# Varadhan’s Decomposition of Shift-Invariant Closed Uniform Forms for Large
Scale Interacting Systems on General Crystal Lattices

Kenichi Bannai∗⋄<EMAIL_ADDRESS>and  Makiko Sasada⋆⋄
<EMAIL_ADDRESS>∗Department of Mathematics, Faculty of Science and
Technology, Keio University, 3-14-1 Hiyoshi, Kouhoku-ku, Yokohama 223-8522,
Japan.  ⋆Department of Mathematics, University of Tokyo, 3-8-1 Komaba, Meguro-
ku, Tokyo 153-8914, Japan.  ⋄Mathematical Science Team, RIKEN Center for
Advanced Intelligence Project (AIP),1-4-1 Nihonbashi, Chuo-ku, Tokyo 103-0027,
Japan.

(Date: (Version $\beta$ – for limited distribution))

###### Abstract.

We prove a uniform version of Varadhan decomposition for shift-invariant
closed uniform forms associated to large scale interacting systems on general
crystal lattices. In particular, this result includes the case of translation
invariant processes on Euclidean lattices $\mathbb{Z}^{d}$ with finite range.
Our result generalizes the result of [BKS20] which was valid for systems on
transferable graphs. In subsequent research [BS22L2], we will use the result
of this article to prove Varadhan’s decomposition of closed $L^{2}$-forms for
large scale interacting systems on general crystal lattices.

###### 2010 Mathematics Subject Classification:

Primary: 82C22, Secondary: 05C63, 55N91, 60J60, 70G40

This work was supported in part by JST CREST Grant Number JPMJCR1913, KAKENHI
18H05233, and the UTokyo Global Activity Support Program for Young
Researchers.

###### Contents

  1. 1 Introduction
  2. 2 Crystal Lattices
  3. 3 Criterion for Uniformity
  4. 4 Decomposition for Closed Uniform Forms

##  1\. Introduction

_Varadhan’s Decomposition of shift-invariant closed $L^{2}$-forms_ has played
an important role in the strategy of proof of the hydrodynamic limit for _non-
gradient_ models [KL99]*§7. See the introduction of [BKS20] for a short
overview of hydrodynamic limits and Varadhan’s strategy. In this article, we
prove a _uniform version_ of such decomposition for large scale interacting
systems on a general _crystal lattice_. In particular, our result is valid for
various translation invariant processes on Euclidean lattices $\mathbb{Z}^{d}$
with finite range. This generalizes our result in [BKS20] for systems on
locales which are transferable or weakly transferable. The simple case of the
exclusion process was previously announced in [KS16]. However, whilst the
rough outline of proof remains similar, many new ideas, such as the origin of
the conserved quantities and the notion for an interaction to be irreducibly
quantified, as well as the strategy to reduce the proof of the key theorem
Theorem 4.5 to the case of essentially Euclidean crystal lattices were
required to grasp and generalize many of the arguments in [KS16] which a
priori relied on properties specific to the exclusion process. Due to such
conceptual advancements, the result of this article is valid for far general
interactions, including models with multiple conserved quantities. Although
there has been some research concerning the hydrodynamic limit on crystal
lattices ([T12], more recently [F20, G20]), the known cases for interacting
systems on non-Euclidean lattices are all for the gradient case – there are no
known examples in the non-gradient case.

A _crystal lattice_ , or a _topological crystal_ in the terminology of
[Sun13]*§6.2, is any connected locally finite multi-graph $\mathcal{X}=(X,E)$
for the set of vertices $X$ and set of edges $E$, with a free action of a
finitely generated abelian group $G$ of rank $>0$ such that the quotient graph
$\mathcal{X}/G=(X/G,E/G)$ is a strictly symmetric finite multi-graph. See §2.1
for terminology concerning graphs. The dimension $\dim\mathcal{X}$ is defined
to be the rank of $G$. The _Euclidean lattices_ , _triangular lattice_ ,
_hexagonal lattice_ , and the _diamond lattice_ are all examples of crystal
lattices.

We consider a non-empty set $S$ which we call the _set of states_ , and
$S^{X}\coloneqq\prod_{x\in X}S$ the _configurations_ on $X$. We fix a map
$\phi\colon S\times S\rightarrow S\times S$ called an _interaction_ satisfying
a certain symmetry condition which specifies the interaction of states on
adjacent vertices. Models such at the _generalized exclusion process_ or the
_multi-species exclusion process_ may be described using such interaction (see
[BKS20]*Example 2.18 for other examples). One important invariant of the
interaction is the _conserved quantity_ , which is a normalized map $\xi\colon
S\rightarrow\mathbb{R}$ satisfying

$\xi(s^{\prime}_{1})+\xi(s^{\prime}_{2})=\xi(s_{1})+\xi(s_{2}),$

where $(s^{\prime}_{1},s^{\prime}_{2})=\phi(s_{1},s_{2})$ for any
$(s_{1},s_{2})\in S\times S$. We denote by ${\mathrm{Consv}}^{\phi}(S)$ the
$\mathbb{R}$-linear space of conserved quantities. Then
$c_{\phi}\coloneqq\dim_{\mathbb{R}}{\mathrm{Consv}}^{\phi}(S)$ gives the
number of independent conserved quantities of our large scale interacting
system.

Via the interaction, for any edge $e\in E$, a configuration
$\eta=(\eta_{x})\in S^{X}$ transitions to a configuration
$\eta^{e}=(\eta^{e}_{x})\in S^{X}$ whose component differs only on the origin
and target of the edge $e$. Such transitions give a structure of a multi-graph
on $S^{X}$. We denote by $C(S^{X})$ and $C^{1}(S^{X})$ the space of
$\mathbb{R}$-valued functions and forms on $S^{X}$, where a form is a system
of functions $\omega=(\omega_{e})_{e\in E}$ satisfying some minor conditions,
where $\omega_{e}$ for each $e\in E$ is a function on $S^{X}$.

We may define the differential $\partial_{\mathcal{X}}\colon
C(S^{X})\rightarrow C^{1}(S^{X})$ by
$\partial_{\mathcal{X}}f\coloneqq(\nabla_{\\!e}f)_{e\in E}$, where

$\nabla_{\\!e}f(\eta)\coloneqq f(\eta^{e})-f(\eta),\qquad\forall\eta\in
S^{X}.$

Functions and forms are _local_ if it depends only on states at a finite
number of vertices of $X$. In our previous work [BKS20] we extended the notion
of local functions to that of _uniform functions_ , which allow for infinite
sums of normalized local functions. Let $C_{\mathrm{unif}}(S^{X})$ and
$C^{1}_{\mathrm{unif}}(S^{X})$ be the space of uniform functions and forms. If
we denote by $Z^{1}_{\mathrm{unif}}(S^{X})$ the space of closed uniform form,
consisting of uniform forms whose integration over a path depends only on the
origin and the target of the path, then the differential induces the
differential

$\partial_{\mathcal{X}}\colon C_{\mathrm{unif}}(S^{X})\rightarrow
Z^{1}_{\mathrm{unif}}(S^{X}).$

The group $G$ acts naturally on $C_{\mathrm{unif}}(S^{X})$ and
$Z^{1}_{\mathrm{unif}}(S^{X})$, and the action is compatible with the
differential $\partial_{\mathcal{X}}$. We let

$\displaystyle\mathcal{C}$ $\displaystyle\coloneqq
Z^{1}_{\mathrm{unif}}(S^{X})^{G},$ $\displaystyle\mathcal{E}$
$\displaystyle\coloneqq\partial_{\mathcal{X}}(C_{\mathrm{unif}}(S^{X})^{G})$

the space of shift-invariant closed uniform forms and shift-invariant uniform
exact forms.

We say that an interaction is _irreducibly quantified_ if the following
condition holds for any finite $\mathcal{X}=(X,E)$: If two configurations
$\eta=(\eta_{x}),\eta^{\prime}=(\eta^{\prime}_{x})\in S^{X}$ satisfy
$\sum_{x\in X}\xi(\eta_{x})=\sum_{x\in X}\xi(\eta^{\prime}_{x})$ for any
conserved quantity $\xi\in{\mathrm{Consv}}^{\phi}(S)$, then there exists a
path from $\eta$ to $\eta^{\prime}$ in $S^{X}$. We say that an interaction is
_simple_ , if $c_{\phi}=1$, and for any non-zero
$\xi\in{\mathrm{Consv}}^{\phi}(S)$, the monoid generated by
$\xi(S)\subset\mathbb{R}$ is isomorphic to $\mathbb{N}$ or $\mathbb{Z}$.

Our main theorem is as follows.

######  Theorem 1.1 (=Theorem 4.3).

Let $\mathcal{X}=(X,E)$ be a crystal lattice of dimension $d=\dim\mathcal{X}$,
and consider an interaction which is irreducibly quantified. We assume in
addition that the interaction is simple if $d=1$. Then we have a decomposition

$\mathcal{C}=\mathcal{E}\oplus\partial_{\mathcal{X}}\mathcal{V},$

where $\partial_{\mathcal{X}}\mathcal{V}\subset\mathcal{C}$ is a certain
subspace given in Lemma 4.2 of dimension $c_{\phi}d$, where
$c_{\phi}=\dim_{\mathbb{R}}{\mathrm{Consv}}^{\phi}(S)$.

Theorem 4.3 is a generalization of [BKS20]*Theorem 5,17 to the case of an
arbitrary crystal lattice. In a subsequent article [BS22L2], will use this
result to prove the decomposition of shift-invariant closed $L^{2}$-forms on
models over a general crystal lattice, generalizing the result of [BS21L2] for
the case of the Euclidean lattice
$\mathbb{Z}^{d}=(\mathbb{Z}^{d},\mathbb{E}^{d})$. Such result will play a
crucial role in the proof of the hydrodynamic limit for large scale
interacting systems on crystal lattices. In particular, we expect this result
would allow for the proof of the hydrodynamic limit for translation invariant
processes on $\mathbb{Z}^{d}$ with finite range, extending results which were
up until now known only for nearest neighbor interactions.

##  2\. Crystal Lattices

In this section, we review some terminology concerning multi-graphs and
crystal lattices. We will then introduce the notion of uniform functions and
forms on a configuration space with transition structure.

###  2.1. Multigraphs and Crystal Lattices

In this subsection, we introduce terminology concerning multi-graphs and
crystal lattices. Our main reference is [Sun13]*§3.1. Since crystal lattices
are in general multi-graphs instead of graphs, we extend our previous works
[BKS20, BS21, BS21L2] to the case when the underlying space is a multi-graph,
which allows for multiple edges between two vertices. A _graph_ in
[Sun13]*§3.1 corresponds to a _strictly symmetric multi-graph_ in our
terminology. We will always assume that a multi-graph is symmetric.

######  Definition 2.1.

We define a _symmetric directed multi-graph_ , which we refer to simply as a
_multi-graph_ , to be the quintuple $\mathcal{X}=(X,E_{X},o,t,\iota)$, where
$X$ is a set which we call the _set of vertices_ , $E_{X}$ is a set which we
call the _set of edges_ , $o$ and $t$ are maps $o\colon E_{X}\rightarrow X$
and $t\colon E_{X}\rightarrow X$ which we call the _origin_ and _target_ , and
$\iota\colon E_{X}\rightarrow E_{X}$, $e\mapsto\bar{e}$ is a bijection which
we call the _inversion_ satisfying $o(\bar{e})=t(e)$ and $t(\bar{e})=o(e)$. We
say that $\mathcal{X}$ is _strictly symmetric_ , if $\bar{e}\neq e$ for any
$e\in E_{X}$.

We will often denote $E_{X}$ by $E$ for simplicity. Suppose
$\mathcal{X}=(X,E,o,t,\iota)$ is a multi-graph. For any edge $e\in E$, we call
$o(e)$ and $t(e)$ the origin and target of $e$. In literature, the map
$(o,t)\colon E\rightarrow X\times X$ is referred to as an _incidence map_. We
will often denote $(X,E,o,t,\iota)$ simply as $(X,E)$. We say that a multi-
graph $\mathcal{X}=(X,E)$ _is locally finite_ if for any $x\in X$, the set
$E_{x}\coloneqq\\{e\in E\mid o(e)=x\\}$ is finite for any $x\in X$, _is
finite_ if its set of vertices $X$ and the set of edges $E$ are both finite,
and _is infinite_ if it is not finite. If a multi-graph $(X,E)$ is locally
finite, then it is finite if and only if the set of vertices $X$ is finite.

We define a _path_ on a multi-graph $\mathcal{X}=(X,E)$ to be a finite
sequence $\vec{p}\coloneqq(e^{1},\ldots,e^{N})$ of edges satisfying
$t(e^{i})=o(e^{i+1})$ for any $0<i<N$. We say that $\vec{p}$ is a path from
$o(\vec{p})\coloneqq o(e^{1})$ to $t(\vec{p})\coloneqq t(e^{N})$ of length
${\mathrm{len}}(\vec{p})\coloneqq N$. For any $x,x^{\prime}\in X$, we let
$P(x,x^{\prime})$ the set of paths from $x$ to $x^{\prime}$. We define the
_distance_ from $x$ to $x^{\prime}$ by

$d_{\mathcal{X}}(x,x^{\prime})\coloneqq\inf_{\vec{p}\in
P(x,x^{\prime})}{\mathrm{len}}(\vec{p})$

if $P(x,x^{\prime})\neq\emptyset$ and
$d_{\mathcal{X}}(x,x^{\prime})\coloneqq\infty$ otherwise. For any $Y\subset
X$, we say that $Y$ is _connected_ , if for any $y,y^{\prime}\in Y$, there
exists a path from $y$ to $y^{\prime}$ consisting of edges whose origin and
target are all elements of $Y$. If $X$ is connected, then we say that the
multi-graph $\mathcal{X}=(X,E)$ is connected.

Any $e\in E$ such that $o(e)=t(e)$ is referred to as a _loop_. A multi-graph
$\mathcal{X}=(X,E)$ is a _graph_ , if the incidence map $E\rightarrow X\times
X$ is injective. For the case of graphs, we will identify $E$ with its image
in $X\times X$ so that $E\subset X\times X$, and the maps $o$ and $t$ are
simply the first and second projections. Using this convention, the inversion
of a graph $\mathcal{X}$ coincides with the bijection given by exchanging the
components of $E\subset X\times X$, and any loop $e\in E$ satisfies
$\bar{e}=e$. A _simple graph_ is a graph without any loops. Graphs are
referred to as a combinatorial graph in [Sun13]. We may associate to a
multigraph $\mathcal{X}=(X,E)$ a graph $\mathcal{X}^{\prime}=(X,E^{\prime})$
by taking $E^{\prime}\subset X\times X$ to be the image of the incidence map,
which we refer to a _graph associated to $\mathcal{X}$_. The _simple graph
$\mathcal{X}^{\prime\prime}$ associated to $\mathcal{X}$_ is the simple graph
obtained by removing the loops from $\mathcal{X}^{\prime}$.

In this article, we allow for the locale underlying the large scale
interacting system to be a multi-graph, which may not necessarily be simple.
Hence we extend the definition of locales given in [BKS20] to multi-graphs as
follows.

######  Definition 2.2.

We let the _locale_ to be any connected locally finite multi-graph.

Next, we define the notion of morphisms and automorphisms of multi-graphs.

######  Definition 2.3.

Let $\mathcal{X}=(X,E_{X})$ and $\mathcal{Y}=(Y,E_{Y})$ be multi-graphs. A
morphism of a multi-graph $u\colon\mathcal{X}\rightarrow\mathcal{Y}$ is a pair
$(u,u_{E})$ consisting of maps of sets $u\colon X\rightarrow Y$ and
$u_{E}\colon E_{X}\rightarrow E_{Y}$ compatible with the incidence map and the
inversion. In other words, we have commutative diagrams

$\textstyle{E_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u_{E}}$$\scriptstyle{(o,t)}$$\textstyle{E_{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(o,t)}$$\textstyle{E_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u_{E}}$$\scriptstyle{\iota}$$\textstyle{E_{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\textstyle{X\times
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u\times
u}$$\textstyle{Y\times
Y}$$\textstyle{E_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u_{E}}$$\textstyle{E_{Y},}$

where the vertical maps are the incidence and inversion maps. Note that if
$\mathcal{X}$ is a graph, then the map $u_{E}$ is determined uniquely by $u$.

An _automorphism_ of a multi-graph $\mathcal{X}=(X,E_{X})$ is a morphism
$(\sigma,\sigma_{E})\colon\mathcal{X}\rightarrow\mathcal{X}$ such that the
induced maps $\sigma\colon X\rightarrow X$ and $\sigma_{E}\colon
E_{X}\rightarrow E_{X}$ are bijective. We denote by
${\mathrm{Aut}}(\mathcal{X})$ the set of automorphisms of $\mathcal{X}$. The
set ${\mathrm{Aut}}(\mathcal{X})$ has a structure of a group whose operation
is the composition of automorphisms.

######  Definition 2.4.

Let $\mathcal{X}=(X,E_{X})$ and $\mathcal{Y}=(Y,E_{Y})$ be multi-graphs. We
say that a morphism $u\colon\mathcal{X}\rightarrow\mathcal{Y}$ of multi-graphs
is a _covering_ , if $u$ and $u_{E}$ are both surjective, and for any $x\in
X$, if we let $E_{x}\coloneqq\\{e\in E_{X}\mid o(e)=x\\}$ and
$E_{u(x)}\coloneqq\\{e\in E_{Y}\mid o(e)=u(x)\\}$, then the map
$E_{x}\rightarrow E_{u(x)}$ induced by $u_{E}$ is bijective.

If $u\colon\mathcal{X}\rightarrow\mathcal{Y}$ is a covering, then
$\mathcal{X}$ is locally finite if and only if $\mathcal{Y}$ is locally
finite. For any multi-graph $\mathcal{X}=(X,E)$, an action of a group $G$ on
$\mathcal{X}$ is defined to be a group homomorphism
$G\rightarrow{\mathrm{Aut}}(\mathcal{X})$. We define the _quotient multi-graph
$\mathcal{X}/G$_ of $\mathcal{X}$ for the action of $G$ as follows. Let $X/G$
and $E/G$ be the set of orbits which we refer also to as a _quotient_ of $X$
and $E$ with respect to the action of $G$. For any $e\in E$ and $\sigma\in G$,
we have

$(o(\sigma_{E}(e)),t(\sigma_{E}(e)))=(\sigma(o(e)),\sigma(t(e)),$

hence the origin and target maps induces maps $o,t\colon E/G\rightarrow X/G$
on the quotient.

######  Proposition 2.5.

The data $\mathcal{X}/G=(X/G,E/G,o,t,\iota)$ is again a multi-graph, which we
call the quotient of $\mathcal{X}$ with respect to the action of $G$. If the
action of $G$ on $X$ is free, in other words, if $\sigma(x)=x$ for some $x\in
X$ implies that $\sigma={\mathrm{id}}$, then the natural map
$\mathcal{X}\rightarrow\mathcal{X}/G$ is a covering.

###### Proof.

The fact that $\mathcal{X}/G$ is a multi-graph follows immediately from the
definition. Suppose the action of $G$ on $\mathcal{X}$ is free. Let $x\in X$
and denote by $x_{0}$ its image in $X/G$. Then we have a map

(1) $E_{x}=\\{e\in E\mid o(e)=x\\}\rightarrow E_{x_{0}}=\\{e_{0}\in E/G\mid
o(e_{0})=x_{0}\\}.$

If $e_{0}\in E_{x_{0}}$, then there exists a representative $e^{\prime}\in E$
such that $o(e^{\prime})=\sigma(x)$ for some $\sigma\in G$. If we let
$e\coloneqq\sigma^{-1}_{E}(e^{\prime})$, then $e\in E_{x}$ and maps to $e_{0}$
in $E_{x_{0}}$. This shows that (1) is surjective. Suppose now that we have
$e,e^{\prime}\in E$ such that $o(e)=o(e^{\prime})=x$, and suppose $e$ and
$e^{\prime}$ coincides in $E/G$. Then $e=\sigma_{E}(e^{\prime})$ for some
$\sigma\in G$. Then compatibility of the action of $G$ with $o$ shows that
$x=o(e)=o(\sigma_{E}(e^{\prime}))=\sigma(o(e^{\prime}))=\sigma(x)$. Since the
action of $G$ on $X$ is free, we see that $\sigma={\mathrm{id}}$, which shows
that $e=e^{\prime}$ hence that (1) is injective as desired. ∎

We see that any path in $\mathcal{X}$ defines a path in $\mathcal{X}/G$, hence
if $\mathcal{X}$ is connected, then $\mathcal{X}/G$ is also connected. For a
covering $u\colon\mathcal{X}\rightarrow\mathcal{Y}$, we say that an
automorphism $\sigma$ of $\mathcal{X}$ is a _covering transformation_ if
$u\circ\sigma=u$, in other words, that we have a commutative diagram

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
6.75pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
32.50006pt\raise 4.50694pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{\sigma}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 68.25pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
8.83076pt\raise-22.9236pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{u}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 30.75pt\raise-30.20856pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
34.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 68.25pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
56.16205pt\raise-22.9236pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{u}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
44.25002pt\raise-30.20197pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
30.75pt\raise-36.83331pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\mathcal{Y}}$}}}}}}}{\hbox{\kern
72.0pt\raise-36.83331pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

We denote by ${\mathrm{Aut}}(\mathcal{X}/\mathcal{Y})$ the group of covering
transformations of $\mathcal{X}$ over $\mathcal{Y}$. By [Sun13]*Theorem 5.2,
if we let $\mathcal{Y}=\mathcal{X}/G$, then
$G\cong{\mathrm{Aut}}(\mathcal{X}/\mathcal{Y})$.

We define a crystal lattice as follows.

######  Definition 2.6.

  1. (i)

We define a _crystal_ to be a strictly symmetric finite multi-graph. This
coincides with what is referred to as a _finite graph_ in [Sun13].

  2. (ii)

We say that a connected multi-graph $\mathcal{X}=(X,E)$ is a _crystal lattice_
, if it has a free action by a finitely generated abelian group $G$ of rank
$>0$, such that $\mathcal{X}_{0}\coloneqq\mathcal{X}/G$ is a crystal.

Note that a crystal lattice is locally finite since
$\mathcal{X}\rightarrow\mathcal{X}/G$ is a covering by Proposition 2.5. Hence
in particular, a crystal lattice is a locale in the sense of Definition 2.2.
Since ${\mathrm{rank}\,}G>0$, a crystal lattice is an infinite milti-graph. We
refer to the crystal $\mathcal{X}_{0}=X/G$ as the _base graph_ or the _seed
crystal_ of the crystal lattice $\mathcal{X}$. Note that a seed crystal is
connected since we have assumed that $\mathcal{X}$ is connected. We define the
dimension of $\mathcal{X}$ denoted $\dim\mathcal{X}$ to be the rank of the
abelian group $G$.

######  Example 2.7.

  1. (i)

The most typical example of a crystal lattice is the _Euclidean lattice_
$\mathcal{X}=(\mathbb{Z}^{d},\mathbb{E}^{d})$ for any integer $d\geq 1$, where
$\mathbb{Z}^{d}$ is the $d$-fold product of $\mathbb{Z}$ and

$\mathbb{E}^{d}\coloneqq\\{(x,y)\in\mathbb{Z}^{d}\times\mathbb{Z}^{d}\mid|x-y|=1\\},$

where $|x|\coloneqq\sum_{j=1}^{d}|x_{j}|$ for $x=(x_{j})\in\mathbb{Z}^{d}$,
with the action of $G=\mathbb{Z}^{d}$ given by translation. The quotient
$\mathcal{X}_{0}=\mathcal{X}/G$ is the finite multi-graph
$\mathcal{X}_{0}=(X_{0},E_{0},o,t)$ such that $X_{0}\coloneqq\\{x_{0}\\}$
consists of a single vertex, $E_{0}\coloneqq\\{e_{j},\bar{e}_{j}\mid
j=1,\ldots,d\\}$, and $o,t\colon E_{0}\rightarrow X_{0}$ are both constant
maps with image $x_{0}$.

  2. (ii)

Let $\mathcal{X}_{0}$ be a connected crystal. Then by [Sun13]*Theorem 6.6, the
maximal abelian covering $\mathcal{X}^{\mathrm{ab}}$ of $\mathcal{X}_{0}$ is a
connected simple graph. The group of covering transformations
${\mathrm{Aut}}(\mathcal{X}^{\mathrm{ab}}/\mathcal{X}_{0})$ of
$\mathcal{X}^{\mathrm{ab}}$ over $\mathcal{X}_{0}$ is isomorphic to the
maximal abelian quotient $\pi_{1}^{\mathrm{ab}}(\mathcal{X}_{0})$ of the
fundamental group of $\mathcal{X}_{0}$, which shows that
$\mathcal{X}/\pi_{1}^{\mathrm{ab}}(\mathcal{X}_{0})=\mathcal{X}_{0}$, hence
that $\mathcal{X}$ is a crystal latttice for the action of
$\pi_{1}^{\mathrm{ab}}(\mathcal{X}_{0})$.

  3. (iii)

Let $\mathcal{X}$ be a crystal lattice for the action by a finitely generated
abelian group $G$. If we let $\mathcal{X}^{\mathrm{ab}}$ be the maximal
abelian covering of the seed crystal $\mathcal{X}_{0}\coloneqq\mathcal{X}/G$,
then we have $\mathcal{X}=\mathcal{X}^{\mathrm{ab}}/H$ for some subgroup
$H\subset\pi_{1}^{\mathrm{ab}}(\mathcal{X}_{0})$, and we have
$G\cong\pi_{1}^{\mathrm{ab}}(\mathcal{X}_{0})/H$.

See Example 3.12 for more examples of crystal lattices.

###  2.2. Configuration Space and Uniform Functions

In order to consider the large scale interacting system on a locale, we next
consider the configuration space with transition structure.

######  Definition 2.8.

We define an _interaction_ to be a pair $(S,\phi)$ consisting of a set $S$
which we call the _set of states_ and a map $\phi\colon S\times S\rightarrow
S\times S$ satisfying

(2) $\bar{\phi}\circ\phi(s_{1},s_{2})=(s_{1},s_{2})$

for any $(s_{1},s_{2})\in S\times S$ such that
$\phi(s_{1},s_{2})\neq(s_{1},s_{2})$. Here,
$\bar{\phi}\coloneqq\iota\circ\phi\circ\iota$ for the bijection $\iota\colon
S\times S\rightarrow S\times S$ obtained by exchanging the components of
$S\times S$.

In what follows, we let $(S,\phi)$ be an interaction. For any multi-graph
$\mathcal{X}=(X,E)$, we define the _configuration space_ on $\mathcal{X}$ by

$S^{X}\coloneqq\prod_{x\in X}S.$

We call any $\eta\in S^{X}$ a _configuration_. For any configuration
$\eta=(\eta_{x})\in S^{X}$ and $e\in E$, we denote by $\eta^{e}$ the element
$\eta^{e}=(\eta^{e}_{x})\in S^{X}$ such that
$(\eta^{e}_{o(e)},\eta^{e}_{t(e)})\coloneqq\phi(\eta_{o(e)},\eta_{t(e)})$ if
$o(e)\neq t(e)$, and $\eta^{e}_{x}=\eta_{x}$ if $x\neq o(e),t(e)$ or
$x=o(e)=t(e)$. Furthermore, we let

$\Phi_{X}\coloneqq S^{X}\times E,$

and we define the maps $o,t\colon\Phi_{X}\rightarrow S^{X}\times S^{X}$ by

(3) $\displaystyle o((\eta,e))$ $\displaystyle=\eta,$ $\displaystyle
t((\eta,e))$ $\displaystyle=\eta^{e}$

for any $(\eta,e)\in\Phi_{X}$. We have the following.

######  Lemma 2.9.

For any $e\in E$ and $\eta\in S^{X}$, if $\eta^{e}\neq\eta$, then
$(\eta^{e})^{\bar{e}}=\eta$.

###### Proof.

Let $e\in E$ and $\eta=(\eta_{x})\in S^{X}$. Note that
$(\eta_{o(e)},\eta_{t(e)})\in S\times S$. Suppose that $\eta^{e}\neq\eta$.
Then $\phi(\eta_{o(e)},\eta_{t(e)})\neq(\eta_{o(e)},\eta_{t(e)})$. By
definition of $(\eta^{e})^{\bar{e}}$, we see that

$((\eta^{e})^{\bar{e}}_{o(e)},(\eta^{e})^{\bar{e}}_{t(e)})=\bar{\phi}\circ\phi(\eta_{o(e)},\eta_{t(e)}).$

Our assertion now follows from (2). ∎

By Lemma 2.9, we have $(\eta^{e})^{\bar{e}}=\eta$ if $\eta^{e}\neq\eta$ for
any $(\eta,e)\in\Phi_{X}$. We define the inversion
$\iota\colon\Phi_{X}\rightarrow\Phi_{X}$ by

(4) $\iota((\eta,e))\coloneqq\begin{cases}(\eta,e)&\eta^{e}=\eta\\\
(\eta^{e},\bar{e})&\eta^{e}\neq\eta\end{cases}$

for any $(\eta,e)\in\Phi_{X}$. Then by construction, if $\eta^{e}=\eta$, then
$\iota\circ\iota(\eta,e)=\iota(\eta,e)=(\eta,e)$ and if $\eta^{e}\neq\eta$,
then
$\iota\circ\iota(\eta,e)=\iota(\eta^{e},\bar{e})=((\eta^{e})^{\bar{e}},e)=(\eta,e)$.
This shows that $\iota\colon\Phi_{X}\rightarrow\Phi_{X}$ is a bijection such
that if we denote $\bar{\varphi}=\iota(\varphi)$ for any $\varphi\in\Phi_{X}$,
then we have $o(\bar{\varphi})=t(\varphi)$ and $t(\bar{\varphi})=o(\varphi)$.
This shows that $(S^{X},\Phi_{X},o,t,\iota)$ is a multi-graph.

######  Definition 2.10.

The multi-graph $S^{\mathcal{X}}_{\phi}\coloneqq(S^{X},\Phi_{X},o,t,\iota),$
often denoted simply as $S^{\mathcal{X}}_{\phi}=(S^{X},\Phi_{X})$, defines a
multi-graph which we call the _configuration space with transition structure_
associated to the multi-graph $\mathcal{X}$ and interaction $(S,\phi)$.

######  Remark 2.11.

In our previous articles [BKS20, BS21, BS21L2], we defined the configuration
space with transition structure simply to be a graph and not a multi-graph. We
note that $S^{\mathcal{X}}_{\phi}$ is in general not strictly symmetric.

Let $\mathcal{X}=(X,E)$ be a locally finite multi-graph, and let
$(S^{X},\Phi_{X})$ be the configuration space with transition structure
associated to some interaction $(S,\phi)$. For any $\Lambda\subset X$, let
$C(S^{\Lambda})$ be the $\mathbb{R}$-linear space of maps from $S^{\Lambda}$
to $\mathbb{R}$. For any $\Lambda\subset\Lambda^{\prime}\subset X$, the
natural projection $S^{\Lambda^{\prime}}\rightarrow S^{\Lambda}$ induces an
injective $\mathbb{R}$-linear homomorphism $C(S^{\Lambda})\hookrightarrow
C(S^{\Lambda^{\prime}})$. Denote by $\mathscr{I}_{X}$ the set of finite
subsets of $X$. We define the space of _local functions_
$C_{\mathrm{loc}}(S^{X})$ on $S^{X}$ by

$C_{\mathrm{loc}}(S^{X})\coloneqq\bigcup_{\Lambda\in\mathscr{I}_{X}}C(S^{\Lambda})\subset
C(S^{X}).$

Next, we fix a point $*\in S$, which we call the _base state_ , and we let
$S^{X}_{*}$ be the configurations $\eta=(\eta_{x})\in S^{X}$ such that
$\eta_{x}=*$ for all but finite $x\in X$. For any $\Lambda\in\mathscr{I}_{X}$,
the natural projection $S^{X}_{*}\rightarrow S^{\Lambda}$ induces an injective
$\mathbb{R}$-linear homomorphism $C(S^{\Lambda})\hookrightarrow C(S^{X}_{*})$,
which induces an embedding

$C_{\mathrm{loc}}(S^{X})\hookrightarrow C(S^{X}_{*}),$

which is compatible with the projection $C(S^{X})\rightarrow C(S^{X}_{*})$.

######  Proposition 2.12 ([BKS20]*Proposition 3.3).

For any $\Lambda\in\mathscr{I}_{X}$, let

$C_{\Lambda}(S^{X})\coloneqq\\{f\in C(S^{\Lambda})\mid f(\eta)=0\text{ if
$\eta=(\eta_{x})$ satisfies $\eta_{x}=*$ for some $x\in\Lambda$}\\}.$

Then any $f\in C(S^{X}_{*})$ has a unique expansion of the form

(5) $f=\sum_{\Lambda\in\mathscr{I}_{X}}f_{\Lambda},$

where $(f_{\Lambda})_{\Lambda\in\mathscr{I}_{X}}$ is a set of functions such
that $f_{\Lambda}\in C_{\Lambda}(S^{X})$ for any $\Lambda\in\mathscr{I}_{X}$.

Suppose we are given a set of functions
$(f_{\Lambda})_{\Lambda\in\mathscr{I}_{X}}$ such that $f_{\Lambda}\in
C_{\Lambda}(S^{X})$ for any $\Lambda\in\mathscr{I}_{X}$. Then for any $\eta\in
S^{X}_{*}$, we have $f_{\Lambda}(\eta)\neq 0$ for all but finite
$\Lambda\in\mathscr{I}_{X}$. This shows that the sum
$\sum_{\Lambda\in\mathscr{I}_{X}}f_{\Lambda}(\eta)$ is well defined for any
$\eta\in S^{X}_{*}$, hence defines a function on $S^{X}_{*}$ (see
[BKS20]*Lemma 3.2 for details). For any $R\geq 0$, we let
$\mathscr{I}_{\mathcal{X},R}$ be the subset of $\mathscr{I}_{X}$ consisting of
$\Lambda\in\mathscr{I}_{X}$ such that ${\mathrm{diam}}(\Lambda)\leq R$, where
${\mathrm{diam}}(\Lambda)\coloneqq\sup_{x,x^{\prime}\in\Lambda}d_{\mathcal{X}}(x,x^{\prime})$.
Since we have assumed that $\mathcal{X}$ is locally finite, any
$\Lambda\subset X$ with finite diameter is a finite set.

In our previous article [BKS20, BS21, BS21L2], we assumed that the underlying
locale is a simple graph. Multi-edges in our current usage do not typically
have a big effect, and various assertions for the associated simple graph may
naturally be extended to the case of multi-graphs. For example, the distance
$d_{\mathcal{X}}$ on $X$ coincides for a multi-graph and its associated simple
graph.

######  Definition 2.13.

We say that a function $f\in C(S^{X}_{*})$ is a _uniform function on $S^{X}$_,
if there exists $R\geq 0$ such that

$f=\sum_{\Lambda\in\mathscr{I}_{\mathcal{X},R}}f_{\Lambda}$

for some set of functions such that $f_{\Lambda}\in C_{\Lambda}(S^{X})$ for
any $\Lambda\in\mathscr{I}_{\mathcal{X},R}$. We denote by
$C_{\mathrm{unif}}(S^{X})$ the $\mathbb{R}$-linear space of uniform functions
on $S^{X}$.

The notion of uniformity a priori depends on the choice of $E$ for the
multigraph $\mathcal{X}=(X,E)$. We will later show that the notion is
independent of the choice of $E$ for crystal lattices. The most basic example
of uniform functions is given by the conserved quantites.

######  Definition 2.14.

We say that a map $\xi\colon S\rightarrow\mathbb{R}$ is a _conserved quantity_
for the pair $(S,\phi)$, if $\xi(*)=0$ for the base state $*\in S$, and

$\xi(s_{1})+\xi(s_{2})=\xi(s^{\prime}_{1})+\xi(s^{\prime}_{2}),$

where $(s_{1}^{\prime},s_{2}^{\prime})=\phi(s_{1},s_{2})$ for any
$(s_{1},s_{2})\in S\times S$.

Let $\xi\in{\mathrm{Consv}}^{\phi}(S)$. For any $x\in X$, let $\xi_{x}$ be the
function in $C(S^{\\{x\\}})\subset C(S^{X})$ such that
$\xi_{x}(\eta)=\xi(\eta_{x})$ for any $\eta=(\eta_{x})\in S^{X}$. If we let
$\xi_{X}\coloneqq\sum_{x\in X}\xi_{x}\in C(S^{X})$, then $\xi_{X}$ is a
uniform function in $C_{\mathrm{unif}}(S^{X})$. The correspondence
$\xi\mapsto\xi_{X}$ gives an injective $\mathbb{R}$-linear homomorphism
${\mathrm{Consv}}^{\phi}(S)\hookrightarrow C_{\mathrm{unif}}(S^{X})$.

We use the conserved quantities to define the notion that an interaction is
irreducibly quantified. Let $(\Lambda,E_{\Lambda})$ be a finite locale. In
this case, we have $\xi_{\Lambda}=\sum_{x\in\Lambda}\xi_{x}\in
C(S^{\Lambda})$. Consider the canonical map

(6) $\boldsymbol{\xi}_{\\!\Lambda}\colon
S^{\Lambda}\rightarrow{\mathrm{Hom}}_{\mathbb{R}}({\mathrm{Consv}}^{\phi}(S),\mathbb{R}),\qquad\eta\mapsto(\xi\mapsto\xi_{\Lambda}(\eta)).$

Note that a choice of basis of ${\mathrm{Consv}}^{\phi}(S)$ gives an
isomorphism
${\mathrm{Hom}}_{\mathbb{R}}({\mathrm{Consv}}^{\phi}(S),\mathbb{R})\cong\mathbb{R}^{c_{\phi}}$.
For any $\eta,\eta^{\prime}\in S^{\Lambda}$, we have
$\boldsymbol{\xi}_{\\!\Lambda}(\eta)=\boldsymbol{\xi}_{\\!\Lambda}(\eta^{\prime})$
if and only if $\xi_{\Lambda}(\eta)=\xi_{\Lambda}(\eta^{\prime})$ for any
conserved quantity $\xi\in{\mathrm{Consv}}^{\phi}(S)$.

######  Definition 2.15.

We say that an interaction $(S,\phi)$ is _irreducibly quantified_ , if for any
finite locale $(\Lambda,E_{\Lambda})$ and $\eta,\eta^{\prime}\in S^{\Lambda}$
such that
$\boldsymbol{\xi}_{\\!\Lambda}(\eta)=\boldsymbol{\xi}_{\\!\Lambda}(\eta^{\prime})$,
there exists a path in $S^{\Lambda}$ from $\eta$ to $\eta^{\prime}$.

Irreducibly quantified is an important assumption for our main results.

###  2.3. Forms and Uniform Forms

In this subsection, we review the notion of forms and uniform forms. We keep
the notation of §2.2. Let

$\Phi_{X}^{*}\coloneqq\\{\varphi\in\Phi_{X}\mid o(\varphi),t(\varphi)\in
S^{X}_{*}\\},$

so that $(S^{X}_{*},\Phi^{*}_{X},o,t,\iota)$ for maps $o,t,\iota$ as in (3)
and (4) is again a multi-graph. We let

$C^{1}(S^{X}_{*})\coloneqq{\mathrm{Map}}^{\mathrm{alt}}(\Phi_{X}^{*},\mathbb{R})$

be the $\mathbb{R}$-linear space of maps
$\omega^{*}\colon\Phi_{X}\rightarrow\mathbb{R}$, satisfying the condition that
$\omega(\bar{\varphi})=-\omega(\varphi)$ for any $\varphi\in\Phi_{X}^{*}$. We
call such $\omega$ a _form_ on $S^{X}_{*}$. We have a differential

$\partial_{\mathcal{X}}\colon C(S^{X}_{*})\rightarrow C^{1}(S^{X}_{*})$

defined by $\partial_{\mathcal{X}}f(\varphi)\coloneqq f(\eta^{e})-f(\eta)$ for
any $\varphi=(\eta,e)\in\Phi^{*}_{X}$.

For any $e\in E$, a form $\omega\in C^{1}(S^{X}_{*})$ defines a map
$\omega_{e}\colon S^{X}_{*}\rightarrow\mathbb{R}$ by
$\omega_{e}(\eta)\coloneqq\omega((\eta,e))$, which gives an inclusion

(7) $C^{1}(S^{X}_{*})\hookrightarrow\prod_{e\in
E}C(S^{X}_{*}),\qquad\omega\mapsto(\omega_{e})_{e\in E}.$

A system $(\omega_{e})_{e\in E}$ is a form if and only if

(8)
$\omega_{e}(\eta)=\begin{cases}-\omega_{\bar{e}}(\eta^{e})&\eta^{e}\neq\eta,\\\
0&\eta^{e}=\eta.\end{cases}$

If $f\in C(S^{X}_{*})$, then $\partial_{\mathcal{X}}f=(\nabla_{\\!e}f)_{e\in
E}$, where $\nabla_{\\!e}f$ is the function in $C(S^{X}_{*})$ given by
$\nabla_{\\!e}f(\eta)\coloneqq f(\eta^{e})-f(\eta)$ for any $\eta\in
S^{X}_{*}$.

For any path $\vec{\gamma}=(\varphi_{1},\ldots,\varphi_{N})$ in $S^{X}_{*}$,
we define the integral of $\omega\in C^{1}(S^{X}_{*})$ with respect to
$\vec{\gamma}$ by

$\int_{\vec{\gamma}}\omega\coloneqq\sum_{i=1}^{N}\omega(\varphi_{i}).$

We say that a form $\omega\in C^{1}(S^{X}_{*})$ is _closed_ , if
$\int_{\vec{\gamma}}\omega=0$ for any closed path $\vec{\gamma}\in S^{X}_{*}$.
We denote by $Z^{1}(S^{X}_{*})$ the space of closed forms. As in [BKS20]*Lemma
2.15 and 2.16, we may prove the following.

######  Lemma 2.16.

A form $\omega\in C^{1}(S^{X}_{*})$ is closed if and only if it is exact, in
other words, if there exists $f\in C(S^{X}_{*})$ such that
$\partial_{\mathcal{X}}f=\omega$. In particular, the differential
$\partial_{\mathcal{X}}$ gives an isomorphism
$C(S^{X}_{*})/{\mathrm{Ker}\,}\partial_{\mathcal{X}}\cong Z^{1}(S^{X}_{*})$.

For any $R\geq 0$, we define the space of $R$-uniform forms by

$C^{1}_{R}(S^{X})\coloneqq C^{1}(S^{X}_{*})\cap\prod_{e\in
E}C\bigl{(}S^{B(o(e),R)}\bigr{)},$

where $B(o(e),R)\coloneqq\\{x\in X\mid d_{\mathcal{X}}(o(e),x)\leq R\\}$ is
the ball in $X$ with center $o(e)$ and radius $R$. Note that $B(o(e),R)$ is
finite since we have assumed that the multi-graph $\mathcal{X}$ is locally
finite. We define the $\mathbb{R}$-linear space of uniform forms on $S^{X}$ by

$C^{1}_{\mathrm{unif}}(S^{X})\coloneqq\bigcup_{R\geq 0}C^{1}_{R}(S^{X}),$

and the $\mathbb{R}$-linear space of closed uniform forms on $S^{X}$ by

$Z^{1}_{\mathrm{unif}}(S^{X})\coloneqq C^{1}_{\mathrm{unif}}(S^{X})\cap
Z^{1}(S^{X}).$

The differential induces an $\mathbb{R}$-linear map

(9) $\partial_{\mathcal{X}}\colon C_{\mathrm{unif}}(S^{X})\rightarrow
Z^{1}_{\mathrm{unif}}(S^{X})$

(see [BKS20]*Lemma 5.1). The key in the proof of our main result is the
investigation of the kernel and the cokernel of this differential. The kernel
may be expressed via the space of conserved quantities as stated in Theorem
2.17 below, when the interaction is irreducibly quantified.

Let $\star\in S^{X}_{*}$ be the configuration whose components are all at base
state, and we let $C^{0}_{\mathrm{unif}}(S^{X}_{*})\coloneqq\\{f\in
C_{\mathrm{unif}}(S^{X})\mid f(\star)=0\\}$. By construction, we have
$\xi_{X}\in C^{0}_{\mathrm{unif}}(S^{X})$ for any
$\xi\in{\mathrm{Consv}}^{\phi}(S)$. Since a conserved quantity is preserved
with respect to interactions, for any $\eta\in S^{X}_{*}$, we have
$\xi_{X}(\eta^{e})=\xi_{X}(\eta)$ for any $e\in E$. This shows that we have
$\partial_{\mathcal{X}}\xi_{X}=(\nabla_{\\!e}\xi_{X})_{e\in E}=0$ in
$Z^{1}_{\mathrm{unif}}(S^{X})$, hence we have
$\xi_{X}\in{\mathrm{Ker}\,}\partial_{\mathcal{X}}$. One of the main results of
[BKS20] is the following, which will be used in the proof of our main theorem.

######  Theorem 2.17.

Let $\mathcal{X}=(X,E)$ be a _locale_ , in other words, a locally finite
multi-graph which is connected. Suppose in addition that the interaction
$(S,\phi)$ is irreducibly quantified. Then the homomorphism
${\mathrm{Consv}}^{\phi}(S)\hookrightarrow C^{0}_{\mathrm{unif}}(S^{X})$
induces an isomorphism

${\mathrm{Consv}}^{\phi}(S)\cong{\mathrm{Ker}\,}\partial_{\mathcal{X}}\subset
C^{0}_{\mathrm{unif}}(S^{X})$

for the differential $\partial_{\mathcal{X}}\colon
C^{0}_{\mathrm{unif}}(S^{X})\rightarrow Z^{1}_{\mathrm{unif}}(S^{X})$.

For the rest of this subsection, we prepare some lemmas concerning functions
whose differentials are uniform. Let $\mathcal{X}$ be a locally finite multi-
graph, and consider an interaction $(S,\phi)$ which is irreducibly quantified.
For any $f\in C(S^{X}_{*})$ with expansion
$f=\sum_{\Lambda^{\prime\prime}\in\mathscr{I}_{X}}f_{\Lambda^{\prime\prime}}$,
we let

$\iota^{\Lambda}f\coloneqq\sum_{\Lambda^{\prime\prime}\subset\Lambda}f_{\Lambda^{\prime\prime}}$

for $\Lambda\in\mathscr{I}_{X}$, which gives a function in $C(S^{\Lambda})$.
If $\eta\in S^{X}_{*}$, we have

$\iota^{\Lambda}f(\eta)=f(\eta|_{\Lambda})$

for any $\Lambda\in\mathscr{I}_{X}$, where $\eta|_{\Lambda}\in S^{X}_{*}$ is
the configuration whose component coincides with that of $\eta$ for
$x\in\Lambda$ and is at base state for $x\not\in\Lambda$.

For any $\Lambda\subset X$ and $R\geq 0$, we let
$B(\Lambda,R)\coloneqq\bigcup_{x\in\Lambda}B(x,R)$.

######  Lemma 2.18.

Let $f\in C(S^{X}_{*})$ and suppose $\partial_{\mathcal{X}}f\in
C^{1}_{R}(S^{X})$ for some $R>0$. Let $Y\subset X$ be a connected subset. Then
for any $W\subset X$ containing $B(Y,R)$, we see that the value

$(f-\iota^{W}f)(\eta)$

for $\eta\in S^{X}_{*}$ depends only on $\boldsymbol{\xi}_{Y}(\eta)$ and
$\eta|_{X\setminus Y}$.

###### Proof.

Consider $\eta^{\prime},\eta^{\prime\prime}\in S^{X}_{*}$ which coincide
outside $Y$ and satisfies
$\boldsymbol{\xi}_{Y}(\eta^{\prime})=\boldsymbol{\xi}_{Y}(\eta^{\prime\prime})$.
Since the interaction is irreducibly quantified, there exists a path
$\vec{\gamma}|_{S^{Y}}=(\varphi_{1}|_{S^{Y}},\ldots,\varphi_{N}|_{S^{Y}})$
from $\eta^{\prime}|_{S^{Y}}$ to $\eta^{\prime\prime}|_{S^{Y}}$ in $S^{Y}$. We
let $e_{1},\ldots,e_{N}\in E_{Y}$ be such that
$\varphi_{i}|_{S^{Y}}=(\tilde{\eta}_{i},e_{i})$, where
$\tilde{\eta}_{1}=\eta^{\prime}|_{S^{Y}}$ and
$\tilde{\eta}_{i+1}=\tilde{\eta}_{i}^{e_{i}}$ for $i=1,\ldots,N-1$. If we let
$\varphi_{i}\coloneqq(\eta_{i},e_{i})$, where $\eta_{1}=\eta^{\prime}$ and
$\eta_{i+1}=\eta_{i}^{e_{i}}$ for $i=1,\ldots,N-1$, then
$\vec{\gamma}\coloneqq(\varphi_{1},\ldots,\varphi_{N})$ gives a path from
$\eta^{\prime}$ to $\eta^{\prime\prime}$. Then we have

$\displaystyle f(\eta^{\prime\prime})-f(\eta^{\prime})$
$\displaystyle=\sum_{i=1}^{N}\nabla_{e_{i}}f(\eta_{i}),$ $\displaystyle
f(\eta^{\prime\prime}|_{W})-f(\eta^{\prime}|_{W})$
$\displaystyle=\sum_{i=1}^{N}\nabla_{e_{i}}f(\eta_{i}|_{W}).$

By our assumption, we have $\nabla_{\\!e}f\in
C\bigl{(}S^{B(o(e),R)}\bigr{)}\subset C(S^{W}_{*})$ for any $e\in E_{Y}$. This
shows that

$\nabla_{\\!e}f(\eta|_{W})=\nabla_{\\!e}f(\eta)$

for any $e\in E_{Y}$ and $\eta\in S^{X}_{*}$, hence
$f(\eta^{\prime\prime})-f(\eta^{\prime})=f(\eta^{\prime\prime}|_{W})-f(\eta^{\prime}|_{W}),$
which proves that

$f(\eta^{\prime\prime})-f(\eta^{\prime\prime}|_{W})=f(\eta^{\prime})-f(\eta^{\prime}|_{W})$

as desired. ∎

For any $\Lambda\in\mathscr{I}_{X}$, let $\boldsymbol{\xi}_{\\!\Lambda}\colon
S^{\Lambda}\rightarrow{\mathrm{Hom}}_{\mathbb{R}}({\mathrm{Consv}}^{\phi}(S),\mathbb{R})$
be the canonical map given in (6). The image of $S^{\Lambda}$ by
$\boldsymbol{\xi}$ depends only on the order of $|\Lambda|$, so we define
$\mathcal{M}_{|\Lambda|}\coloneqq\boldsymbol{\xi}_{\\!\Lambda}(S^{\Lambda})$
for any $\Lambda\in\mathscr{I}_{X}$.

######  Proposition 2.19.

Let $f\in C(S^{X}_{*})$ such that $\partial_{\mathcal{X}}f\in
C^{1}_{R}(S^{X})$ for some $R>0$. Then for any finite connected
$\Lambda_{1},\Lambda_{2}\subset X$ such that
$d_{\mathcal{X}}(\Lambda_{1},\Lambda_{2})>R$, there exists a function
$h^{\Lambda_{1},\Lambda_{2}}_{f}\colon\mathcal{M}_{|\Lambda_{1}|}\times\mathcal{M}_{|\Lambda_{2}|}\rightarrow\mathbb{R}$
such that

(10)
$\iota^{\Lambda_{1}\cup\Lambda_{2}}f(\eta)-\iota^{\Lambda_{1}}f(\eta)-\iota^{\Lambda_{2}}f(\eta)=h^{\Lambda_{1},\Lambda_{2}}_{f}(\boldsymbol{\xi}_{\\!\Lambda_{1}}(\eta),\boldsymbol{\xi}_{\\!\Lambda_{2}}(\eta))$

for all $\eta\in S^{X}_{*}$.

###### Proof.

It is sufficient to prove the statement for
$\eta=\eta|_{\Lambda_{1}\cup\Lambda_{2}}$. Let $Y=\Lambda_{1}$ and
$W=B(\Lambda_{1},R)$. Since $\Lambda_{2}\cap W=\emptyset$, we have
$\eta|_{W}=\eta|_{\Lambda_{1}}$ and

$(f-\iota^{W}f)(\eta|_{\Lambda_{1}\cup\Lambda_{2}})=f(\eta|_{\Lambda_{1}\cup\Lambda_{2}})-f(\eta|_{\Lambda_{1}})=\iota^{\Lambda_{1}\cup\Lambda_{2}}f(\eta)-\iota^{\Lambda_{1}}f(\eta).$

This equality and Lemma 2.18 show that the left hand side of (10) depends only
on $\boldsymbol{\xi}_{\\!\Lambda_{1}}(\eta)$ and $\eta|_{\Lambda_{2}}$. By
replacing the roles of $\Lambda_{1}$ and $\Lambda_{2}$, we see that the left
hand side of (10) depends only on
$\boldsymbol{\xi}_{\\!\Lambda_{1}}(\eta)\in\mathcal{M}_{|\Lambda_{1}|}$ and
$\boldsymbol{\xi}_{\\!\Lambda_{2}}(\eta)\in\mathcal{M}_{|\Lambda_{2}|}$. ∎

By definition, for any finite connected $\Lambda_{1},\Lambda_{2}\subset X$ and
$\alpha\in\mathcal{M}_{|\Lambda_{1}|}$ and
$\beta\in\mathcal{M}_{|\Lambda_{2}|}$, we have

(11)
$h^{\Lambda_{1},\Lambda_{2}}_{f}(\alpha,\beta)=h^{\Lambda_{2},\Lambda_{1}}_{f}(\beta,\alpha).$

##  3\. Criterion for Uniformity

In this section, we introduce the notion of an essentially Euclidean crystal
lattice, and to prove a criterion Proposition 3.7 for ensuring that a function
on $S^{X}_{*}$ for such crystal lattice is uniform. This criterion will be
used in the next section to prove Theorem 4.8 concerning the surjectivity of
the differential from the space of uniform functions to closed uniform forms.
In §3.3, we will give examples of essentially Euclidean crystal lattices.

###  3.1. Pairings for Essentially Euclidean Crystal Lattices

In what follows, let $\mathcal{X}=(X,E)$ be a crystal lattice for the action
of a finitely generated abelian group $G$. In this subsection, we introduce
the notion of _essentially Euclidean crystal lattices_ , and construct certain
pairings associated to each function $f\in C(S^{X}_{*})$ whose differential is
uniform. We first introduce a Euclidean type coordinate on $\mathcal{X}$,
which we call a _block coordinate_.

We let $F\subset G$ be a free subgroup of finite index in $G$. Note that in
this case, the rank of $F$ coincides with that of $G$. We say that
$\Lambda_{F}$ is a _fundamental domain_ of $\mathcal{X}$ for the action of
$F$, if $\Lambda_{F}$ is connected, and $\Lambda_{F}$ gives a representative
of the orbits of $X$ with respect to the action of $F$, so that for any $x\in
X$, there exists a unique $\sigma\in F$ such that $x\in\sigma(\Lambda_{F})$.
Note that $\Lambda_{F}$ is finite since $\mathcal{X}/F$ is finite. We let
$\mathcal{S}^{+}=\\{\sigma_{1},\ldots,\sigma_{d}\\}$ be a free generator of
$F$. We call the pair $(\Lambda_{F},\mathcal{S}^{+})$ a _coordinate_ of $X$.
For any $\tau\in F$, we let
$(n_{1}(\tau),\ldots,n_{d}(\tau))\in\mathbb{Z}^{d}$ be such that
$\tau=\sigma_{1}^{n_{1}(\tau)}\cdots\sigma_{d}^{n_{d}(\tau)}$. We define the
_block coordinate_ of $x\in X$ for $(\Lambda_{F},\mathcal{S}^{+})$ to be
$(n_{1}(\tau),\ldots,n_{d}(\tau))\in\mathbb{Z}^{d}$, where $\tau\in F$ is such
that $x\in\tau(\Lambda_{F})$. We let $u_{\mathcal{S}^{+}}\colon
X\rightarrow\mathbb{Z}^{d}$ mapping any $x\in X$ to its block coordinate. For
the set of edges

(12)
$\mathbb{E}_{\mathcal{S}}\coloneqq\\{(\bar{x},\bar{x}^{\prime})\in\mathbb{Z}^{d}\times\mathbb{Z}^{d}\mid\exists
e\in
E,u_{\mathcal{S}^{+}}(o(e))=\bar{x},u_{\mathcal{S}^{+}}(t(e))=\bar{x}^{\prime}\\},$

the pair
$\mathbb{Z}_{\mathcal{S}}\coloneqq(\mathbb{Z}^{d},\mathbb{E}_{\mathcal{S}})$
form a connected locally finite graph such that the block coordinate gives a
morphism of multi-graphs
$u_{\mathcal{S}^{+}}\colon\mathcal{X}\rightarrow\mathbb{Z}_{\mathcal{S}}$.

######  Definition 3.1.

  1. (i)

We define the _block distance_ on $\mathcal{X}=(X,E)$ to be the function
$d_{\mathcal{S}}\colon X\times X\rightarrow\mathbb{R}$ given by

$d_{\mathcal{S}}(x,x^{\prime})\coloneqq
d_{\mathbb{Z}_{\mathcal{S}}}(u_{\mathcal{S}^{+}}(x),u_{\mathcal{S}^{+}}(x^{\prime}))$

for any $x,x^{\prime}\in X$, where $d_{\mathbb{Z}_{\mathcal{S}}}$ denotes the
graph distance on $\mathbb{Z}_{\mathcal{S}}$. Note that by definition, we have

$d_{\mathcal{X}}(x,x^{\prime})\leq{\mathrm{diam}}(\Lambda_{F})d_{\mathcal{S}}(x,x^{\prime}).$

The block distance is not a distance in the usual sense since we may have
$d_{\mathcal{S}}(x,x^{\prime})=0$ even if $x\neq x^{\prime}$.

  2. (ii)

We define the _block ball_ in $\mathcal{X}=(X,E)$ with center $x\in X$ and
radius $R\geq 0$ by

$\mathscr{B}(x,R)\coloneqq\\{x^{\prime}\in X\mid
d_{\mathcal{S}}(x,x^{\prime})\leq R\\},$

which is a finite connected subset of $X$.

  3. (iii)

For any $\Lambda\in\mathscr{I}_{X}$, we define the _block diameter_ of
$\Lambda$ by

${\mathrm{diam}}_{\mathcal{S}}(\Lambda)\coloneqq\inf_{x,x^{\prime}\in\Lambda}d_{\mathcal{S}}(x,x^{\prime}).$

######  Definition 3.2.

We say that a crystal lattice $\mathcal{X}=(X,E)$ is _essentially Euclidean_
for the action of $G$, if there exists a free subgroup $F\subset G$ of finite
index with coordinate $(\Lambda_{F},\mathcal{S}^{+})$ such that for the graph
$\mathbb{Z}_{\mathcal{S}}=(\mathbb{Z}^{d},\mathbb{E}_{\mathcal{S}})$, we have

(13) $d_{\mathbb{Z}_{\mathcal{S}}}=d_{\mathbb{Z}^{d}}$

on $\mathbb{Z}^{d}$, where $d_{\mathbb{Z}^{d}}$ denotes the graph distance on
the Euclidean lattice $\mathbb{Z}^{d}=(\mathbb{Z}^{d},\mathbb{E}^{d})$. In
this case, $\mathbb{Z}_{\mathcal{S}}$ coincides with $\mathbb{Z}^{d}$ except
for possibly simple loops in $\mathbb{Z}_{\mathcal{S}}$.

If $\mathcal{X}$ is a crystal lattice which is essentially Euclidean for the
action of $G$, and if we fix a coordinate $(\Lambda_{F},\mathcal{S}^{+})$
satisfying (13), then by definition, the topology of
$\mathbb{Z}_{\mathcal{S}}$ corresponds to that of the Euclidean lattice
$\mathbb{Z}^{d}=(\mathbb{Z}^{d},\mathbb{E}^{d})$. Any subset
$\bar{\Lambda}\subset\mathbb{Z}^{d}$ is connected if and only if the inverse
image

$\Lambda\coloneqq u^{-1}_{\mathcal{S}^{+}}(\bar{\Lambda})$

is connected in $\mathcal{X}$.

######  Lemma 3.3.

Suppose $\mathcal{X}$ is essentially Euclidean for the action of $G$, and fix
a coordinate $(\Lambda_{F},\mathcal{S}^{+})$ satisfying (13). For any $x\in X$
and $R\geq 0$, the complement $X\setminus\mathscr{B}(x,R)$ is a connected
infinite multi-graph if $\dim\mathcal{X}>1$, and has exactly two connected
components which are both infinite multi-graphs if $\dim\mathcal{X}=1$.

###### Proof.

This follows from the assertion for the Euclidean lattice $\mathbb{Z}^{d}$,
noting that the connected components of $X\setminus\mathscr{B}(x,R)$
corresponds bijectively via $u_{\mathcal{S}^{+}}$ with the connected
components of
$\mathbb{Z}^{d}\setminus\\{\tilde{x}^{\prime}\in\mathbb{Z}^{d}\mid
d_{\mathbb{Z}^{d}}(\tilde{x}^{\prime},u_{\mathcal{S}^{+}}(x))\leq R\\}$. ∎

For the rest of this section, we assume that $\mathcal{X}=(X,E)$ is a crystal
lattice which is essentially Euclidean. Let
$\mathcal{M}\coloneqq\boldsymbol{\xi}_{\\!X}(S^{X}_{*})$, where

(14) $\boldsymbol{\xi}_{\\!X}\colon
S^{X}_{*}\rightarrow{\mathrm{Hom}}_{\mathbb{R}}({\mathrm{Consv}}^{\phi}(S),\mathbb{R}),\qquad\eta\mapsto(\xi\mapsto\xi_{X}(\eta))$

for $\xi_{X}\coloneqq\sum_{x\in X}\xi_{x}\in C^{0}_{\mathrm{unif}}(S^{X})$. We
will construct a certain pairing
$h_{f}\colon\mathcal{M}\times\mathcal{M}\rightarrow\mathbb{R}$ for functions
$f\in C(S^{X}_{*})$ whose differential is uniform. This pairing will used in
§3.2 to give a criterion for $f$ to be a uniform function. For an integer
$r\in\mathbb{N}$, let

$\mathscr{D}^{r}\coloneqq\\{\mathscr{B}(x,r^{\prime})\mid x\in
X,r^{\prime}\in\mathbb{N},|\mathscr{B}(x,r^{\prime})|\geq r\\}$

and

$\mathscr{B}^{r}_{R}\coloneqq\\{(D_{1},D_{2})\mid
D_{1},D_{2}\in\mathscr{D}^{r},d_{\mathcal{S}}(D_{1},D_{2})>R\\},$

where $d_{\mathcal{S}}(D_{1},D_{2})\coloneqq\min_{x\in D_{1},x^{\prime}\in
D_{2}}d_{\mathcal{S}}(x,x^{\prime})$. We define a relation in
$\mathscr{B}^{r}_{R}$ by $(D_{1},D_{2})\leftrightarrow(D_{1},D_{2}^{\prime})$
if $D_{2}$ and $D_{2}^{\prime}$ are in the same connected component of
$X\setminus\mathscr{B}(D_{1},R)$, and similarly
$(D_{1},D_{2})\leftrightarrow(D_{1}^{\prime},D_{2})$ if $D_{1}$ and
$D_{1}^{\prime}$ are in the same connected component of
$X\setminus\mathscr{B}(D_{2},R)$.

For any $\Lambda\subset X$ and $R\geq 0$, we let
$\mathscr{B}(\Lambda,R)\coloneqq\bigcup_{x\in X}\mathscr{B}(x,R)$.

######  Lemma 3.4.

Let $f\in C(S^{X}_{*})$ such that $\partial_{\mathcal{X}}f\in
C^{1}_{R}(S^{X})$ for some $R>0$. For any finite connected
$\Lambda_{1},\Lambda_{2},\Lambda_{2}^{\prime}$, suppose $\Lambda_{2}$ and
$\Lambda_{2}^{\prime}$ are in the same connected component of
$X\setminus\mathscr{B}(\Lambda_{1},R)$. Then the functions
$h^{\Lambda_{1},\Lambda_{2}}$ and $h^{\Lambda_{1},\Lambda_{2}^{\prime}}$ of
Proposition 2.19 satisfy

$h^{\Lambda_{1},\Lambda_{2}}_{f}(\alpha,\beta)=h^{\Lambda_{1},\Lambda_{2}^{\prime}}_{f}(\alpha,\beta)$

for any $\alpha\in\mathcal{M}_{|\Lambda_{1}|}$ and
$\beta\in\mathcal{M}_{|\Lambda_{2}|}\cap\mathcal{M}_{|\Lambda_{2}^{\prime}|}=\mathcal{M}_{\min\\{|\Lambda_{2}|,|\Lambda_{2}^{\prime}|\\}}$.

###### Proof.

Let $\alpha\in\mathcal{M}_{|\Lambda_{1}|}$ and
$\beta\in\mathcal{M}_{|\Lambda_{2}|}\cap\mathcal{M}_{|\Lambda_{2}^{\prime}|}=\mathcal{M}_{\min\\{|\Lambda_{2}|,|\Lambda_{2}^{\prime}|\\}}$.
Since
$\Lambda_{1}\cap\Lambda_{2}=\Lambda_{1}\cap\Lambda_{2}^{\prime}=\emptyset$,
there exists configurations $\eta,\eta^{\prime}\in S^{X}_{*}$ such that

$\displaystyle\eta$ $\displaystyle=\eta|_{\Lambda_{1}\cup\Lambda_{2}},$
$\displaystyle\eta^{\prime}$
$\displaystyle=\eta^{\prime}|_{\Lambda_{1}\cup\Lambda_{2}^{\prime}},$
$\displaystyle\eta|_{\Lambda_{1}}$
$\displaystyle=\eta^{\prime}|_{\Lambda_{1}},$
$\displaystyle\boldsymbol{\xi}_{\\!\Lambda_{1}}(\eta)$
$\displaystyle=\boldsymbol{\xi}_{\\!\Lambda_{1}}(\eta^{\prime})=\alpha,$
$\displaystyle\boldsymbol{\xi}_{\\!\Lambda_{2}}(\eta)$
$\displaystyle=\boldsymbol{\xi}_{\\!\Lambda_{2}^{\prime}}(\eta^{\prime})=\beta.$

Let $Y\subset X\setminus\mathscr{B}(\Lambda_{1},R)$ be a finite connected
subset such that $\Lambda_{2}\cup\Lambda_{2}^{\prime}\subset Y$. By
construction,
$\boldsymbol{\xi}_{Y}(\eta)=\boldsymbol{\xi}_{Y}(\eta^{\prime})=\beta$ and
$\eta|_{X\setminus Y}=\eta^{\prime}|_{X\setminus Y}$. If we let $W=B(Y,R)$,
then by Lemma 2.18, we have

$(f-\iota^{W}f)(\eta)=(f-\iota^{W}f)(\eta^{\prime}).$

Since $W\cap(\Lambda_{1}\cup\Lambda_{2})=\Lambda_{2}$ and
$W\cap(\Lambda_{1}\cup\Lambda_{2}^{\prime})=\Lambda_{2}^{\prime}$, we have

$\displaystyle(f-\iota^{W}f)(\eta)$
$\displaystyle=(f-\iota^{W}f)(\eta|_{\Lambda_{1}\cup\Lambda_{2}})=\iota^{\Lambda_{1}\cup\Lambda_{2}}f(\eta)-\iota^{\Lambda_{2}}f(\eta)$
$\displaystyle(f-\iota^{W}f)(\eta^{\prime})$
$\displaystyle=(f-\iota^{W}f)(\eta^{\prime}|_{\Lambda_{1}\cup\Lambda_{2}^{\prime}})=\iota^{\Lambda_{1}\cup\Lambda_{2}^{\prime}}f(\eta^{\prime})-\iota^{\Lambda_{2}}f(\eta^{\prime}).$

Combined with the fact that
$\iota^{\Lambda_{1}}f(\eta)=\iota^{\Lambda_{1}}f(\eta^{\prime})$, we see that

$h_{f}^{\Lambda_{1},\Lambda_{2}}(\alpha,\beta)=h_{f}^{\Lambda_{1},\Lambda_{2}}(\boldsymbol{\xi}_{\\!\Lambda_{1}}(\eta),\boldsymbol{\xi}_{\\!\Lambda_{2}}(\eta))=h_{f}^{\Lambda_{1},\Lambda_{2}^{\prime}}(\boldsymbol{\xi}_{\\!\Lambda_{1}}(\eta^{\prime}),\boldsymbol{\xi}_{\\!\Lambda_{2}^{\prime}}(\eta^{\prime}))=h_{f}^{\Lambda_{1},\Lambda_{2}^{\prime}}(\alpha,\beta)$

as desired. ∎

######  Definition 3.5.

Let $f\in C(S^{X}_{*})$ such that $\partial_{\mathcal{X}}f\in
C^{1}_{R}(S^{X})$ for some $R>0$. We define the paring
$h_{f}\colon\mathcal{M}\times\mathcal{M}\rightarrow\mathbb{R}$ on
$\mathcal{M}=\bigcup_{k\in\mathbb{N}}\mathcal{M}_{k}$ as follows. We let
$k\in\mathbb{N}$ such that $\alpha,\beta\in\mathcal{M}_{k}$. By taking an
arbitrary $(D_{1},D_{2})\in\mathscr{B}^{k}_{R}$, we let

$h_{f}(\alpha,\beta)\coloneqq h^{D_{1},D_{2}}_{f}(\alpha,\beta).$

If $(D_{1},D_{2}),(D_{1}^{\prime},D_{2}^{\prime})\in\mathscr{B}^{k}_{R}$, if
we take $D^{\prime\prime}_{2}\in\mathscr{D}^{k}$ such that
$D_{2}^{\prime\prime}\subset
X\setminus(\mathscr{B}(D_{1},R)\cup\mathscr{B}(D_{1}^{\prime},R))$, then we
have

$(D_{1},D_{2})\leftrightarrow(D_{1},D_{2}^{\prime\prime})\leftrightarrow(D_{1}^{\prime},D_{2}^{\prime\prime})\leftrightarrow(D_{1}^{\prime},D_{2}^{\prime}).$

Using Lemma 3.4 and (11), we may prove that

$h^{D_{1},D_{2}}_{f}(\alpha,\beta)=h^{D_{1},D_{2}^{\prime\prime}}_{f}(\alpha,\beta)=h^{D_{1}^{\prime},D_{2}^{\prime\prime}}_{f}(\alpha,\beta)=h^{D_{1}^{\prime},D_{2}^{\prime}}_{f}(\alpha,\beta),$

hence that the pairing $h_{f}$ is independent of the choice of
$(D_{1},D_{2})\in\mathscr{B}^{k}_{R}$.

######  Proposition 3.6.

Let $f\in C(S^{X}_{*})$ such that $\partial_{\mathcal{X}}f\in
C^{1}_{R}(S^{X})$ for some $R>0$. Then the pairing
$h_{f}\colon\mathcal{M}\times\mathcal{M}\rightarrow\mathbb{R}$ satisfies the
cocycle condition

(15)
$h_{f}(\alpha,\beta)+h_{f}(\alpha+\beta,\gamma)=h_{f}(\beta,\gamma)+h_{f}(\alpha,\beta+\gamma)$

for any $\alpha,\beta,\gamma\in\mathcal{M}$.

###### Proof.

Take $k>0$ sufficiently large so that $\alpha,\beta,\gamma\in\mathcal{M}_{k}$.
We take $D_{1},D_{2},D_{3}\in\mathscr{D}^{k}$ sufficiently apart so that
$(D_{1},D_{2}),(D_{1},D_{3}),(D_{2},D_{3})\in\mathscr{B}^{k}_{R}$ and
$(D_{1},D_{2})\leftrightarrow(D_{1},D_{3})\leftrightarrow(D_{2},D_{3})$. We
may take $\bar{D}_{1},\bar{D}_{2},\bar{D}_{3}$ to be balls of radius $k$ in
the Euclidean lattice with center $(-2R-2k-2,0,\ldots,0)$, $(0,\ldots,0)$,
$(2R+2k+2,0,\ldots,0)\in\mathbb{Z}^{d}$, and we let $D_{i}\coloneqq
u_{\mathcal{S}^{+}}^{-1}(\bar{D}_{i})$ for $i=1,2,3$. We let $\eta\in
S^{X}_{*}$ be such that $\eta=\eta|_{D_{1}\cup D_{2}\cup D_{2}}$ and
$\boldsymbol{\xi}_{\\!D_{1}}(\eta)=\alpha$,
$\boldsymbol{\xi}_{\\!D_{2}}(\eta)=\beta$,
$\boldsymbol{\xi}_{\\!D_{3}}(\eta)=\gamma$. Next, take any
$D_{12},D_{23}\in\mathscr{D}^{k}$ such that

$\displaystyle D_{1}\cup D_{2}$ $\displaystyle\subset D_{12}\subset
X\setminus\mathscr{B}(D_{3},R),$ $\displaystyle D_{2}\cup D_{3}$
$\displaystyle\subset D_{23}\subset X\setminus\mathscr{B}(D_{1},R).$

We may take for example $D_{12}\coloneqq
u_{\mathcal{S}^{+}}^{-1}(\bar{D}_{12})$ and $D_{23}\coloneqq
u_{\mathcal{S}^{+}}^{-1}(\bar{D}_{23})$, where $\bar{D}_{12}$ and
$\bar{D}_{23}$ are balls of radius $R+2k+1$ with centers
$(-R-k-1,0,\ldots,0)$, $(R+k+1,0,\ldots,0)$ in $\mathbb{Z}^{d}$. Then
$\boldsymbol{\xi}_{\\!D_{12}}(\eta)=\alpha+\beta$ and
$\boldsymbol{\xi}_{\\!D_{23}}(\eta)=\beta+\gamma$. By Lemma 3.4 and the
definition of $h_{f}$, we have

$\displaystyle h_{f}(\alpha,\beta)$
$\displaystyle=h_{f}^{D_{1},D_{2}}(\alpha,\beta)=\iota^{D_{1}\cup
D_{2}}f(\eta)-\iota^{D_{1}}f(\eta)-\iota^{D_{2}}f(\eta)$ $\displaystyle
h_{f}(\alpha+\beta,\gamma)$
$\displaystyle=h_{f}^{D_{12},D_{3}}(\alpha+\beta,\gamma)=\iota^{D_{12}\cup
D_{3}}f(\eta)-\iota^{D_{12}}f(\eta)-\iota^{D_{3}}f(\eta)$ $\displaystyle
h_{f}(\beta,\gamma)$
$\displaystyle=h_{f}^{D_{2},D_{3}}(\alpha,\beta)=\iota^{D_{2}\cup
D_{3}}f(\eta)-\iota^{D_{2}}f(\eta)-\iota^{D_{3}}f(\eta)$ $\displaystyle
h_{f}(\alpha,\beta+\gamma)$
$\displaystyle=h_{f}^{D_{1},D_{23}}(\alpha,\beta+\gamma)=\iota^{D_{1}\cup
D_{23}}f(\eta)-\iota^{D_{1}}f(\eta)-\iota^{D_{23}}f(\eta).$

Our assertion follows from the fact that

$\displaystyle\iota^{D_{12}\cup D_{3}}f(\eta)$ $\displaystyle=\iota^{D_{1}\cup
D_{23}}f(\eta)=\iota^{D_{1}\cup D_{2}\cup D_{3}}f(\eta)$

and $\iota^{D_{12}}f(\eta)=\iota^{D_{1}\cup D_{2}}f(\eta)$,
$\iota^{D_{23}}f(\eta)=\iota^{D_{2}\cup D_{3}}f(\eta)$. ∎

###  3.2. Criterion for Uniform Locality

In this subsection, we prove a criterion for uniformity. Let
$\mathcal{X}=(X,E)$ be a crystal lattice for the action of a group $G$ which
is essentially Euclidean, and we fix a coordinate
$(\Lambda_{F},\mathcal{S}^{+})$ satisfying (13). Assume that $(S,\phi)$ is an
interaction which is irreducibly quantified.

######  Proposition 3.7.

Let $f\in C(S^{X}_{*})$ such that $\partial_{\mathcal{X}}f\in
C^{1}_{R}(S^{X})$ for some $R>0$. If the pairing
$h_{f}\colon\mathcal{M}\times\mathcal{M}\rightarrow\mathbb{R}$ satisfies
$h_{f}\equiv 0$, then $f$ is uniform.

We will give a proof of Proposition 3.7 at the end of this section. We first
give a condition for a function to be uniform.

######  Lemma 3.8.

Suppose $f\in C(S^{X}_{*})$. If there exists integer $R>0$ such that for any
finite $\Lambda\subset X$ and $x\in X$, we have

(16)
$\iota^{\Lambda}f-\iota^{\Lambda\setminus\mathscr{B}(x,0)}f=\iota^{\Lambda\cap\mathscr{B}(x,R)}f-\iota^{\Lambda\cap\mathscr{B}^{*}(x,R)}f,$

then $f$ is a uniform function in $C_{\mathrm{unif}}(S^{X})$. Here,
$\mathscr{B}^{*}(x,R)\coloneqq\mathscr{B}(x,R)\setminus\mathscr{B}(x,0)$.

###### Proof.

First observe that for any $f\in C(S^{X}_{*})$, we have

$\displaystyle\iota^{\Lambda}f$
$\displaystyle=\sum_{\Lambda^{\prime\prime}\subset\Lambda}f_{\Lambda^{\prime\prime}}=\sum_{\Lambda^{\prime\prime}\subset\Lambda\setminus\mathscr{B}(x,0)}f_{\Lambda^{\prime\prime}}+\sum_{\begin{subarray}{c}\Lambda^{\prime\prime}\subset\Lambda,\\\
\Lambda^{\prime\prime}\cap\mathscr{B}(x,0)\neq\emptyset\end{subarray}}f_{\Lambda^{\prime\prime}}$
$\displaystyle=\iota^{\Lambda\setminus\mathscr{B}(x,0)}f+\sum_{\begin{subarray}{c}\Lambda^{\prime\prime}\subset\Lambda,\\\
\Lambda^{\prime\prime}\cap\mathscr{B}(x,0)\neq\emptyset\end{subarray}}f_{\Lambda^{\prime\prime}}.$

If (16) holds, then we have

(17) $\sum_{\begin{subarray}{c}\Lambda^{\prime\prime}\subset\Lambda,\\\
\Lambda^{\prime\prime}\cap\mathscr{B}(x,0)\neq\emptyset\end{subarray}}f_{\Lambda^{\prime\prime}}=\sum_{\begin{subarray}{c}\Lambda^{\prime\prime}\subset\Lambda\cap\mathscr{B}(x,R),\\\
\Lambda^{\prime\prime}\cap\mathscr{B}(x,0)\neq\emptyset\end{subarray}}f_{\Lambda^{\prime\prime}}.$

Let $\mathscr{I}^{\prime}\coloneqq\\{\Lambda\in\mathscr{I}_{X}\mid
f_{\Lambda}\neq 0,{\mathrm{diam}}_{\mathcal{S}}(\Lambda)>R\\}$, and assume
that $\mathscr{I}^{\prime}\neq\emptyset$. We take $\Lambda$ to be a minimal
element in $\mathscr{I}^{\prime}$ with respect to inclusions, so that if
$\Lambda^{\prime\prime}\subsetneq\Lambda$, then we have
$f_{\Lambda^{\prime\prime}}=0$ or
${\mathrm{diam}}_{\mathcal{S}}(\Lambda^{\prime\prime})\leq R$. For such
$\Lambda$, take $x\in\Lambda$ such that there exists $y\in\Lambda$ with
$d_{\mathcal{S}}(x,y)>R$. Then we have

$\sum_{\begin{subarray}{c}\Lambda^{\prime\prime}\subset\Lambda,\\\
\Lambda^{\prime\prime}\cap\mathscr{B}(x,0)\neq\emptyset\end{subarray}}f_{\Lambda^{\prime\prime}}=f_{\Lambda}+\sum_{\begin{subarray}{c}\Lambda^{\prime\prime}\subsetneq\Lambda,\\\
\Lambda^{\prime\prime}\cap\mathscr{B}(x,0)\neq\emptyset\end{subarray}}f_{\Lambda^{\prime\prime}}=f_{\Lambda}+\sum_{\begin{subarray}{c}\Lambda^{\prime\prime}\subset\Lambda\cap\mathscr{B}(x,R),\\\
\Lambda^{\prime\prime}\cap\mathscr{B}(x,0)\neq\emptyset\end{subarray}}f_{\Lambda^{\prime\prime}}.$

Comparing this with equality (17), we see that $f_{\Lambda}\equiv 0$. This
contradicts the fact that $\Lambda\in\mathscr{I}^{\prime}$, hence that
$\mathscr{I}^{\prime}=\emptyset$. In other words, any
$\Lambda\in\mathscr{I}_{X}$ such that $f_{\Lambda}\neq 0$ satisfies
${\mathrm{diam}}_{\mathcal{S}}(\Lambda)\leq R$. If we let
$R^{\prime}\coloneqq{\mathrm{diam}}(\Lambda_{0})R$, then
${\mathrm{diam}}(\Lambda)\leq{\mathrm{diam}}(\Lambda_{0}){\mathrm{diam}}_{\mathcal{S}}(\Lambda)\leq
R^{\prime}$. We see that

$f=\sum_{\Lambda\in\mathscr{I}_{\mathcal{X},R^{\prime}}}f_{\Lambda},$

which proves that $f$ is uniform as desired. ∎

######  Proposition 3.9.

Let $f\in C(S^{X}_{*})$ such that $\partial_{\mathcal{X}}f\in
C^{1}_{R}(S^{X})$ for some $R>0$. If the pairing
$h_{f}\colon\mathcal{M}\times\mathcal{M}\rightarrow\mathbb{R}$ satisfies
$h_{f}\equiv 0$, then for any $x\in X$ and $\Lambda\in\mathscr{I}_{X}$ such
that $d_{\mathcal{S}}(x,\Lambda)>R$, we have

$\iota^{\Lambda\cup\mathscr{B}(x,R)}f-\iota^{\Lambda\cup\mathscr{B}^{*}(x,R)}f=\iota^{\mathscr{B}(x,R)}f-\iota^{\mathscr{B}^{*}(x,R)}f.$

###### Proof.

Suppose $\eta=\eta|_{\Lambda\cup\mathscr{B}(x,R)}$. If we let $Y\coloneqq
X\setminus\mathscr{B}(x,R)$, then $Y$ is connected if $\dim\mathcal{X}>1$, and
$Y=Y_{1}\cup Y_{2}$ for two infinite connected components $Y_{1},Y_{2}$ if
$\dim\mathcal{X}=1$. Let $\Lambda_{i}=Y_{i}\cap\Lambda$ for $i=1,2$ so that
$\Lambda=\Lambda_{1}\cup\Lambda_{2}$, with the convention
$Y_{2}=\Lambda_{2}=\emptyset$ if $\dim\mathcal{X}>1$. For
$\mathscr{D}\coloneqq\bigcup_{k\in\mathbb{N}}\mathscr{D}^{k}$, let
$D_{1}^{\prime}\in\mathscr{D}$ be such that
$\mathscr{B}(\Lambda,R)\cup\mathscr{B}(x,R)\subset D_{1}^{\prime}$, and let
$D_{1}\in\mathscr{D}$ such that

$\displaystyle d_{\mathcal{S}}(\Lambda,D_{1})$ $\displaystyle>R,$
$\displaystyle D_{1}$
$\displaystyle\subset(X\setminus\mathscr{B}(D^{\prime}_{1},R))\cap Y_{1},$
$\displaystyle|D_{1}|$ $\displaystyle\geq|\Lambda_{1}|.$

Such $D_{1}$ exists since $Y_{1}$ is an infinite set. Choose $\eta^{\prime}\in
S^{X}_{*}$ such that $\eta^{\prime}$ coincides with $\eta$ outside
$\Lambda_{1}\cup D_{1}\subset Y_{1}$, is at base state on $\Lambda_{1}$, and
$\boldsymbol{\xi}_{\\!D_{1}}(\eta^{\prime})=\boldsymbol{\xi}_{\\!\Lambda_{1}}(\eta)$.
This implies that
$\boldsymbol{\xi}_{\\!Y_{1}}(\eta^{\prime})=\boldsymbol{\xi}_{\\!Y_{1}}(\eta)$.
Hence by Lemma 2.18 applied to $Y_{1}$ and $W=X\setminus\\{x\\}$, we have

$f(\eta)-f(\eta|_{W})=f(\eta^{\prime})-f(\eta^{\prime}|_{W}).$

Since $\eta=\eta|_{D_{1}^{\prime}}$ and
$\eta^{\prime}=\eta^{\prime}|_{D_{1}\cup D_{1}^{\prime}}$, we see that

(18)
$f(\eta|_{D_{1}^{\prime}})-f(\eta|_{D_{1}^{\prime}\setminus\mathscr{B}(x,0)})=f(\eta^{\prime}|_{D_{1}\cup
D_{1}^{\prime}})-f(\eta^{\prime}|_{D_{1}\cup(D_{1}^{\prime}\setminus\mathscr{B}(x,0)}).$

By construction, $d_{\mathcal{S}}(D_{1},D^{\prime}_{1})>R$, hence the
calculation of $h_{f}$ via the pairs
$(D_{1},D^{\prime}_{1})\in\mathscr{B}_{R}$ and
$(D_{1},D_{1}^{\prime}\setminus\mathscr{B}(x,0))$ as well as our assumption
that $h_{f}\equiv 0$ gives

$\displaystyle f(\eta^{\prime}|_{D_{1}\cup D_{1}^{\prime}})$
$\displaystyle=f(\eta^{\prime}|_{D_{1}})+f(\eta^{\prime}|_{D_{1}^{\prime}}),$
$\displaystyle
f(\eta^{\prime}|_{D_{1}\cup(D_{1}^{\prime}\setminus\mathscr{B}(x,0))})$
$\displaystyle=f(\eta^{\prime}|_{D_{1}})+f(\eta^{\prime}|_{D_{1}^{\prime}\setminus\mathscr{B}(x,0)}).$

Hence (18) gives

$f(\eta|_{D_{1}^{\prime}})-f(\eta|_{D_{1}^{\prime}\setminus\mathscr{B}(x,0)})=f(\eta^{\prime}|_{D_{1}^{\prime}})-f(\eta^{\prime}|_{D_{1}^{\prime}\setminus\mathscr{B}(x,0)}).$

In particular, since $\eta=\eta|_{\Lambda\cup\mathscr{B}(x,R)}$ and
$\eta^{\prime}=\eta^{\prime}|_{D_{1}\cup\Lambda_{2}\cup\mathscr{B}(x,R)}$, we
have

$f(\eta|_{\Lambda\cup\mathscr{B}(x,R)})-f(\eta|_{\Lambda\cup\mathscr{B}^{*}(x,R)})=f(\eta^{\prime}|_{\Lambda_{2}\cup\mathscr{B}(x,R)})-f(\eta^{\prime}|_{\Lambda_{2}\cup\mathscr{B}^{*}(x,R)}).$

Since $\eta^{\prime}|_{X\setminus Y_{1}}=\eta|_{X\setminus Y_{1}}$, this gives

$f(\eta|_{\Lambda\cup\mathscr{B}(x,R)})-f(\eta|_{\Lambda\cup\mathscr{B}^{*}(x,R)})=f(\eta|_{\Lambda_{2}\cup\mathscr{B}(x,R)})-f(\eta|_{\Lambda_{2}\cup\mathscr{B}^{*}(x,R)}).$

This gives our assertion for $\dim\mathcal{X}>1$, since
$\Lambda_{2}=\emptyset$ in this case. The proof for the case
$\dim\mathcal{X}=1$ follows by applying the same argument as that of
$\Lambda_{1}$ to $\Lambda_{2}$. ∎

We may now prove Proposition 3.7.

###### Proof of Proposition 3.7.

Suppose $f\in C(S^{X}_{*})$ such that $\partial_{\mathcal{X}}f\in
C^{1}_{R}(S^{X})$ for some $R>0$, and assume that $h_{f}\equiv 0$ for the
paring of Definition 3.5. By Lemma 3.8, it is sufficient to prove that for any
$\Lambda\in\mathscr{I}_{X}$ and $x\in\Lambda$, we have

$\iota^{\Lambda}f-\iota^{\Lambda\setminus\mathscr{B}(x,0)}f=\iota^{\Lambda\cap\mathscr{B}(x,R)}f-\iota^{\Lambda\cap\mathscr{B}^{*}(x,R)}f.$

Let $\Lambda^{\prime}\coloneqq\Lambda\setminus(\Lambda\cap\mathscr{B}(x,R))$.
By Proposition 3.9 applied to $\Lambda^{\prime}$, we have

(19)
$\iota^{\Lambda^{\prime}\cup\mathscr{B}(x,R)}f-\iota^{\Lambda^{\prime}\cup\mathscr{B}^{*}(x,0)}f=\iota^{\mathscr{B}(x,R)}f-\iota^{\mathscr{B}^{*}(x,R)}f.$

By applying $\iota^{\Lambda^{\prime}\cup(\Lambda\cap\mathscr{B}(x,R))}$ to
both sides of (19), we have

$\iota^{\Lambda^{\prime}\cup(\Lambda\cap\mathscr{B}(x,R))}f-\iota^{\Lambda^{\prime}\cup(\Lambda\cap\mathscr{B}^{*}(x,0))}f=\iota^{\Lambda\cap\mathscr{B}(x,R)}f-\iota^{\Lambda\cap\mathscr{B}^{*}(x,R)}f.$

Since $\Lambda=\Lambda^{\prime}\cup(\Lambda\cap\mathscr{B}(x,R))$, this
coincides with the desired equality. ∎

###  3.3. Examples of Essentially Euclidean Crystal Lattices

In this subsection, we give examples of essentially Euclidean crystal
lattices. In particular, we will give a criterion for a crystal lattice to be
essentially Euclidean, and show that the maximal abelian covering of a crystal
is an essentially Euclidean lattice. The results of this subsection are not
necessary for the proof of our main theorem.

In what follows, for any $\Lambda\in\mathscr{I}_{X}$, we let
$E_{\Lambda}\coloneqq\\{e\in E\mid o(e),t(e)\in\Lambda\\}$ and $\partial
E_{\Lambda}\coloneqq\\{e\in E\mid o(e)\in\Lambda,t(e)\not\in\Lambda\\}$.

######  Lemma 3.10.

Suppose $\mathcal{X}=(X,E)$ is a crystal lattice for the action of a finitely
generated free group $G$. Let $d\coloneqq\dim\mathcal{X}$ be the rank of $G$.
For $\mathcal{X}/G=(X_{0},E_{0})$, if

(20) $d=1-|X_{0}|+\frac{1}{2}|E_{0}|,$

then $\mathcal{X}$ is essentially Euclidean.

###### Proof.

Let $\Lambda_{0}$ be a fundamental domain of $\mathcal{X}$ for the action of
$G$. In order for the graph $\Lambda_{0}$ consisting of $|\Lambda_{0}|$
vertices to be connected in the sense of a symmetric graphs, one needs at
least $|\Lambda_{0}|-1$ edges to connect distinct vertices. Considering
symmetry, this would require $2(|\Lambda_{0}|-1)$ edges. This implies that
$|E_{\Lambda_{0}}|\geq 2(|\Lambda_{0}|-1)$, which shows that

$1-|\Lambda_{0}|+\frac{1}{2}|E_{\Lambda_{0}}|\geq 0.$

Moreover, we have $|X_{0}|=|\Lambda_{0}|$ and
$|E_{0}|=|E_{\Lambda_{0}}|+|\partial E_{\Lambda_{0}}|$, hence we see from (20)
that

$|\partial
E_{\Lambda_{0}}|=2d-2\Bigl{(}1-|X_{0}|+\frac{1}{2}|E_{\Lambda_{0}}|\Bigr{)}\leq
2d.$

Let $\mathcal{S}=\\{\sigma\in G\mid\exists e\in\partial
E_{\Lambda_{0}},t(e)\in\sigma(\Lambda_{0})\\}$. We show that $\mathcal{S}$
generates $G$. Fix $x_{0}\in\Lambda_{0}$. For any $\tau\in G$, since $X$ is
connected, there exists a path $\vec{\gamma}=(e^{1},\ldots,e^{N})$ from
$x_{0}$ to $\tau(x_{0})$. We let $\tau_{0}={\mathrm{id}}$, $\tau_{i}\in G$ be
such that $t(e^{i})\in\tau_{i}(\Lambda_{0})$ for $i=1,\ldots,N$, and
$\sigma_{e^{i}}\in G$ be such that
$t(e^{i})=\sigma_{e^{i}}\tau_{i-1}(\Lambda_{0})$. Then we have
$\sigma_{e^{i}}={\mathrm{id}}$ or $\sigma_{e^{i}}\in\mathcal{S}$. Then

$\tau_{N}=\sigma_{e^{N}}\tau_{N-1}=\sigma_{e^{N}}\sigma_{e^{N-1}}\tau_{N-2}=\cdots=\sigma_{e^{N}}\cdots\sigma_{e^{1}},$

which shows that $\tau=\tau_{N}$ is in the subgroup generated by
$\mathcal{S}$. Note that if $\sigma\in\mathcal{S}$, then
$\sigma^{-1}\in\mathcal{S}$. This shows that $|\mathcal{S}|\geq 2d$ since $d$
is the rank of $G$.

For any $e\in\partial E_{\Lambda_{0}}$, let $\sigma_{e}\in\mathcal{S}$ be such
that $t(e)\in\sigma_{e}(\Lambda_{0})$. Then $e\mapsto\sigma_{e}$ gives a map
$\partial E_{\Lambda_{0}}\rightarrow\mathcal{S}$ which is surjective from the
definition of $\mathcal{S}$. Since $|\partial E_{\Lambda_{0}}|\leq 2d$, we
must have $|\mathcal{S}|=|\partial E_{\Lambda_{0}}|=2d$. If we fix any
$\mathcal{S}^{+}\subset\mathcal{S}$ such that either
$\sigma\in\mathcal{S}^{+}$ or $\sigma^{-1}\in\mathcal{S}^{+}$ for any
$\sigma\in\mathcal{S}$, then $\mathcal{S}^{+}$ is a free generator of $G$, and
$(\Lambda_{0},\mathcal{S}^{+})$ gives a coordinate of $\mathcal{X}$ which
satisfies (13). ∎

######  Lemma 3.11.

Let $\mathcal{X}_{0}=(X_{0},E_{0})$ be a connected crystal, in other words, a
strictly symmetric finite connected multi-graph. Then the maximal abelian
covering $\mathcal{X}^{\mathrm{ab}}$ of $\mathcal{X}_{0}$ has a free action of
the finitely generated abelian group
$G=\pi_{1}^{\mathrm{ab}}(\mathcal{X}_{0})$, where
$\pi_{1}^{\mathrm{ab}}(\mathcal{X}_{0})$ is the maximal abelian quotient of
the fundamental group $\pi_{1}(\mathcal{X}_{0})$ of $\mathcal{X}_{0}$. Then
$\mathcal{X}^{\mathrm{ab}}$ for the action of $G$ satisfies the conditions of
Lemma 3.10, hence is a crystal lattice which is essentially Euclidean.

###### Proof.

Since $\mathcal{X}_{0}$ is connected, the homology group
$H_{1}(\mathcal{X}_{0},\mathbb{Z})$ of the crystal $\mathcal{X}_{0}$ is a free
abelian group of rank

$d=1-|X_{0}|+\frac{1}{2}|E_{0}|.$

The standard isomorphism $\pi^{\mathrm{ab}}_{1}(\mathcal{X}_{0})\cong
H_{1}(\mathcal{X}_{0},\mathbb{Z})$ shows that
$\pi^{\mathrm{ab}}_{1}(\mathcal{X})$ is a free abelian group of rank $d$.
Hence $\mathcal{X}^{\mathrm{ab}}=(X^{\mathrm{ab}},E^{\mathrm{ab}})$ is a
connected locally finite multi-graph with a free action of the finitely
generated abelian group $G\coloneqq\pi_{1}^{\mathrm{ab}}(\mathcal{X}_{0})$,
such that $\mathcal{X}^{\mathrm{ab}}/G=\mathcal{X}_{0}$. This shows in
particular that $\mathcal{X}^{\mathrm{ab}}$ satisfies the conditions of Lemma
3.10. ∎

######  Example 3.12.

  1. (i)

The _hexagonal lattice_ $\mathcal{X}$ gives an example of a crystal lattice
which is essentially Euclidean. It has a free action of $G=\mathbb{Z}^{2}$
given by translation, and the quotient $\mathcal{X}/G=(X_{0},E_{0},o,t,\iota)$
is given by $X_{0}=\\{x_{0},x_{1}\\}$,
$E_{0}=\\{e_{0},\bar{e}_{0},e_{1},\bar{e}_{1},e_{2},\bar{e}_{2}\\}$,
$o(e_{j})=x_{0}$, $t(e_{j})=x_{1}$, and $\iota(e_{j})=\bar{e}_{j}$,
$\iota(\bar{e}_{j})=e_{j}$ for $j=1,2,3$. In fact, $\mathcal{X}$ is the
maximal abelian covering of $\mathcal{X}/G$.

  2. (ii)

The _triangular lattice_ $\mathcal{X}$ gives an example of a crystal lattice
which is _not_ essentially Euclidean. $\mathcal{X}$ has an action of
$G=\mathbb{Z}^{2}$ by translation. For any coordinate
$(\Lambda_{F},\mathcal{S}^{+})$, the graph
$\mathbb{Z}_{\mathcal{S}}=(\mathbb{Z}^{2},\mathbb{E}_{\mathcal{S}})$ contains
extra diagonal edges, hence the graph distance $d_{\mathcal{S}}$ does not
coincide with $d_{\mathbb{Z}^{2}}$.

Figure 1. The Hexagonal and Triangular Lattices

  3. (iii)

For integers $d\geq 1$ and $n>0$, the Euclidean lattice with nearest
$n$-neighbor $\mathbb{Z}^{d}_{n}=(\mathbb{Z}^{d},\mathbb{E}_{n})$, where

$\mathbb{E}_{n}\coloneqq\bigl{\\{}(x,y)\in\mathbb{Z}^{d}\times\mathbb{Z}^{d}\,\,\big{|}\,\,0<|x-y|\leq
n\bigr{\\}},$

is a crystal lattice for the action of the group $G=\mathbb{Z}^{d}$ by
translation.

Figure 2. The Euclidean Lattices $\mathbb{Z}^{2}$ and with nearest
$2$-neighbor $\mathbb{Z}^{2}_{2}$

The lattice $\mathbb{Z}^{d}_{n}$ for $d>1$ and $n>1$ give examples of crystal
lattices which are _not_ essentially Euclidean.

##  4\. Decomposition for Closed Uniform Forms

In this section, we prove our main result, Theorem 4.3, which gives the
decomposition of shift-invariant closed uniform forms on configuration spaces
with transition structure over general crystal lattices. Theorem 4.3 is
obtained from Theorem 4.5 concerning the surjectivity of the differential from
the space of uniform functions to closed uniform forms. Theorem 4.5 is proved
by reducing to the essentially Euclidean case, Theorem 4.8.

###  4.1. The Main Theorem

In this subsection, we will state our main theorem. Let $\mathcal{X}=(X,E)$ be
a crystal lattice of dimension $\dim\mathcal{X}$ for the action of an abelian
group $G$. We let $(S,\phi)$ be an interaction which is irreducibly
quantified. The action of $G$ on $\mathcal{X}$ induces actions on the spaces
$C^{0}_{\mathrm{unif}}(S^{X})$, $C^{1}_{\mathrm{unif}}(S^{X})$,
$Z^{1}_{\mathrm{unif}}(S^{X})$ compatible with the differential (9). We denote
by $C^{0}_{\mathrm{unif}}(S^{X})^{G}$, $C^{1}_{\mathrm{unif}}(S^{X})^{G}$,
$Z^{1}_{\mathrm{unif}}(S^{X})^{G}$ the $G$-invariant subspaces of
$C^{0}_{\mathrm{unif}}(S^{X})$, $C^{1}_{\mathrm{unif}}(S^{X})$,
$Z^{1}_{\mathrm{unif}}(S^{X})$ consisting of functions or forms invariant with
respect to the action of $G$.

Again let $d=\dim\mathcal{X}$. We fix a generator
$\sigma_{1},\ldots,\sigma_{d}$ of the free part $F\subset G$ which gives an
isomorphism $F\cong\mathbb{Z}^{d}$. We may extend this isomorphism to a
surjective homomorphism $G\rightarrow\mathbb{Z}^{d}$ by mapping the torsion
subgroup of $G$ to _zero_. For any $\tau\in G$, we let
$(n_{1}(\tau),\ldots,n_{d}(\tau))\in\mathbb{Z}^{d}$ be the image of $\tau$
with respect to the above surjection. We fix a fundamental domain
$\Lambda_{0}$ of $X$ for the action of $G$.

######  Lemma 4.1.

For any conserved quantity $\xi\in{\mathrm{Consv}}^{\phi}(S)$ and
$j=1,\ldots,d$, let

$\mathfrak{A}^{j}_{\xi}\coloneqq\sum_{\tau\in
G}n_{j}(\tau)\xi_{\tau(\Lambda_{0})}\in C^{0}_{\mathrm{unif}}(S^{X}).$

Then for any $j,k=1,\ldots,d$, the functions $\mathfrak{A}^{j}_{\xi}$
satisfies

$(1-\sigma_{k})\mathfrak{A}^{j}_{\xi}=\begin{cases}\xi_{X}&j=k,\\\ 0&j\neq
k.\end{cases}$

In particular, we have $\partial_{\mathcal{X}}\mathfrak{A}^{j}_{\xi}\in
Z^{1}_{\mathrm{unif}}(S^{X})^{G}$.

###### Proof.

By Definition 2.13, $\mathfrak{A}^{j}_{\xi}=\sum_{\tau\in
G}n_{j}(\tau)\sum_{x\in\tau(\Lambda_{0})}\xi_{x}$ is a uniform function, hence

$\partial_{\mathcal{X}}\mathfrak{A}^{j}_{\xi}\in
Z^{1}_{\mathrm{unif}}(S^{X}).$

For any $\sigma\in G$, we have

$\displaystyle(1-\sigma)\mathfrak{A}^{j}_{\xi}$ $\displaystyle=\sum_{\tau\in
G}n_{j}(\tau)\xi_{\tau(\Lambda_{0})}-\sum_{\tau\in
G}n_{j}(\tau)\xi_{\sigma\tau(\Lambda_{0})}=\sum_{\tau\in
G}n_{j}(\tau)\xi_{\tau(\Lambda_{0})}-\sum_{\tau\in
G}n_{j}(\sigma^{-1}\tau)\xi_{\tau(\Lambda_{0})}$ $\displaystyle=\sum_{\tau\in
G}n_{j}(\tau)\xi_{\tau(\Lambda_{0})}-\sum_{\tau\in
G}(n_{j}(\tau)-n_{j}(\sigma))\xi_{\tau(\Lambda_{0})}=n_{j}(\sigma)\sum_{\tau\in
G}\xi_{\tau(\Lambda_{0})}=n_{j}(\sigma)\xi_{X}$

for $j=1,\ldots,d$. Since $\partial_{\mathcal{X}}\xi_{X}=0$ by [BKS20]*Lemma
2.20 (see also Theorem 4.8), the compatibility of the group action with
respect to the differential gives
$(1-\sigma)\partial_{\mathcal{X}}\mathfrak{A}^{j}_{\xi}=\partial_{\mathcal{X}}\xi_{X}=0$,
which shows that $\partial_{\mathcal{X}}\mathfrak{A}^{j}_{\xi}\in
Z^{1}_{\mathrm{unif}}(S^{X})^{G}$ as desired. ∎

Denote by

$\mathcal{V}\coloneqq{\mathrm{Span}}_{\mathbb{R}}\\{\mathfrak{A}^{j}_{\xi}\mid\xi\in{\mathrm{Consv}}^{\phi}(S),j=1,\ldots,d\\}$

the $\mathbb{R}$-linear subspace of $C^{0}_{\mathrm{unif}}(S^{X})$ spanned by
$\mathfrak{A}^{j}_{\xi}$ for $\xi\in{\mathrm{Consv}}^{\phi}(S)$ and
$j=1,\ldots,d$. Note that the space $\mathcal{V}$ depends on the choice of the
fundamental domain $\Lambda_{0}$. We let $\mathcal{C}$ and $\mathcal{E}$ be
the spaces of shift-invariant closed and exact uniform forms given by

$\displaystyle\mathcal{C}$ $\displaystyle\coloneqq
Z^{1}_{\mathrm{unif}}(S^{X})^{G},$ $\displaystyle\mathcal{E}$
$\displaystyle\coloneqq\partial_{\mathcal{X}}(C^{0}_{\mathrm{unif}}(S^{X})^{G}).$

######  Lemma 4.2.

Let $\partial_{\mathcal{X}}\mathcal{V}$ be the $\mathbb{R}$-linear space

$\partial_{\mathcal{X}}\mathcal{V}\coloneqq{\mathrm{Span}}_{\mathbb{R}}\\{\partial_{\mathcal{X}}\mathfrak{A}^{j}_{\xi}\mid\xi\in{\mathrm{Consv}}^{\phi}(S),j=1,\ldots,d\\}.$

If we map
$(\zeta^{(1)},\ldots,\zeta^{(d)})\in\bigoplus_{j=1}^{d}{\mathrm{Consv}}^{\phi}(S)$
to
$\mathfrak{A}\coloneqq\sum_{j=1}^{d}\partial_{\mathcal{X}}\mathfrak{A}^{j}_{\zeta^{(j)}}$,
then we have an $\mathbb{R}$-linear isomorphism

$\bigoplus_{j=1}^{d}{\mathrm{Consv}}^{\phi}(S)\cong\partial_{\mathcal{X}}\mathcal{V}.$

In particular, we have
$\dim_{\mathbb{R}}\partial_{\mathcal{X}}\mathcal{V}=c_{\phi}d$ for
$c_{\phi}=\dim_{\mathbb{R}}{\mathrm{Consv}}^{\phi}(S)$.

###### Proof.

The map is $\mathbb{R}$-linear and surjective by the definition of
$\mathfrak{A}^{j}_{\xi}$ and $\partial_{\mathcal{X}}\mathcal{V}$. It is
sufficient to prove that the map is injective. Suppose we have
$\partial_{\mathcal{X}}\mathfrak{A}=0$ for

$\displaystyle\mathfrak{A}=\sum_{j=1}^{d}\mathfrak{A}^{j}_{\zeta^{(j)}},\qquad(\zeta^{(1)},\ldots,\zeta^{(d)})\in\bigoplus_{j=1}^{d}{\mathrm{Consv}}^{\phi}(S).$

By Theorem 2.17, we have
${\mathrm{Consv}}^{\phi}(S)\cong{\mathrm{Ker}\,}\partial_{\mathcal{X}}$, hence
$\mathfrak{A}=\xi_{X}$ for some $\xi\in{\mathrm{Consv}}^{\phi}(S)$. Since
$\xi_{X}(\eta)=\xi_{X}(\sigma(\eta))$ for any $\sigma\in G$, we see that
$\xi_{X}\in C^{0}_{\mathrm{unif}}(S^{X})^{G}$. This shows that

$(1-\sigma_{j})\mathfrak{A}=\zeta^{(j)}_{X}=0,\qquad j=1,\ldots,d,$

where $\zeta^{(j)}_{X}$ is the image of
$\zeta^{(j)}\in{\mathrm{Consv}}^{\phi}(S)$ in $C^{0}_{\mathrm{unif}}(S^{X})$
through the isomorphism
${\mathrm{Consv}}^{\phi}(S)\cong{\mathrm{Ker}\,}\partial_{\mathcal{X}}$ of
Theorem 2.17. This proves that $\zeta^{(j)}=0$ for any $j=1,\ldots,d$, hence
that our map is injective as desired. ∎

We say that an interaction $(S,\phi)$ is simple, if $c_{\phi}=1$, and for any
nonzero conserved quantity $\xi\in{\mathrm{Consv}}^{\phi}(S)$, the monoid
generated by $\xi(S)$ via addition in $\mathbb{R}$ is isomorphic to
$\mathbb{N}$ or $\mathbb{Z}$. Our main theorem is as follows.

######  Theorem 4.3.

Let $\mathcal{X}=(X,E)$ be a crystal lattice for a finitely generated abelian
group $G$. Assume that the interaction $(S,\phi)$ is irreducibly quantified,
and assume in addition that $(S,\phi)$ is simple if $\dim\mathcal{X}=1$. Then
we have a decomposition

$\mathcal{C}\cong\mathcal{E}\oplus\partial_{\mathcal{X}}\mathcal{V}.$

######  Remark 4.4.

In [BKS20]*Theorem 5, we proved the theorem when the locale $\mathcal{X}$ is
weakly transferable or transferable. Unfortunately, a crystal lattice does not
in general satisfy this condition. We modify the method of [BKS20] to prove
Theorem 4.3.

###  4.2. Surjectivity of the Differential

In this subsection, we will prove the following Theorem 4.5 for a general
crystal lattice $\mathcal{X}=(X,E)$. This theorem plays a crucial role in the
proof of our main theorem.

######  Theorem 4.5.

Let $\mathcal{X}=(X,E)$ be a crystal lattice, and let $(S,\phi)$ be an
interaction which is irreducibly quantified. We assume in addition that
$(S,\phi)$ is simple if $\dim\mathcal{X}=1$. Then the differential

$\partial_{\mathcal{X}}\colon C_{\mathrm{unif}}(S^{X})\rightarrow
Z^{1}_{\mathrm{unif}}(S^{X})$

of (9) is surjective.

We will give the proof of Theorem 4.5 at the end of this section. The proof is
obtained by reducing to the essentially Euclidean case. We first prove the
essentially Euclidean case. Assume now that $\mathcal{X}=(X,E)$ is a crystal
lattice which is essentially Euclidean, and let the notations be as in §3.1
and §3.2. The following algebraic lemma will be used in the proof.

######  Lemma 4.6.

Let $\mathcal{M}$ be a commutative monoid satisfying the cancellation
property. In other words, for any $\alpha,\beta,\gamma\in\mathcal{M}$, if
$\alpha+\beta=\alpha+\gamma$, then we have $\beta=\gamma$. Consider a pairing
$H\colon\mathcal{M}\times\mathcal{M}\rightarrow\mathbb{R}$ satisfying the
cocycle condition

$H(\alpha,\beta)+H(\alpha+\beta,\gamma)=H(\beta,\gamma)+H(\alpha,\beta+\gamma).$

Assume either of the following conditions.

  1. (i)

$H$ is _symmetric_. In other words $H(\alpha,\beta)=H(\beta,\alpha)$ for any
$\alpha,\beta\in\mathcal{M}$.

  2. (ii)

$\mathcal{M}\cong\mathbb{N}$ or $\mathcal{M}\cong\mathbb{Z}$ viewed as a
commutative monoid with respect to addition.

Then there exists a map

$h\colon\mathcal{M}\rightarrow\mathbb{R}$

such that $H(\alpha,\beta)=h(\alpha)+h(\beta)-h(\alpha+\beta)$ for any
$\alpha,\beta\in\mathcal{M}$.

###### Proof.

(i) is exactly [BKS20]*Lemma 5.5 and (ii) is exactly [BKS20]*Lemma 5.6. The
idea behind the proof is that the conditions implies that the extension of
monoids

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

defined by the cocycle $H$ is split. See also [BKS20]*Remark 5.7. ∎

If $\dim\mathcal{X}>1$, then we may prove that the pairing $h_{f}$ of (10) is
symmetric as follows:

######  Lemma 4.7.

Assume that $\mathcal{X}$ is essentially Euclidean and $\dim\mathcal{X}>1$.
Then for any $f\in C(S^{X}_{*})$ such that $\partial_{\mathcal{X}}f\in
C^{1}_{\mathrm{unif}}(S^{X})$, the pairing

$h_{f}\colon\mathcal{M}\times\mathcal{M}\rightarrow\mathbb{R}$

of Definition 3.5 is symmetric.

###### Proof.

Let $R>0$ be an integer such that $\partial_{\mathcal{X}}f\in
C^{1}_{R}(S^{X})$. For any integer $k>0$, consider
$\alpha,\beta\in\mathcal{M}_{k}$. If we take $D_{1},D_{2},D_{3}$ such that
$d_{\mathcal{S}}(D_{i},D_{j})>R$ for $i\neq j$, then since
$\dim\mathcal{X}>1$, we see that $(D_{i},D_{j})\in\mathscr{D}^{k}_{R}$ for
$i\neq j$, and

$(D_{1},D_{2})\leftrightarrow(D_{1},D_{3})\leftrightarrow(D_{2},D_{3})\leftrightarrow(D_{2},D_{1}).$

By Lemma 3.4 and (11), we see that

$h^{D_{1},D_{2}}_{f}(\alpha,\beta)=h^{D_{1},D_{3}}_{f}(\alpha,\beta)=h^{D_{2},D_{3}}_{f}(\alpha,\beta)=h^{D_{3},D_{2}}_{f}(\beta,\alpha)=h^{D_{1},D_{2}}_{f}(\beta,\alpha).$

The definition of $h_{f}$ gives

$h_{f}(\alpha,\beta)=h^{D_{1},D_{2}}_{f}(\alpha,\beta)=h^{D_{1},D_{2}}_{f}(\beta,\alpha)=h_{f}(\beta,\alpha)$

as desired. ∎

We may now prove the following theorem, which is Theorem 4.5 for essentially
Euclidean crystal lattices.

######  Theorem 4.8.

Let $\mathcal{X}=(X,E)$ be a crystal lattice which is essentially Euclidean,
and let $(S,\phi)$ be an interaction which is irreducibly quantified. We
assume in addition that $(S,\phi)$ is simple if $\dim\mathcal{X}=1$. Then the
differential

$\partial_{\mathcal{X}}\colon C_{\mathrm{unif}}(S^{X})\rightarrow
Z^{1}_{\mathrm{unif}}(S^{X})$

is surjective.

###### Proof.

Suppose $\omega\in Z_{\mathrm{unif}}(S^{X})\subset Z^{1}(S^{X}_{*})$. Since
$\omega\in Z^{1}(S^{X}_{*})$, by Lemma 2.16, there exists $f\in C(S^{X}_{*})$
such that $\partial_{\mathcal{X}}f=\omega$. Furthermore, since $\omega\in
C^{1}_{\mathrm{unif}}(S^{X})$, there exists $R>0$ such that
$\partial_{\mathcal{X}}f=\omega\in C^{1}_{R}(S^{X})$. Let
$h_{f}\colon\mathcal{M}\times\mathcal{M}\rightarrow\mathbb{R}$ be the pairing
of Definition 3.5 which by Proposition 3.6 satisfying the cocycle condition
(15). Note that if $\dim\mathcal{X}>1$, then by Lemma 4.7, the pairing $h_{f}$
is symmetric. Hence by Lemma 4.6, for both the cases $\dim\mathcal{X}=1$ and
$\dim\mathcal{X}>1$, there exists $h\colon\mathcal{M}\rightarrow\mathbb{R}$
such that

(21) $h_{f}(\alpha,\beta)=h(\alpha)+h(\beta)-h(\alpha+\beta)$

for any $\alpha,\beta\in\mathcal{M}$. We define the function $g\in
C(S^{X}_{*})$ by

$g\coloneqq f+h\circ\boldsymbol{\xi}_{\\!X}.$

We will prove that $g$ is uniform. By Theorem 2.17, for any
$\xi\in{\mathrm{Consv}}^{\phi}(S)$, we have $\partial_{\mathcal{X}}\xi_{X}=0$.
This implies that $\nabla_{\\!e}\xi_{X}=0$, hence
$\xi_{X}(\eta^{e})=\xi_{X}(\eta)$ for any $e\in E$. This shows that
$h\circ\boldsymbol{\xi}_{\\!X}(\eta^{e})=h\circ\boldsymbol{\xi}_{\\!X}(\eta)$
for any $e\in E$, hence $\nabla_{\\!e}(h\circ\boldsymbol{\xi}_{\\!X})=0$,
which implies that $\partial_{\mathcal{X}}(h\circ\boldsymbol{\xi}_{\\!X})=0$.
This gives $\partial_{\mathcal{X}}g=\partial_{\mathcal{X}}f=\omega$.
Furthermore, noting that
$\iota^{D_{1}}(h\circ\boldsymbol{\xi}_{\\!X}(\eta))=h\circ\boldsymbol{\xi}_{\\!D_{1}}(\eta)$
for any $D_{1}\subset X$, we have

$\displaystyle\iota^{D_{1}\cup
D_{2}}g(\eta)-\iota^{D_{1}}g(\eta)-\iota^{D_{2}}g(\eta)$
$\displaystyle=\iota^{D_{1}\cup
D_{2}}f(\eta)-\iota^{D_{1}}f(\eta)-\iota^{D_{2}}f(\eta)$
$\displaystyle\qquad+h\circ\boldsymbol{\xi}_{\\!D_{1}\cup
D_{2}}(\eta)-h\circ\boldsymbol{\xi}_{\\!D_{1}}(\eta)-h\circ\boldsymbol{\xi}_{\\!D_{2}}(\eta)=0$

for any $(D_{1},D_{2})$ such that $d_{\mathcal{S}}(D_{1},D_{2})>R$ and
$\eta\in S^{X}$, where we have used the coboundary condition (21) and the fact
that $\boldsymbol{\xi}_{\\!D_{1}\cup
D_{2}}(\eta)=\boldsymbol{\xi}_{\\!D_{1}}(\eta)+\boldsymbol{\xi}_{\\!D_{2}}(\eta)$.
From the definition of the pairing given in Definition 3.5, we see that
$h_{g}\equiv 0$. Hence by Proposition 3.7, we see that $g\in
C_{\mathrm{unif}}(S^{X})$. Our assertion is now proved by replacing $f$ by
$g$. ∎

We now return to the case of a general crystal lattice $\mathcal{X}=(X,E)$. In
order to reduce the proof of Theorem 4.5 to the essentially Euclidean case, we
first prove the existence of a canonical isomorphism between the spaces of
uniform functions for equivalent locales, as well as between the spaces of
closed uniform forms.

######  Definition 4.9.

Given two locales $\mathcal{X}=(X,E)$ and
$\mathcal{X}^{\prime}=(X^{\prime},E^{\prime})$ we say that $\mathcal{X}$ and
$\mathcal{X}^{\prime}$ are _equivalent_ , if $X=X^{\prime}$ and there exists
constants $C,C^{\prime}\geq 1$ such that

(22) $\displaystyle d_{\mathcal{X}}(x,y)$ $\displaystyle\leq
Cd_{\mathcal{X}^{\prime}}(x,y),$ $\displaystyle\qquad
d_{\mathcal{X}^{\prime}}(x,y)$ $\displaystyle\leq
C^{\prime}d_{\mathcal{X}}(x,y)$

for any $x,y\in X$.

We next prove that the spaces of uniform functions and uniform closed forms
are isomorphic for equivalent locales.

######  Lemma 4.10.

Let $\mathcal{X}=(X,E)$ and $\mathcal{X}^{\prime}=(X^{\prime},E^{\prime})$ be
locales which are equivalent in the sense of Definition 4.9. Assume in
addition that $(S,\phi)$ is an interaction which is irreducibly quantified.
Then there exists canonical isomorphisms

$\displaystyle C^{0}_{\mathrm{unif}}(S^{X})$ $\displaystyle\cong
C^{0}_{\mathrm{unif}}(S^{X^{\prime}}),$ $\displaystyle
Z^{1}_{\mathrm{unif}}(S^{X})$ $\displaystyle\cong
Z^{1}_{\mathrm{unif}}(S^{X^{\prime}})$

which are compatible with the differentials $\partial_{\mathcal{X}}$ and
$\partial_{X^{\prime}}$. In other words, we have a commutative diagram

$\textstyle{C^{0}_{\mathrm{unif}}(S^{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{\mathcal{X}}}$$\scriptstyle{\cong}$$\textstyle{Z^{1}_{\mathrm{unif}}(S^{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{C^{0}_{\mathrm{unif}}(S^{X^{\prime}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{\mathcal{X}^{\prime}}}$$\textstyle{Z^{1}_{\mathrm{unif}}(S^{X^{\prime}}).}$

###### Proof.

Since $\mathcal{X}$ and $\mathcal{X}^{\prime}$ are equivalent, we have
$X=X^{\prime}$ and there exists constants $C,C\geq 1$ such that
$d_{\mathcal{X}}(x,y)\leq Cd_{\mathcal{X}^{\prime}}(x,y)$ and
$d_{\mathcal{X}^{\prime}}(x,y)\leq C^{\prime}d_{\mathcal{X}}(x,y)$ for any
$x,y\in X$. This gives

$\displaystyle{\mathrm{diam}}_{\mathcal{X}}(\Lambda)$ $\displaystyle\leq
C{\mathrm{diam}}_{\mathcal{X}^{\prime}}(\Lambda),$
$\displaystyle{\mathrm{diam}}_{\mathcal{X}^{\prime}}(\Lambda)$
$\displaystyle\leq C^{\prime}{\mathrm{diam}}_{\mathcal{X}}(\Lambda)$

for any $\Lambda\in\mathscr{I}_{X}$, which shows that
$C^{0}_{\mathrm{unif}}(S^{X})=C^{0}_{\mathrm{unif}}(S^{X^{\prime}})$ by
definition of uniform functions.

Next, let $E^{\prime\prime}\coloneqq E\cup E^{\prime}$ be the disjoint union
of $E$ and $E^{\prime}$, and we let $o,t\colon E^{\prime\prime}\rightarrow X$
be the map induced from the origin and target maps on $E$ and $E^{\prime}$.
Then $E^{\prime\prime}$ has a natural inversion $\iota$ induced from the
inversions on $E$ and $E^{\prime}$, and we see that
$\mathcal{X}^{\prime\prime}=(X,E^{\prime\prime},o,t,\iota)$ is a locale. Since
we have natural inclusions $E,E^{\prime}\hookrightarrow E^{\prime\prime}$, we
have

$d_{\mathcal{X}^{\prime\prime}}(x,y)\leq
d_{\mathcal{X}}(x,y),d_{\mathcal{X}^{\prime}}(x,y)$

for any $x,y\in X$. Moreover, by construction, we have

$\displaystyle d_{\mathcal{X}}(x,y)$ $\displaystyle\leq
Cd_{\mathcal{X}^{\prime\prime}}(x,y),$ $\displaystyle
d_{\mathcal{X}^{\prime}}(x,y)$ $\displaystyle\leq
C^{\prime}d_{\mathcal{X}^{\prime\prime}}(x,y)$

for any $x,y\in X$, hence $\mathcal{X}^{\prime\prime}$ is equivalent to both
$\mathcal{X}$ and $\mathcal{X}^{\prime}$. This shows in particular that
$C^{0}_{\mathrm{unif}}(S^{X^{\prime\prime}})=C^{0}_{\mathrm{unif}}(S^{X})=C^{0}_{\mathrm{unif}}(S^{X^{\prime}})$
for $X^{\prime\prime}=X$. The natural morphisms
$\mathcal{X},\mathcal{X}^{\prime}\hookrightarrow\mathcal{X}^{\prime\prime}$ of
multi-graphs induces projections

(23) $\displaystyle C^{1}(S^{X^{\prime\prime}})$ $\displaystyle\rightarrow
C^{1}(S^{X}),$ $\displaystyle C^{1}(S^{X^{\prime\prime}})$
$\displaystyle\rightarrow C^{1}(S^{X^{\prime}})$

given through (7) by the natural projections $\prod_{e\in
E^{\prime\prime}}C(S^{X^{\prime\prime}}_{*})\rightarrow\prod_{e\in
E}C(S^{X}_{*})$ and $\prod_{e\in
E^{\prime\prime}}C(S^{X^{\prime\prime}}_{*})\rightarrow\prod_{e\in
E^{\prime}}C(S^{X^{\prime}}_{*})$ which are compatible with the differentials.

By definition of the differential, the subspaces
${\mathrm{Ker}\,}\partial_{\mathcal{X}},{\mathrm{Ker}\,}\partial_{\mathcal{X}^{\prime}},{\mathrm{Ker}\,}\partial_{\mathcal{X}^{\prime\prime}}$
of $C^{0}(S^{X}_{*})$ coincide with the functions on $S^{X}_{*}$ which are
constant on the connected components of respectively $S^{X}_{*}$,
$S^{X^{\prime}}_{*}$, $S^{X^{\prime\prime}}_{*}$. If we let

$\boldsymbol{\xi}_{\\!X}\colon
S^{X}_{*}\rightarrow{\mathrm{Hom}}_{\mathbb{R}}({\mathrm{Consv}}^{\phi}(S),\mathbb{R}),\qquad\eta\mapsto(\xi\mapsto\xi_{X}(\eta))$

be the map given in (14), then since we have assumed that the interaction
$(S,\phi)$ is irreducibly quantified, the connected components of $S^{X}_{*}$
corresponds bijectively with the elements in
$\mathcal{M}=\boldsymbol{\xi}_{X}(S^{X}_{*})$ (see [BKS20]*Remark 2.33). Since
the definition of the map $\boldsymbol{\xi}_{\\!X}$ depends only on $X$ and is
independent of the edges of the locales $\mathcal{X}$, $\mathcal{X}^{\prime}$,
$\mathcal{X}^{\prime\prime}$, we see that the connected components of
$S^{X}_{*}$, $S^{X^{\prime}}_{*}$, $S^{X^{\prime\prime}}_{*}$ coincide. This
implies that

${\mathrm{Ker}\,}\partial_{\mathcal{X}}={\mathrm{Ker}\,}\partial_{\mathcal{X}^{\prime}}={\mathrm{Ker}\,}\partial_{\mathcal{X}^{\prime\prime}}.$

Using this fact, Lemma 2.16 shows that the projections (23) induce
isomorphisms

$Z^{1}(S^{X^{\prime}}_{*})\cong
C(S^{X^{\prime}}_{*})/{\mathrm{Ker}\,}\partial_{\mathcal{X}^{\prime}}=C(S^{X^{\prime\prime}}_{*})/{\mathrm{Ker}\,}\partial_{\mathcal{X}^{\prime\prime}}=C(S^{X}_{*})/{\mathrm{Ker}\,}\partial_{\mathcal{X}}\cong
Z^{1}(S^{X}_{*}).$

Our assertion for uniform forms follows from the fact that since $\mathcal{X}$
and $\mathcal{X}^{\prime}$ are equivalent, for any $R\geq 0$, there exists
$R^{\prime}\geq 0$ sufficiently large that any ball of radius $R\geq 0$ in
$\mathcal{X}$ is included in a ball with same center of radius $R^{\prime}\geq
0$ in $\mathcal{X}^{\prime}$, and vice versa. ∎

Crystal lattices for the same group $G$ are equivalent.

######  Lemma 4.11.

Let $\mathcal{X}=(X,E)$ and $\mathcal{X}^{\prime}=(X^{\prime},E^{\prime})$ be
crystal lattices for the action of $G$ such that $X^{\prime}=X$. Then
$\mathcal{X}$ and $\mathcal{X}^{\prime}$ are equivalent.

###### Proof.

We choose a representative $\Lambda_{0}$ of the orbits of $X$ with respect to
the action of $G$. Then the set $E_{\Lambda_{0}}\cup\partial E_{\Lambda_{0}}$
gives a representative of the orbits of $E$ for the action of $G$, and the set
$E^{\prime}_{\Lambda_{0}}\cup\partial E^{\prime}_{\Lambda_{0}}$ gives a
representative of the orbits of $E^{\prime}$ for the action of $G$. We let

$\displaystyle C$
$\displaystyle\coloneqq\max\\{d_{\mathcal{X}}(o(e^{\prime}),t(e^{\prime}))\mid
e^{\prime}\in E^{\prime}_{\Lambda_{0}}\cup\partial
E^{\prime}_{\Lambda_{0}}\\},$ $\displaystyle C^{\prime}$
$\displaystyle\coloneqq\max\\{d_{\mathcal{X}^{\prime}}(o(e),t(e))\mid e\in
E_{\Lambda_{0}}\cup\partial E_{\Lambda_{0}}\\}.$

Then for any $x,y\in X$, we have

$\displaystyle d_{\mathcal{X}}(x,y)$ $\displaystyle\leq
Cd_{\mathcal{X}^{\prime}}(x,y),$ $\displaystyle d_{\mathcal{X}^{\prime}}(x,y)$
$\displaystyle\leq C^{\prime}d_{\mathcal{X}}(x,y),$

which proves that $\mathcal{X}$ and $\mathcal{X}^{\prime}$ are equivalent. ∎

We now show that any crystal lattice is equivalent to a crystal lattice which
is essentially Euclidean.

######  Lemma 4.12.

Let $\mathcal{X}=(X,E)$ be a crystal lattice for the action of a group $G$,
Then if we let $F\subset G$ be the free part of $G$, then there exists a
crystal lattice $\mathcal{X}^{\prime}=(X^{\prime},E^{\prime})$ for the action
of $F$ which is essentially Euclidean.

###### Proof.

We let $X^{\prime}=X$. Note that the action of $F$ on $\mathcal{X}$ induces an
action of $F$ on $X$ and $X\times X$. We we fix a fundamental domain
$\Lambda_{F}$ of the action of $F$ on $X$. We fix a free generator
$\mathcal{S}^{+}=\\{\sigma_{1},\ldots,\sigma_{d}\\}$ of $F$, and we let
$\mathcal{S}\coloneqq\\{\sigma_{1}^{\pm},\ldots,\sigma_{d}^{\pm}\\}$. We let
$E^{\prime}_{\Lambda_{F}}\coloneqq\\{e=(x,y)\in\Lambda_{F}\times\Lambda_{F}\mid
x\neq y\\}$ so that $(\Lambda_{F},E^{\prime}_{\Lambda_{F}})$ is a complete
graph, and we let

$\partial E^{\prime}_{\Lambda_{F}}\coloneqq\\{e=(x,y)\in X\times X\mid
x\in\Lambda_{F},\exists\sigma\in\mathcal{S},\,y\in\sigma(\Lambda_{F})\\}.$

Consider the disjoint union

$E^{\prime}\coloneqq\bigcup_{\tau\in
F}\tau\bigl{(}E^{\prime}_{\Lambda_{F}}\cup\partial
E^{\prime}_{\Lambda_{F}}\bigr{)}\subset X\times X$

with natural map $o,t\colon E^{\prime}\rightarrow X$ given by the first and
second projections. Then by construction,

$\mathcal{X}^{\prime}=(X,E^{\prime})$

is a graph which is a crystal lattice for the action of $F$. Moreover, by
construction, $\mathcal{X}^{\prime}$ satisfies (13) for the coordinate
$(\Lambda_{F},\mathcal{S}^{+})$, hence is essentially Euclidean. By Lemma
4.11, $\mathcal{X}$ is equivalent to $\mathcal{X}^{\prime}$ since both
$\mathcal{X}$ and $\mathcal{X}^{\prime}$ are crystal lattices for the action
of $F$. This proves our assertion. ∎

We are now ready to prove Theorem 4.5.

###### Proof of Theorem 4.5.

By Lemma 4.12, there exists a crystal lattice
$\mathcal{X}^{\prime}=(X^{\prime},E^{\prime})$ which is essentially Euclidean
and equivalent to $\mathcal{X}$. Since $\mathcal{X}^{\prime}$ is essentially
Euclidean, by Theorem 4.8, the differential $\partial_{\mathcal{X}^{\prime}}$
of $\mathcal{X}^{\prime}$ is surjective. Our assertion now follows from Lemma
4.10. ∎

###  4.3. Proof of the Main Theorem

We are now ready prove our main theorem, Theorem 4.3. Let the notations be as
in §4.1. In particular, let $\mathcal{X}=(X,E)$ be a general crystal lattice
for the action of a finite generated abelian group $G$, and let $(S,\phi)$ be
an interaction which is irreducibly quantified. We assume in addition that
$(S,\phi)$ is simple if $\dim\mathcal{X}=1$.

######  Lemma 4.13.

We have
$\mathcal{E}\oplus\partial_{\mathcal{X}}\mathcal{V}\subset\mathcal{C}$.

###### Proof.

Let $\omega\in\mathcal{E}\cap\partial_{\mathcal{X}}\mathcal{V}$. Since
$\omega\in\mathcal{E}$, there exists $f\in C^{0}_{\mathrm{unif}}(S^{X})^{G}$
such that $\partial_{\mathcal{X}}f=\omega$. Moreover, since
$\omega\in\partial_{\mathcal{X}}\mathcal{V}$, by Lemma 4.2, there exists
$\zeta^{(1)},\ldots,\zeta^{(d)}\in{\mathrm{Consv}}^{\phi}(S)$ such that
$\partial_{\mathcal{X}}f=\partial_{\mathcal{X}}\mathfrak{A}$ for
$\mathfrak{A}=\sum_{j=1}^{d}\mathfrak{A}^{j}_{\zeta^{(j)}}\in\mathcal{V}$.
Since $\partial_{\mathcal{X}}(f-\mathfrak{A})=0$, by Theorem 2.17, we have
$f-\mathfrak{A}=\xi_{X}$ for some $\xi\in{\mathrm{Consv}}^{\phi}(S)$. We have
$\xi_{X}\in C^{0}_{\mathrm{unif}}(S^{X})^{G}$ since $\xi_{X}$ is invariant
with respect to the action of $G$. Noting that $f$ is also invariant with
respect to the action of $G$, we have

$\zeta^{(j)}_{X}=(1-\sigma_{j})\mathfrak{A}=(\sigma_{j}-1)(f-\mathfrak{A})=(\sigma_{j}-1)\xi_{X}=0,$

which implies that $\zeta^{(j)}=0$ for $j=1,\ldots,d$, hence that
$\omega=\partial_{\mathcal{X}}\mathfrak{A}=0$ as desired. ∎

We now prove Theorem 4.3.

###### Proof of Theorem 4.3.

Let $\omega\in\mathcal{C}=Z^{1}_{\mathrm{unif}}(S^{X})^{G}$ be a shift-
invariant closed uniform form. By Theorem 4.5, there exists $f\in
C^{0}_{\mathrm{unif}}(S^{X})$ such that $\partial_{\mathcal{X}}f=\omega$. For
any $j=1,\ldots,d$, we have

$\partial_{\mathcal{X}}((1-\sigma_{j})f)=(1-\sigma_{j})\partial_{\mathcal{X}}f=(1-\sigma_{j})\omega=0$

since $\omega$ is a shift-invariant form. By Theorem 2.17, we have

$(1-\sigma_{j})f\in{\mathrm{Ker}\,}\partial_{\mathcal{X}}\cong{\mathrm{Consv}}^{\phi}(S),$

hence $(1-\sigma_{j})f=\zeta^{(j)}_{X}$ for some
$\zeta^{(j)}\in{\mathrm{Consv}}^{\phi}(S)$. If we let

$g\coloneqq f-\sum_{j=1}^{d}\mathfrak{A}^{j}_{\zeta^{(j)}}\in
C^{0}_{\mathrm{unif}}(S^{X}),$

then by Lemma 4.1, we have

$(1-\sigma_{k})g=(1-\sigma_{k})f-\sum_{j=1}^{d}(1-\sigma_{k})\mathfrak{A}^{j}_{\zeta^{(j)}}=\zeta^{(k)}_{X}-\zeta^{(k)}_{X}=0$

for any $k=1,\ldots,d$. This shows that $g\in
C^{0}_{\mathrm{unif}}(S^{X})^{G}$, hence we have

$\omega=\partial_{\mathcal{X}}f=\partial_{\mathcal{X}}g+\sum_{j=1}^{d}\mathfrak{A}^{j}_{\zeta^{(j)}}\in\mathcal{E}\oplus\partial_{\mathcal{X}}\mathcal{V}.$

This proves our assertion. ∎

######  Remark 4.14.

By Theorem 2.17 and Theorem 4.5, for any $\mathcal{X}=(X,E)$ and $(S,\phi)$
satisfying the conditions of Theorem 4.5, we have an exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathrm{Consv}}^{\phi}(S)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{0}_{\mathrm{unif}}(S^{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{\mathcal{X}}}$$\textstyle{Z^{1}_{\mathrm{unif}}(S^{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

One may prove that the long exact sequence associated to the group cohomology
of $G$ gives an exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathrm{Consv}}^{\phi}(S)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{0}_{\mathrm{unif}}(S^{X})^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{\mathcal{X}}}$$\textstyle{Z^{1}_{\mathrm{unif}}(S^{X})^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{H^{1}(G,{\mathrm{Consv}}^{\phi}(S))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Since $G$ acts trivially on ${\mathrm{Consv}}^{\phi}(S)$, we have

$H^{1}(G,{\mathrm{Consv}}^{\phi}(S))\cong{\mathrm{Hom}}(G,{\mathrm{Consv}}^{\phi}(S))\cong\bigoplus_{j=1}^{d}{\mathrm{Consv}}^{\phi}(S).$

The isomorphism of Lemma 4.2

$\bigoplus_{j=1}^{d}{\mathrm{Consv}}^{\phi}(S)\xrightarrow{\cong}\partial_{\mathcal{X}}\mathcal{V}\subset
Z^{1}_{\mathrm{unif}}(S^{X})^{G}$

gives a section of $\delta$ with image $\partial_{\mathcal{X}}\mathcal{V}$
giving the decomposition
$\mathcal{C}=\mathcal{E}\oplus\partial_{\mathcal{X}}\mathcal{V}$ of Theorem
4.3.

## References

  * Bibliography