1 Introduction

The quantum double models were introduced by Kitaev in [6]. They have substantial importance as the first examples of topological phases. An astonishing property of these models is the existence of quasi-particles, called anyons. Such anyons can be mathematically formulated as superselection sectors [3, 5, 7,8,9]. In this sense, the mathematical structures of topological phases look quite similar to that of algebraic quantum field theory (AQFT) [1, 2, 4]. In spite of such similarity, there is some significant difference. In this paper, we consider the local von Neumann algebra on convex cone areas of the frustration-free ground state of abelian quantum double models. We show they are of type \(II_\infty \). This is in contrast to the situation of AQFT, where all the local algebras are of type III.

Let us introduce the quantum double model. See [5] for a more detailed description. Let G be a finite group. We denote by \(\hat{\mathbb {Z}}^2\) the set of all edges of \({\mathbb {Z}}^2\). All the horizontal edges have an orientation from left to right, and all the vertical edges have an orientation from down to up. We denote by \(v_{e,-1}\), \(v_{e,+1}\) the vertices corresponding to the origin and target of the edge e, with respect to this orientation.

By a path of edges, we mean a sequence of edges associated with direction

$$\begin{aligned} {{\mathfrak {p}}}=(e_1,{\sigma _1})(e_2,{\sigma _2})\cdots (e_m,\sigma _m) \end{aligned}$$

such that

$$\begin{aligned} v_{e_{i}, \sigma _i}=v_{e_{i+1}, -\sigma _{i+1}},\quad i=1,\ldots , m-1. \end{aligned}$$

Here, \(e_i\in {\hat{{\mathbb {Z}}}}^2\) and \(\sigma _i=+1\) (resp. \(\sigma _i=-1\)) if, along the path, we proceed in the direction (resp. opposite direction) of the given orientation of \(e_i\). We call \(v_{e_1,-\sigma _1}\) the origin and \(v_{e_m,\sigma _m}\) the target of \({{\mathfrak {p}}}\). The path \({{\mathfrak {p}}}\) is self-avoiding if all of \(v_{e_i,\pm 1}\) are different. It is a loop if \(v_{e_1,-\sigma _1}=v_{e_m,\sigma _m}\).

For a path \({{\mathfrak {p}}}\), the symbol \({{\mathfrak {p}}}^{-1}\) represents the inverse path of \({{\mathfrak {p}}}\). If the target of \({{\mathfrak {p}}}_1\) is the same as the origin of \({{\mathfrak {p}}}_2\), then \({{\mathfrak {p}}}_1{{\mathfrak {p}}}_2\) indicates the path obtained by connecting them.

For \(\Lambda \subset {\mathbb {Z}}^2\), let \({\mathcal {P}}_\Lambda \) be the set of squares included in \(\Lambda \). We set \({\tilde{\Lambda }}:=\cup _{p\in {\mathcal {P}}_\Lambda }\{ \text {edges in } p\}\), the set of all edges forming a part of the squares in \(\Lambda \). The boundary of \(\Lambda \subset {\mathbb {Z}}^2\) is

$$\begin{aligned} \partial \Lambda :=&\Big \{ v\in \Lambda \mid \text {there exists } \;\;e\in {\hat{{\mathbb {Z}}}}^2 \;\;\text {such that}\;\; \left( v_{e,-1}=v,\text {and}\;v_{e,+1}\in \Lambda ^c \right) \;\;\nonumber \\&\text {or}\;\; \left( v_{e,+1}=v,\text {and}\;v_{e,-1}\in \Lambda ^c\right) \Big \}. \end{aligned}$$
(1)

For all \(S\subset {\mathbb {Z}}^2\), \(T\subset {\hat{{\mathbb {Z}}}}^2\),

$$\begin{aligned} G^{S}:=\{{\varvec{g}}=(g_v)_{v\in S}\mid g_v\in G\}, \quad G^{T}:=\{{\varvec{g}}=(g_e)_{e\in T}\mid g_e\in G\} \end{aligned}$$

denote the set of all the configurations on S, T, respectively. If \(S_1\subset S_2\) and \({\varvec{h}}\in G^{S_2}\), we set \({\varvec{h}}_{S_1}:={\varvec{h}}\vert _{S_1}\).

For each path \({{\mathfrak {p}}}=(e_1,{\sigma _1})(e_2,{\sigma _2})\cdots (e_m,\sigma _m)\) where \(e_i\in {\hat{{\mathbb {Z}}}}^2\) with its direction \(\sigma _i=\pm 1\) and configuration on \({{\mathfrak {p}}}\), \({\varvec{h}}_{{{\mathfrak {p}}}}=(h_e)_{e\in {{\mathfrak {p}}}}\in G^{{\mathfrak {p}}}\), we set

$$\begin{aligned} \Psi _{{\mathfrak {p}}}({\varvec{h}}_{{\mathfrak {p}}})=h_{e_1}^{\sigma _1}h_{e_2}^{\sigma _2}\cdots h_{e_m}^{\sigma _m}. \end{aligned}$$
(2)

For each \((a,b)\in {\mathbb {Z}}^2\), we denote by \(S_{(a,b)}\) the square given by four points \((a,b)-(a+1,b)-(a+1,b+1)-(a,b+1)\). We denote by \({{\mathfrak {p}}}_{S_{(a,b)}}\) the path of edges obtained by going around \(S_{(a,b)}\) in the order \((a,b)-(a+1,b)-(a+1,b+1)-(a,b+1)\). We say a configuration \({\varvec{g}}_{{\hat{\Lambda }}}\in G^{{\hat{\Lambda }}}\) on \({\hat{\Lambda }}\subset {\hat{{\mathbb {Z}}}}^2\) is admissible if for any unit square \(S_{(a,b)}=(a,b)-(a+1,b)-(a+1,b+1)-(a,b+1)\) formed by 4-edges in \({\hat{\Lambda }}\), we have

$$\begin{aligned} g_{(a,b)-(a+1,b)}g_{(a+1,b)-(a+1,b+1)}=g_{(a,b)-(a,b+1)}g_{(a,b+1)-(a+1,b+1)}. \end{aligned}$$

The set of all admissible configurations on \({\hat{\Lambda }}\subset {\hat{{\mathbb {Z}}}}^2\) is denoted by \(C_{{\hat{\Lambda }}}\).

For \({\hat{S}}\subset {\hat{{\mathbb {Z}}}}^2\), set \({\mathcal {B}}_{{\hat{S}}}:=\bigotimes _{{\hat{S}}} B(l^2(G))\). Here, \(B(l^2(G))\) denotes the set of all bounded linear operators on Hilbert space \(l^2(G)\) with CONS \(\{\left| g \right\rangle \mid g\in G\}\). For finite \({\hat{S}}\subset {\hat{{\mathbb {Z}}}}^2\) and \({\varvec{k}}\in G^{{\hat{S}}}\), we set \(\left| {\varvec{k}} \right\rangle :=\otimes _{e\in {\hat{S}}}\left| k_e \right\rangle \). The algebra, \({\mathcal {B}}:={\mathcal {B}}_{{\hat{{\mathbb {Z}}}}^2}\), is our 2-dimensional quantum spin system. For \(S_1\subset S_2\), \({\mathcal {B}}_{S_1}\) can be regarded as a sub-\(C^*\)-algebra of \({\mathcal {B}}_{S_2}\) naturally. The left regular representation of G on \(l^2(G)\) is denoted by \(L_g\), and the right regular representation of G on \(l^2(G)\) is denoted by \(R_g\):

$$\begin{aligned} \begin{aligned} L_g\left| h \right\rangle =\left| gh \right\rangle ,\quad R_g\left| h \right\rangle =\left| h g^{-1} \right\rangle ,\quad g,h \in G. \end{aligned} \end{aligned}$$
(3)

For each vertex \(v\in {\mathbb {Z}}^2\), we set

$$\begin{aligned} A_v^{(g)}:= \bigotimes _{e\in {\hat{{\mathbb {Z}}}}^2: v_{e,-1}=v}L_{g}^{e} \bigotimes _{e\in {\hat{{\mathbb {Z}}}}^2: v_{e,+1}=v}R_{g^{-1}}^{e}. \end{aligned}$$
(4)

We then set

$$\begin{aligned} A_v:=\frac{1}{|G|}\sum _{g\in G}A_v^{(g)}. \end{aligned}$$
(5)

For each unit square (plaquette) p, we set

$$\begin{aligned} B_{p}:=\sum _{{\varvec{k}}\in C_{p}}\left| {\varvec{k}} \right\rangle \left\langle {\varvec{k}} \right| . \end{aligned}$$
(6)

From [5], we know the following:

Proposition 1.1

(Proposition 2.1 [5]). There exists a unique state \(\omega _0\) on \({\mathcal {B}}\) which satisfies \(\omega _0(A_v)=\omega _0(B_p)=1\) for all vertices v and squares p in \({\mathbb {Z}}^2\).

Now, we are ready to state our main theorem.

Theorem 1.2

Suppose G is abelian. Let \(({\mathcal {H}}_0,\pi _0,\Omega _0)\) be the GNS triple of \(\omega _0\). Let \(\Gamma \subset {\mathbb {R}}^2\) be a convex cone. Then, the local von Neumann algebra \(\pi _0\left( {\mathcal {B}}_{\widetilde{\left( \Gamma \cap {\mathbb {Z}}^2\right) }}\right) ''\) is a type \(II_\infty \) factor.

From the general theory in Lemma 5.5 [9], with the existence of nontrivial superselection sectors and the Haag duality [5], we know that \(\pi _0\left( {\mathcal {B}}_{\widetilde{\left( \Gamma \cap {\mathbb {Z}}^2\right) }}\right) ''\) is either type \(II_\infty \) or type III factor. The question here is which of these occurs.

Theorem is restricted to abelian groups G. This is because we use the result from [5], the Haag duality and the existence of nontrivial superselection sectors. All other parts of the proof work for non-abelian cases. Therefore, if the Haag duality and existence of nontrivial superselection sectors are proven for non-abelian quantum double models, our proof immediately shows that the local algebras for non-abelian quantum double models are type \(II_\infty \) factors.

The strategy of the proof is as follows. We first show that the state \(\omega _0\) has an explicit representation of its restrictions to finite rectangles (Lemma 2.5). For each rectangle, the support of the reduced density matrices is in the subspace with admissible configurations, with restrictions on the boundary configuration (Lemma 2.1). We show that the reduced density matrix is uniformly distributed on its support, when we consider the intersection of the rectangles with cone-shaped areas, by a combinatorial argument. In order to do so, one needs to figure out how the restrictions on the boundary configurations behave. If it is only half-plane, that is easy, but general cones require much more arguments. This is done in Sects. 3, 4, and 5. Out of that, in Sect. 6, we derive a nonzero finite projection as a limit of the support of the reduced density matrices.

2 The Frustration-Free Ground State \(\omega _0\) of the Quantum Double Model

In this section, we use the notation from Sect. 1, but we do not assume G to be abelian. For each \(N\in {\mathbb {N}}\) and \(n_0,m_0\in {\mathbb {N}}\), let \(\Lambda _N^{(n_0,m_0)}:=( [-Nn_0,Nn_0]\times [-Nm_0,Nm_0])\cap {\mathbb {Z}}^2\) and \({\hat{\Lambda }}_N^{(n_0,m_0)}\) the set of all edges in \(\Lambda _N^{(n_0,m_0)}\). Let \(\partial {\hat{\Lambda }}_N^{(n_0,m_0)}\) be the set of edges forming the boundary lines of \(\Lambda _N^{(n_0,m_0)}=([-Nn_0,Nn_0]\times [-Nm_0,Nm_0])\cap {\mathbb {Z}}^2\). In particular, we set \(\Lambda _N:=\Lambda _N^{(1,1)}\) and \({\hat{\Lambda }}_{N}:={\hat{\Lambda }}_N^{(1,1)}\), \(\partial {\hat{\Lambda }}_N:=\partial {\hat{\Lambda }}_N^{(1,1)}\). In this section, we derive a concrete expression of the restriction of \(\omega _0\) onto \({\mathcal {B}}_{{\hat{\Lambda }}_N}\).

First we construct a state \(\varphi _0\) by some concrete formula and show that this \(\varphi _0\) satisfies the condition \(\varphi _0(A_v)=\varphi _0(B_p)=1\) for all vertex v and squares p in \({\mathbb {Z}}^2\). From the uniqueness in Proposition 1.1, this means \(\varphi _0=\omega _0\). For an event \({\mathcal {S}}\), we set \(\chi ({\mathcal {S}})\) to be 1 if \({\mathcal {S}}\) holds and 0 if \({\mathcal {S}}\) does not hold. For a finite set T, \(\left| T\right| \) denotes the cardinality of T.

Lemma 2.1

There is a state \(\varphi _0\) on \({\mathcal {B}}_{{\hat{{\mathbb {Z}}}}^2}\) such that

$$\begin{aligned} \begin{aligned} \varphi _0\left( \left| {\varvec{h}} \right\rangle _{{\hat{\Lambda }}_N}\left\langle {\varvec{k}} \right| \right) =\frac{1}{|C_{{\hat{\Lambda }}_N}|} \chi \left( {{\varvec{k}}, {\varvec{h}}\in C_{{\hat{\Lambda }}_N}}\right) \chi \left( {\varvec{k}}_{\partial {{\hat{\Lambda }}_{N}}}={\varvec{h}}_{\partial {{\hat{\Lambda }}_{N}}}\right) , \end{aligned} \end{aligned}$$
(7)

for all \(N\in {\mathbb {N}}\), \({\varvec{k}}, {\varvec{h}}\in G^{{\hat{\Lambda }}_N}\).

Proof

For each \(N\in {\mathbb {N}}\) and \({\varvec{a}}\in G^{\partial {\hat{\Lambda }}_N}\), set

$$\begin{aligned} {\mathcal {I}}_{{\varvec{a}}}^{(N)}:=\left\{ {\varvec{k}} \in C_{{\hat{\Lambda }}_{N}}\mid {\varvec{k}}_{\partial {\hat{\Lambda }}_N}={\varvec{a}}\right\} . \end{aligned}$$
(8)

We denote by \({\mathbb A}_{{\hat{\Lambda }}_N}\) the set of all \({\varvec{a}}\in G^{\partial {\hat{\Lambda }}_N}\) such that \({\mathcal {I}}_{{\varvec{a}}}^{(N)}\ne \emptyset \). For \({\varvec{a}}\in {\mathbb A}_{{\hat{\Lambda }}_N}\), we set

$$\begin{aligned} \psi _{{\varvec{a}}}:= \sum _{{\varvec{k}}\in {\mathcal {I}}_{{\varvec{a}}}^{(N)}} \left| {\varvec{k}} \right\rangle _{{\hat{\Lambda }}_{N}},\quad \quad {\hat{\psi }}_{{\varvec{a}}}:=\frac{1}{|{\mathcal {I}}_{{\varvec{a}}}^{(N)}|^{\frac{1}{2}}}\psi _{{\varvec{a}}}. \end{aligned}$$
(9)

Then,

$$\begin{aligned} \begin{aligned} \varphi _{{\hat{\Lambda }}_{N}}(B):=\frac{1}{|C_{{\hat{\Lambda }}_{N}}|} \sum _{{\varvec{a}}\in {\mathbb A}_{{\hat{\Lambda }}_N}} \left\langle \psi _{{\varvec{a}}}, B\psi _{{\varvec{a}}}\right\rangle ,\quad B\in {\mathcal {B}}_{{\hat{\Lambda }}_{N}} \end{aligned} \end{aligned}$$
(10)

defines a state on \({\mathcal {B}}_{{\hat{\Lambda }}_{N}}\). For \({\varvec{k}}, {\varvec{h}}\in G^{{\hat{\Lambda }}_N}\), we have

$$\begin{aligned} \begin{aligned} \varphi _{{\hat{\Lambda }}_{N}}\left( \left| {\varvec{h}} \right\rangle _{{\hat{\Lambda }}_N}\left\langle {\varvec{k}} \right| \right) =\frac{1}{|C_{{\hat{\Lambda }}_N}|} \chi \left( {{\varvec{k}}, {\varvec{h}}\in C_{{\hat{\Lambda }}_N}}\right) \chi \left( {\varvec{k}}_{\partial {{\hat{\Lambda }}_{N}}}={\varvec{h}}_{\partial {{\hat{\Lambda }}_{N}} }\right) . \end{aligned} \end{aligned}$$
(11)

We have

$$\begin{aligned} \varphi _{{\hat{\Lambda }}_{N+1}}\vert _{{\mathcal {B}}_{{\hat{\Lambda }}_{N}}}=\varphi _{{\hat{\Lambda }}_{N}}. \end{aligned}$$
(12)

In order to show this, note that for each \({\varvec{k}} \in C_{{\hat{\Lambda }}_{N}}\), there are \(|G|^{4(2N+2)}\)-number of \({\varvec{m}}\in G^{{\hat{\Lambda }}_{N+1}{\setminus } {\hat{\Lambda }}_{N}}\) such that

$$\begin{aligned} \begin{aligned} \left( {\varvec{k}},{\varvec{m}}\right) \in C_{{\hat{\Lambda }}_{N+1}}. \end{aligned} \end{aligned}$$
(13)

In fact, fix \({\varvec{k}} \in C_{{\hat{\Lambda }}_{N}}\). We freely fix \({\varvec{m}}_i\in G^{L_i}\), \(i=1,2,3,4\) with

$$\begin{aligned} \begin{aligned}&L_1:=\text {edges in}\;\; \left( \left( [-N,N+1]\times \{N+1\}\right) \cup \left( \{-N\}\times [N,N+1]\right) \right) ,\\&L_2:=\text {edges in}\;\; \left( \left( \{N+1\}\times [-N-1,N]\right) \cup \left( [N,N+1]\times \{N\}\right) \right) \\&L_3:=\text {edges in}\; \;\left( \left( [-N-1,N]\times \{-N-1\}\right) \cup \left( \{N\}\times [-N-1,-N]\right) ,\right) \\&L_4:=\text {edges in}\;\; \left( \left( \{-N-1\}\times [-N,N+1]\right) \cup [-N-1, -N]\times \{-N\}\right) . \end{aligned} \end{aligned}$$
(14)

Then, the configuration of all the other edges is determined uniquely so that all the admissibility conditions in \({\hat{\Lambda }}_{N+1}\) are satisfied. Hence, the possible number of \({\varvec{m}}\in G^{{\hat{\Lambda }}_{N+1}{\setminus } {\hat{\Lambda }}_{N}}\) satisfying (13) for a given \({\varvec{k}}\in C_{{\hat{\Lambda }}_{N}}\) is

$$\begin{aligned} \begin{aligned} |G|^{|L_1|+|L_2|+|L_3|+|L_4|} =|G|^{4(2N+2)}. \end{aligned} \end{aligned}$$
(15)

In particular, we have

$$\begin{aligned} |G|^{4(2N+2)}|C_{{\hat{\Lambda }}_{N}}|=|C_{{\hat{\Lambda }}_{N+1}}|. \end{aligned}$$
(16)

Hence from (11) for \(N+1\), for any \({\varvec{h}}, {\varvec{k}}\in G^{{\hat{\Lambda }}_N}\) we have

$$\begin{aligned} \begin{aligned}&\varphi _{{\hat{\Lambda }}_{N+1}}\left( \left| {\varvec{h}} \right\rangle _{{\hat{\Lambda }}_N}\left\langle {\varvec{k}} \right| \right) \\&=\sum _{{\varvec{m}}\in G^{{\hat{\Lambda }}_{N+1}\setminus {\hat{\Lambda }}_{N}}}\varphi _{{\hat{\Lambda }}_{N+1}}\left( \left| {\varvec{h}},{\varvec{m}} \right\rangle _{{\hat{\Lambda }}_{N+1}}\left\langle {\varvec{k}},{\varvec{m}} \right| \right) \\&=\sum _{{\varvec{m}}\in G^{{\hat{\Lambda }}_{N+1}\setminus {\hat{\Lambda }}_{N}}}\frac{1}{|C_{{\hat{\Lambda }}_{N+1}}|} \chi \left( {\left( {\varvec{k}},{\varvec{m}}\right) , \left( {\varvec{h}},{\varvec{m}}\right) \in C_{{\hat{\Lambda }}_{N+1}}}\right) \\&\quad \chi \left( \left( {\varvec{k}},{\varvec{m}}\right) _{\partial {{\hat{\Lambda }}_{N+1}} } = \left( {\varvec{h}},{\varvec{m}}\right) _{\partial {{\hat{\Lambda }}_{N+1}}}\right) \\&=\sum _{{\varvec{m}}\in G^{{\hat{\Lambda }}_{N+1}\setminus {\hat{\Lambda }}_{N}}}\frac{1}{|C_{{\hat{\Lambda }}_{N+1}}|} \chi \left( {\left( {\varvec{k}},{\varvec{m}}\right) , \left( {\varvec{h}},{\varvec{m}}\right) \in C_{{\hat{\Lambda }}_{N+1}}}\right) \\&= \chi \left( { {\varvec{k}}, {\varvec{h}}\in C_{{\hat{\Lambda }}_{N}}}\right) \chi \left( {\varvec{k}}_{\partial {{\hat{\Lambda }}_{N}}} ={\varvec{h}}_{\partial {{\hat{\Lambda }}_{N}}} \right) \sum _{{\varvec{m}}\in G^{{\hat{\Lambda }}_{N+1}\setminus {\hat{\Lambda }}_{N}}}\frac{1}{|C_{{\hat{\Lambda }}_{N+1}}|}\chi \left( { \left( {\varvec{h}},{\varvec{m}}\right) \in C_{{\hat{\Lambda }}_{N+1}}}\right) \\&=\frac{1}{|C_{{\hat{\Lambda }}_{N+1}}|}\chi \left( { {\varvec{k}}, {\varvec{h}}\in C_{{\hat{\Lambda }}_{N}}}\right) \chi \left( {\varvec{k}}_{\partial {{\hat{\Lambda }}_{N}}} = {\varvec{h}}_{\partial {{\hat{\Lambda }}_{N}}} \right) \left| \left\{ {\varvec{m}}\mid \left( {\varvec{h}},{\varvec{m}}\right) \in C_{{\hat{\Lambda }}_{N+1}} \right\} \right| \\&=\frac{1}{|C_{{\hat{\Lambda }}_{N}}|}\chi \left( { {\varvec{k}}, {\varvec{h}}\in C_{{\hat{\Lambda }}_{N}}}\right) \chi \left( {\varvec{k}}_{\partial {{\hat{\Lambda }}_{N}}} = {\varvec{h}}_{\partial {{\hat{\Lambda }}_{N}}} \right) \\&=\varphi _{{\hat{\Lambda }}_{N}}\left( \left| {\varvec{h}} \right\rangle _{{\hat{\Lambda }}_N}\left\langle {\varvec{k}} \right| \right) . \end{aligned} \end{aligned}$$
(17)

In the third equality, we noted \(\left( {\varvec{k}},{\varvec{m}}\right) _{\partial {{\hat{\Lambda }}_{N+1}} }={\varvec{m}}_{\partial {{\hat{\Lambda }}_{N+1}} } = \left( {\varvec{h}},{\varvec{m}}\right) _{\partial {{\hat{\Lambda }}_{N+1}}}\). In the fourth equality, we noted the fact that \(\left( {\varvec{k}},{\varvec{m}}\right) , \left( {\varvec{h}},{\varvec{m}}\right) \in C_{{\hat{\Lambda }}_{N+1}}\) implies \({\varvec{k}}_{\partial {{\hat{\Lambda }}_{N}}} ={\varvec{h}}_{\partial {{\hat{\Lambda }}_{N}}}\), because the labels of all the edges of \({\varvec{k}}\) and \({\varvec{h}}\) on \({\partial {{\hat{\Lambda }}_{N}}}\) are determined by \({\varvec{m}}\) uniquely by the admissibility condition. Furthermore, if \(\left( {\varvec{h}},{\varvec{m}}\right) \in C_{{\hat{\Lambda }}_{N+1}}\), \({\varvec{k}}_{\partial {{\hat{\Lambda }}_{N}}} ={\varvec{h}}_{\partial {{\hat{\Lambda }}_{N}}}\), \( {\varvec{k}}, {\varvec{h}}\in C_{{\hat{\Lambda }}_{N+1}}\), then \(\left( {\varvec{k}},{\varvec{m}}\right) \in C_{{\hat{\Lambda }}_{N}}\) holds because all the admissibility conditions of \(\left( {\varvec{k}},{\varvec{m}}\right) \in C_{{\hat{\Lambda }}_{N+1}}\) which are not included in \({\varvec{k}}\in C_{{\hat{\Lambda }}_{N}}\) are the ones including only \({\varvec{k}}_{\partial {{\hat{\Lambda }}_{N}}} ={\varvec{h}}_{\partial {{\hat{\Lambda }}_{N}}}\) and \({\varvec{m}}\). In the sixth equality, we used the fact \(\left| \left\{ {\varvec{m}}\mid \left( {\varvec{h}},{\varvec{m}}\right) \in C_{{\hat{\Lambda }}_{N+1}} \right\} \right| =|G|^{4(2N+2)}= \frac{|C_{{\hat{\Lambda }}_{N+1}}|}{|C_{{\hat{\Lambda }}_{N}}|} \), observed above.

This proves the claim (12). Hence, the consistency condition holds and we can extend the states \(\varphi _{{\hat{\Lambda }}_{N}}\) to a state on \({\mathcal {B}}\), obtaining the desired state \(\varphi _0\). \(\square \)

We will need the following version later.

Lemma 2.2

For all \(N\in {\mathbb {N}}\), \({\varvec{k}}, {\varvec{h}}\in G^{{\hat{\Lambda }}_N^{(n_0,m_0)}}\),

$$\begin{aligned} \varphi _0\left( \left| {\varvec{h}} \right\rangle _{{\hat{\Lambda }}_N^{(n_0,m_0)}}\left\langle {\varvec{k}} \right| \right)&=\frac{1}{|C_{{\hat{\Lambda }}_N^{(n_0,m_0)}}|} \chi \left( {{\varvec{k}}, {\varvec{h}}\in C_{{\hat{\Lambda }}_N^{(n_0,m_0)}}}\right) \nonumber \\&\quad \chi \left( {\varvec{k}}_{\partial {{\hat{\Lambda }}_{N}^{(n_0,m_0)}}} ={\varvec{h}}_{\partial {{\hat{\Lambda }}_{N}^{(n_0,m_0)}}}\right) . \end{aligned}$$
(18)

Proof

We consider the case that \(n_0\le m_0\). The proof for \(n_0>m_0\) is the same. As in the proof of Lemma 2.1, from the simple counting for any \(3\le N\in {\mathbb {N}}\) and \({\varvec{h}}\in C_{{\hat{\Lambda }}_N^{(n_0,m_0)}}\), we have

$$\begin{aligned} \begin{aligned} \left| \left\{ {\varvec{m}}\in G^{{\hat{\Lambda }}_{Nm_0}\setminus {\hat{\Lambda }}_{N}^{(n_0,m_0)}}\mid \left( {\varvec{h}},{\varvec{m}}\right) \in C_{{\hat{\Lambda }}_{Nm_0}} \right\} \right| =|G|^{2\left( 2m_0N+1\right) \left( m_0-n_0\right) N}. \end{aligned} \end{aligned}$$
(19)

In particular, we have

$$\begin{aligned} \begin{aligned} \left| C_{{\hat{\Lambda }}_{N}^{(n_0,m_0)}}\right| \cdot |G|^{2\left( 2m_0N+1\right) \left( m_0-n_0\right) N }=\left| C_{{\hat{\Lambda }}_{Nm_0}}\right| . \end{aligned} \end{aligned}$$
(20)

For any \(N\in {\mathbb {N}}\) and \({\varvec{k}}, {\varvec{h}}\in G^{{\hat{\Lambda }}_N^{(n_0,m_0)}}\),

$$\begin{aligned} \begin{aligned}&\varphi _{0}\left( \left| {\varvec{h}} \right\rangle _{{\hat{\Lambda }}_N^{(n_0,m_0)}}\left\langle {\varvec{k}} \right| \right) \\&=\sum _{{\varvec{m}}\in G^{{\hat{\Lambda }}_{Nm_0}\setminus {\hat{\Lambda }}_{N}^{(n_0,m_0)}}}\varphi _{0}\left( \left| {\varvec{h}},{\varvec{m}} \right\rangle _{{\hat{\Lambda }}_{Nm_0}}\left\langle {\varvec{k}},{\varvec{m}} \right| \right) \\&= \sum _{{\varvec{m}}\in G^{{\hat{\Lambda }}_{Nm_0}\setminus {\hat{\Lambda }}_{N}^{(n_0,m_0)}}} \frac{1}{|C_{{\hat{\Lambda }}_{Nm_0}}|} \\ {}&\qquad \quad \chi \left( {\left( {\varvec{k}},{\varvec{m}}\right) , \left( {\varvec{h}},{\varvec{m}}\right) \in C_{{\hat{\Lambda }}_{Nm_0}}}\right) \chi \left( \left( {\varvec{k}},{\varvec{m}}\right) _{\partial {{\hat{\Lambda }}_{N m_0}}}= \left( {\varvec{h}},{\varvec{m}}\right) _{\partial {{\hat{\Lambda }}_{Nm_0}}}\right) \\&= \sum _{{\varvec{m}}\in G^{{\hat{\Lambda }}_{Nm_0}\setminus {\hat{\Lambda }}_{N}^{(n_0,m_0)}}} \frac{1}{|C_{{\hat{\Lambda }}_{Nm_0}}|} \\ {}&\quad \qquad \chi \left( {{\varvec{k}}, {\varvec{h}}\in C_{{\hat{\Lambda }}_{N}^{(n_0,m_0)}}}\right) \chi \left( {\varvec{k}}_{\partial {{\hat{\Lambda }}_{N}^{(n_0,m_0)}}}= {\varvec{h}}_{\partial {{\hat{\Lambda }}_{N}^{(n_0,m_0)}}}\right) \\&\quad \qquad \chi \left( {\left( {\varvec{h}},{\varvec{m}}\right) \in C_{{\hat{\Lambda }}_{Nm_0}}}\right) \\ {}&\quad \qquad \chi \left( {\left( {\varvec{k}},{\varvec{m}}\right) , \left( {\varvec{h}},{\varvec{m}}\right) \in C_{{\hat{\Lambda }}_{Nm_0}}}\right) \chi \left( \left( {\varvec{k}},{\varvec{m}}\right) _{\partial {{\hat{\Lambda }}_{N m_0}}}= \left( {\varvec{h}},{\varvec{m}}\right) _{\partial {{\hat{\Lambda }}_{Nm_0}}}\right) \\&= \sum _{{\varvec{m}}\in G^{{\hat{\Lambda }}_{Nm_0}\setminus {\hat{\Lambda }}_{N}^{(n_0,m_0)}}} \frac{1}{|C_{{\hat{\Lambda }}_{Nm_0}}|} \chi \left( {{\varvec{k}}, {\varvec{h}}\in C_{{\hat{\Lambda }}_{N}^{(n_0,m_0)}}}\right) \\ {}&\quad \qquad \chi \left( {\varvec{k}}_{\partial {{\hat{\Lambda }}_{N}^{(n_0,m_0)}}}= {\varvec{h}}_{\partial {{\hat{\Lambda }}_{N}^{(n_0,m_0)}}}\right) \chi \left( {\left( {\varvec{h}},{\varvec{m}}\right) \in C_{{\hat{\Lambda }}_{Nm_0}}}\right) \\&=\frac{1}{|C_{{\hat{\Lambda }}_{Nm_0}}|} \chi \left( {{\varvec{k}}, {\varvec{h}}\in C_{{\hat{\Lambda }}_{N}^{(n_0,m_0)}}}\right) \\ {}&\quad \qquad \chi \left( {\varvec{k}}_{\partial {{\hat{\Lambda }}_{N}^{(n_0,m_0)}}}= {\varvec{h}}_{\partial {{\hat{\Lambda }}_{N}^{(n_0,m_0)}}}\right) \left| \left\{ {\varvec{m}}\in G^{{\hat{\Lambda }}_{Nm_0}\setminus {\hat{\Lambda }}_{N}^{(n_0,m_0)}}\mid \left( {\varvec{h}},{\varvec{m}}\right) \in C_{{\hat{\Lambda }}_{Nm_0}} \right\} \right| \\&=\frac{1}{|C_{{\hat{\Lambda }}_{N}^{(n_0,m_0)}}|} \\ {}&\quad \qquad \chi \left( {{\varvec{k}}, {\varvec{h}}\in C_{{\hat{\Lambda }}_{N}^{(n_0,m_0)}}}\right) \chi \left( {\varvec{k}}_{\partial {{\hat{\Lambda }}_{N}^{(n_0,m_0)}}}= {\varvec{h}}_{\partial {{\hat{\Lambda }}_{N}^{(n_0,m_0)}}}\right) . \end{aligned} \end{aligned}$$
(21)

Here, we used (19) and (20) for the last equality.

\(\square \)

Lemma 2.3

Let \(v\in \Lambda _{N-1}\) with \(N\ge 3\). Then for each \(g\in G\), there exists a bijection \(T_g^{(Nv)}: C_{{\hat{\Lambda }}_{N}}\rightarrow C_{{\hat{\Lambda }}_{N}}\) such that

$$\begin{aligned} A_v^{(g)}\left| {\varvec{k}} \right\rangle _{{\hat{\Lambda }}_{N}} =\left| T_g^{(Nv)}{\varvec{k}} \right\rangle _{{\hat{\Lambda }}_{N}},\quad {\varvec{k}}\in C_{{\hat{\Lambda }}_{N}} \end{aligned}$$
(22)

and

$$\begin{aligned} {\varvec{k}}_{\partial {{\hat{\Lambda }}_{N}} }=\left( T_g^{(Nv)}{\varvec{k}}\right) _{\partial {{\hat{\Lambda }}_{N}}} . \end{aligned}$$
(23)

Proof

Let \(v=(a, b)\in \Lambda _{N-1}\). Then, we have

$$\begin{aligned} A_v^{(g)}\left| {\varvec{k}} \right\rangle _{{\hat{\Lambda }}_{N}} =\left| T_g^{(Nv)}{\varvec{k}} \right\rangle _{{\hat{\Lambda }}_{N}},\quad {\varvec{k}}\in C_{{\hat{\Lambda }}_{N}} \end{aligned}$$
(24)

with

$$\begin{aligned} \begin{aligned} \left( T_g^{(Nv)}{\varvec{k}}\right) _{e} = \left\{ \begin{gathered} k_{(a-1,b), (a,b)}g^{-1},\quad e=((a-1,b), (a,b)),\\ k_{(a,b-1), (a,b)}g^{-1},\quad e=((a,b-1), (a,b)),\\ gk_{(a,b), (a,b+1)},\quad e=((a,b), (a,b+1)),\\ gk_{(a,b), (a+1,b)},\quad e=((a,b), (a+1,b)),\\ k_e,\quad \text {otherwise} \end{gathered} \right. . \end{aligned} \end{aligned}$$
(25)

We then have

$$\begin{aligned} \begin{aligned}&\left( T_g^{(Nv)}{\varvec{k}}\right) _{(a,b)-(a+1,b)}\left( T_g^{(Nv)}{\varvec{k}}\right) _{(a+1,b)-(a+1,b+1)} \\ {}&\quad = gk_{(a,b), (a+1,b)}k_{(a+1,b)-(a+1,b+1)}\\&\quad =gk_{(a,b), (a,b+1)}k_{(a,b+1)-(a+1,b+1)} \\ {}&\quad = \left( T_g^{(Nv)}{\varvec{k}}\right) _{(a,b)-(a,b+1)}\left( T_g^{(Nv)}{\varvec{k}}\right) _{(a,b+1)-(a+1,b+1)},\\&\quad \left( T_g^{(Nv)}{\varvec{k}}\right) _{(a,b-1)-(a+1,b-1)}\left( T_g^{(Nv)}{\varvec{k}}\right) _{(a+1,b-1)-(a+1,b)} \\ {}&\quad = k_{(a,b-1)-(a+1,b-1)}k_{(a+1,b-1)-(a+1,b)}\\&\quad =k_{(a,b-1), (a,b)}g^{-1}gk_{(a,b), (a+1,b)} \\ {}&\quad = \left( T_g^{(Nv)}{\varvec{k}}\right) _{(a,b-1)-(a,b)}\left( T_g^{(Nv)}{\varvec{k}}\right) _{(a,b)-(a+1,b)},\\&\quad \left( T_g^{(Nv)}{\varvec{k}}\right) _{(a-1,b)-(a,b)}\left( T_g^{(Nv)}{\varvec{k}}\right) _{(a,b)-(a,b+1)} \\ {}&\quad = k_{(a-1,b), (a,b)}g^{-1}gk_{(a,b), (a,b+1)}\\&\quad =k_{(a-1,b)-(a-1,b+1)}k_{(a-1,b+1)-(a,b+1)} \\ {}&\quad = \left( T_g^{(Nv)}{\varvec{k}}\right) _{(a-1,b)-(a-1,b+1)}\left( T_g^{(Nv)}{\varvec{k}}\right) _{(a-1,b+1)-(a,b+1)},\\&\quad \left( T_g^{(Nv)}{\varvec{k}}\right) _{(a-1,b-1)-(a,b-1)}\left( T_g^{(Nv)}{\varvec{k}}\right) _{(a,b-1)-(a,b)} \\ {}&\quad =k_{(a-1,b-1)-(a,b-1)}k_{(a,b-1), (a,b)}g^{-1},\\&\quad =k_{(a-1,b-1)-(a-1,b)}k_{(a-1,b), (a,b)}g^{-1} \\ {}&\quad = \left( T_g^{(Nv)}{\varvec{k}}\right) _{(a-1,b-1)-(a-1,b)}\left( T_g^{(Nv)}{\varvec{k}}\right) _{(a-1,b)-(a,b)}, \end{aligned}\nonumber \\ \end{aligned}$$
(26)

and we have \(T_g^{(Nv)}{\varvec{k}}\in C_{{\hat{\Lambda }}_{N}}\). Because \(\left( A_v^{(g)}\right) ^{-1}=A_v^{(g^{-1})}\), \(T_g^{(Nv)}\) is a bijection. Because \(A_v^{(g)}\), \(v\in \Lambda _{N-1}\) does not change the boundary edge, we have

$$\begin{aligned} {\varvec{k}}_{\partial {{\hat{\Lambda }}_{N}}}=\left( T_g^{(Nv)}{\varvec{k}}\right) _{\partial {{\hat{\Lambda }}_{N}} }. \end{aligned}$$
(27)

\(\square \)

The state \(\varphi _0\) is the frustration-free ground state of the quantum double model.

Lemma 2.4

We have

$$\begin{aligned} \begin{aligned} \varphi _0\left( A_v^{(g)} \right) =1 \end{aligned} \end{aligned}$$
(28)

for all vertices \(v\in {\mathbb {Z}}^2\) and \(g\in G\). In particular, for \(A_v=\frac{1}{|G|}\sum _{g\in G}A_v^{(g)}\), we have

$$\begin{aligned} \varphi _0(A_v)=1 \end{aligned}$$
(29)

for all vertices \(v\in {\mathbb {Z}}^2\) and

$$\begin{aligned} \varphi _0(B_p)=1 \end{aligned}$$
(30)

for all squares p.

Proof

By (7), we have

$$\begin{aligned} \begin{aligned} \varphi _0\left( \left| {\varvec{k}} \right\rangle _{{\hat{\Lambda }}_{N}}\left\langle {\varvec{k}} \right| \right) =0,\quad {\varvec{k}}\in G^{{\hat{\Lambda }}_{N}}\setminus C_{{\hat{\Lambda }}_{N}}, \end{aligned} \end{aligned}$$
(31)

and hence, \(\varphi _0(B_p)=1\).

For any \(v\in {\mathbb {Z}}^2\), choose \(3\le N\in {\mathbb {N}}\) so that \(v\in \Lambda _{N-1}\). For any \(g\in G\), we have

$$\begin{aligned} \begin{aligned} \varphi _0\left( A_v^{(g)} \right)&=\sum _{{\varvec{k}}\in G^{{\hat{\Lambda }}_{N}}}\varphi _0\left( A_v^{(g)}\left| {\varvec{k}} \right\rangle _{{\hat{\Lambda }}_N}\left\langle {\varvec{k}} \right| \right) \qquad =\sum _{{\varvec{k}}\in C_{{{\hat{\Lambda }}_{N}}}}\varphi _0\left( A_v^{(g)}\left| {\varvec{k}} \right\rangle _{{\hat{\Lambda }}_N}\left\langle {\varvec{k}} \right| \right) \\&=\sum _{{\varvec{k}}\in C_{{{\hat{\Lambda }}_{N}}}}\varphi _0\left( \left| T_g^{(Nv)}{\varvec{k}} \right\rangle _{{\hat{\Lambda }}_N}\left\langle {\varvec{k}} \right| \right) \\&=\frac{1}{|C_{{\hat{\Lambda }}_N}|}\sum _{{\varvec{k}}\in C_{{{\hat{\Lambda }}_{N}}}} \chi \left( {{\varvec{k}}, T_g^{(Nv)}{\varvec{k}}\in C_{{\hat{\Lambda }}_N}}\right) \chi \left( {\varvec{k}}_{\partial {{\hat{\Lambda }}_{N}}}= \left( T_g^{(Nv)}{\varvec{k}}\right) _{\partial {{\hat{\Lambda }}_{N}}}\right) =1. \end{aligned} \nonumber \\ \end{aligned}$$
(32)

We used (31) for the second equality, and Lemma 2.3 for the third and fifth equality. \(\square \)

Hence, our \(\varphi _0\) is the frustration-free ground state of the quantum double model.

Lemma 2.5

We have \(\omega _0=\varphi _0\). Namely, the restriction of \(\omega _0\) onto \({\mathcal {B}}_{{\hat{\Lambda }}_{N}}\) is given by the formula (7).

3 Admissible Configurations on Layers of Squares

In this section, we consider admissible configuration on layers of squares. More precisely, we consider the following shape. We use the notation from Sect. 1, but we do not assume G to be abelian.

Definition 3.1

We consider l-layers \({\mathfrak {S}}\) of a sequence of squares \(m=1,\ldots , l\) in \({\mathbb {Z}}^2\), with m-th layer

$$\begin{aligned} S_{(x_1^{(m)}, y+m-1)}, S_{(x_2^{(m)}, y+m-1)}, \ldots , S_{(x_{n_m}^{(m)}, y+m-1)}, \end{aligned}$$

with \(x_k^{(m)}=x_1^{(m)}+k-1\in {\mathbb {Z}}\), \(y\in {\mathbb {Z}}\). We say this layer satisfies the condition S if

$$\begin{aligned} \begin{aligned} z^{(m)}:=\max \{x_{1}^{(m)}, x_1^{(m+1)}\}<\min \{x_{n_m}^{(m)}, x_{n_{m+1}}^{(m+1)} \}+1=:w^{(m)}, \end{aligned} \end{aligned}$$
(33)

for all \(m=1,\ldots ,l-1\).

Let us consider l-layers \({\mathfrak {S}}\) of a sequence of squares with notations in Definition 3.1, satisfying the condition S. We set

$$\begin{aligned} \begin{aligned}&M^{(m)}:=w^{(m)}-z^{(m)}\in {\mathbb {N}}, \\&{\tilde{z}}^{(m)}:=\min \{x_{1}^{(m)}, x_1^{(m+1)}\},\\&{\tilde{w}}^{(m)}:=\max \{x_{n_m}^{(m)}, x_{n_{m+1}}^{(m+1)} \}+1, \end{aligned} \end{aligned}$$
(34)

for \(m=1,\ldots ,l-1\). We name the interior horizontal edges between m-th and \(m+1\)-th layer as

$$\begin{aligned} \begin{aligned}&{\varvec{e}}_{k}^{({m})}:=(z^{(m)}+k-1, y+{m})-(z^{({m})}+k, y+{m}),\quad k=1,\ldots , M^{(m)}-1\\ \end{aligned} \end{aligned}$$
(35)

for \(1\le m\le l-1\) and

$$\begin{aligned} \begin{aligned} {\varvec{f}}^{(m)}:=(w^{(m)}-1, y+{m})-(w^{(m)}, y+{m}). \end{aligned} \end{aligned}$$
(36)

for \(1\le m\le l-1\). We denote interior vertical edges at the m-th layer as

$$\begin{aligned} \begin{aligned}&{\varvec{{\tilde{f}}}}^{(m)}_{k}:=(x_{k}^{(m)}+1, y+m-1)-(x_{k}^{(m)}+1, y+m),\quad k=1,\ldots , n_m-1 \end{aligned} \end{aligned}$$
(37)

for \(1\le m\le l\). The edges on the boundary are

$$\begin{aligned} \begin{aligned}&{\varvec{b}}_k^{(1)}:=(x_k^{(1)},y)-(x_{k}^{(1)}+1,y),\quad k=1,\ldots , n_1,\\&{\varvec{b}}_k^{(l+1)}:=(x_k^{(l)},y+l)-(x_{k}^{(l)}+1,y+l),\quad k=1,\ldots , n_l, \end{aligned} \end{aligned}$$
(38)

corresponding to the top and bottom lines and

$$\begin{aligned} \begin{aligned}&{\varvec{b}}_k^{(m L)}:=(\tilde{z}^{(m)}+k-1, y+{m})-(\tilde{z}^{(m)}+k, y+{m}),\quad k=1,\ldots , z^{(m)}-{\tilde{z}}^{(m)}\\&{\varvec{b}}_k^{(m R)}:=(w^{(m)}+k-1, y+{m})-(w^{(m)}+k, y+{m}),\quad k=1,\ldots , {\tilde{w}}^{(m)}-w^{(m)} \end{aligned} \end{aligned}$$
(39)

corresponding to the left and right segment of the horizontal line between m-th and \(m+1\)-th layer \(m=1,\ldots ,l-1\), and

$$\begin{aligned} \begin{aligned}&\tilde{{\varvec{b}}}^{(m L)}:=(x_1^{(m)},y+m-1)-(x_1^{(m)},y+m),\\&\tilde{{\varvec{b}}}^{(m R)}:=(x_{n_m}^{(m)}+1,y+m-1)-(x_{n_m}^{(m)}+1,y+m), \end{aligned} \end{aligned}$$
(40)

\(m=1,\ldots ,l\) corresponding to the vertical line.

We set

$$\begin{aligned} \begin{aligned}&{\mathcal {E}}^{(1)}\left( {{\mathfrak {S}}}\right) := \left\{ {\varvec{e}}_{k}^{({m})} \right\} ,\quad {\mathcal {E}}^{(2)}\left( {{\mathfrak {S}}}\right) := \left\{ {\varvec{f}}^{(m)} \right\} ,\quad {\mathcal {E}}^{(3)}\left( {{\mathfrak {S}}}\right) := \left\{ \tilde{{\varvec{f}}}^{(m)}_{k} \right\} ,\\&{\mathcal {E}}^{(4)}\left( {{\mathfrak {S}}}\right) := \left\{ {\varvec{b}}_k^{(1)}, \;{\varvec{b}}_k^{(l+1)},\;{\varvec{b}}_k^{(m L)},\; {\varvec{b}}_k^{(m R)},\; \tilde{{\varvec{b}}}^{(m L)},\;\tilde{{\varvec{b}}}^{(m R)} \right\} . \end{aligned} \end{aligned}$$
(41)

Then, we have

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}\left( {\mathfrak {S}}\right) :=\cup _{p\in {\mathfrak {S}}}\{\text {edge of}\;p\} = {\mathcal {E}}^{(1)}\left( {{\mathfrak {S}}}\right) \dot{\cup } {\mathcal {E}}^{(2)}\left( {{\mathfrak {S}}}\right) \dot{\cup } {\mathcal {E}}^{(3)}\left( {{\mathfrak {S}}}\right) \dot{\cup } {\mathcal {E}}^{(4)}\left( {{\mathfrak {S}}}\right) . \end{aligned} \end{aligned}$$
(42)

We also set

$$\begin{aligned}&\mathbb V\left( {\mathfrak {S}} \right) :=\cup _{e\in \mathbb E\left( {\mathfrak S}\right) }\{ v_{e,+1}, v_{e,-1}\}. \end{aligned}$$
(43)

Definition 3.2

Let \({\mathfrak {S}}\) be l-layers of a sequence of squares given in Definition 3.1, satisfying the condition S. We denote by \({{\mathfrak {p}}}^{(r)}\) the self-avoiding path in the boundary \({\mathcal {E}}^{(4)}\left( {{\mathfrak {S}}}\right) \) with origin \((x_1^{(1)},y)\) target \((x_{n_{l}}^{({l})}+1, y+l)\) starting as \((x_1^{(1)}, y)-(x_1^{(1)}+1, y)-\cdots \) and following the boundary. We denote by \({{\mathfrak {p}}}^{(l)}\) the self-avoiding path in \({\mathcal {E}}^{(4)}\left( {\mathfrak S}\right) \) with origin \((x_1^{(1)},y)\) target \((x_{n_{l}}^{({l})}+1, y+l)\) starting as \((x_1^{(1)}, y)-(x_1^{(1)}, y+1)-\cdots \) and following the boundary. Because of condition S, \({{\mathfrak {p}}}^{(r)}\) and \({{\mathfrak {p}}}^{(l)}\) intersect only at the origin \((x_1^{(1)},y)\) and the target \((x_{n_{l}}^{({l})}+1, y+l)\).

Lemma 3.3

Let \({\mathfrak {S}}\) be l-layers of a sequence of squares given in Definition 3.1. Suppose that the \({\mathfrak {S}}\) satisfies the condition S. Then, for any \(v_0,v_1\in \mathbb V\left( {\mathfrak {S}} \right) \) and paths \({{\mathfrak {p}}},\tilde{{{\mathfrak {p}}}}\) in \({\mathbb {E}}\left( {\mathfrak {S}}\right) \) with origin \(v_0\) and target \(v_1\), there exists a finite sequence of paths \({{\mathfrak {p}}}_i \) in \({\mathbb {E}}({\mathfrak {S}})\) with origin \(v_0\) and target \(v_1\) \(i=1,\ldots , n\), such that \({{\mathfrak {p}}}_1={{\mathfrak {p}}}\), \({{\mathfrak {p}}}_n={\tilde{{{\mathfrak {p}}}}}\) and \(\left( {{\mathfrak {p}}}_i\right) ^{-1}{{\mathfrak {p}}}_{i+1}={{\mathfrak {p}}}_{S_i}\) or \(\left( {{\mathfrak {p}}}_i\right) ^{-1}{{\mathfrak {p}}}_{i+1}={{\mathfrak {p}}}_{S_i}^{-1}\) for some square \(S_i\) in \({\mathfrak {S}}\) or \(\left( {{\mathfrak {p}}}_i\right) ^{-1}{{\mathfrak {p}}}_{i+1}={{\mathfrak {q}}}_i{{\mathfrak {q}}}_i^{-1}\) for a path \({{\mathfrak {q}}}_i\) in \({\mathbb {E}}\left( {\mathfrak {S}}\right) \), \(i=1,\ldots ,n-1\).

Remark 3.4

Let \({{{\mathfrak {S}}}'}\) be a set of squares and \({\mathbb {E}}\left( {{\mathfrak S}'}\right) \) the set of all edges of the squares in \({{{\mathfrak {S}}}'}\). For paths \({{\mathfrak {p}}},\tilde{{{\mathfrak {p}}}}\) in \({\mathbb {E}}\left( {{{\mathfrak {S}}}'}\right) \) with common origin and target, we say \({{\mathfrak {p}}}\) can be deformed into \({\tilde{{{\mathfrak {p}}}}}\) in \({{{\mathfrak {S}}}'}\) if there is a sequence of paths in \({\mathbb {E}}\left( {{{\mathfrak {S}}}'}\right) \) as in Lemma 3.3.

Proof

We consider the following proposition for each \(l\in {\mathbb {N}}\).

\(P_l\) : For l-layers \({\mathfrak {S}}\) of a sequence of squares satisfying the condition S, for any \(v_0,v_1\in \mathbb V\left( {\mathfrak {S}} \right) \) and paths \({{\mathfrak {p}}},\tilde{{{\mathfrak {p}}}}\) in \({\mathbb {E}}\left( {\mathfrak {S}}\right) \) with origin \(v_0\) and target \(v_1\), there exists a finite sequence of paths \({{\mathfrak {p}}}_i \) in \({\mathbb {E}}\left( \mathfrak S\right) \) with origin \(v_0\) and target \(v_1\), \(i=1,\ldots , n\), such that \({{\mathfrak {p}}}_1={{\mathfrak {p}}}\), \({{\mathfrak {p}}}_n={\tilde{{{\mathfrak {p}}}}}\) and \(\left( {{\mathfrak {p}}}_i\right) ^{-1}{{\mathfrak {p}}}_{i+1}={{\mathfrak {p}}}_{S_i}\) or \(\left( {{\mathfrak {p}}}_i\right) ^{-1}{{\mathfrak {p}}}_{i+1}={{\mathfrak {p}}}_{S_i}^{-1}\) for some square \(S_i\) in \({\mathfrak {S}}\) or \(\left( {{\mathfrak {p}}}_i\right) ^{-1}{{\mathfrak {p}}}_{i+1}={{\mathfrak {q}}}_i{{\mathfrak {q}}}_i^{-1}\) for a path \({{\mathfrak {q}}}_i\) in \({\mathbb {E}}\left( {\mathfrak {S}}\right) \) , \(i=1,\ldots ,n-1\).

\(P_1\) is true because any path from \(v_0=(a,b)\) to \(v_1=(c,d)\) in 1-layer of squares can be deformed into a path

\((a,b)\rightarrow (c,b)\) horizontally \((c,b)\rightarrow (c,d)\) vertically.

Suppose \(P_l\) is true. Let \({\mathfrak {S}}\) be \(l+1\)-layers of a sequence of squares satisfying the condition S. Let \(v_0,v_1\in \mathbb V\left( {\mathfrak {S}} \right) \), and let \({{\mathfrak {p}}},\tilde{{{\mathfrak {p}}}}\) be paths in \({\mathbb {E}}\left( {\mathfrak {S}}\right) \) with origin \(v_0\) and target \(v_1\). We consider the case \(v_0\) is in the first l-layers and \(v_1\) is in the \(l+1\)-th layers. The proof is the same for other cases. The path \({{\mathfrak {p}}}\) can be split into a sequence of paths

$$\begin{aligned} {{\mathfrak {p}}}_1, {\hat{{{\mathfrak {p}}}}}_1, {{\mathfrak {p}}}_2, {\hat{{{\mathfrak {p}}}}}_2,\cdots {{\mathfrak {p}}}_L, {\hat{{{\mathfrak {p}}}}}_L, \end{aligned}$$

where \({{\mathfrak {p}}}_k\) is a path inside of the first l-layers and \({\hat{{{\mathfrak {p}}}}}_k\) is a path inside of the \(l+1\)-th layer. The origin of \({{\mathfrak {p}}}_1\) is \(v_0\), and the target of \(\hat{{{{\mathfrak {p}}}}}_{ L}\) is \(v_1\). The target of \({{\mathfrak {p}}}_k\) is origin of \(\hat{{{\mathfrak {p}}}}_k\), \(k=1,\ldots ,L\) and the target of \(\hat{{{\mathfrak {p}}}}_k\) is the origin of \({{\mathfrak {p}}}_{k+1}\), \(k=1,\ldots ,L-1\). For each k, let \({\mathfrak {q}}_k\) \(k=1,\ldots , L\) (resp \(\hat{{\mathfrak {q}}}_k\), \(k=1,\ldots , L-1\)) be a horizontal path in \({\mathfrak {S}}\) in the line between l-th and \(l+1\)-th layer such that

$$\begin{aligned} \text {origin of}\; {\mathfrak {q}}_k=\text {target of}\; {{\mathfrak {p}}}_k=\text {origin of}\; {\hat{{{\mathfrak {p}}}}}_k\quad \text {target of}\;{\mathfrak {q}}_k=(x_1^{(l+1)}, y+l) \\ \text {origin of}\; \hat{{\mathfrak {q}}}_k=\text {target of}\; \hat{{{\mathfrak {p}}}}_k=\text {origin of}\;{{\mathfrak {p}}}_{k+1},\quad \text {target of}\;\hat{{\mathfrak {q}}}_k=(x_1^{(l+1)}, y+l). \end{aligned}$$

The path \(\tilde{{{\mathfrak {p}}}}\) can be split into a sequence of paths

$$\begin{aligned} {\tilde{{{\mathfrak {p}}}}}_1, \hat{\tilde{{{\mathfrak {p}}}}}_1, {\tilde{{{\mathfrak {p}}}}}_2, \hat{\tilde{{{\mathfrak {p}}}}}_2,\cdots {\tilde{{{\mathfrak {p}}}}}_{{\tilde{L}}}, \hat{\tilde{{{\mathfrak {p}}}}}_{{\tilde{L}}}, \end{aligned}$$

where \({\tilde{{{\mathfrak {p}}}}}_k\) is a path inside of the first l-layers and \(\hat{\tilde{{{\mathfrak {p}}}}}_k\) is a path inside of the \(l+1\)-th layer. The target of \({\tilde{{{\mathfrak {p}}}}}_k\) is origin of \(\hat{{\tilde{{{\mathfrak {p}}}}}}_k\), \(k=1,\ldots ,{\tilde{L}}\), and the target of \(\hat{{\tilde{{{\mathfrak {p}}}}}}_k\) is the origin of \({{\tilde{{{\mathfrak {p}}}}}}_{k+1}\), \(k=1,\ldots ,{\tilde{L}}-1\). The origin of \({\tilde{{{\mathfrak {p}}}}}_1\) is \(v_0\), and the target of \(\hat{\tilde{{{\mathfrak {p}}}}}_{{\tilde{L}}}\) is \(v_1\). For each k, let \(\tilde{{\mathfrak {q}}}_k\) \(k=1,\ldots , {\tilde{L}}\) (resp. \(\hat{\tilde{{\mathfrak {q}}}}_k\) \(k=1,\ldots , {\tilde{L}}-1\)) be a horizontal path in \({\mathfrak {S}}\) in the line between l-th and \(l+1\)-th layer such that

$$\begin{aligned} \text {origin of}\; {\tilde{{\mathfrak {q}}}_k}=\text {target of}\; {\tilde{{{\mathfrak {p}}}}}_k =\text {origin of}\;{ \hat{\tilde{{{\mathfrak {p}}}}}}_k,\quad \text {target of}\;{\tilde{{\mathfrak {q}}}_k}=(x_1^{(l+1)}, y+l) \\ \text {origin of}\; \hat{\tilde{{\mathfrak {q}}}}_k=\text {target of}\; \hat{{\tilde{{{\mathfrak {p}}}}}}_k =\text {origin of}\;{\tilde{{{\mathfrak {p}}}}}_{k+1},\quad \text {target of}\;\hat{\tilde{{\mathfrak {q}}}}_k=(x_1^{(l+1)}, y+l). \end{aligned}$$

We fix a horizontal loop \({\mathfrak {l}}\) in \({\mathfrak {S}}\), in the line between l-th and \(l+1\)-th layer, with origin \((x_1^{(l+1)}, y+l)\) and target \((x_1^{(l+1)}, y+l)\).

Both of \({{\mathfrak {p}}}_1{\mathfrak {q}}_1\) and \(\tilde{{{\mathfrak {p}}}}_1\tilde{\mathfrak q}_1\) are paths inside of the first l-layers with origin \(v_0\) and target \((x_1^{(l+1)}, y+l)\). Hence by \(P_l\), \({{\mathfrak {p}}}_1{\mathfrak {q}}_1\) can be deformed into \(\tilde{{{\mathfrak {p}}}}_1\tilde{{\mathfrak {q}}}_1\) in the first l-layers. All of \({{\mathfrak {q}}}_j^{-1} {\hat{{{\mathfrak {p}}}}}_j\hat{{\mathfrak {q}}}_j\), \(j=1,\ldots , L-1\) and \(\tilde{{\mathfrak {q}}}_k^{-1} \hat{\tilde{{{\mathfrak {p}}}}}_k\hat{\tilde{\mathfrak q}}_k\) \(k=1,\ldots , {\tilde{L}}-1\) are paths inside of the \(l+1\)-th layer with origin \((x_1^{(l+1)}, y+l)\) and target \((x_1^{(l+1)}, y+l)\). Hence from \(P_1\), they can be deformed into \({\mathfrak {l}}\) in the \(l+1\)-th layer. All of \(\hat{{\mathfrak {q}}}_{j-1}^{-1} {{\mathfrak {p}}}_j{\mathfrak {q}}_{j}\), \(j=2,\ldots , L\) and \(\hat{\tilde{{\mathfrak {q}}}}_{k-1}^{-1} {\tilde{{{\mathfrak {p}}}}}_k\tilde{{\mathfrak {q}}}_{k}\) \(k=2,\ldots , {\tilde{L}}\) are paths inside of the first l-layers with origin \((x_1^{(l+1)}, y+l)\) and target \((x_1^{(l+1)}, y+l)\). Hence by \(P_l\), they all can be deformed into \({\mathfrak {l}}^{-1}\). Both of \({{\mathfrak {q}}}_L^{-1}{\hat{{{\mathfrak {p}}}}}_L\) and \(\tilde{{{\mathfrak {q}}}}_{\tilde{L}}^{-1}\hat{{\tilde{{{\mathfrak {p}}}}}}_{{\tilde{L}}}\) are paths in \(l+1\)-th layer with origin \((x_1^{(l+1)}, y+l)\) target \(v_1\). Hence by \(P_1\), \({{\mathfrak {q}}}_L^{-1}{\hat{{{\mathfrak {p}}}}}_L\) can be deformed into \(\tilde{{{\mathfrak {q}}}}_{\tilde{L}}^{-1}\hat{{\tilde{{{\mathfrak {p}}}}}}_{{\tilde{L}}}\) in the \(l+1\)-th layer.

Hence, \({{\mathfrak {p}}}\) can be deformed into

$$\begin{aligned} \begin{aligned} \tilde{{{\mathfrak {p}}}}_1\tilde{{\mathfrak {q}}}_1, {\mathfrak {l}}, {\mathfrak {l}}^{-1},\cdots , {\mathfrak {l}}, {\mathfrak {l}}^{-1}, \tilde{{{\mathfrak {q}}}}_{{\tilde{L}}}^{-1}\hat{{\tilde{{{\mathfrak {p}}}}}}_{{\tilde{L}}} \end{aligned} \end{aligned}$$
(44)

with \(L-1\) number of \( {\mathfrak {l}}, {\mathfrak {l}}^{-1}\) and \({\tilde{{{\mathfrak {p}}}}}\) can be deformed into

$$\begin{aligned} \begin{aligned} \tilde{{{\mathfrak {p}}}}_1\tilde{{\mathfrak {q}}}_1, {\mathfrak {l}}, {\mathfrak {l}}^{-1},\cdots , {\mathfrak {l}}, {\mathfrak {l}}^{-1}, \tilde{{{\mathfrak {q}}}}_{{\tilde{L}}}^{-1}\hat{{\tilde{{{\mathfrak {p}}}}}}_{{\tilde{L}}}, \end{aligned} \end{aligned}$$
(45)

with \({\tilde{L}}-1\) number of \( {\mathfrak {l}}, {\mathfrak {l}}^{-1}\). Therefore, \({{\mathfrak {p}}}\) can be deformed into \({\tilde{{{\mathfrak {p}}}}}\) in \({\mathfrak {S}}\). \(\square \)

Definition 3.5

Let \({\mathfrak {S}}\) be l-layers of a sequence of squares satisfying condition S. Any two vertices in \(v_0,v\in {\mathbb {V}}({\mathfrak {S}})\) can be connected via a path in \({{\mathbb {E}}({\mathfrak {S}})}\). By Lemma 3.3, for any paths \({{\mathfrak {p}}},{\tilde{{{\mathfrak {p}}}}}\) in \({\mathbb {E}}({\mathfrak {S}})\) with origin \(v_0\) and target \(v\in {\mathbb {V}}({\mathfrak {S}})\) can be deformed into each other in \({\mathfrak {S}}\). Note (with notation in Lemma 3.3) that for each i-th step of the deformation, we have \(\Psi _{{{\mathfrak {p}}}_i}({\varvec{h}}_{{{\mathfrak {p}}}_i})=\Psi _{{{\mathfrak {p}}}_{i+1}}({\varvec{h}}_{{{\mathfrak {p}}}_{i+1}})\) for any \({\varvec{h}}\in C_{{\mathbb {E}}({\mathfrak {S}})}\), by the admissibility condition. Hence for any \({\varvec{h}}\in C_{{\mathbb {E}}({\mathfrak {S}})}\) and \(v_0\in {\mathbb {V}}({\mathfrak {S}})\), any paths \({{\mathfrak {p}}},{\tilde{{{\mathfrak {p}}}}}\) in \({\mathbb {E}}({\mathfrak {S}})\) with origin \(v_0\) and target \(v\in {\mathbb {V}}({\mathfrak {S}})\), we have

$$\begin{aligned} \Psi _{{\mathfrak {p}}}({\varvec{h}}_{{\mathfrak {p}}})=\Psi _{{\tilde{{{\mathfrak {p}}}}}}({\varvec{h}}_{{\tilde{{{\mathfrak {p}}}}}}) \end{aligned}$$
(46)

and we may define

$$\begin{aligned} \left( \Psi ^{v_0}({\varvec{h}})\right) _v:=\Psi _{ {{\mathfrak {p}}}}({\varvec{h}}_{ {{\mathfrak {p}}}}),\quad {\varvec{h}}\in C_{{\mathbb {E}}({\mathfrak {S}})}, \; v\in {\mathbb V\left( {\mathfrak {S}} \right) }, \end{aligned}$$
(47)

independent of the choice of the path \({{\mathfrak {p}}}\) in \({\mathfrak {S}}\) with origin \(v_0\) and target v.

We would like to count the number of admissible configurations on \({\mathbb {E}}\left( {\mathfrak {S}}\right) \) under a given boundary condition. We have the following lemma.

Lemma 3.6

Let \({\mathfrak {S}}\) be l-layers of a sequence of squares. Suppose that \({\mathfrak {S}}\) satisfies the condition S. Let \({{\mathfrak {p}}}^{(l)}\), \({{\mathfrak {p}}}^{(r)}\) be the paths in Definition 3.2 for this \({\mathfrak {S}}\). For \({\varvec{a}}=(a_v)_{v\in {\mathcal {E}}^{(4)}\left( {\mathfrak S}\right) }\in G^{{\mathcal {E}}^{(4)}\left( {{\mathfrak {S}}}\right) }\), the followings are equivalent.

  1. (i)

    \(\Psi _{{{\mathfrak {p}}}^{(r)}}({\varvec{a}})=\Psi _{{{\mathfrak {p}}}^{(l)}}({\varvec{a}})\).

  2. (ii)

    There is a

    $$\begin{aligned} {\varvec{g}}=\left( g_e\right) _{e\in {{\mathcal {E}}^{(1)}\left( {{\mathfrak {S}}}\right) }}\in G^{{\mathcal {E}}^{(1)}\left( {{\mathfrak {S}}}\right) } \end{aligned}$$

    which allows unique \({\varvec{b}}^{(2)}\in G^{{\mathcal {E}}^{(2)}\left( {\mathfrak S}\right) }\), \({\varvec{b}}^{(3)}\in G^{{\mathcal {E}}^{(3)}\left( {{\mathfrak {S}}}\right) }\) such that

    $$\begin{aligned} ({\varvec{a}}, {\varvec{g}}, {\varvec{b}}^{(2)}, {\varvec{b}}^{(3)})\in C_{{\mathbb {E}}\left( \mathfrak S\right) }. \end{aligned}$$
  3. (iii)

    For any

    $$\begin{aligned} {\varvec{g}}=\left( g_e\right) _{e\in {{\mathcal {E}}^{(1)}\left( {{\mathfrak {S}}}\right) }}\in G^{{\mathcal {E}}^{(1)}\left( {{\mathfrak {S}}}\right) }, \end{aligned}$$

    there is unique \({\varvec{b}}^{(2)}\in G^{{\mathcal {E}}^{(2)}\left( {\mathfrak S}\right) }\), \({\varvec{b}}^{(3)}\in G^{{\mathcal {E}}^{(3)}\left( {{\mathfrak {S}}}\right) }\) such that

    $$\begin{aligned} ({\varvec{a}}, {\varvec{g}}, {\varvec{b}}^{(2)}, {\varvec{b}}^{(3)})\in C_{{\mathbb {E}}\left( \mathfrak S\right) }. \end{aligned}$$

Proof

We use the notation in Definition 3.1 to describe \(\mathfrak S\). (iii)\(\Rightarrow \) (ii) is trivial. (ii)\(\Rightarrow \) (i) holds from the admissibility condition of \(({\varvec{a}}, {\varvec{g}}, {\varvec{b}}^{(2)}, {\varvec{b}}^{(3)})\in C_{{\mathbb {E}}\left( {\mathfrak {S}}\right) }\), because from Lemma 3.3 we can deform the path \({{\mathfrak {p}}}^{(l)}\) into \({{\mathfrak {p}}}^{(r)}\) via paths inside of \({\mathbb {E}}\left( {\mathfrak {S}}\right) \) connecting \((x_1^{(1)},y)\) to \((x_{n_{l}}^{({l})}+1, y+{l})\).

Now, we prove (i)\(\Rightarrow \) (iii). Fix any \({\varvec{g}}=\left( g_e\right) _{e\in {{\mathcal {E}}^{(1)}\left( {{\mathfrak {S}}}\right) }}\in G^{{{\mathcal {E}}^{(1)}\left( {{\mathfrak {S}}}\right) }}\). We show that the label of rest of the edges \({\mathcal {E}}^{(2)}\left( {\mathfrak S}\right) \cup {\mathcal {E}}^{(3)}\left( {{\mathfrak {S}}}\right) \) is determined uniquely by the admissibility condition. For each \(m=1,\ldots , l\), we denote by \({\mathfrak {S}}^{(m)}\) the m-th layer of the squares of \(\mathfrak S\).

(1) We start from \({\mathfrak {S}}^{(1)}\). Suppose \(l\ge 2\). The condition \({\varvec{g}}=\left( g_e\right) _{e\in {{\mathcal {E}}^{(1)}\left( {\mathfrak S}\right) }}\in G^{{{\mathcal {E}}^{(1)}\left( {{\mathfrak {S}}}\right) }}\) and \({\varvec{a}}=(a_v)_{v\in {\mathcal {E}}^{(4)}\left( {{\mathfrak {S}}}\right) }\in G^{{\mathcal {E}}^{(4)}\left( {{\mathfrak {S}}}\right) }\) set all the labels of edges in \(\mathfrak S^{(1)}\): The edges whose labels are not determined yet are

$$\begin{aligned} \begin{aligned} {\varvec{f}}^{(1)}:=(w^{(1)}-1, y+{1})-(w^{(1)}, y+{1}). \end{aligned} \end{aligned}$$
(48)

and

$$\begin{aligned} \begin{aligned}&{\varvec{{\tilde{f}}}}^{(1)}_{k}:=(x_{k}^{(1)}+1, y)-(x_{k}^{(1)}+1, y+1),\quad k=1,\ldots , n_1-1. \end{aligned} \end{aligned}$$
(49)

If \(n_1\ge 2\) and \(x_1^{(1)}+1\le w^{(1)}-1\), \( {\varvec{\tilde{f}}}^{(1)}_{1}\) belongs to a square \(S_{(x_1^{(1)}, y)}\). All other edges in \(S_{(x_1^{(1)}, y)}\) are in \({\mathcal {E}}^{(4)}\left( {\mathfrak S}\right) \cup {\mathcal {E}}^{(1)}\left( {{\mathfrak {S}}}\right) \). The labels of these edges are determined from \({\varvec{g}}\) and \({\varvec{a}}\). Therefore, by the admissibility condition, the label of \( {\varvec{{\tilde{f}}}}^{(1)}_{1}\) is determined automatically. Next if \(x_{2}^{(1)}+1\le w^{(1)}-1\), then \( {\varvec{{\tilde{f}}}}^{(1)}_{1}\) and \( {\varvec{{\tilde{f}}}}^{(2)}_{1}\) belong to the same square \(S_{(x_2^{(1)}, y)}\). Two other edges from this square \(S_{(x_2^{(1)}, y)}\) belong to \({\mathcal {E}}^{(4)}\left( {{\mathfrak {S}}}\right) \cup {\mathcal {E}}^{(1)}\left( {{\mathfrak {S}}}\right) \). Hence, the label of three edges in this square is already determined. By the admissibility condition, the label of the last one \( {\varvec{\tilde{f}}}^{(2)}_{1}\) is uniquely determined. We can continue this \(w^{(1)}-x_1^{(1)}-1\) times and determine labels of \( {\varvec{\tilde{f}}}^{(1)}_{1},\cdot ,{\varvec{{\tilde{f}}}}^{(1)}_{w^{(1)}-x_1^{(1)}-1}\) uniquely from the admissibility conditions on \(S_{(x_1^{(1)}, y)}, S_{(x_2^{(1)}, y)},\ldots , S_{(w^{(1)}-2, y)}\). If \(w^{(1)}\le x_{n_1}^{(1)}\), then \(x_{n_2}^{(1)}\le x_{n_1}^{(1)}\) and \({\varvec{{\tilde{f}}}}^{(1)}_{n_1-1}\) belongs to a square \(S_{(x_{n_1}^{(1)}, y)}\). All other edges of \(S_{(x_{n_1}^{(1)}, y)}\) are in \({\mathcal {E}}^{(4)}\left( {{\mathfrak {S}}}\right) \). The labels of these edges are determined from \({\varvec{a}}\). Therefore, by the admissibility condition, the label of \( {\varvec{{\tilde{f}}}}^{(1)}_{n_1-1}\) is determined automatically. By the same procedure as before, all the labels of \({\varvec{{\tilde{f}}}}^{(1)}_{k}\), \(k=n_1-1, n_1-2,\ldots , w^{(1)}-x_1^{(1)}\) can be decided uniquely so that the admissibility condition on

$$\begin{aligned} \begin{aligned} S_{(w^{(1)}, y)},\ldots , S_{(x_{n_1}^{(1)}, y)} \end{aligned} \end{aligned}$$
(50)

is satisfied. Hence, the admissibility condition on all the squares in \({\mathfrak {S}}^{(1)}\) but \(S_{(w^{(1)}-1, y)}\) is satisfied, and all the edges in \({\mathfrak {S}}^{(1)}\) but \({\varvec{f}}^{(1)}\) are labeled. But \({\varvec{f}}^{(1)}\) belongs to the square \(S_{(w^{(1)}, y)}\).Three other edges of \(S_{(w^{(1)}, y)}\) are already labeled. Hence from the admissibility condition of \(S_{(w^{(1)}, y)}\), this label \({\varvec{f}}^{(1)}\) is also determined uniquely.

(2) This procedure continues. By the m-th step, all the labels of edges in \({\mathfrak {S}}^{(1)},\ldots , {\mathfrak {S}}^{(m-1)}\) are determined uniquely, so that the admissibility conditions of all the squares in \({\mathfrak {S}}^{(1)},\ldots , {\mathfrak {S}}^{(m-1)}\) are satisfied. In particular, all the edges in \({\mathfrak {S}}^{(m)}\) except for \({\varvec{f}}^{(m)}\), \({\varvec{{\tilde{f}}}}^{(m)}_{k}:=(x_{k}^{(m)}+1, y+m-1)-(x_{k}^{(m)}+1, y+m)\), \(k=1,\ldots , n_m-1\) are already labeled. By the same procedure as in (1), we can decide the label of \({\varvec{{\tilde{f}}}}^{(m)}_{k}\), \(k=1,\ldots , n_m-1\) uniquely so that the admissibility condition of

$$\begin{aligned} \begin{aligned}&S_{(x_1^{(m)}, y+m-1)}, S_{(x_2^{(m)}, y+m-1)}, \ldots , S_{(w^{(m)}-2, y+m-1)},\\ {}&\qquad \qquad S_{(w^{(m)}, y+m-1)},\ldots , S_{(x_{n_m}^{(m)}, y+m-1)} \end{aligned} \end{aligned}$$
(51)

(i.e., all the squares in \({\mathfrak {S}}^{(m)}\) but \(S_{(w^{(m)}-1, y+m-1)}\)) is satisfied. The label of \({\varvec{f}}^{(m)}\) is the determined by the admissibility condition on \(S_{(w^{(m)}-1, y+m-1)}\) uniquely, from the labels of edges already determined.

(3) Hence, we label all the edges in \({\mathfrak {S}}^{(1)},\ldots , {\mathfrak {S}}^{(l-1)}\), uniquely so that the admissibility conditions hold for all squares in \({\mathfrak {S}}^{(1)},\ldots , \mathfrak S^{(l-1)}\). In \({\mathfrak {S}}^{(l)}\), the only edges without labels are \({\varvec{{\tilde{f}}}}^{(l)}_{k}\), \(k=1,\ldots , n_l-1\). We proceed in the order

$$\begin{aligned} \begin{aligned} S_{(x_1^{(l)}, y+l-1)}, S_{(x_2^{(l)}, y+l-1)},\ldots , S_{(x_{n_l}^{(l)}-1, y+l-1)} \end{aligned} \end{aligned}$$
(52)

to define the labels \({\varvec{{\tilde{f}}}}^{(l)}_{k}\), \(k=1,\ldots , n_l-1\) uniquely via the admissibility condition of these squares. Hence, we obtain the unique label \({\varvec{b}}^{(2)}\in {\mathcal {E}}^{(2)}\left( {{\mathfrak {S}}}\right) \), \({\varvec{b}}^{(3)}\in {\mathcal {E}}^{(3)}\left( {{\mathfrak {S}}}\right) \) which satisfies the admissibility conditions of all the squares in \({\mathfrak {S}}\) but \(S_{(x_{n_l}^{(l)}, y+l-1)}\). The necessary and sufficient condition for the admissibility condition of \(S_{(x_{n_l}^{(l)}, y+l-1)}\) to hold for the obtained label \({\varvec{k}}:=({\varvec{a}}, {\varvec{g}}, {\varvec{b}}^{(2)}, {\varvec{b}}^{(3)})\in G^{{\mathbb {E}}({\mathfrak {S}})}\) is

$$\begin{aligned} \begin{aligned}&a_{(x_{n_{l}}^{({l})}+1,y+l-1)-(x_{n_{l}}^{({l})}+1,y+l )}\left( \Psi _{{{\mathfrak {p}}}^{(r)}}({\varvec{a}})\right) ^{-1} \Psi _{{{\mathfrak {p}}}^{(l)}}({\varvec{a}}) \left( a_{(x_{n_{l}}^{({l})},y+l )-(x_{n_{l}}^{({l})}+1,y+l )}\right) ^{-1}\\&\quad = \left( k_{(x_{n_l}^{(l)}, y+l-1)-(x_{n_l}+1, y+l-1)}\right) ^{-1}k_{(x_{n_l}^{(l)}, y+l-1)-(x_{n_l}, y+l) }\\&\quad = a_{(x_{n_{l}}^{({l})}+1,y+l-1)-(x_{n_{l}}^{({l})}+1,y+l )} \left( a_{(x_{n_{l}}^{({l})},y+l )-(x_{n_{l}}^{({l})}+1,y+l )}\right) ^{-1}. \end{aligned} \end{aligned}$$
(53)

For the first equality, we used the fact that \({\varvec{k}}:=({\varvec{a}}, {\varvec{g}}, {\varvec{b}}^{(2)}, {\varvec{b}}^{(3)})\in G^{{\mathbb {E}}({\mathfrak {S}})}\) satisfies the admissibility conditions of all the squares but \(S_{(x_{n_l}^{(l)}, y+l-1)}\). This condition (53) is equivalent to (i). Hence, all the admissibility conditions are satisfied.

\(\square \)

4 Admissible Configurations in Areas Surrounded by a Certain Type of Loops

In this section, we count the number of admissible configurations in the area surrounded by a certain type of loops. We use the notation from Sects. 1, 2, 3 but we do not assume G to be abelian.

First we specify the kind of loops we consider. By a path with origin \((x,y)\in {\mathbb {Z}}^2\) going up-right direction, we mean a path of edges \({\mathfrak {p}}\) proceeding as follows

$$\begin{aligned} \begin{aligned}&(x, y)-(x, y+1)-\cdots -(x, y+m_1),\quad \text {move vertically up},\\&(x, y+m_1)-(x+1, y+m_1)-\cdots - (x+l_1, y+m_1)\quad \text {move horizontally right},\\&(x+l_1, y+m_1)-(x+l_1, y+m_1+1)-\cdots -(x+l_1, y+m_1+m_2), \quad \text {move vertically up},\\&\cdots \\&(x+l_1+\cdots l_k, y+m_1+\cdots m_k) -(x+l_1+\cdots l_k, y+m_1+\cdots m_k+1)\\&-\cdots -(x+l_1+\cdots l_k, y+m_1+\cdots m_k+m_{k+1}) \quad \text {move vertically up},\\&(x+l_1+\cdots l_k, y+m_1+\cdots m_k+m_{k+1})- (x+l_1+\cdots l_k+1, y+m_1+\cdots m_k+m_{k+1}) \\ {}&-\cdots - (x+l_1+\cdots l_k+l_{k+1}, y+m_1+\cdots m_k+m_{k+1}) \quad \text {move horizontally right},\\&\cdots \end{aligned} \end{aligned}$$
(54)

with \(m_1\in {\mathbb {N}}\cup \{0\}\), \(l_1\in {\mathbb {N}}\cup \{0\}\) \(m_k\in {\mathbb {N}}\), \(l_k\in {\mathbb {N}}\) \(k\ge 2\). When \(m_1=0\), we understand that the path first moves horizontally right before going vertically up. If \(l_1=0\), we understand that the path never goes up horizontally and parallel to y-axis. Note that all the edges have \(+1\) directions along the path. We denote by \({\mathfrak {P}}_{u,r}(x,y)\) the set of all infinitely long paths with origin \((x,y)\in {\mathbb {Z}}^2\) going up-right direction. We define the set of all paths with origin \((x,y)\in {\mathbb {Z}}^2\) going up-left direction \({\mathfrak {P}}_{u,l}(x,y)\), down-right direction \({\mathfrak {P}}_{d,r}(x,y)\), down-left direction \({\mathfrak {P}}_{d,l}(x,y)\) analogously. Set \(\mathfrak P(x,y):={\mathfrak {P}}_{u,r}(x,y)\cup {\mathfrak {P}}_{u,l}(x,y)\cup {\mathfrak {P}}_{d,r}(x,y)\cup {\mathfrak {P}}_{d,l}(x,y)\). For each \({{\mathfrak {p}}}\in {\mathfrak {P}}(x,y)\), we denote the corresponding parameters \(m_k\), \(l_k\)s above by \(m_k^{{{\mathfrak {p}}}}\), \(l_k^{{{\mathfrak {p}}}}\).

Let \({\mathfrak {p}}\in {\mathfrak {P}}_{u,r}(x,y)\) with description (54). We attach it a sequence of squares from below and above. The sequence from below is defined as follows. For each portion of \({{\mathfrak {p}}}\) in (54)

$$\begin{aligned} \begin{aligned}&(x+l_1+\cdots l_k, y+m_1+\cdots m_k) -(x+l_1+\cdots l_k, y+m_1+\cdots m_{k}+1)\\&-\cdots -(x+l_1+\cdots l_k, y+m_1+\cdots m_k+m_{k+1}) \quad \text {move vertically},\\ \end{aligned} \end{aligned}$$
(55)

we attach squares

$$\begin{aligned} \begin{aligned}&S_{(x+l_1+\cdots l_k, y+m_1+\cdots m_k)}- S_{(x+l_1+\cdots l_k, y+m_1+\cdots m_{k}+1)}\\&-\cdots -S_{(x+l_1+\cdots l_k, y+m_1+\cdots m_k+m_{k+1}-1)}. \end{aligned} \end{aligned}$$
(56)

For each portion of \({{\mathfrak {p}}}\) in (54)

$$\begin{aligned} \begin{aligned}&(x+l_1+\cdots l_k, y+m_1+\cdots m_k+m_{k+1}) \\ {}&- (x+l_1+\cdots l_k+1, y+m_1+\cdots m_k+m_{k+1}) \\ {}&-\cdots - (x+l_1+\cdots l_k+l_{k+1}, y+m_1+\cdots m_k+m_{k+1}) \quad \text {move horizontally}, \end{aligned} \end{aligned}$$
(57)

we attach squares

$$\begin{aligned} \begin{aligned}&S_{(x+l_1+\cdots l_k+1, y+m_1+\cdots m_k+m_{k+1}-1)} -S_{(x+l_1+\cdots l_k+2, y+m_1+\cdots m_k+m_{k+1}-1)}\\&\quad -\cdots - S_{(x+l_1+\cdots l_k+l_{k+1}-1, y+m_1+\cdots m_k+m_{k+1}-1)}\\ {}&\quad - S_{(x+l_1+\cdots l_k+l_{k+1}, y+m_1+\cdots m_k+m_{k+1}-1)}. \end{aligned} \end{aligned}$$
(58)

We denote by \(S_i^{b, {{\mathfrak {p}}}}\), \(i\in {\mathbb {N}}\) the sequence obtained in this way from \({{\mathfrak {p}}}\in {\mathfrak {P}}_{(u,r)}(x,y)\).

For the sequence from above, we set as follows. For each portion of \({{\mathfrak {p}}}\) in (54)

$$\begin{aligned} \begin{aligned}&(x+l_1+\cdots l_k, y+m_1+\cdots m_k) -(x+l_1+\cdots l_k, y+m_1+\cdots m_{k}+1)\\&\quad -\cdots -(x+l_1+\cdots l_k, y+m_1+\cdots m_k+m_{k+1}) \quad \text {move vertically},\\ \end{aligned} \end{aligned}$$
(59)

we attach squares

$$\begin{aligned} \begin{aligned}&S_{(x+l_1+\cdots l_k-1, y+m_1+\cdots m_k)}- S_{(x+l_1+\cdots l_k-1, y+m_1+\cdots m_{k}+1)}\\&\quad -\cdots -S_{(x+l_1+\cdots l_k-1, y+m_1+\cdots m_k+m_{k+1}-1)}. \end{aligned} \end{aligned}$$
(60)

For each portion of \({{\mathfrak {p}}}\) in (54)

$$\begin{aligned} \begin{aligned}&(x+l_1+\cdots l_k, y+m_1+\cdots m_k+m_{k+1}) \\ {}&\quad - (x+l_1+\cdots l_k+1, y+m_1+\cdots m_k+m_{k+1}) \\ {}&\quad -\cdots - (x+l_1+\cdots l_k+l_{k+1}, y+m_1+\cdots m_k+m_{k+1}) \quad \text {move horizontally}, \end{aligned} \end{aligned}$$
(61)

we attach squares

$$\begin{aligned} \begin{aligned}&S_{(x+l_1+\cdots l_k-1, y+m_1+\cdots m_k+m_{k+1})} -S_{(x+l_1+\cdots l_k, y+m_1+\cdots m_k+m_{k+1})}\\&\quad -\cdots - S_{(x+l_1+\cdots l_k+l_{k+1}-2, y+m_1+\cdots m_k+m_{k+1})}. \end{aligned} \end{aligned}$$
(62)

We denote by \(S_i^{a, {{\mathfrak {p}}}}\), \(i\in {\mathbb {N}}\) the sequence obtained in this way from \({{\mathfrak {p}}}\in {\mathfrak {P}}_{(u,r)}(x,y)\).

Analogously, for each \({{\mathfrak {p}}}\in {\mathfrak {P}}_{u,l}(x,y)\cup \mathfrak P_{d,r}(x,y)\cup {\mathfrak {P}}_{d,l}(x,y)\), we obtain sequences of squares from below and above \(S_i^{b, {{\mathfrak {p}}}}\), \(i\in {\mathbb {N}}\), \(S_i^{a, {{\mathfrak {p}}}}\), \(i\in {\mathbb {N}}\).

Now, we fix \(v_0=(x,y)\in {\mathbb {Z}}^2\) and take \({{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2\in \mathfrak P(x,y)\). Each of \({{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2\) is associated with a sequence of squares \(\{S_i^{b, {{\mathfrak {p}}}_1}\}_i\), \(\{S_i^{a, {{\mathfrak {p}}}_1}\}_i\),\(\{S_i^{b, {{\mathfrak {p}}}_2}\}_i\), \(\{S_i^{a, {{\mathfrak {p}}}_2}\}_i\) as above. Let \(n_0,m_0\in {\mathbb {N}}\) and \(N\in {\mathbb {N}}\) be large enough so that \(v_0\in \Lambda _{N-3}^{(n_0,m_0)}\). Then, each of \({{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2\) intersects with \(\partial {\hat{\Lambda }}_N^{(n_0,m_0)}\). Let \(w_{{{\mathfrak {p}}}_1}^{(N)}\) (resp. \(w_{{{\mathfrak {p}}}_2}^{(N)}\)) be the first vertex that \({{\mathfrak {p}}}_1\) (resp.\({{\mathfrak {p}}}_2\)) intersects with \(\partial \hat{\Lambda }_N^{(n_0,m_0)}\) when we proceed from \(v_0=(x,y)\). We denote by \({{\mathfrak {p}}}_1^{(N)}\) (resp. \({{\mathfrak {p}}}_2^{(N)}\)) the portion of \({{\mathfrak {p}}}_1\) (resp. \({{\mathfrak {p}}}_2\)) from \(v_0\) to \(w_{{{\mathfrak {p}}}_1}^{(N)}\) (resp \(w_{{{\mathfrak {p}}}_2}^{(N)}\)). We denote by \({\mathfrak {l}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\) (resp. \(\mathfrak l_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,-1}^{(N)}\)) the path of edges on \(\partial {{{\hat{\Lambda }}_{N}}^{(n_0,m_0)}}\) with origin \(w_{{{\mathfrak {p}}}_1}^{(N)}\) and the terminal \(w_{{{\mathfrak {p}}}_2}\), given by proceeding along \(\partial {{{\hat{\Lambda }}_{N}}^{(n_0,m_0)}}\) from \(w_{{{\mathfrak {p}}}_1}^{(N)}\) to \(w_{{{\mathfrak {p}}}_2}^{(N)}\) counterclockwise (resp. clockwise). We obtain a closed loop of edges \({\mathfrak {c}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,\sigma }^{(N)}\), \(\sigma =\pm 1\) as follows: Start from \(v_0\). Proceed along \({{\mathfrak {p}}}_1^{(N)}\) from \(v_0\) to \(w_{{{\mathfrak {p}}}_1}^{(N)}\). Proceed along \(\mathfrak l_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,\sigma }^{(N)}\) from \(w_{{{\mathfrak {p}}}_1}^{(N)}\) to \(w_{{{\mathfrak {p}}}_2}^{(N)}\). Proceed along \({{{{\mathfrak {p}}}_2}^{(N)}}^{-1}\), from \(w_{{{\mathfrak {p}}}_2}^{(N)}\) to \(v_0\).

Definition 4.1

Let \(v_0=(x,y)\in {\mathbb {Z}}^2\) and \(N_0, n_0,m_0\in {\mathbb {N}}\), \(\sigma =\pm 1\). We say two paths \({{\mathfrak {p}}}_1, {{\mathfrak {p}}}_2\in {\mathfrak {P}}(x,y)\) of edges are well separated with respect to \(N_0, n_0,m_0,\sigma \) if for any \(N_0\le N\in {\mathbb {N}}\), \((x,y)\in {\Lambda _N^{(n_0,m_0)}}\) holds and the loop \({\mathfrak {c}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,\sigma }^{(N)}= {{\mathfrak {p}}}_1^{(N)}{\mathfrak {l}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,\sigma }^{(N)}\left( {{\mathfrak {p}}}_2^{(N)}\right) ^{-1}\) is a simple closed loop and the area inside of it consists of a set of squares satisfying the condition S. We denote by \({\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,\sigma }^{(N)}\) the set of all squares inside of this closed loop \(\mathfrak c_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,\sigma }^{(N)}={{\mathfrak {p}}}_1^{(N)}\mathfrak l_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,\sigma }^{(N)}\left( {{\mathfrak {p}}}_2^{(N)}\right) ^{-1}\).

Setting 4.2

Let \(v_0=(x,y)\in {\mathbb {Z}}^2\) and \(N_0, n_0,m_0\in {\mathbb {N}}\). We consider three cases:

  1. (1)

    Well-separated \({{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2\in {\mathfrak {P}}_{(u,r)}(x,y)\) with respect to \(N_0, n_0,m_0,+1\). The path \({{\mathfrak {p}}}_2\) is above \({{\mathfrak {p}}}_1\), i.e., for all vertex \((s,t_1)\) on \({{\mathfrak {p}}}_1\), any vertex on \({{\mathfrak {p}}}_2\) of the form \((s,t_2)\) satisfies \(t_1<t_2\). Note that because \({{\mathfrak {p}}}_1\),\({{\mathfrak {p}}}_2\) are well separated, \(m_{{{\mathfrak {p}}}_1}=0\) and \(m_{{{\mathfrak {p}}}_2}\ne 0\). Because \((x, y-1)\notin {\mathbb {V}}\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\right) \), we have \((x,y)\in \partial {\mathbb {V}}\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\right) \). The set of squares

    $$\begin{aligned} \begin{aligned} \left( \{S_i^{b,{{\mathfrak {p}}}_1}\}_{i}\cup \{S_{(x,y-1)}\}\cup \{S_{(x-1,y-1)}\}\cup \{S_i^{a, {{\mathfrak {p}}}_2}\}_{i}\right) \cap {\mathcal {P}}_{{\Lambda _N^{(n_0,m_0)}}} \end{aligned} \end{aligned}$$
    (63)

    forms a sequence of squares \(\mathfrak T_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}:=(S_i^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)})\) along the path \(\left( {{\mathfrak {p}}}_1^{(N)}\right) ^{-1}{{\mathfrak {p}}}_2^{(N)}\).

  2. (2)

    Well-separated \({{\mathfrak {p}}}_1\in {\mathfrak {P}}_{(u,r)}(x,y)\), \({{\mathfrak {p}}}_2\in {\mathfrak {P}}_{(u,l)}(x,y)\), with respect to \(N_0, n_0,m_0,+1\). Note that \(m_{{{\mathfrak {p}}}_1}=0\) or \(m_{{{\mathfrak {p}}}_2}=0\). Because \((x, y-1)\notin {\mathbb {V}}\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\right) \), we have \((x,y)\in \partial {\mathbb {V}}\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\right) \). The set of squares

    $$\begin{aligned} \begin{aligned} \left( \{S_i^{b,{{\mathfrak {p}}}_1}\}_{i}\cup \{S_{(x,y-1)}\}\cup \{S_{(x-1,y-1)}\}\cup \{S_i^{b, {{\mathfrak {p}}}_2}\}_{i}\right) \cap {\mathcal {P}}_{{\Lambda _N^{(n_0,m_0)}}}, \end{aligned} \end{aligned}$$
    (64)

    forms a sequence of squares \(\mathfrak T_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}:=(S_i^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)})\) along the path \(\left( {{\mathfrak {p}}}_1^{(N)}\right) ^{-1}{{\mathfrak {p}}}_2^{(N)}\).

  3. (3)

    Well-separated \({{\mathfrak {p}}}_1\in {\mathfrak {P}}_{(u,r)}(x,y)\), \({{\mathfrak {p}}}_2\in {\mathfrak {P}}_{(d,l)}(x,y)\) with respect to \(N_0, n_0,m_0,+1\). We assume \(m_{{{\mathfrak {p}}}_2}=0\). Because of \(m_{{{\mathfrak {p}}}_2}=0\), \((x, y-1)\notin {\mathbb {V}}\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\right) \), and we have \((x,y)\in \partial {\mathbb {V}}\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\right) \). The set of squares

    $$\begin{aligned} \begin{aligned} \left( \{S_i^{b,{{\mathfrak {p}}}_1}\}_{i}\cup \{S_{(x,y-1)}\}\cup \{S_{(x-1,y-1)}\}\cup \{S_i^{b, {{\mathfrak {p}}}_2}\}_{i}\right) \cap {\mathcal {P}}_{{\Lambda _N^{(n_0,m_0)}}} \end{aligned} \end{aligned}$$
    (65)

    forms a sequence of squares \(\mathfrak T_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}:=(S_i^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)})_{i=1}^{M_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}\) along the path \(\left( {{\mathfrak {p}}}_1^{(N)}\right) ^{-1}{{\mathfrak {p}}}_2^{(N)}\).

We introduce

$$\begin{aligned} \begin{aligned}&{\mathfrak {B}}{\mathfrak {E}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}:= \left\{ e\in \hat{\Lambda }_N^{(n_0,m_0)}\Bigg | \begin{array}{l} \left( v_{e,-1}\in \mathbb V\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) , v_{e,+1}\notin \mathbb V\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) \right) \\ \text {or} \left( v_{e,+1}\in \mathbb V\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) , v_{e.-1}\notin \mathbb V\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) \right) \end{array} \right\} ,\\ \end{aligned} \end{aligned}$$
(66)

and

$$\begin{aligned} \begin{aligned}&{\mathfrak {O}}{\mathfrak {E}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}:= \hat{\Lambda }_N^{(n_0,m_0)}\setminus \left( {\mathfrak {B}}{\mathfrak {E}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}\cup \mathbb E\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) \right) \end{aligned} \end{aligned}$$
(67)

If all the vertices of a square S in \(\Lambda _N^{(n_0, m_0)}\) belong to \(\left( {\mathbb {V}}\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) \right) ^c\), then all the edges of S belong to \(\mathfrak {OE}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}\).

Remark 4.3

Under Setting 4.2, the following can be checked from the construction.

  1. (A)

    For each \(i=1,\ldots , M_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}-1\), \(S_i^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\) and \(S_{i+1}^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\) shares exactly one edge \( e_{i}^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\). This edge \( e_{i}^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\) belongs to \(\mathfrak {BE}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}\) and \(e_{i}^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\notin \partial {{{\hat{\Lambda }}_{N}}^{(n_0,m_0)}}\).

  2. (B)

    There exists exactly one edge \(e_{ M_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\) in \(S_{M_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\cap \partial {{{\hat{\Lambda }}_{N}}^{(n_0,m_0)}}\) satisfying the condition \(e_{ M_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\in {\mathfrak {B}}{\mathfrak {E}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}\). There exists exactly one edge \(e_0^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\) in \(S_{1}^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\cap \partial {{{\hat{\Lambda }}_{N}}^{(n_0,m_0)}}\) satisfying the condition \(e_{ 0}^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\in {\mathfrak {B}}{\mathfrak {E}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}\).

  3. (C)

    For \(i+1<j\), \(S_i^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\) and \(S_{j}^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\) do not share any edge. In particular, combining with (A), (B), \(e_{i}^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\ne e_{i-1}^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\) for \(1\le i\le M_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\).

  4. (D)

    Any square \(S_i^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\) in \(\mathfrak T_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}\) has exactly two distinct edges \(e_{i-1}^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}, e_{i}^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\) in \({\mathfrak {B}}{\mathfrak {E}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}\), and two other edges of \(S_i^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\) are in \(\mathbb E\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) \cup \mathfrak {OE}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)} \). Furthermore, we have \(\mathfrak {BE}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}=\{ e_{i}^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\}_{i=0}^{M_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}\).

  5. (E)

    For any \(v\in \partial \mathbb V\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) \setminus \partial {\Lambda _N^{(n_0,m_0)}}\), there exists a path \({{\mathfrak {p}}}_v\) in \({{{\hat{\Lambda }}_{N}}^{(n_0,m_0)}}{\setminus } \mathbb E\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) \) with origin v and target \(v_0=(x,y)\).

  6. (F)

    For \(v\in \partial {\Lambda _N^{(n_0,m_0)}}\), there is a path \({{\mathfrak {p}}}_v^1\) along \(\partial {{{\hat{\Lambda }}_{N}}^{(n_0,m_0)}}\) with origin v and target \((Nn_0,-Nm_0)\). There is a path \({{\mathfrak {p}}}_0\) in \({{{\hat{\Lambda }}_{N}}^{(n_0,m_0)}}{\setminus } \mathbb E\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) \) with origin \((Nn_0,-Nm_0)\) target \(v_0=(x,y)\).

  7. (G)

    Any square which is not in \(\mathbb E\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) \) nor \((S_i^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)})_{i=1}^{M_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}\) does not have vertices from \({\mathbb {V}}\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) \).

Lemma 4.4

Consider Setting 4.2. Given \({\varvec{m}}\in C_{\mathfrak {OE}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)} }\), \( {\varvec{k}}\in C_{\mathbb E\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }\) and \(g\in G\), there exists a unique \({\varvec{o}}:={\varvec{o}}\left( {\varvec{m}}, {\varvec{k}}, g\right) \in C_{{{{\hat{\Lambda }}_{N}}^{(n_0,m_0)}}}\) such that

$$\begin{aligned} \begin{aligned} {\varvec{o}}\vert _{{\mathfrak {O}}{\mathfrak {E}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}}={\varvec{m}},\quad {\varvec{o}}\vert _{{{\mathbb {E}}\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }}={\varvec{k}},\quad {\varvec{o}}\vert _{e_0^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}=g. \end{aligned} \end{aligned}$$
(68)

Furthermore, if

$$\begin{aligned} \left. \Psi ^{(x,y)}({\varvec{k}})\right| _{\partial {\mathbb V\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }} =\left. \Psi ^{(x,y)}({\varvec{h}})\right| _{\partial {\mathbb V\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }} \end{aligned}$$
(69)

for \( {\varvec{k}}, {\varvec{h}}\in C_{\mathbb E\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }\), then

$$\begin{aligned} \left. {\varvec{o}}\left( {\varvec{m}}, {\varvec{k}}, g\right) \right| _{\mathfrak {BE}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}} =\left. {\varvec{o}}\left( {\varvec{m}}, {\varvec{h}}, g\right) \right| _{{\mathfrak {B}}{\mathfrak {E}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}} \end{aligned}$$
(70)

for any \({\varvec{m}} \in C_{{\mathfrak {O}}{\mathfrak {E}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)} } \).

Remark 4.5

Recall Definition 3.5 for (69).

Proof

Existence

Fix \({\varvec{m}}\in C_{{\mathfrak {O}}{\mathfrak {E}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}} \), \( {\varvec{k}}\in C_{\mathbb E\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }\) and \(g\in G\) labeling edges in \({\mathfrak {O}}{\mathfrak {E}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)} \), \(\mathbb E\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) \), and \(e_0^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\), respectively. With this configuration, the admissibility condition holds on any squares in \(\mathfrak {OE}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)} \) or \(\mathbb E\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) \). Note that the squares in \({{{\hat{\Lambda }}_{N}}^{(n_0,m_0)}}\) which are not in \(\mathfrak {OE}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}\) nor in \(\mathbb E\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) \) are exactly \(S_i^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\), \(i=1,\ldots , {M_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}\). The only edges in \({{{\hat{\Lambda }}_{N}}^{(n_0,m_0)}}\) which are not yet labeled are \(\mathfrak {BE}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}{\setminus }\{e_0^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\}=\{ e_i^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\}_{i=1}^{{M_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}}\). We have to decide labels on these edges in a way that the admissibility condition holds on any of \(S_i^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\), \(i=1,\ldots , {M_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}\). But the admissibility conditions on \(S_i^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\), \(i=1,\ldots , {M_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}\) determine the labels on these edges \(\mathfrak {BE}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}{\setminus }\{e_0^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\}=\{ e_i^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\}_{i=1}^{{M_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}}\) uniquely: On the square \(S_1^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\) two edges of \(S_1^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\) are in \( {\mathbb E\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }\cup \mathfrak {OE}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)} \). Therefore, their label is already fixed by \({\varvec{m}}\) and \({\varvec{k}}\). One of the rest edges is \(e_0^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\), whose label is already fixed as g. The label of the last edge \(e_1^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\) is then determined uniquely by the admissibility condition. Next proceed to \(S_2^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\). Two edges of \(S_2^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\) are in \( {\mathbb E\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }\cup {\mathfrak {O}}{\mathfrak {E}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)} \), and the other edges are \(e_1^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)},e_2^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\). Hence, the first three of them have already fixed labels. The admissibility condition determines the label of \(e_2^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\) uniquely. Continuing this, we obtain the labels on \(\mathfrak {BE}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}{\setminus }\{e_0^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\}=\{ e_i^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\}_{i=1}^{{M_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}}\) satisfying the admissibility condition on \(S_i^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\), \(i=1,\ldots , {M_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}\).

Uniqueness and the last statement

Suppose that \( {\varvec{k}}, {\varvec{h}}\in C_{\mathbb E\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }\) satisfy

$$\begin{aligned} \left. \Psi ^{(x,y)}({\varvec{k}})\right| _{\partial {\mathbb V\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }} =\left. \Psi ^{(x,y)}({\varvec{h}})\right| _{\partial {\mathbb V\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }} \end{aligned}$$
(71)

and \({\varvec{o}}, \tilde{{\varvec{o}}}\in C_{{{{\hat{\Lambda }}_{N}}^{(n_0,m_0)}}}\) satisfy

$$\begin{aligned} \begin{aligned} {\varvec{o}}\vert _{{\mathfrak {O}}{\mathfrak {E}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)} }={\varvec{m}},\quad {\varvec{o}}\vert _{{{\mathbb {E}}\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }}={\varvec{k}},\quad {\varvec{o}}\vert _{e_0^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}=g,\\ \tilde{{\varvec{o}}}\vert _{{\mathfrak {O}}{\mathfrak {E}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)} }={\varvec{m}},\quad \tilde{{\varvec{o}}}\vert _{{{\mathbb {E}}\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }}={\varvec{h}},\quad \tilde{{\varvec{o}}}\vert _{e_0^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}=g. \end{aligned} \end{aligned}$$
(72)

By the above argument, from the admissibility, the label of \({\mathfrak {B}}{\mathfrak {E}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}{\setminus }\{e_0^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\}=\{ e_i^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\}_{i=1}^{{M_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}}\) for \({\varvec{o}}\), \(\tilde{{\varvec{o}}}\) is determined by this condition uniquely. We claim

$$\begin{aligned} {\varvec{o}} \vert _{e_i^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}=\tilde{{\varvec{o}}} \vert _{e_i^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}},\quad i=0,\ldots , {M_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}} \end{aligned}$$
(73)

corresponding to (70). From the construction of \(S_i^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\), \(e_i^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\), we note the following.

  1. (i)

    Let \(v, v'\in \partial {\mathbb V\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }\) be vertices next to each other, and let \(e\in {\hat{{\mathbb {Z}}}}^2\) be the edge \(e=v-v'\). Because of (71), the label of e in \({\varvec{o}}\), \(\tilde{{\varvec{o}}}\) coincides:

    $$\begin{aligned} {\varvec{o}}\vert _{e}= \left( \left. \Psi ^{(x,y)}({\varvec{k}})\right| _{v}\right) ^{-1}\left. \Psi ^{(x,y)}({\varvec{k}})\right| _{v'} =\left( \left. \Psi ^{(x,y)}({\varvec{h}})\right| _{v}\right) ^{-1}\left. \Psi ^{(x,y)}({\varvec{h}})\right| _{v'} =\tilde{{\varvec{o}}}\vert _e. \end{aligned}$$
    (74)
  2. (ii)

    \({\varvec{o}}\vert _{{\mathfrak {O}}{\mathfrak {E}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)} }={\varvec{m}}=\tilde{{\varvec{o}}}\vert _{{\mathfrak {O}}{\mathfrak {E}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)} }\).

  3. (iii)

    If \(e_i^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\) and \(e_{i-1}^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\) are parallel, then the edges of the square \(S_i^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\) are \(e_i^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\), \(e_{i-1}^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\), and one edge in \({\mathbb E\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }\), (with vertices in \(\partial {\mathbb {V}}\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) \)) one edge in \(\mathfrak {OE}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)} \). From (i) and (ii), it means all the edges in \(S_i^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\) except for \(e_i^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\), \(e_{i-1}^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\) have the same label for \(\tilde{{\varvec{o}}}\) and \({\varvec{o}}\). By the admissibility condition on \(S_i^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\), it means that if \({\varvec{o}}\vert _{e_{i-1}^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}=\tilde{{\varvec{o}}}\vert _{e_{i-1}^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}\), then we have \({\varvec{o}}\vert _{e_i^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}=\tilde{{\varvec{o}}}\vert _{e_i^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}\).

  4. (iv)

    If \(e_i^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\) and \(e_{i-1}^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\) share a point in \(\partial {\mathbb V\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }\), the edges in \(S_i^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\) other than \(e_i^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\), \(e_{i-1}^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\) belong to \(\mathfrak {OE}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)} \). From (ii), it means all the edges in \(S_i^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\) except for \(e_i^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\), \(e_{i-1}^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\) have the same label for \(\tilde{{\varvec{o}}}\) and \({\varvec{o}}\). By the admissibility condition, it means that if \({\varvec{o}}\vert _{e_{i-1}^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}=\tilde{{\varvec{o}}}\vert _{e_{i-1}^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}\), then we have \({\varvec{o}}\vert _{e_i^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}=\tilde{{\varvec{o}}}\vert _{e_i^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}\).

  5. (v)

    If \(e_i^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\) and \(e_{i-1}^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\) share a point in \({\Lambda _N^{(n_0,m_0)}}{\setminus } {\mathbb V\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) } \), the edges \(e,{\tilde{e}}\) in \(S_i^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\) other than \(e_i^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\), \(e_{i-1}^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\) belong to \({{\mathbb E\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }}\). We may assume e and \(e_{i-1}^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\) share a point \(v\in \partial \mathbb V\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) \), and \({\tilde{e}}\) and \(e_{i}^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\) share a point \(v'\in \partial \mathbb V\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) \). Because of (71), by the admissibility condition, we have

    $$\begin{aligned}{} & {} \left( \left( {\varvec{o}}\right) _{{e_{i-1}^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}} \right) ^{\sigma _{i-1}} \left( \left( {\varvec{o}}\right) _{{e_{i}^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}} \right) ^{\sigma _{i}} =\left( \left. \Psi ^{(x,y)}({\varvec{k}})\right| _{v}\right) ^{-1} \left. \Psi ^{(x,y)}({\varvec{k}})\right| _{v'}\nonumber \\{} & {} \quad =\left( \left. \Psi ^{(x,y)}({\varvec{h}})\right| _{v}\right) ^{-1} \left. \Psi ^{(x,y)}({\varvec{h}})\right| _{v'} =\left( \left( \tilde{{\varvec{o}}}\right) _{{e_{i-1}^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}} \right) ^{\sigma _{i-1}} \left( \left( \tilde{{\varvec{o}}}\right) _{{e_{i}^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}} \right) ^{\sigma _{i}},\nonumber \\ \end{aligned}$$
    (75)

    with some \(\sigma _{i-1},\sigma _i=\pm 1\). It means that if \({\varvec{o}}\vert _{e_{i-1}^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}=\tilde{{\varvec{o}}}\vert _{e_{i-1}^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}\), then we have \({\varvec{o}}\vert _{e_i^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}=\tilde{{\varvec{o}}}\vert _{e_i^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}\).

We have \({\varvec{o}} \vert _{e_0^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}=g=\tilde{{\varvec{o}}} \vert _{e_0^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}\). Then with the above observation, we have \({\varvec{o}}\vert _{e_i^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}=\tilde{{\varvec{o}}}\vert _{e_i^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}\) for all \(i=0,\ldots ,{M_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}\) inductively, proving the claim (73). \(\square \)

Lemma 4.6

Consider Setting 4.2. For \({\varvec{h}}, {\varvec{k}}\in G^{\mathbb E\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }\), the followings are equivalent.

  1. (i)

    For any \({\varvec{m}}\in C_{{\mathfrak {O}}{\mathfrak {E}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)} }\), and \(g_0\in G^{e_0^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}\), there exists a unique \(\left( g_{i}\right) _{i=1,\ldots ,{M_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}}\in G^{\{ e_i^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\}_{i=1}^{{M_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}}}\), such that

    $$\begin{aligned} \begin{aligned} \left( {\varvec{k}},{\varvec{m}}, {\varvec{g}}\right) ,\quad \left( {\varvec{h}},{\varvec{m}}, {\varvec{g}}\right) \in C_{{{{\hat{\Lambda }}_{N}}^{(n_0,m_0)}}}, \end{aligned} \end{aligned}$$
    (76)

    and

    $$\begin{aligned} \begin{aligned} \left( {\varvec{k}},{\varvec{m}}, {\varvec{g}} \right) _{\partial {{{{\hat{\Lambda }}_{N}}^{(n_0,m_0)}}} } =\left( {\varvec{h}},{\varvec{m}}, {\varvec{g}}\right) _{\partial {{{{\hat{\Lambda }}_{N}}^{(n_0,m_0)}}}}. \end{aligned} \end{aligned}$$
    (77)

    Here, we set \({\varvec{g}}:=(g_{i})_{i=0}^{{M_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}}\in G^{{\mathfrak {B}}{\mathfrak {E}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}}\).

  2. (ii)

    There exist \({\varvec{m}}\in C_{{\mathfrak {O}}{\mathfrak {E}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)} }\), and \(g_0\in G^{e_0^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}\), which allow unique \(\left( g_{i}\right) _{i=1,\ldots ,{M_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}}\in G^{\{ e_i^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\}_{i=1}^{{M_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}}}\), such that

    $$\begin{aligned} \begin{aligned} \left( {\varvec{k}},{\varvec{m}}, {\varvec{g}}\right) ,\quad \left( {\varvec{h}},{\varvec{m}}, {\varvec{g}}\right) \in C_{{{{\hat{\Lambda }}_{N}}^{(n_0,m_0)}}}, \end{aligned} \end{aligned}$$
    (78)

    and

    $$\begin{aligned} \begin{aligned} \left( {\varvec{k}},{\varvec{m}}, {\varvec{g}} \right) _{\partial {{{{\hat{\Lambda }}_{N}}^{(n_0,m_0)}}} } =\left( {\varvec{h}},{\varvec{m}}, {\varvec{g}}\right) _{\partial {{{{\hat{\Lambda }}_{N}}^{(n_0,m_0)}}}}. \end{aligned} \end{aligned}$$
    (79)

    Here, we set \({\varvec{g}}:=(g_{i})_{i=0}^{{M_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}}\in G^{{\mathfrak {B}}{\mathfrak {E}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}}\).

  3. (iii)

    \( {\varvec{k}}, {\varvec{h}}\in C_{\mathbb E\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }\) and \( \Psi ^{(x,y)}({\varvec{h}})\vert _{{\partial {\mathbb V\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }}}= \Psi ^{(x,y)}({\varvec{k}})\vert _{{\partial {\mathbb V\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }}} \).

Remark 4.7

In (iii), \(\Psi ^{(x,y)}({\varvec{h}})\), \( \Psi ^{(x,y)}({\varvec{k}})\) are well defined because \( {\varvec{k}}, {\varvec{h}}\in C_{\mathbb E\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }\).

Proof

(i)\(\Rightarrow \)(ii): Note that \(C_{\mathfrak {OE}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}}\) is non-empty. For example, we may set all the labels in \({\mathfrak {O}}{\mathfrak {E}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}\) to be the unit of the group G. Hence, (i) implies (ii).

(ii)\(\Rightarrow \)(iii): Let \({\varvec{m}}\in C_{\mathfrak {OE}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)} }\), and \(g_0\in G^{e_0^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}\) satisfying (ii) for \({\varvec{k}},{\varvec{h}}\). The first part \( {\varvec{k}}, {\varvec{h}}\in C_{{\mathbb E\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }}\) is trivial. We would like to show \( \Psi ^{(x,y)}({\varvec{h}})\vert _{{{\partial {\mathbb V\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }}}}= \Psi ^{(x,y)}({\varvec{k}})\vert _{{{\partial {\mathbb V\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }}}} \). For any \(v\in \partial \mathbb V\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) {\setminus }\partial {\Lambda _N^{(n_0,m_0)}}\), as in (E) Remark 4.3 there exists a path \({{\mathfrak {p}}}_v\) in \({{{\hat{\Lambda }}_{N}}^{(n_0,m_0)}}{\setminus } \mathbb E\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) \) with origin v and target \(v_0=(x,y)\). Note because of \( {\left( {\varvec{k}},{{\varvec{m}}, {\varvec{g}}}\right) , \left( {\varvec{h}},{{\varvec{m}}, {\varvec{g}}}\right) \in C_{{\hat{\Lambda }}_{N}^{(n_0,m_0)}}} \) that

$$\begin{aligned} \begin{aligned} \Psi ^{(x,y)}({\varvec{h}})_{v}=\Psi _{{{\mathfrak {p}}}_v^{-1}}\left( {\varvec{h}},{{\varvec{m}}, {\varvec{g}}}\right) ,\\ \Psi ^{(x,y)}({\varvec{k}})_{v}=\Psi _{{{\mathfrak {p}}}_v^{-1}}\left( {\varvec{k}},{{\varvec{m}}, {\varvec{g}}}\right) .\\ \end{aligned} \end{aligned}$$
(80)

Because the path \({{\mathfrak {p}}}_v\) is in \({{{\hat{\Lambda }}_{N}}^{(n_0,m_0)}}{\setminus } \mathbb E\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) \), we have

$$\begin{aligned} \left( {\varvec{h}},{{\varvec{m}}, {\varvec{g}}}\right) _{{{\mathfrak {p}}}_v}=\left( {\varvec{m}}, {\varvec{g}}\right) _{{{\mathfrak {p}}}_v}=\left( {\varvec{k}},{{\varvec{m}}, {\varvec{g}}}\right) _{{{\mathfrak {p}}}_v}. \end{aligned}$$
(81)

Hence from (80), we obtain

$$\begin{aligned} \begin{aligned} \Psi ^{(x,y)}({\varvec{h}})_{v}=\Psi ^{(x,y)}({\varvec{k}})_{v},\quad v\in \partial \mathbb V\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) \setminus \partial {\Lambda _N^{(n_0,m_0)}},\quad . \end{aligned} \end{aligned}$$
(82)

For \(v\in \partial {\Lambda _N^{(n_0,m_0)}}\), as in (F) Remark 4.3, there is a path \({{\mathfrak {p}}}_v^1\) along \(\partial {{{\hat{\Lambda }}_{N}}^{(n_0,m_0)}}\), with origin v and target \((Nn_0,-Nm_0)\). There is a path \({{\mathfrak {p}}}_0\) in \({{{\hat{\Lambda }}_{N}}^{(n_0,m_0)}}{\setminus } \mathbb E\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) \) with origin \((Nn_0,-Nm_0)\) and target \(v_0=(x,y)\). On the path \({{\mathfrak {p}}}_v^1\), the value of \(({\varvec{k}},{{\varvec{m}}, {\varvec{g}}})\) and \( ({\varvec{h}},{{\varvec{m}}, {\varvec{g}}})\) is the same because of (79). Because the path \({{\mathfrak {p}}}_0\) is in \({{{\hat{\Lambda }}_{N}}^{(n_0,m_0)}}{\setminus } \mathbb E\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) \), we again have

$$\begin{aligned} \left( {\varvec{h}},{{\varvec{m}}, {\varvec{g}}}\right) _{{{\mathfrak {p}}}_0}=\left( {\varvec{m}}, {\varvec{g}}\right) _{{{\mathfrak {p}}}_0}=\left( {\varvec{k}},{{\varvec{m}}, {\varvec{g}}}\right) _{{{\mathfrak {p}}}_0}. \end{aligned}$$
(83)

Hence from (78), we get

$$\begin{aligned} \begin{aligned}&\Psi ^{(x,y)}({\varvec{h}})_{v} =\Psi _{\left( {{\mathfrak {p}}}_v^1{{\mathfrak {p}}}_0\right) ^{-1}}\left( {\varvec{h}},{{\varvec{m}}, {\varvec{g}}}\right) =\Psi _{\left( {{\mathfrak {p}}}_v^1{{\mathfrak {p}}}_0\right) ^{-1}}\left( {\varvec{k}},{{\varvec{m}}, {\varvec{g}}}\right) =\Psi ^{(x,y)}({\varvec{k}})_{v},\\ {}&\quad v\in \partial {\Lambda _N^{(n_0,m_0)}}. \end{aligned} \end{aligned}$$
(84)

This completes the proof.

(iii) \(\Rightarrow \) (i): Fix any \({\varvec{m}}\in C_{\mathfrak {OE}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)} }\), and \(g_0\in G^{e_0^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}\). Let \({\varvec{o}}\left( {\varvec{m}}, {\varvec{k}}, g_0\right) , {\varvec{o}}\left( {\varvec{m}}, {\varvec{h}}, g_0\right) \in C_{{{{\hat{\Lambda }}_{N}}^{(n_0,m_0)}}}\) be the configurations obtained in Lemma 4.4 from \({\varvec{m}}\in C_{\mathfrak {OE}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}}\), \(g_0\in G^{e_0^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,(N)}}\), \( {\varvec{k}},{\varvec{h}}\in C_{\mathbb E\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }\), respectively. By Lemma 4.4, (iii) implies

$$\begin{aligned} {\varvec{g}}:=\left. {\varvec{o}}\left( {\varvec{m}}, {\varvec{k}}, g_0\right) \right| _{\mathfrak {BE}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}} =\left. {\varvec{o}}\left( {\varvec{m}}, {\varvec{h}}, g_0\right) \right| _{{\mathfrak {B}}{\mathfrak {E}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}} \in G^{{\mathfrak {B}}{\mathfrak {E}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}}, \end{aligned}$$
(85)

and

$$\begin{aligned} \begin{aligned} \left( {\varvec{k}},{\varvec{m}}, {\varvec{g}}\right) ={\varvec{o}}\left( {\varvec{m}}, {\varvec{k}}, g_0\right) ,\quad \left( {\varvec{h}},{\varvec{m}}, {\varvec{g}}\right) ={\varvec{o}}\left( {\varvec{m}}, {\varvec{h}}, g_0\right) \in C_{{{{\hat{\Lambda }}_{N}}^{(n_0,m_0)}}}. \end{aligned} \end{aligned}$$
(86)

By Lemma 4.4, this \(\left( g_{i}\right) _{i=1,\ldots ,{M_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}}\in G^{\{ e_i^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}\}_{i=1}^{{M_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}}}\) is the unique configuration satisfying (76). Furthermore for any \(e\in \partial {{{{\hat{\Lambda }}_{N}}^{(n_0,m_0)}}}\cap { {\mathbb E\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }}\), e is of the form \(e=v-v'\) with \(v, v'\in {\partial {\mathbb V\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }}\). Therefore, by (iii), we have

$$\begin{aligned} \begin{aligned} \left( {\varvec{k}},{\varvec{m}}, {\varvec{g}}\right) _{e} =\left( \Psi ^{(x,y)}({\varvec{k}})\right) _{v}^{-1} \Psi ^{(x,y)}({\varvec{k}})_{v'} =\left( \Psi ^{(x,y)}({\varvec{h}})\right) _{v}^{-1} \Psi ^{(x,y)}({\varvec{h}})_{v'} =\left( {\varvec{h}},{\varvec{m}}, {\varvec{g}}\right) _{e}. \end{aligned} \end{aligned}$$
(87)

For any \(e\in \partial {{{{\hat{\Lambda }}_{N}}^{(n_0,m_0)}}}{\setminus } { {\mathbb E\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }}\), we have \(e\in {\mathfrak {O}}{\mathfrak {E}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}\cup \mathfrak {BE}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}\). But we have

$$\begin{aligned} \begin{aligned} \left( {\varvec{k}},{\varvec{m}}, {\varvec{g}}\right) \vert _{\mathfrak {BE}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}}={\varvec{o}}\left( {\varvec{m}}, {\varvec{k}}, g_0\right) \vert _{{\mathfrak {B}}{\mathfrak {E}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}} ={\varvec{o}}\left( {\varvec{m}}, {\varvec{h}}, g_0\right) \vert _{{\mathfrak {B}}{\mathfrak {E}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}}=\left( {\varvec{h}},{\varvec{m}}, {\varvec{g}}\right) \vert _{{\mathfrak {B}}{\mathfrak {E}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}}, \end{aligned} \end{aligned}$$
(88)

from (85), and

$$\begin{aligned} \begin{aligned}&\left( {\varvec{k}},{\varvec{m}}, {\varvec{g}}\right) \vert _{\mathfrak {OE}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}}={\varvec{o}}\left( {\varvec{m}}, {\varvec{k}}, g_0\right) \vert _{{\mathfrak {O}}{\mathfrak {E}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}} ={\varvec{m}}\vert _{{\mathfrak {O}}{\mathfrak {E}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}} \\ {}&\quad ={\varvec{o}}\left( {\varvec{m}}, {\varvec{h}}, g_0\right) \vert _{{\mathfrak {O}}{\mathfrak {E}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}}=\left( {\varvec{h}},{\varvec{m}}, {\varvec{g}}\right) \vert _{{\mathfrak {O}}{\mathfrak {E}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}}. \end{aligned} \end{aligned}$$
(89)

Hence, we have

$$\begin{aligned} \begin{aligned} \left( {\varvec{k}},{\varvec{m}}, {\varvec{g}} \right) _e =\left( {\varvec{h}},{\varvec{m}}, {\varvec{g}}\right) _e,\quad e \in \partial {{{{\hat{\Lambda }}_{N}}^{(n_0,m_0)}}}\setminus { {\mathbb E\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }}. \end{aligned} \end{aligned}$$
(90)

Hence, we obtain (77) \( \left( {\varvec{k}},{\varvec{m}}, {\varvec{g}} \right) _{\partial {{{{\hat{\Lambda }}_{N}}^{(n_0,m_0)}}} } =\left( {\varvec{h}},{\varvec{m}}, {\varvec{g}}\right) _{\partial {{{{\hat{\Lambda }}_{N}}^{(n_0,m_0)}}}} \).

\(\square \)

Consider Setting 4.2. Recall the loop \(\mathfrak c_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}={{\mathfrak {p}}}_1^{(N)}\mathfrak l_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\left( {{\mathfrak {p}}}_2^{(N)}\right) ^{-1}\) from Definition 4.1. Because \({{\mathfrak {p}}}_1\) and \({{\mathfrak {p}}}_2\) are well separated, the \({\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\), the squares inside of the loop \({\mathfrak {c}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\) satisfy the condition S. Therefore, there are two distinct self-avoiding paths \({{\mathfrak {p}}}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}^{(l)}\), \({{\mathfrak {p}}}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}^{(r)}\) given in Definition 3.2 for \({\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \). Note that \(\mathfrak c_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}=\left( {{\mathfrak {p}}}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}^{(l)} \right) ^{-1}{{\mathfrak {p}}}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}^{(r)}\). We recall from Sect. 3 that \({\mathcal {E}}^{(4)}\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\right) \) denotes the set of all edges in \({\mathfrak {c}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\). We denote by \({\mathbb {V}}\left( {\mathfrak {c}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\right) \) the set of all vertices in \({\mathfrak {c}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\). We also set

$$\begin{aligned} \begin{aligned}&{J}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2},{N} }:={\mathbb {V}}\left( \mathfrak c_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\right) \setminus {{\partial {\mathbb V\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }}}. \end{aligned} \end{aligned}$$
(91)

Because \(v_0\in {{\partial {\mathbb V\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }}}\), we have \(v_0\notin {J}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2},{N} }\). We consider the following set of configurations

$$\begin{aligned} \begin{aligned}&{\mathcal {F}}{\mathcal {B}}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}} :=\left\{ {\varvec{a}}\in G^{{\mathcal {E}}^{(4)}\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\right) }\mid \Psi _{{{\mathfrak {p}}}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}^{(l)}}({\varvec{a}})=\Psi _{{{\mathfrak {p}}}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}^{(r)}}({\varvec{a}}) \right\} ,\\&\mathcal{P}\mathcal{T}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}:= \left\{ {\varvec{t}}\in G^{\partial \mathbb V\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }\mid t_{v_0}=e \right\} . \end{aligned} \end{aligned}$$
(92)

Note for any \({\varvec{a}}\in \mathcal{F}\mathcal{B}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}\) and any \(w\in {\mathbb {V}}\left( {\mathfrak {c}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\right) \), the value \(\Psi _{\tilde{{{\mathfrak {q}}}}_w}({\varvec{a}})\) is independent of the choice of the path \(\tilde{{{\mathfrak {q}}}}_w\) in \({\mathcal {E}}^{(4)}\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\right) \) with origin \(v_0\) and target w. We denote this value by \(\left( \Psi _{{\mathcal {E}}^{(4)}\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\right) }^{(x, y)}({\varvec{a}})\right) _w\) and set

$$\begin{aligned} \begin{aligned} \Psi _{{\mathcal {E}}^{(4)}\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\right) }^{(x, y)}({\varvec{a}}):= \left( \left( \Psi _{{\mathcal {E}}^{(4)}\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\right) }^{(x, y)}({\varvec{a}})\right) _w\right) _{w\in {\mathbb {V}}\left( {\mathfrak {c}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\right) }. \end{aligned} \end{aligned}$$
(93)

The restriction of this to \({{\partial {\mathbb V\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }}}\) is denoted by

$$\begin{aligned} \Psi _{{{\partial {\mathbb V\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }}}}^{(x, y)}({\varvec{a}}) := \left. \Psi _{{\mathcal {E}}^{(4)}\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\right) }^{(x, y)}({\varvec{a}}) \right| _{{{\partial {\mathbb V\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }}}}. \end{aligned}$$
(94)

Lemma 4.8

Consider Setting 4.2. Then, there is a bijection

$$\begin{aligned} \begin{aligned} \Xi _{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}: \mathcal{P}\mathcal{T}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}} \times G^{\left| {J}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}\right| }\rightarrow \mathcal{F}\mathcal{B}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}} \end{aligned} \end{aligned}$$
(95)

such that

$$\begin{aligned} \begin{aligned} \Psi _{{{\partial {\mathbb V\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }}}}^{(x, y)}\left( \Xi _{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}({\varvec{t}}, {\varvec{u}})\right) = {\varvec{t}} ,\quad \left( {\varvec{t}},{\varvec{u}}\right) \in \mathcal{P}\mathcal{T}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}} \times G^{\left| {J}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}\right| }. \end{aligned} \end{aligned}$$
(96)

Remark 4.9

If \({J}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2},{N} }=\emptyset \), \(\mathcal{P}\mathcal{T}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}} \times G^{\left| {J}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}\right| }\) should be understood as \(\mathcal{P}\mathcal{T}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}\).

Proof

1. First we consider the case \({J}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2},{N} }\ne \emptyset \). We fix some \({\check{v}}\in {J}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2},{N} }\). Let \(\check{f}_1\), \({\check{f}}_2\) be the only two edges in \({\mathcal {E}}^{(4)}\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\right) \) which have \({\check{v}}\) as their vertices. Let \({\check{v}}_1\) (resp. \({\check{v}}_2\) ) be the vertex of \({\check{f}}_1\) (resp. \({\check{f}}_2\)) which is not \({\check{v}}\). We may assume that as we proceed with respect to the direction given by the loop \({\mathfrak {c}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\), we encounter \({\check{v}}_1, {\check{v}}, {\check{v}}_2\) in succession. For \(e\in {\mathcal {E}}^{(4)}\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\right) \), set \(\sigma _{e}:=+1\) (resp. \(\sigma _{e}:=-1\)) if \(\mathfrak c_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\) goes through e in the positive (resp. negative) direction. We also denote by \({\check{{{\mathfrak {q}}}}}\) the path of edges obtained from the loop \({\mathfrak {c}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\) by removing edges \({\check{f}}_1\), \({\check{f}}_2\). The direction of \({\check{{{\mathfrak {q}}}}}\) is inherited from that of \(\mathfrak c_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\).

For each \({\varvec{z}}\in \mathcal{P}\mathcal{T}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}\times G^{{J}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2},{N} }{\setminus } \{{\check{v}}\}}\), define

$$\begin{aligned} {\varvec{s}}({\varvec{z}}):=(s_e({\varvec{z}}))_{e\in {\check{{{\mathfrak {q}}}}} }\;\in G^{{\check{{{\mathfrak {q}}}}}} \end{aligned}$$

by

$$\begin{aligned} \begin{aligned} s_e({\varvec{z}}):=\left( \left( z_{v_{e,-\sigma _e}}\right) ^{-1}z_{v_{e,\sigma _e}}\right) ^{\sigma _e},\quad e\in {\check{{{\mathfrak {q}}}}}. \end{aligned} \end{aligned}$$
(97)

Furthermore, for each \( {\varvec{z}}\in \mathcal{P}\mathcal{T}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}\times G^{{J}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2},{N} }{\setminus } \{{\check{v}}\}}\) and \(u\in G^{{\check{f}}_1}\), we set \({\varvec{a}}({\varvec{z}},u):=(a_e({\varvec{z}}, u))_{e\in {\mathcal {E}}^{(4)}\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\right) }\in G^{{\mathcal {E}}^{(4)}\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\right) }\) by

$$\begin{aligned} \begin{aligned} a_{e}({\varvec{z}}, u) :=\left\{ \begin{gathered} s_e({\varvec{z}}),\quad \text {if}\;\;e\in {\check{{{\mathfrak {q}}}}} ,\\ u,\quad \text {if}\;\;e={\check{f}}_1,\\ \left( \Psi _{{\check{{{\mathfrak {q}}}}}}\left( {\varvec{s}}({\varvec{z}})\right) u^{\sigma _{{\check{f}}_1}} \right) ^{-\sigma _{{\check{f}}_2}} ,\quad \text {if}\;\;e={\check{f}}_2 \end{gathered} \right. ,\quad \quad e\in {\mathcal {E}}^{(4)}\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\right) . \end{aligned} \end{aligned}$$
(98)

We claim \({\varvec{a}}({\varvec{z}}, u)\in \mathcal{F}\mathcal{B}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}\). Note that

$$\begin{aligned} \begin{aligned} \Psi _{{\check{{{\mathfrak {q}}}}}}\left( {\varvec{s}}({\varvec{z}})\right) u^{\sigma _{{\check{f}}_1}}\cdot \left( \left( \Psi _{{\check{{{\mathfrak {q}}}}}}\left( {\varvec{s}}({\varvec{z}})\right) u^{\sigma _{{\check{f}}_1}} \right) ^{-\sigma _{{\check{f}}_2}} \right) ^{\sigma _{{\check{f}}_2}} =e. \end{aligned} \end{aligned}$$
(99)

By the choice of \(a_{{\check{f}}_2}({\varvec{z}}, u)\), this means \(\Psi _{{{\mathfrak {p}}}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}^{(l)}}({\varvec{a}}({\varvec{z}}, u))=\Psi _{{{\mathfrak {p}}}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}^{(r)}}({\varvec{a}}({\varvec{z}}, u))\). Hence, we prove the claim \({\varvec{a}}({\varvec{z}}, u)\in \mathcal{F}\mathcal{B}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}\) and obtain a map

$$\begin{aligned} \begin{aligned} \Xi _{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}: \mathcal{P}\mathcal{T}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}\times G^{{J}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2},{N} }\setminus \{{\check{v}}\}}\times G^{{\check{f}}_1}\ni \left( {\varvec{t}},{\varvec{r}}, u\right) \mapsto {\varvec{a}}({\varvec{t}},{\varvec{r}}, u)\in \mathcal{F}\mathcal{B}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}. \end{aligned} \end{aligned}$$
(100)

We regard \(G^{{J}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2},{N} }{\setminus } \{\check{v}\}}\times G^{{\check{f}}_1}\) as \( G^{\left| {J}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}\right| }\) and obtain the map (95).

Because \(\Xi _{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}({\varvec{t}},{\varvec{r}}, u)={\varvec{a}}({\varvec{t}},{\varvec{r}}, u)\in \mathcal{F}\mathcal{B}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}} \), from (93), we can define \( \Psi _{{\mathcal {E}}^{(4)}\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\right) }^{(x, y)}\left( \Xi _{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}({\varvec{t}},{\varvec{r}}, u) \right) \). In particular, for any \(w\in {\mathbb {V}}\left( \mathfrak c_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\right) {\setminus }\{{\check{v}}\}\), let \({{\mathfrak {q}}}_w\) be the portion of \({\mathfrak {c}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\) from \(v_0=(x,y)\) to w which does not go through \({\check{v}}\). Then, we have

$$\begin{aligned} \begin{aligned}&\Psi _{{\mathcal {E}}^{(4)}\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\right) }^{(x, y)}\left( \Xi _{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}({\varvec{t}},{\varvec{r}}, u) \right) _w =\Psi _{{{\mathfrak {q}}}_w}\left( \Xi _{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}({\varvec{t}},{\varvec{r}}, u) \right) _w \\&=\left( \left( {\varvec{t}},{\varvec{r}}\right) _{v_0}\right) ^{-1}\left( {\varvec{t}},{\varvec{r}}\right) _w=\left( {\varvec{t}},{\varvec{r}}\right) _w, \end{aligned} \end{aligned}$$
(101)

by the definition of \({\varvec{s}}(\left( {\varvec{t}},{\varvec{r}}\right) )\) and the fact that \(\left( {\varvec{t}},{\varvec{r}}\right) _{v_0}=e\). In particular, (96) holds.

The map \( \Xi _{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}\) is an injection. In fact, if \({\varvec{a}}({\varvec{t}}^{(1)},{\varvec{r}}^{(1)}, u^{(1)}) ={\varvec{a}}({\varvec{t}}^{(2)},{\varvec{r}}^{(2)}, u^{(2)})\), then clearly \(u^{(1)}=u^{(2)}\) and (101) implies \(\left( {\varvec{t}}^{(1)},{\varvec{r}}^{(1)}\right) =\left( {\varvec{t}}^{(2)},{\varvec{r}}^{(2)}\right) \).

It is also a surjection. In fact, for any \({\varvec{a}}=(a_e)_{e\in {\mathcal {E}}^{(4)}\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\right) }\in \mathcal{F}\mathcal{B}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}\), set

$$\begin{aligned} \begin{aligned}&z_w:=\Psi _{{{\mathfrak {q}}}_w}({\varvec{a}}),\quad w\in {\mathbb {V}}\left( \mathfrak c_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\right) \setminus \{{\check{v}}\} ,\\&u:=a_{{\check{f}}_1} \end{aligned} \end{aligned}$$
(102)

with \({{\mathfrak {q}}}_w\) above. Then, \(z_{v_0}=e\) and we have \({\varvec{z}}:=(z_w)_{ w\in {\mathbb {V}}\left( \mathfrak c_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\right) {\setminus }\{{\check{v}}\}}\in \mathcal{P}\mathcal{T}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}\times G^{{J}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2},{N} }{\setminus } \{{\check{v}}\}}\). For this \({\varvec{z}}\), for any \(e\in {\mathcal {E}}^{(4)}\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\right) {\setminus }\{{\check{f}}_1,{\check{f}}_2\}\), we have

$$\begin{aligned} \begin{aligned} s_e({\varvec{z}})=\left( \left( z_{v_{e,-\sigma _e}}\right) ^{-1}z_{v_{e,\sigma _e}}\right) ^{\sigma _e} =\left( \left( \Psi _{{{\mathfrak {q}}}_{v_{e,-\sigma _e}}}({\varvec{a}})\right) ^{-1} \Psi _{{{\mathfrak {q}}}_{v_{e,\sigma _e}}}({\varvec{a}})\right) ^{\sigma _e} =a_e. \end{aligned} \end{aligned}$$
(103)

Furthermore, from (103), we have

$$\begin{aligned} \begin{aligned}&\left( \Psi _{{\check{{{\mathfrak {q}}}}}}\left( {\varvec{s}}({\varvec{z}})\right) u^{\sigma _{{\check{f}}_1}} \right) ^{-\sigma _{{\check{f}}_2}} = \left( \Psi _{{\check{{{\mathfrak {q}}}}}}\left( {\varvec{a}}\right) \left( a_{\check{f}_1}\right) ^{\sigma _{{\check{f}}_1}} \right) ^{-\sigma _{{\check{f}}_2}} =\left( a_{{\check{f}}_2}^{-\sigma _{{\check{f}}_2}}\right) ^{-\sigma _{\check{f}_2}}= a_{{\check{f}}_2}. \end{aligned} \end{aligned}$$
(104)

Here, we used \(\Psi _{{\mathfrak {c}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}}\left( {\varvec{a}}\right) =e\), from \({\varvec{a}}\in \mathcal{F}\mathcal{B}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}\). Hence, we obtain \(\Xi _{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}} ({\varvec{z}}, u) ={\varvec{a}} \), proving the surjectivity.

2. Next we consider the case \({J}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2},{N} }=\emptyset \). For each \({\varvec{z}}\in \mathcal{P}\mathcal{T}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}\), define

$$\begin{aligned} {\varvec{a}}({\varvec{z}}):=(a_e({\varvec{z}}))_{e\in {\mathcal {E}}^{(4)}\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\right) }\;\in G^{{\mathcal {E}}^{(4)}\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\right) } \end{aligned}$$

by

$$\begin{aligned} \begin{aligned} a_e({\varvec{z}}):=\left( \left( z_{v_{e,-\sigma _e}}\right) ^{-1}z_{v_{e,\sigma _e}}\right) ^{\sigma _e},\quad e\in {\mathcal {E}}^{(4)}\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\right) . \end{aligned} \end{aligned}$$
(105)

By this definition, we have

$$\begin{aligned} \begin{aligned}&\left( \Psi _{{{\mathfrak {p}}}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}^{(l)}}({\varvec{a}}({\varvec{z}}, u))\right) ^{-1} \Psi _{{{\mathfrak {p}}}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}^{(r)}}({\varvec{a}}({\varvec{z}}, u)) =\left( a_{e_1}({\varvec{z}})\right) ^{\sigma _{e_1}} \left( a_{e_2}({\varvec{z}})\right) ^{\sigma _{e_2}}\cdots \\&=\left( \left( \left( z_{v_{e_1,-\sigma _{e_1}}}\right) ^{-1}z_{v_{e_1,\sigma _{e_1}}}\right) ^{\sigma _{e_1}}\right) ^{\sigma _{e_1}} \left( \left( \left( z_{v_{e_2,-\sigma _{e_2}}}\right) ^{-1}z_{v_{e_2,\sigma _{e_2}}}\right) ^{\sigma _{e_2}}\right) ^{\sigma _{e_2}} \\&\cdots \left( \left( \left( z_{v_{e_L,-\sigma _{e_L}}}\right) ^{-1}z_{v_{e_L,\sigma _{e_L}}}\right) ^{\sigma _{e_L}}\right) ^{\sigma _{e_L}}\\&= \left( z_{v_{e_1,-\sigma _{e_1}}}\right) ^{-1}z_{v_{e_1,\sigma _{e_1}}} \left( z_{v_{e_2,-\sigma _{e_2}}}\right) ^{-1}z_{v_{e_2,\sigma _{e_2}}} \cdots \left( z_{v_{e_L,-\sigma _{e_L}}}\right) ^{-1}z_{v_{e_L,\sigma _{e_L}}}\\&=\left( z_{v_{e_1,-\sigma _{e_1}}}\right) ^{-1}z_{v_{e_L,\sigma _{e_L}}} =e. \end{aligned} \end{aligned}$$
(106)

Here, we labeled the edges in \(\left( {{\mathfrak {p}}}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}^{(l)} \right) ^{-1}{{\mathfrak {p}}}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}^{(r)}\) in order as \(\{e_i\}_{i=1}^L\). Note that \(v_{e_{i}, \sigma _i}=v_{e_{i+1}, -\sigma _{i+1}}\), \(i=1,\ldots , L\) and \(v_{e_{L}, \sigma _L}=v_{e_{1}, -\sigma _{1}}\) because \({\mathfrak {c}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\) is a loop. Hence, we prove the claim \({\varvec{a}}({\varvec{z}}, u)\in \mathcal{F}\mathcal{B}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}\) and obtain the map (95)

$$\begin{aligned} \begin{aligned} \Xi _{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}: \mathcal{P}\mathcal{T}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}\ni \left( {\varvec{t}}\right) \mapsto {\varvec{a}}({\varvec{t}})\in \mathcal{F}\mathcal{B}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}. \end{aligned} \end{aligned}$$
(107)

As in the case 1., (96) holds. By (96), \(\Xi _{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}\) is injective. To see the surjectivity, let \({\varvec{a}}\in \mathcal{F}\mathcal{B}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}\). For any \(w\in {\mathbb {V}}\left( \mathfrak c_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\right) ={{\partial {\mathbb V\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }}}\), set \( z_w:=\Psi _{\tilde{{{\mathfrak {q}}}}_w}({\varvec{a}}) \) with a path \(\tilde{{{\mathfrak {q}}}_w}\) from \(v_0\) to w. This value is independent of the choice of \(\tilde{{{\mathfrak {q}}}_w}\) because \({\varvec{a}}\in \mathcal{F}\mathcal{B}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}\). By the definition of \(\Xi _{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}\), we obtain \(\Xi _{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}({\varvec{z}})={\varvec{a}}\).

\(\square \)

Lemma 4.10

Consider Setting 4.2. For any \({\varvec{t}}\in G^{\partial \mathbb V\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }\), we have

$$\begin{aligned} \begin{aligned} \left| \left\{ {\varvec{k}}\in C_{{\mathbb E\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }}\Bigg | \Psi ^{(x,y)}({\varvec{k}})\vert _{{{\partial {\mathbb V\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }}}}={\varvec{t}} \right\} \right| =\left| G\right| ^{\left| {J}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2},{N} }\right| } |G|^{\left| {\mathcal {E}}_{\mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}}^{(1)}\right| } \end{aligned} \end{aligned}$$
(108)

In particular, the number (108) does not depend on the choice of \({\varvec{t}}\in G^{\partial \mathbb V\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }\).

Proof

For any \({\varvec{t}}\in \mathcal{P}\mathcal{T}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}\), set

$$\begin{aligned} \begin{aligned} \mathcal{F}\mathcal{B}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}^{{\varvec{t}}} :=\left\{ {\varvec{a}} \in \mathcal{F}\mathcal{B}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}\mid \Psi _{{{\partial {\mathbb V\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }}}}^{(x, y)}({\varvec{a}})={\varvec{t}} \right\} . \end{aligned} \end{aligned}$$
(109)

Then, we have

$$\begin{aligned} \begin{aligned}&\left\{ {\varvec{k}}\in C_{{{\mathbb E\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }}}\Bigg | \Psi ^{(x,y)}\left( {\varvec{k}} \right) \vert _{\partial \mathbb V\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }={\varvec{t}} \right\} \\&=\dot{\bigcup }_{\begin{array}{c} {\varvec{a}}\in \mathcal{F}\mathcal{B}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}} ,\\ \end{array} } \left\{ {\varvec{k}}\in C_{{{\mathbb E\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }}}\Bigg |\;\; {\varvec{k}} \vert _{{\mathcal {E}}^{(4)}\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\right) }={\varvec{a}},\; \Psi ^{(x,y)}\left( {\varvec{k}} \right) \vert _{\partial \mathbb V\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }={\varvec{t}} \right\} \\&=\dot{\bigcup }_{\begin{array}{c} {\varvec{a}}\in \mathcal{F}\mathcal{B}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}^{{\varvec{t}}} ,\\ \end{array} } \left\{ {\varvec{k}}\in C_{{{\mathbb E\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }}}\Bigg |\;\; {\varvec{k}} \vert _{{\mathcal {E}}^{(4)}\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\right) }={\varvec{a}} \right\} \\&={\dot{\bigcup }}_{{\varvec{u}}\in G^{\left| {J}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}\right| }} \left\{ {\varvec{k}}\in C_{{{\mathbb E\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }}}\Bigg |\;\; {\varvec{k}} \vert _{{\mathcal {E}}^{(4)}\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\right) }= \Xi _{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}({\varvec{t}}, {\varvec{u}}) \right\} . \end{aligned} \end{aligned}$$
(110)

Here, we used Lemma 4.8. By Lemma 3.6, there exists \(1-|G|^{\left| {\mathcal {E}}_{{\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}}^{(1)}\right| } \)-correspondence between \(\mathcal{F}\mathcal{B}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}\) and \(C_{{\mathbb E\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }}\). Combining this fact and (110), for any \({\varvec{t}}\in \mathcal{P}\mathcal{T}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}\), we have

$$\begin{aligned} \begin{aligned}&\left| \left\{ {\varvec{k}}\in C_{{{\mathbb E\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }}}\Bigg | \Psi ^{(x,y)}\left( {\varvec{k}} \right) \vert _{\partial \mathbb V\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }={\varvec{t}} \right\} \right| \\&=\sum _{{\varvec{u}}\in G^{\left| {J}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}\right| }} \left| \left\{ {\varvec{k}}\in C_{{{\mathbb E\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }}}\Bigg |\;\; {\varvec{k}} \vert _{{\mathcal {E}}^{(4)}\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}\right) } =\Xi _{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}({\varvec{t}}, {\varvec{u}}) \right\} \right| \\&=\sum _{{\varvec{u}}\in G^{\left| {J}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}\right| }} |G|^{\left| {\mathcal {E}}_{{\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}}^{(1)}\right| } =\left| G\right| ^{\left| {J}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2},{N} }\right| } |G|^{\left| {\mathcal {E}}_{\mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}}^{(1)}\right| }. \end{aligned} \end{aligned}$$
(111)

\(\square \)

5 Restriction of \(\omega _0\) to a Cone Shape Area

We use the notation from Sects. 1, 2, 3 but we do not assume G to be abelian.

Let \(\Gamma \) be a convex cone in \({\mathbb {R}}^2\) with apex \({\varvec{a}}_\Gamma \in [0,1]\times [0,1]\). Because of the \({\mathbb {Z}}^2\)-translation invariance of the model, it suffices to consider this case. The boundary of \(\Gamma \) consists of two lines \(L_1^\Gamma :={\varvec{a}}_\Gamma +{\mathbb {R}}_{\ge 0}{\varvec{e}}_{\theta _1}\), \(L_2^\Gamma :={\varvec{a}}_\Gamma +{\mathbb {R}}_{\ge 0}{\varvec{e}}_{\theta _2}\), with \(0<\theta _2-\theta _1\le \pi \), because \(\Gamma \) is convex. Here, \({\varvec{e}}_\theta :=(\cos \theta ,\sin \theta )\). We concentrate on the case \(0\le \theta _1<\frac{\pi }{2}\). The proof of other cases is the same, just \(\frac{\pi }{2},\pi ,\frac{3\pi }{2}\)-rotate the following argument.

Note that three cases can occur:

  1. (1)

    \(0\le \theta _1<\theta _2\le \frac{\pi }{2}\),

  2. (2)

    \(0\le \theta _1<\frac{\pi }{2}\le \theta _2\le \pi \),

  3. (3)

    \(0\le \theta _1<\frac{\pi }{2}<\pi<\theta _2\le \pi +\theta _1<\frac{3\pi }{2}\).

For each case, we fix \(n_0,m_0\in {\mathbb {N}}\) so that

(1):

\(\frac{m_0}{n_0}< \tan \theta _2\),

(2):

\(n_0=m_0=1\),

(3):

\(\frac{m_0}{n_0}< \tan \theta _2\).

Then, for case (1),(2) \(\Gamma \) is in the upper-half-plane and there exists \(M_0,N_0\in {\mathbb {N}}\) such that

$$\begin{aligned} \begin{aligned} 3 \le \left| \left\{ x\in {\mathbb {Z}}\mid S_{(x,M)}\subset \Gamma \cap \Lambda _N^{(n_0,m_0)} \right\} \right| \end{aligned} \end{aligned}$$
(112)

for all \(M,N\in {\mathbb {N}}\) with \(M_0\le M\le N\), \(N_0\le N\). For case (3), there exists \(N_0\in {\mathbb {N}}\) such that

$$\begin{aligned} \begin{aligned} 3 \le \left| \left\{ x\in {\mathbb {Z}}\mid S_{(x,M)}\subset \Gamma \cap \Lambda _N^{(n_0,m_0)} \right\} \right| \end{aligned} \end{aligned}$$
(113)

for all \(M,N\in {\mathbb {N}}\) with \(-Nm_0\le M\le Nm_0\), \(N_0\le N\). For (3) case, we set \(M_0:=-\infty \). For \(M_0\in {\mathbb {Z}}\cup \{-\infty \}\), set

$$\begin{aligned} H^U_{M_0}:= \left\{ (x,y)\in {\mathbb {Z}}^2\mid y\ge M_0 \right\} . \end{aligned}$$
(114)

Consider the set of squares inside of

$$\begin{aligned} \begin{aligned} \Gamma \cap {\mathbb {Z}}^2\cap H^U_{M_0}. \end{aligned} \end{aligned}$$
(115)

By considering the boundary of the set given by these squares, for each case (1)-(3), we obtain a point \(v_0=(x,y)\in {\mathbb {Z}}^2\), and \({{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2\in {\mathfrak {P}}(x,y)\) a pair of paths well separated with respect to \(N_0,n_0, m_0,+1\) satisfying (1)-(3) in Setting 4.2, respectively. We use the notation from Sect. 4 for this \(v_0=(x,y)\in {\mathbb {Z}}^2\), and \({{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2\in {\mathfrak {P}}(x,y)\) freely.

Recall that \({\mathcal {P}}_{\left( \Gamma \cap {\mathbb {Z}}^2\cap H^U_{M_0}\right) }\) denotes all the squares in \(\left( \Gamma \cap {\mathbb {Z}}^2\cap H^U_{M_0}\right) \) and \(\widetilde{\left( \Gamma \cap {\mathbb {Z}}^2\cap H^U_{M_0}\right) }\) denotes the edges inside of these squares. We set

$$\begin{aligned} \begin{aligned}&{\hat{\Gamma }}:=\widetilde{\left( \Gamma \cap {\mathbb {Z}}^2\cap H^U_{M_0}\right) },\\&{{\hat{\Gamma }}_N}:=\widetilde{\left( \Gamma \cap \Lambda _N^{(n_0,m_0)}\cap H^U_{M_0}\right) }=\mathbb E\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) . \end{aligned} \end{aligned}$$
(116)

In this section, we obtain an explicit formula for the restriction of \(\omega _0=\varphi _0\) to \({\mathcal {B}}_{{\hat{\Gamma }}_N}\).

Lemma 5.1

With the notation above, for any \(N\ge N_0\), there is some number \(c_{\Gamma ,N}>0\) such that

$$\begin{aligned} \begin{aligned}&\varphi _0\left( \left| {\varvec{h}} \right\rangle _{{{\hat{\Gamma }}_N}}\left\langle {\varvec{k}} \right| \right) =c_{\Gamma ,N}\cdot \chi \left( {\varvec{k}}_{}, {\varvec{h}}_{}\in C_{{\mathbb {E}}\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) } \right) \\&\chi \left( \Psi ^{v_0}({\varvec{h}}_{})\vert _{\partial {\mathbb V\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }}= \Psi ^{v_0}({\varvec{k}}_{})\vert _{\partial {\mathbb V\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }} \right) \\ \end{aligned} \end{aligned}$$
(117)

for any \({\varvec{h}}, {\varvec{k}}\in G^{{\hat{\Gamma }}_N}=G^{ {{\mathbb {E}}\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }}\).

Remark 5.2

Note because \({\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \) satisfies condition S, \(\Psi ^{v_0}({\varvec{h}}_{})\) is well defined for any \({\varvec{h}}_{}\in C_{{\mathbb {E}}\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }\).

Proof

By Lemma 2.2, for any \({\varvec{h}}, {\varvec{k}}\in G^{{{\hat{\Gamma }}_N}}\),

$$\begin{aligned} \begin{aligned}&\varphi _{0}\left( \left| {\varvec{h}} \right\rangle _{{{\hat{\Gamma }}_N}}\left\langle {\varvec{k}} \right| \right) \\&= \sum _{{\varvec{m}}\in G^{{\mathfrak {O}}{\mathfrak {E}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}}} \sum _{{\varvec{g}}\in G^{{\mathfrak {B}}{\mathfrak {E}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}}} \varphi _{0}\left( \left| {\varvec{h}},{\varvec{m}},{\varvec{g}} \right\rangle _{{\hat{\Lambda }}_N^{(n_0,m_0)}}\left\langle {\varvec{k}},{\varvec{m}},{\varvec{g}} \right| \right) \\&=\sum _{{\varvec{m}}\in G^{{\mathfrak {O}}{\mathfrak {E}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}}} \sum _{{\varvec{g}}\in G^{{\mathfrak {B}}{\mathfrak {E}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}}} \frac{1}{|C_{{{\hat{\Lambda }}_N^{(n_0,m_0)}}}|} \\&\quad \chi \left( {\left( {\varvec{k}},{\varvec{m}},{\varvec{g}}\right) , \left( {\varvec{h}},{\varvec{m}},{\varvec{g}}\right) \in C_{{{\hat{\Lambda }}_N^{(n_0,m_0)}}}}\right) \\&\quad \chi \left( \left( {\varvec{k}},{\varvec{m}},{\varvec{g}}\right) _{\partial {{{\hat{\Lambda }}_N^{(n_0,m_0)}}} } = \left( {\varvec{h}},{\varvec{m}},{\varvec{g}}\right) _{\partial {{{\hat{\Lambda }}_N^{(n_0,m_0)}}}}\right) \\&=\sum _{{\varvec{m}}\in C_{{\mathfrak {O}}{\mathfrak {E}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}}} \sum _{g\in G^{{e_0^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}}} \sum _{\tilde{{\varvec{g}}}\in G^{{\mathfrak {B}}{\mathfrak {E}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}\setminus \{{e_0^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}\}}} \frac{1}{|C_{{{\hat{\Lambda }}_N^{(n_0,m_0)}}}|} \\&\quad \chi \left( {\left( {\varvec{k}},{\varvec{m}},g, \tilde{{\varvec{g}}}, \right) , \left( {\varvec{h}},{\varvec{m}},g, \tilde{{\varvec{g}}}\right) \in C_{{{\hat{\Lambda }}_N^{(n_0,m_0)}}}}\right) \\&\quad \chi \left( \left( {\varvec{k}},{\varvec{m}},g,\tilde{{\varvec{g}}}\right) _{\partial {{{\hat{\Lambda }}_N^{(n_0,m_0)}}}} =\left( {\varvec{h}},{\varvec{m}},g, \tilde{{\varvec{g}}}\right) _{\partial {{{\hat{\Lambda }}_N^{(n_0,m_0)}}} }\right) \\&\quad \chi \left( {\varvec{k}}, {\varvec{h}}\in C_{{\mathbb E\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }} \right) \\&\quad \chi \left( \Psi ^{(x,y)}({\varvec{h}})\vert _{{{\partial {\mathbb V\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }}}}= \Psi ^{(x,y)}({\varvec{k}})\vert _{{{\partial {\mathbb V\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }}}} \right) . \end{aligned} \end{aligned}$$
(118)

Here, we used Lemma 4.6 equivalence of (ii),(iii) for the last line. Recall from Lemma 4.6 that if \({\varvec{k}}, {\varvec{h}}\in C_{{\mathbb E\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }}\), \(\Psi ^{(x,y)}({\varvec{h}})\vert _{{{\partial {\mathbb V\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }}}} = \Psi ^{(x,y)}({\varvec{k}})\vert _{{{\partial {\mathbb V\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }}}}\), for any \({\varvec{m}}\in C_{{{\mathfrak {O}}{\mathfrak {E}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}}}\) and \(g\in G^{{e_0^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}}\), there exists a unique \(\tilde{{\varvec{g}}}\in G^{{\mathfrak {B}}{\mathfrak {E}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}{\setminus } \{{e_0^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}\}}\) satisfying

$$\begin{aligned} \begin{aligned} \left( {\varvec{k}},{\varvec{m}},g, \tilde{{\varvec{g}}}, \right) , \left( {\varvec{h}},{\varvec{m}},g, \tilde{{\varvec{g}}}\right) \in C_{{{\hat{\Lambda }}_N^{(n_0,m_0)}}} \end{aligned} \end{aligned}$$
(119)

and

$$\begin{aligned} \begin{aligned} \left( {\varvec{k}},{\varvec{m}},g,\tilde{{\varvec{g}}}\right) _{\partial {{{\hat{\Lambda }}_N^{(n_0,m_0)}}}} = \left( {\varvec{h}},{\varvec{m}},g, \tilde{{\varvec{g}}}\right) _{\partial {{\hat{\Lambda }_N^{(n_0,m_0)}}}}. \end{aligned} \end{aligned}$$
(120)

We have

$$\begin{aligned} \begin{aligned} (118)=&\sum _{{\varvec{m}}\in C_{\mathfrak {OE}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}}} \sum _{g\in G^{{e_0^{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2, (N)}}}} \frac{1}{|C_{{{\hat{\Lambda }}_N^{(n_0,m_0)}}}|} \chi \left( {\varvec{k}}, {\varvec{h}}\in C_{{\mathbb E\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }} \right) \\&\quad \chi \left( \Psi ^{(x,y)}({\varvec{h}})\vert _{{{\partial {\mathbb V\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }}}}= \Psi ^{(x,y)}({\varvec{k}})\vert _{{{\partial {\mathbb V\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }}}} \right) \\ =&\frac{\left| C_{{\mathfrak {O}}{\mathfrak {E}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2}^{(N)}}\right| \left| G\right| }{{|C_{{{\hat{\Lambda }}_N^{(n_0,m_0)}}}|}} \chi \left( {\varvec{k}}, {\varvec{h}}\in C_{{\mathbb E\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }} \right) \\&\chi \left( \Psi ^{(x,y)}({\varvec{h}})\vert _{{{\partial {\mathbb V\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }}}}= \Psi ^{(x,y)}({\varvec{k}})\vert _{{{\partial {\mathbb V\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }}}} \right) \\ \end{aligned} \end{aligned}$$
(121)

\(\square \)

6 Tracial State

We use the notation from Sects. 1, 2, 3, 4, 5, but we do not assume G to be abelian.

We first consider the same setting as in Sect. 5 and use notations from Sect. 4. For each \({\varvec{t}}\in \mathcal{P}\mathcal{T}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}\), set

$$\begin{aligned} {\mathcal {J}}_{{\varvec{t}}}^{(\Gamma , N)} :=\left\{ {\varvec{k}}\in C_{{\mathbb E\left( {\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }}\mid \Psi ^{(x,y)}({\varvec{k}})\vert _{{{\partial {\mathbb V\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }}}}={\varvec{t}} \right\} . \end{aligned}$$
(122)

By Lemma 4.10, we have

$$\begin{aligned} \begin{aligned} \left| {\mathcal {J}}_{{\varvec{t}}}^{(\Gamma , N)}\right| = \left| G\right| ^{\left| {J}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2},{N} }\right| } \left| {\mathcal {E}}_{\mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}}^{(1)}\right| \end{aligned} \end{aligned}$$
(123)

for any \({\varvec{t}}\in \mathcal{P}\mathcal{T}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}\). For \({\varvec{t}}\in \mathcal{P}\mathcal{T}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}\), we set

$$\begin{aligned}&\psi _{{\varvec{t}}}^{N}:= \sum _{{\varvec{k}}\in {\mathcal {J}}_{{\varvec{t}}}^{(\Gamma , N)}}\left| {\varvec{k}} \right\rangle _{{\hat{\Gamma }}_N}. \end{aligned}$$
(124)

By (123),

$$\begin{aligned} \begin{aligned}&{\hat{\psi }}_{{\varvec{t}}}^{N}:= \frac{1}{\sqrt{ \left| G\right| ^{\left| {J}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2},{N} }\right| } \left| {\mathcal {E}}_{\mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}}^{(1)}\right| }}\; \psi _{{\varvec{t}}}^{N},\quad \quad {\varvec{t}}\in \mathcal{P}\mathcal{T}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}} \end{aligned} \end{aligned}$$
(125)

are unit vectors. We set

$$\begin{aligned} \begin{aligned} Q_N:=\sum _{{\varvec{t}}\in \mathcal{P}\mathcal{T}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}} \left| {\hat{\psi }}_{{\varvec{t}}}^{N} \right\rangle \left\langle {\hat{\psi }}_{{\varvec{t}}}^{N} \right| . \end{aligned} \end{aligned}$$
(126)

Lemma 6.1

For any \(N\ge N_0\), \(Q_N\) is the support of \(\varphi _0\) on \({\mathcal {B}}_{{{\hat{\Gamma }}_N}}\) and \(\varphi _0\vert _{Q_N{\mathcal {B}}_{{{\hat{\Gamma }}_N}}Q_N}\) is a tracial state. Namely, we have

$$\begin{aligned} \begin{aligned} \varphi _0\left( Q_N A Q_N B Q_N\right) =\varphi _0\left( Q_N B Q_N A Q_N\right) ,\quad A,B\in {\mathcal {B}}_{{{\hat{\Gamma }}_N}}. \end{aligned} \end{aligned}$$
(127)

Proof

From Lemma 5.1, for any \({\varvec{k}},{\varvec{h}}\in G^{{\hat{\Gamma }}_N}\), we have

$$\begin{aligned} \begin{aligned}&\varphi _0\left( \left| {\varvec{h}} \right\rangle _{{{\hat{\Gamma }}_N}}\left\langle {\varvec{k}} \right| \right) \\&=c_{\Gamma ,N}\cdot \chi \left( {\varvec{k}}, {\varvec{h}}\in C_{{\mathbb E\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }} \right) \chi \left( \Psi ^{(x,y)}({\varvec{h}})\vert _{{{\partial {\mathbb V\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }}}}= \Psi ^{(x,y)}({\varvec{k}})\vert _{{{\partial {\mathbb V\left( \mathfrak S_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)} \right) }}}} \right) \\&=c_{\Gamma ,N} \sum _{{\varvec{t}}\in \mathcal{P}\mathcal{T}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}} \left\langle \psi _{{\varvec{t}}}^{N}, \left( \left| {\varvec{h}} \right\rangle _{_{{{\hat{\Gamma }}_N}}}\left\langle {\varvec{k}} \right| \right) \psi _{{\varvec{t}}}^{N}\right\rangle \\&=c_{\Gamma ,N} \left| G\right| ^{\left| {J}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2},{N} }\right| } \left| {\mathcal {E}}_{{\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}}^{(1)}\right| \sum _{{\varvec{t}}\in \mathcal{P}\mathcal{T}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,N}}} \left\langle {\hat{\psi }}_{{\varvec{t}}}^{N}, \left( \left| {\varvec{h}} \right\rangle _{{{\hat{\Gamma }}_N}}\left\langle {\varvec{k}} \right| \right) {\hat{\psi }}_{{\varvec{t}}}^{N}\right\rangle \\&=c_{\Gamma ,N} \left| G\right| ^{\left| {J}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2},{N} }\right| } \left| {\mathcal {E}}_{{\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}}^{(1)}\right| \mathop {\textrm{Tr}}\nolimits Q_N\left( \left( \left| {\varvec{h}} \right\rangle _{{{\hat{\Gamma }}_N}}\left\langle {\varvec{k}} \right| \right) \right) \\&=c_N\mathop {\textrm{Tr}}\nolimits Q_N\left( \left( \left| {\varvec{h}} \right\rangle _{{{\hat{\Gamma }}_N}}\left\langle {\varvec{k}} \right| \right) \right) , \end{aligned} \end{aligned}$$
(128)

with \(c_N:= c_{\Gamma ,N} \left| G\right| ^{\left| {J}_{{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2},{N} }\right| } \left| {\mathcal {E}}_{{\mathfrak {S}}_{{{\mathfrak {p}}}_1,{{\mathfrak {p}}}_2,+1}^{(N)}}^{(1)}\right| \). We then have

$$\begin{aligned} \begin{aligned}&\varphi _0(B)=c_N\mathop {\textrm{Tr}}\nolimits Q_N B Q_N, \end{aligned} \end{aligned}$$
(129)

for all \(B\in {\mathcal {B}}_{{{\hat{\Gamma }}_N}}\). Hence, we have

$$\begin{aligned} \begin{aligned}&\varphi _0\left( Q_N A Q_N B Q_N\right) =c_N\mathop {\textrm{Tr}}\nolimits \left( Q_N A Q_N B Q_N\right) \\ {}&=c_N\mathop {\textrm{Tr}}\nolimits \left( Q_N B Q_N A Q_N\right) =\varphi _0\left( Q_N B Q_N A Q_N\right) \end{aligned} \end{aligned}$$
(130)

for all \(A, B\in {\mathcal {B}}_{{{\hat{\Gamma }}_N}}\).

\(\square \)

Because \({Q_N}\) is the support of \(\varphi _0\) on \({\mathcal {B}}_{{{\hat{\Gamma }}_N}}\) and \({Q_{N+1}}\) is the support of \(\varphi _0\) on \({\mathcal {B}}_{{\hat{\Gamma }}_{N+1}}\), we have \( 1-{Q_N}\le 1-{Q_{N+1}} \) and hence

$$\begin{aligned} {Q_{N+1}}\le {Q_N}. \end{aligned}$$
(131)

Let \(({\mathcal {H}}_0,\pi _0,\Omega _0)\) be the GNS triple of \(\varphi _0\). Because of (131), there is a projection \(q\in \pi _0({\mathcal {B}}_{{\hat{\Gamma }}})''\) such that

$$\begin{aligned} s-\lim \pi _0\left( {Q_N}\right) =q. \end{aligned}$$
(132)

Because

$$\begin{aligned} \pi _0\left( {Q_N}\right) \Omega _0=\Omega _0, \end{aligned}$$
(133)

we have

$$\begin{aligned} q\Omega _0=\Omega _0. \end{aligned}$$
(134)

In particular, \(q\ne 0\).

Lemma 6.2

Let \(\omega \) be a state on \(q\pi _0({\mathcal {B}}_{{\hat{\Gamma }}})''q\) such that

$$\begin{aligned} \omega (x):=\left\langle \Omega _0, x\Omega _0\right\rangle ,\quad x\in q\pi _0({\mathcal {B}}_{{\hat{\Gamma }}})''q. \end{aligned}$$
(135)

Then, \(\omega \) is a faithful tracial state on \(q\pi _0({\mathcal {B}}_{{\hat{\Gamma }}})''q\).

Proof

We first show for any \(A,B\in {\mathcal {B}}_{{\hat{\Gamma }},\textrm{loc}}:=\cup _{M\in {\mathbb {N}}} {\mathcal {B}}_{{\hat{\Gamma }}_M}\) that

$$\begin{aligned} \begin{aligned} \left\langle \Omega _0,q\pi _0(A)q\pi _0(B)q\Omega _0\right\rangle =\left\langle \Omega _0,q\pi _0(B)q\pi _0(A)q\Omega _0\right\rangle . \end{aligned} \end{aligned}$$
(136)

From Lemma 6.1, for any \(A,B\in {\mathcal {B}}_{{\hat{\Gamma }},\textrm{loc}}\), we have

$$\begin{aligned} \begin{aligned}&\left\langle \Omega _0,\pi _0\left( A\right) \pi _0\left( {Q_N} \right) \pi _0\left( B \right) \Omega _0\right\rangle = \left\langle \Omega _0,\pi _0\left( {Q_N} A {Q_N} B {Q_N}\right) \Omega _0\right\rangle \\&=\varphi _0\left( Q_N A Q_N B Q_N\right) =\varphi _0\left( Q_N B Q_N A Q_N\right) \\&=\left\langle \Omega _0,\pi _0\left( {Q_N} B {Q_N} A {Q_N}\right) \Omega _0\right\rangle =\left\langle \Omega _0,\pi _0\left( B\right) \pi _0\left( {Q_N} \right) \pi _0\left( A \right) \Omega _0\right\rangle \end{aligned} \end{aligned}$$
(137)

for N large enough. Taking \(N\rightarrow \infty \) limit, we obtain

$$\begin{aligned} \begin{aligned} \left\langle \Omega _0,\pi _0\left( A\right) q\pi _0\left( B \right) \Omega _0\right\rangle = \left\langle \Omega _0,\pi _0\left( B\right) q \pi _0\left( A \right) \Omega _0\right\rangle , \end{aligned} \end{aligned}$$
(138)

proving (136). From (136), we have

$$\begin{aligned} \begin{aligned} \omega \left( qxqyq\right) =\left\langle \Omega _0,qxqyq\Omega _0\right\rangle =\left\langle \Omega _0,qyqxq\Omega _0\right\rangle =\omega \left( qyqxq\right) , \end{aligned} \end{aligned}$$
(139)

for any \(x, y\in \pi _0\left( {\mathcal {B}}_{{\hat{\Gamma }}}\right) ''\). Hence, \(\omega \) is a tracial state on \(q\pi _0( {\mathcal {B}}_{{\hat{\Gamma }}})''q\).

In order to show that \(\omega \) is faithful, suppose that \(x\in q\pi _0( {\mathcal {B}}_{{\hat{\Gamma }}})''q\) satisfies \(\omega (x^*x)=0\). Then for any \(y\in q\pi _0( {\mathcal {B}}_{{\hat{\Gamma }}})''q\), we have

$$\begin{aligned} \begin{aligned} \left\| xy\Omega _0\right\| ^2=\omega \left( y^*x^*x y\right) = \omega \left( x^*x yy^*\right) = \omega \left( x yy^*x^*\right) \le \left\| y \right\| ^2\omega \left( x x^*\right) =\left\| y \right\| ^2\omega \left( x^* x\right) =0. \end{aligned} \end{aligned}$$
(140)

This means

$$\begin{aligned} xy\Omega _0=0,\quad \text {for all}\quad y\in q\pi _0( {\mathcal {B}}_{{\hat{\Gamma }}})''q. \end{aligned}$$
(141)

Furthermore, because

$$\begin{aligned} x\in q\pi _0( {\mathcal {B}}_{{\hat{\Gamma }}})''q\subset \pi _0( {\mathcal {B}}_{{\hat{\Gamma }}})'' \subset \pi _0({\mathcal {B}}_{{\hat{{\mathbb {Z}}}}^2\setminus {\hat{\Gamma }}})',\quad q\in \pi _0( {\mathcal {B}}_{{\hat{\Gamma }}})''\subset \pi _0({\mathcal {B}}_{{\hat{{\mathbb {Z}}}}^2\setminus {\hat{\Gamma }}})', \end{aligned}$$
(142)

we have

$$\begin{aligned} \begin{aligned}&x\pi _0\left( A\right) \pi _0\left( B\right) \Omega _0 =\pi _0\left( B\right) x\pi _0\left( A\right) \Omega _0 =\pi _0\left( B\right) xq\pi _0\left( A\right) q\Omega _0=0,\\&A\in {\mathcal {B}}_{{\hat{\Gamma }}}, \quad B\in {\mathcal {B}}_{{\hat{{\mathbb {Z}}}}^2\setminus {\hat{\Gamma }}}. \end{aligned} \end{aligned}$$
(143)

Hence, we obtain \(x=0\), proving the faithfulness. \(\square \)

Lemma 6.3

Let \(\Gamma \) be a convex cone in \({\mathbb {R}}^2\) with apex \({\varvec{a}}_\Gamma \in [0,1]\times [0,1]\) given by half-lines \(L_1^\Gamma :={\varvec{a}}_\Gamma +{\mathbb {R}}_{\ge 0}{\varvec{e}}_{\theta _1}\), \(L_2^\Gamma :={\varvec{a}}_\Gamma +{\mathbb {R}}_{\ge 0}{\varvec{e}}_{\theta _2}\), with \(0\le \theta _1<\frac{\pi }{2}\), \(0<\theta _2-\theta _1<\pi \). Then, \(\pi _0\left( {\mathcal {B}}_{\widetilde{\left( \Gamma \cap {\mathbb {Z}}^2\right) }}\right) ''\) is not type III.

Proof

Let q be the projection in \(\pi _0( {\mathcal {B}}_{{\hat{\Gamma }}})''\) given above. Suppose \(p\in \pi _0( {\mathcal {B}}_{{\hat{\Gamma }}})''\) is a projection such that \(p\sim q\) in \(\pi _0( {\mathcal {B}}_{{\hat{\Gamma }}})''\) and \(p\le q\). We have \(p=v^*v\), \(q=vv^*\) for some \(v\in \pi _0( {\mathcal {B}}_{{\hat{\Gamma }}})''\). Because \(p\le q\), \(v=qvq\in q\pi _0( {\mathcal {B}}_{{\hat{\Gamma }}})''q\). We then have

$$\begin{aligned} \begin{aligned} \omega \left( q\right) =\omega (vv^*)=\omega (v^*v)=\omega (p), \end{aligned} \end{aligned}$$
(144)

and for \(q-p\ge 0\) we have \(\omega (q-p)=0\). By the faithfulness of \(\omega \), we have \(p=q\). Hence, q is finite in \(\pi _0( {\mathcal {B}}_{{\hat{\Gamma }}})''\). Because \(\pi _0( {\mathcal {B}}_{{\hat{\Gamma }}})''\) has a nonzero finite projection, it is not type III. Note the difference between \({\hat{\Gamma }}\) and \(\widetilde{\left( \Gamma \cap {\mathbb {Z}}^2\right) }\) is at most finite. Therefore, \(\pi _0\left( {\mathcal {B}}_{\widetilde{\left( \Gamma \cap {\mathbb {Z}}^2\right) }}\right) ''\) is not type III either.

\(\square \)

The \(\frac{\pi }{2},\pi ,\frac{3\pi }{2}\)-rotation of the above argument and \({\mathbb {Z}}^2\)-translation invariance of the model allow us to extend the result as follows.

Lemma 6.4

Let \(\Gamma \) be a convex cone in \({\mathbb {R}}^2\). Then, \(\pi _0\left( {\mathcal {B}}_{\widetilde{\left( \Gamma \cap {\mathbb {Z}}^2\right) }}\right) ''\) is not type III.

7 Proof of Theorem 1.2

Now, finally we assume G to be abelian.

Theorem 1.2

From [5] Theorem 6.3, there is a nontrivial superselection sector for abelian quantum double model. Furthermore, Haag duality holds for the abelian quantum double model Theorem 4.14 [5]. Therefore, by Lemma 5.5 [9], for any cone \(\Gamma \subset {\mathbb {R}}^2\), \(\pi _0\left( {\mathcal {B}}_{\widetilde{\left( \Gamma \cap {\mathbb {Z}}^2\right) }}\right) ''\) is a type \(II_\infty \) factor or a type III-factor. For a convex \(\Gamma \), Lemma 6.4 then implies that \(\pi _0\left( {\mathcal {B}}_{\widetilde{\left( \Gamma \cap {\mathbb {Z}}^2\right) }}\right) ''\) is a type \(II_\infty \) factor. \(\square \)