{\mathsf{m},1}$ of semi-groups
(see (2.40))

$\displaystyle\mathsf{T}_{\mathsf{m},j}\colon\
\mathcal{W}_{\Lambda}\longrightarrow\mathcal{W}_{\Lambda}^{\vee},\qquad\bigl{(}\mathsf{T}_{\mathsf{m},j}(\nu)\bigr{)}(\nu^{\prime})=\nu^{\prime
t}\cdot\tau_{\mathsf{m},j}\cdot\nu.$

Here $\nu,\nu^{\prime}\in\mathcal{W}_{\Lambda}$ are viewed as column vectors.

Using the vectors $\mathsf{q}_{\mathbf{b}}(e)$ and
$\mathsf{q}_{\mathbf{w}}(e)$ we define maps
$\mathsf{Q}_{\mathbf{b}},\mathsf{Q}_{\mathbf{w}}\colon\mathcal{W}_{\Lambda}\longrightarrow\mathbb{R}^{2}$,

$\displaystyle\mathsf{Q}_{\mathbf{b}}(\nu)=\sum_{e\in{\mathcal{E}_{\Lambda}}}\nu(e)\mathsf{q}_{\mathbf{b}}(e),\qquad\mathsf{Q}_{\mathbf{w}}(\nu)=\sum_{e\in{\mathcal{E}_{\Lambda}}}\nu(e)\mathsf{q}_{\mathbf{w}}(e).$

Formulas (1.3) and (2.51)–(2.52) with
$\omega=(\nu_{1}-\nu_{3},\nu_{2}-\nu_{3})$ and
$\theta=\frac{1}{\deg\nu_{3}}\nu_{3}$ then show that the maps
$\mathsf{Q}_{\mathbf{b}}$, $\mathsf{Q}_{\mathbf{w}}$ can be expressed as
linear combinations of

$\displaystyle\mathsf{T}_{\mathsf{m},0}(\nu_{1}),\
\mathsf{T}_{\mathsf{m},0}(\nu_{2}),\ \mathsf{T}_{\mathsf{m},0}(\nu_{3}),\
\mathsf{T}_{\mathsf{m},1}(\nu_{1}),\ \mathsf{T}_{\mathsf{m},1}(\nu_{2}),\
\mathsf{T}_{\mathsf{m},1}(\nu_{3}).$

Every perfect matching $\mathsf{m}$ yields two matrices $\tau_{\mathsf{m},0}$
and $\tau_{\mathsf{m},1}$ with entries in $\mathbb{Z}_{\geq 0}$ and
determinant $1$. Products and transposes of such matrices also have entries in
$\mathbb{Z}_{\geq 0}$ and determinant $1$.

In this way one obtains lots of maps from $\mathcal{W}_{\Lambda}$ to
$\mathcal{W}_{\Lambda}^{\vee}$. It would be nice if these can be used to built
a correspondence between the toric geometry of
$\mathsf{Proj}(\mathbb{Z}[\mathcal{W}_{\Lambda}])$ on the
$\Gamma_{\\!\\!\Lambda}^{\vee}$-side and the non-commutative algebraic
geometry of $\mathsf{Jac}([\mathcal{F}]_{\Lambda})$ on the
$\Gamma_{\\!\\!\Lambda}$-side.

We leave further analysis of this structure for future research.

##  5 Practical matters

In this section I describe some methods for using a computer to draw the
tiling associated with a Zhegalkin zebra polynomial $\mathcal{F}$, compute the
superpotential $[\mathcal{F}]_{\Lambda}$ and check some conditions. Although
the ideas work quite generally the exposition here is strongly influenced by
my habit of using matlab.

###  5.1 How to draw the picture of the tiling of $\boldsymbol{\mathcal{F}}$

The defining formula for a czzf $\mathcal{F}$ can be rewritten as formula
(5.1) (below) with which one can easily draw the picture of the tiling. For
the description of formula (5.1) we define the function
$\neg\colon\mathbb{R}\rightarrow\\{0,1\\}$ by $\neg(r)=1$ if $r=0$ and
$\neg(r)=0$ if $r\neq 0$, we identify $0,1\in\mathbb{F}_{2}$ with
$0,1\in\mathbb{R}$ and we interpret in the matrix operations the matrix
entries as elements of $\mathbb{R}$.

Extract from the defining formula for $\mathcal{F}$ the $2\times n$-matrix $V$
of which the columns are the used frequency vectors. Put the coordinates of
the points at which the function should be evaluated as rows in a $k\times
2$-matrix $X$. Compute the matrix $2X\cdot V$ and apply the function
$\lfloor\;\rfloor\bmod 2$ to its entries. In short hand notation this can be
summarized as $\lfloor 2X\cdot V\rfloor\bmod 2$.

Extract from the defining formula for $\mathcal{F}$ the $n\times m$-matrix $M$
with entries $0,1$ of which the columns correspond to the monomials in the
formula. Note that a monomial evaluates to $1$ if and only if all variables it
involves have value $1$. This leads to the formula $\neg\bigl{(}(\neg(\lfloor
2X\cdot V\rfloor\bmod 2))\cdot M\bigr{)}$ for evaluating the monomials. In
this formula the function $\neg$ is applied to the entries of the matrices.

The next and final step is to take the sum of the columns (or, equivalently,
multiply on the right by the column vector $\mathbf{1}$ consisting of $m$
$1$’s) and reduce the result modulo $2$. The result is a vector of $0$’s and
$1$’s which gives the value of $\mathcal{F}$ at the points listed in $X$. Thus
the whole evaluation process reads:

$\displaystyle\mathcal{F}(X)=\big{(}\big{(}\neg\big{(}\big{(}\neg(\lfloor
2X\cdot V\rfloor\bmod 2)\big{)}\cdot
M\big{)}\big{)}\cdot\mathbf{1}\big{)}\bmod 2.$ (5.1)

$\begin{array}[]{@{}l}V=(\mathfrak{v}_{2},\mathfrak{v}_{3},\mathfrak{v}_{4},\mathfrak{v}_{6},2\mathfrak{v}_{4})\\\\[4.30554pt]
M=\left(\begin{matrix}1&0&0&0&0\\\ 0&1&0&0&1\\\ 0&0&1&0&0\\\ 0&0&0&1&1\\\
0&0&0&0&1\end{matrix}\right)\end{array}$

Figure 12: Matrices $V$, $M$ and picture for
$\mathcal{F}=\mathsf{Z}^{21}+\mathsf{Z}^{31}+\mathsf{Z}^{41}+\mathsf{Z}^{61}+\mathsf{Z}^{31}\mathsf{Z}^{42}\mathsf{Z}^{61}$.

###  5.2 How to compute the superpotential
$\boldsymbol{[\mathcal{F}]_{\Lambda}}$ and the realization
$\boldsymbol{\omega_{\mathcal{F}}}$

We now explain how one can compute the superpotential
$[\mathcal{F}]_{\Lambda}=({\mathcal{E}_{\Lambda}},\sigma_{0},\sigma_{1})$ from
the defining formula of the czzf $\mathcal{F}$ and the lattice
$\Lambda\subset\mathsf{Aut}(\mathcal{F})$.

5.2.1.  The first step is to multiply the vectors
$\mathfrak{v}_{1},\ldots,\mathfrak{v}_{6}$ by the diagonal matrix
$\operatorname{diag}(2,2/\sqrt{3})$. This results in the new basic frequency
vectors

$\displaystyle\tilde{\mathfrak{v}}_{1}=\left(\begin{matrix}-3\\\
\hphantom{-}1\end{matrix}\right),\quad\tilde{\mathfrak{v}}_{2}=\left(\begin{matrix}-1\\\
\hphantom{-}1\end{matrix}\right),\quad\tilde{\mathfrak{v}}_{3}=\left(\begin{matrix}0\\\
2\end{matrix}\right),\quad\tilde{\mathfrak{v}}_{4}=\left(\begin{matrix}1\\\
1\end{matrix}\right),\quad\tilde{\mathfrak{v}}_{5}=\left(\begin{matrix}3\\\
1\end{matrix}\right),\quad\tilde{\mathfrak{v}}_{6}=\left(\begin{matrix}2\\\
0\end{matrix}\right),$

with which we reinterpret the frequency vectors in the defining formula for
$\mathcal{F}$. This clearly does not change the combinatorial structure of the
picture, but it allows to do most computations with integer arithmetic.

As in Section 5.1 we let the matrix $V$ be such that its columns are the
frequency vectors of $\mathcal{F}$. The edges in the picture lie on lines with
equation $2\mathbf{x}\cdot\mathfrak{v}=m$ with $m\in\mathbb{Z}$ and
$\mathfrak{v}$ a column of $V$. The vertices are intersection points of two
such lines $2\mathbf{x}\cdot\mathfrak{v}=m$ and
$2\mathbf{x}\cdot\mathfrak{v}^{\prime}=m^{\prime}$ with linearly independent
$\mathfrak{v}$ and $\mathfrak{v}^{\prime}$. The coordinates of the
intersection point are then rational numbers with denominators dividing the
number
$2|\det(\mathfrak{v},\mathfrak{v}^{\prime})|/|\mathrm{gcd}(\textrm{entries of
}\mathfrak{v},\mathfrak{v}^{\prime})|$. Let $K$ denote the least common
multiple of these numbers for $(\mathfrak{v},\mathfrak{v}^{\prime})$ running
over all pairs of linearly independent frequency vectors of $\mathcal{F}$.

Fix a sufficiently large444$N$ should at least be so large that the
periodicity lattice which we want to implement later has two basis vectors in
$\mathbb{R}^{2}$ with non-negative coordinates $\leq\frac{1}{3}N$. positive
integer $N$. Let $X$ be the $N^{2}\times 2$ -matrix with set of rows
$\\{(n,m)\in\mathbb{Z}^{2}\,|\,0\leq n,m<N\\}$ such that row $(n,m)$ is above
row $(n^{\prime},m^{\prime})$ if $n+m\sqrt{2}<n^{\prime}+m^{\prime}\sqrt{2}$.

We want to find among the rows of $\frac{1}{K}X$ those which are intersection
points of two lines $2\mathbf{x}\cdot\mathfrak{v}=m$ and
$2\mathbf{x}\cdot\mathfrak{v}^{\prime}=m^{\prime}$ with linearly independent
$\mathfrak{v}$ and $\mathfrak{v}^{\prime}$. First we determine which entries
of the matrix $2X\cdot V$ are divisible by $K$; with the notation of Section
5.1 this means the entries $1$ in the matrix $\neg((2X\cdot V)\bmod K)$. The
intersection points correspond to the rows with a $1$ for at least two
linearly independent frequency vectors. Let $X^{*}$ denote the submatrix of
$X$ given by this selection of rows. Correspondingly we have the two matrices

$\displaystyle 2X^{*}\cdot V\qquad\textrm{and}\qquad\neg((2X^{*}\cdot V)\bmod
K).$

Looking at these two matrices column by column one easily determines what are
the relevant lines and how the points in $X^{*}$ divide these lines into
closed intervals with non-overlapping interiors. For this the initial ordering
of the elements of $X$ is very useful. Since it can happen that the same
interval is produced from two different columns we remove the duplicates
retaining for each interval exactly one copy. We list the intervals thus found
by giving for each the two endpoints.

The above calculations were done with integer arithmetic. In the next steps we
have to work in $\mathbb{R}^{2}$ and must therefore divide for all intervals
in our list the coordinates of the endpoints by $K$. For each interval in the
list, say $I$, take on both sides of the interval a point close to the
midpoint and evaluate $\mathcal{F}$ at these two points using formula (5.1).
Remove $I$ from the list if $\mathcal{F}$ has at these two points the same
value. What is left is a list of intervals separating black and white regions.
For each of these intervals we call one endpoint the source and the other the
target, so that going along the interval from source to target the black
region is on the right.

We make a new list with for each interval $I$ besides the endpoints $s(I)$ and
$t(I)$ also the midpoint $m(I)=\frac{1}{2}(s(I)+t(I))$ and the vector
$\vec{I}=t(I)-s(I)$:

$\displaystyle\bigl{\\{}m(I),s(I),t(I),\vec{I}\,\bigr{\\}}_{I}.$ (5.2)

5.2.2.  We use the list (5.2) to compute the group $\mathsf{Aut}(\mathcal{F})$
of translations which leave the function $\mathcal{F}$ invariant. Let $I_{1}$
be the first item in this list. Determine all $I$ with $\vec{I}=\vec{I_{1}}$
and compute for each of these the vector $T(I)=m(I)-m(I_{1})$. In order to
check which of these vectors $T(I)$ leave $\mathcal{F}$ invariant we take at
random a point $\mathbf{x}$ in the unit square and compute
$\mathcal{F}(\mathbf{x}+T(I))$ and $\mathcal{F}(\mathbf{x})$. We remove $I$ if
$\mathcal{F}(\mathbf{x}+T(I))\neq\mathcal{F}(\mathbf{x})$. The remaining
vectors $T(I)$ are then tested against a new randomly chosen $\mathbf{x}$. The
vectors which are left after repeating this procedure a good number of times
generate $\mathsf{Aut}(\mathcal{F})$. From these generators we choose a basis
for $\mathsf{Aut}(\mathcal{F})$.

5.2.3.  Having a basis for $\mathsf{Aut}(\mathcal{F})$ we can specify the
desired periodicity lattice $\Lambda$ by an integer $(2\times 2)$-matrix with
non-zero determinant. In order to make the reduction modulo $\Lambda$ we fix a
basis for $\Lambda$ and write $m(I)$, $s(I)$, $t(I)$ and $\vec{I}$ in
coordinates with respect to this basis. Reduction modulo $\Lambda$ is achieved
by taking the fractional parts of the coordinates of $m(I)$, $s(I)$, $t(I)$
leaving $\vec{I}$ unchanged. This results in the list

$\displaystyle\bigl{\\{}m(I)-\lfloor m(I)\rfloor,s(I)-\lfloor
s(I)\rfloor,t(I)-\lfloor t(I)\rfloor,\vec{I}\,\bigr{\\}}_{I}.$ (5.3)

The vectors in (5.3) are given by their coordinates with respect to the chosen
basis of $\Lambda$. Converting this back to the original coordinates on
$\mathbb{R}^{2}$ and multiplying by $2K$ turns (5.3) into a list of quadruples
of elements of $\mathbb{Z}^{2}$. The first three elements in these quadruples
have non-negative coordinates $<2K$ and can be made into integers using the
injective map
$\\{0,\ldots,2K-1\\}\times\\{0,\ldots,2K-1\\}\rightarrow\mathbb{N}$,
$(a,b)\mapsto a+2bK$. These integers can be used as labels to identify the
midpoint, source and target of the edge.

The list which thus results from (5.3) contains many duplicates, which we
remove. What remains is a list of labeled edges $e$ with labeled source $s(e)$
and target $t(e)$ and the edge vector $\mathsf{vec}(e)$. It can still happen
that this list contains edges $e$ for which there is only one edge
$e^{\prime}$ with $s(e^{\prime})=t(e)$, which case $e$ and $e^{\prime}$ must
be fused. This will be taken care of in the final part of the next step.

Figure 13: Situation at the target of edge $e$.

Let $e$ be an edge in the list. Take all edges $e^{\prime}$ in the list
satisfying $s(e^{\prime})=t(e)$. Using the vectors $\mathsf{vec}(e)$ and
$\mathsf{vec}(e^{\prime})$ we define for every such $e^{\prime}$ the number

$\displaystyle
C(e^{\prime})=\left(1-\frac{\mathsf{vec}(e)\cdot\mathsf{vec}(e^{\prime})}{|\mathsf{vec}(e)||\mathsf{vec}(e^{\prime})|}\right)\operatorname{sign}\bigl{(}\det(\mathsf{vec}(e),\mathsf{vec}(e^{\prime}))\bigr{)}.$

Finally we define the edges $\sigma_{1}(e)$ and $\sigma_{0}(e)$ by
$s(\sigma_{1}(e))=s(\sigma_{0}(e))=t(e)$ and

$\displaystyle C(\sigma_{1}(e))\leq C(e^{\prime})\leq
C(\sigma_{0}(e))\qquad\text{for all $e^{\prime}$ with $s(e^{\prime})=t(e)$.}$

This yields the permutations $\sigma_{0}$ and $\sigma_{1}$ of the set of edges
in our list.

Edges $e$ for which there is only one edge $e^{\prime}$ with
$s(e^{\prime})=t(e)$ can be recognized as those for which
$\sigma_{1}(e)=\sigma_{0}(e)$. Now repeat the following procedure as long as
there are edges with this undesirable property. Take the first edge $e$ for
which $\sigma_{1}(e)=\sigma_{0}(e)$. Add a new edge $e^{*}$ to the list,
setting $\text{label}(e^{*})=\text{label}(e)$, $s(e^{*})=s(e)$,
$t(e^{*})=t(\sigma_{0}(e))$, $\sigma_{1}(e^{*})=\sigma_{1}^{2}(e)$,
$\sigma_{0}(e^{*})=\sigma_{0}^{2}(e)$,
$\mathsf{vec}(e^{*})=\mathsf{vec}(e)+\mathsf{vec}(\sigma_{0}(e))$ and remove
$e$ and $\sigma_{0}(e)$ from the list.

5.2.4.  After all this we have obtained the list of elements of
${\mathcal{E}_{\Lambda}}$ and for every $e\in{\mathcal{E}_{\Lambda}}$ the edge
vector $\mathsf{vec}(e)$ as well as $\sigma_{0}(e)$ and $\sigma_{1}(e)$. The
superpotential is then
$[\mathcal{F}]_{\Lambda}=({\mathcal{E}_{\Lambda}},\sigma_{0},\sigma_{1})$ and
$\omega_{\mathcal{F}}(e)=\mathsf{vec}(e)$ gives the realization
$\omega_{\mathcal{F}}$ of $[\mathcal{F}]_{\Lambda}$.

######  Example 5.1.

For $\mathcal{F}$ as in Fig. 12 we find
$\mathsf{Aut}(\mathcal{F})=\frac{1}{3}\mathfrak{v}_{3}\mathbb{Z}\oplus\mathfrak{v}_{6}\mathbb{Z}$.
For $\Lambda=\mathsf{Aut}(\mathcal{F})$ the superpotential
$[\mathcal{F}]_{\Lambda}=({\mathcal{E}_{\Lambda}},\sigma_{0},\sigma_{1})$ is
given by

$\displaystyle\sigma_{0}=(1,3,13)(4,6,2)(7,10,11)(9,5,8,12),\qquad\sigma_{1}=(1,5,2)(4,3,11,12)(7,6,8)(9,10,13).$

One can recover the labeling of the edges by matching $\sigma_{0}$ and
$\sigma_{1}$ with the picture in Fig. 12.

###  5.3 How to compute the lattice and the quadrangles of a weight
realization

5.3.1.  From Sections 5.2 and 5.2 we have obtained a list of edge vectors
$\\{\omega_{\mathcal{F}}(e)\\}_{e\in{\mathcal{E}_{\Lambda}}}$ for the
realization $\omega_{\mathcal{F}}$ and a basis $\lambda_{1},\lambda_{2}$ for
the lattice $\Lambda_{\omega_{\mathcal{F}}}$. This implies that there are maps
$\mathsf{n}_{1},\mathsf{n}_{2}\colon{\mathcal{E}_{\Lambda}}\rightarrow\mathbb{Z}$
such that for $j=1,2$:

$\displaystyle*\lambda_{j}=\sum_{e\in{\mathcal{E}_{\Lambda}}}\mathsf{n}_{j}(e)\omega_{\mathcal{F}}(e),$
$\displaystyle\sum_{e\in{\mathcal{E}_{\Lambda}}}\mathsf{n}_{j}(e)e\quad\textrm{is
the homology class of a closed path on}\ \Gamma_{\\!\\!\Lambda};$ (5.4)

see (2.7). A basis $\lambda_{\omega,1},\lambda_{\omega,2}$ of the lattice
$\Lambda_{\omega}$ for any other realization $\omega$ of the superpotential
$[\mathcal{F}]_{\Lambda}$ is then given by

$\displaystyle\lambda_{\omega,j}=\sum_{e\in{\mathcal{E}_{\Lambda}}}\mathsf{n}_{j}(e)\omega(e)\qquad\textrm{for}\quad
j=1,2.$

The following method avoids the explicit computation of $\mathsf{n}_{1}$,
$\mathsf{n}_{2}$ (which are not uniquely determined anyway). Fix
$\mathbf{v}_{0}\in\mathsf{P}_{\\!\Lambda}^{\star}$ and define the matrices
$\Omega_{\mathcal{F}}$ and $\Omega_{\omega}$ with $|{\mathcal{E}_{\Lambda}}|$
rows and $|\mathsf{P}_{\\!\Lambda}^{\star}|+1$ columns by: the first two
columns of $\Omega_{\mathcal{F}}$ are $\omega_{\mathcal{F}}$ and the first two
columns of $\Omega_{\omega}$ are $\omega$; the last
$|\mathsf{P}_{\\!\Lambda}^{\star}|-1$ columns of both $\Omega_{\mathcal{F}}$
and $\Omega_{\omega}$ are $\alpha_{\mathbf{v}}$ with
$\mathbf{v}\in\mathsf{P}_{\\!\Lambda}^{\star}$,
$\mathbf{v}\neq\mathbf{v}_{0}$; here $\alpha_{\mathbf{v}}$ is as in (2.1).
Schematically:

$\displaystyle\Omega_{\mathcal{F}}=[\omega_{\mathcal{F}}|A],\qquad\Omega_{\omega}=[\omega|A].$

Viewing $\mathsf{n}_{1}$ and $\mathsf{n}_{2}$ as row vectors we have for
$j=1,2$

$\displaystyle\mathsf{n}_{j}\cdot\Omega_{\mathcal{F}}=[\lambda_{j}|\mathsf{0}],\qquad\mathsf{n}_{j}\cdot\Omega_{\omega}=[\lambda_{\omega,j}|\mathsf{0}].$

Here we used that (2.5) and (5.4) imply $\mathsf{n}_{j}\cdot A=\mathsf{0}$.
The (column) rank of both matrices $\Omega_{\mathcal{F}}$ and
$\Omega_{\omega}$ is $|\mathsf{P}_{\\!\Lambda}^{\star}|+1$. So there is an
invertible matrix $G$ with entries in $\mathbb{Q}$ such that
$\Omega_{\omega}=\Omega_{\mathcal{F}}\cdot G$. Putting things together we have
for $j=1,2$:

$\displaystyle[\lambda_{\omega,j}|\mathsf{0}]=[\lambda_{j}|\mathsf{0}]\cdot
G.$

5.3.2.  Next we determine the vertices in the tiling of $\mathbb{R}^{2}$
corresponding to a realization $\omega$ of $[\mathcal{F}]_{\Lambda}$. Fix
$\mathbf{v}_{0}\in\mathsf{P}_{\\!\Lambda}^{\star}$ and set
$z_{\mathbf{v}_{0}}=\mathsf{0}\in\mathbb{R}^{2}$. For
$\mathbf{v}\in\mathsf{P}_{\\!\Lambda}^{\star}$ with
$\mathbf{v}\neq\mathbf{v}_{0}$ take a path $\mathbf{p}_{\mathbf{v}}$ in
$\Gamma_{\\!\\!\Lambda}$ starting at $\mathbf{v}_{0}$ and ending at
$\mathbf{v}$, say $\mathbf{p}_{\mathbf{v}}=(e_{1},\ldots,e_{k})$, and set
$z_{\mathbf{v}}=\omega(e_{1})+\dots+\omega(e_{k})$. Then we obviously have:

######  Lemma 5.2.

The set of vertices in the tiling of $\mathbb{R}^{2}$ which lie over the
vertex $\mathbf{v}$ of
$\Gamma_{\\!\\!\Lambda}\subset\mathbb{R}^{2}/\Lambda_{\omega}$ is precisely
$z_{\mathbf{v}}+\Lambda_{\omega}$.

5.3.3.  For a realization $\omega$ and a positive fractional matching $\theta$
we now compute for every polygon in the tiling the point which is the convex
combination specified by $\theta$ of the midpoints of the edges of that
polygon. In order to do this in an efficient way we fix a perfect matching
$\mathsf{m}$. Consider a black polygon
$\mathbf{b}\in\mathsf{P}_{\\!\Lambda}^{\bullet}$. Let $(e_{1},\ldots,e_{q})$
be the corresponding cycle of the permutation $\sigma_{1}$ written such that
$e_{1}\in\mathsf{m}$. Then the vertices of the polygon $\mathbf{b}$ are
located at

$\displaystyle
z_{\mathbf{v}}+\sum_{j=1}^{h}\omega(e_{j})\qquad\textrm{for}\quad 1\leq h\leq
q,$

where $\mathbf{v}=s(e_{1})$ and $z_{\mathbf{v}}$ is as in Section 5.3. The
midpoints of its sides are

$\displaystyle
z_{\mathbf{v}}+\textstyle{\frac{1}{2}}\displaystyle\omega(e_{h})+\sum_{j=1}^{h-1}\omega(e_{j})\qquad\textrm{for}\quad
1\leq h\leq q.$

The convex combination of these midpoints specified by $\theta$ is therefore

$\displaystyle
B_{\mathbf{b},\mathsf{m},\theta,\omega}=z_{\mathbf{v}}+\sum_{h=1}^{q}\theta(e_{h})\left(\textstyle{\frac{1}{2}}\displaystyle\omega(e_{h})+\sum_{j=1}^{h-1}\omega(e_{j})\right).$
(5.5)

For a white polygon $\mathbf{w}$ one can construct in the same way a point
$W_{\mathbf{w},\mathsf{m},\theta,\omega}$.

Using translations from the lattice $\Lambda_{\omega}$ one subsequently
obtains a marked point in every polygon in the tiling of $\mathbb{R}^{2}$ and
vectors connecting this point to the vertices of the polygon.

5.3.4.  Using the map
$\beta_{\mathbf{b}}\colon{\mathcal{E}_{\Lambda}}\rightarrow\mathbb{Z}$ from
(2.2) and the matrix $\rho_{\mathsf{m},1}$ from (2.27) we can rewrite (5.5) as

$\displaystyle
B_{\mathbf{b},\mathsf{m},\theta,\omega}=z_{\mathbf{v}}+\theta^{t}\cdot\mathsf{diag}(\beta_{\mathbf{b}})\cdot\big{(}{-}\textstyle{\frac{1}{2}}\displaystyle\mathbb{I}+\rho_{\mathsf{m},1}\big{)}\cdot\omega,$
(5.6)

where on the right-hand side we view $\theta^{t}$ as a row vector and $\omega$
as a $|{\mathcal{E}_{\Lambda}}|\times 2$-matrix with real entries. The source
point of edge $e_{k}$ of $\mathbf{b}$ is located at

$\displaystyle
z_{\mathbf{v}}+\sum_{j=1}^{k-1}\omega(e_{j})=z_{\mathbf{v}}+\big{(}\textrm{$e_{k}$-th
row of matrix}\ (-\mathbb{I}+\rho_{\mathsf{m},1})\cdot\omega\big{)}.$ (5.7)

The vector from the source point of edge $e_{k}$ of $\mathbf{b}$ to the marked
point $B_{\mathbf{b},\mathsf{m},\theta,\omega}$ in $\mathbf{b}$ is obtained by
subtracting (5.7) from (5.6). The term $z_{\mathbf{v}}$ cancels out. Noticing
that $\mathbf{b}=b(e_{k})$ we are led to introduce the
$|{\mathcal{E}_{\Lambda}}|\times|{\mathcal{E}_{\Lambda}}|$-matrix
$\mathbf{B}_{\mathsf{m},\theta}$ by

$\displaystyle\text{$e$-th row of
}\mathbf{B}_{\mathsf{m},\theta}=\theta^{t}\cdot\mathsf{diag}(\beta_{b(e)})\cdot\big{(}{-}\textstyle{\frac{1}{2}}\displaystyle\mathbb{I}+\rho_{\mathsf{m},1}\big{)}.$
(5.8)

The vector from the source point of edge $e$ to the marked point in the
polygon $b(e)$ is then the $e$-th row of the matrix
$(\mathbf{B}_{\mathsf{m},\theta}+\mathbb{I}-\rho_{\mathsf{m},1})\cdot\omega$.

Proceeding in the same way for the white polygons we define the
$|{\mathcal{E}_{\Lambda}}|\times|{\mathcal{E}_{\Lambda}}|$-matrix
$\mathbf{W}_{\mathsf{m},\theta}$ by

$\displaystyle\text{$e$-th row of
}\mathbf{W}_{\mathsf{m},\theta}=\theta^{t}\cdot\mathsf{diag}(\beta_{w(e)})\cdot\big{(}{-}\textstyle{\frac{1}{2}}\displaystyle\mathbb{I}+\rho_{\mathsf{m},0}\big{)}.$
(5.9)

The following proposition summarizes our findings about the quadrangles.

######  Proposition 5.3.

In the quadrangle corresponding to $e\in{\mathcal{E}_{\Lambda}}$

  1. $(i)$

the vector from $s(\widetilde{e})$ to $b(\widetilde{e})$ is the $e$-th row of
the $|{\mathcal{E}_{\Lambda}}|\times 2$-matrix

$\displaystyle\bigl{(}\mathbf{B}_{\mathsf{m},\theta}+\mathbb{I}-\rho_{\mathsf{m},1}\bigr{)}\cdot\omega,$

  2. $(ii)$

the vector from $s(\widetilde{e})$ to $w(\widetilde{e})$ is the $e$-th row of
the $|{\mathcal{E}_{\Lambda}}|\times 2$-matrix

$\displaystyle\bigl{(}\mathbf{W}_{\mathsf{m},\theta}+\mathbb{I}-\rho_{\mathsf{m},0}\bigr{)}\cdot\omega,$

  3. $(iii)$

the vector from $s(\widetilde{e})$ to $t(\widetilde{e})$ is the $e$-th row of
the $|{\mathcal{E}_{\Lambda}}|\times 2$-matrix $\omega$,

In the application of this result to the weight realization of
$[\mathcal{F}]_{\Lambda}$ given by the triple of positive weight functions
$(\nu_{1},\nu_{2},\nu_{3})$ $($see Definition 1.6) one takes
$\omega=(\nu_{1}-\nu_{3},\nu_{2}-\nu_{3})$ and
$\theta=\frac{1}{\deg\nu_{3}}\nu_{3}$.

5.3.5.  By combining Sections 5.3, 5.3, 5.3 and 5.3 we can draw the quadrangle
tiling as follows. For $e\in{\mathcal{E}_{\Lambda}}$ take $z_{s(e)}$ as in as
in Section 5.3 and draw for every $z\in z_{s(e)}+\Lambda_{\omega}$ the
quadrangle as in Proposition 5.3 with vertex $s(\widetilde{e})$ located at $z$
and with the vectors from $s(\widetilde{e})$ to $b(\widetilde{e})$,
$w(\widetilde{e})$ and $t(\widetilde{e})$ as specified in Proposition 5.3.

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