$\displaystyle=c(g,h,k)\gamma_{g}\left(u(h,k)\right)\cdot
c(g,hk,f)\left(\gamma_{g}\left(u(hk,f)\right)\left(u(g,hkf)\right)\right)$
$\displaystyle=c(g,h,k)c(g,hk,f)\left(\gamma_{g}\left[u(h,k)u(hk,f)\right]\right)\cdot\left(u(g,hkf)\right)$
$\displaystyle=c(g,h,k)c(g,hk,f)c(h,k,f)\gamma_{g}\left(\gamma_{h}\left(u(k,f)\right)\left(u(h,kf)\right)\right)\cdot\left(u(g,hkf)\right)$
$\displaystyle=c(g,h,k)c(g,hk,f)c(h,k,f)\cdot\gamma_{g}\gamma_{h}\left(u(k,f)\right)\cdot\left[\left(\gamma_{g}\left(u(h,kf)\right)\right)u(g,hkf)\right]$
$\displaystyle=c(g,h,k)c(g,hk,f)c(h,k,f)\overline{c(g,h,kf)}\cdot\left\\{\gamma_{g}\gamma_{h}\left(u(k,f)\right)\right)\\}\cdot\left(u(g,h)u(gh,kf)\right).$

Here, (and below) we apply (3.2) for terms in $\left[\cdot\right]$ to get the
succeeding equality. Using (3.1) for $\gamma_{g}\gamma_{h}$ and again (3.2),
we obtain

$\displaystyle(\ref{temp})$
$\displaystyle=c(g,h,k)c(g,hk,f)c(h,k,f)\overline{c(g,h,kf)}\left(\mathop{\mathrm{Ad}}\nolimits\left(u(g,h)\right)\gamma_{gh}\right)\left(u(k,f)\right)\left(u(g,h)u(gh,kf)\right)$
$\displaystyle=c(g,h,k)c(g,hk,f)c(h,k,f)\overline{c(g,h,kf)}\left(u(g,h)\right)\left[\gamma_{gh}\left(u(k,f)\right)\left(u(gh,kf)\right)\right]$
$\displaystyle=c(g,h,k)c(g,hk,f)c(h,k,f)\overline{c(g,h,kf)}\overline{c(gh,k,f)}\left(u(g,h)u(gh,k)u(ghk,f)\right).$

Hence, we obtain

$\displaystyle
c(g,h,k)c(g,hk,f)c(h,k,f)\overline{c(g,h,kf)}\overline{c(gh,k,f)}=1,\quad\text{for
all}\quad g,h,k,f\in G.$

This means $c\in Z^{3}(G,{\mathbb{T}})$. Hence we obtain $[c]_{H^{3}(G,{\rm
U}(1))}\in H^{3}(G,{\rm U}(1))$. Note that resulting $[c]_{H^{3}(G,{\rm
U}(1))}$ does not depend on the choice of $u$. Indeed, if
$\gamma_{g}\gamma_{h}=\mathop{\mathrm{Ad}}\nolimits\left(\tilde{u}(g,h)\right)\gamma_{gh}$
holds for another $\tilde{u}(g,h)\in{\mathcal{M}}$, the difference
$u(g,h)^{*}\tilde{u}(g,h)$ belongs to the center
${\mathcal{M}}\cap{\mathcal{M}}^{\prime}={\mathbb{C}}\mathbb{I}$. Hence there
is some $a(g,h)\in{\rm U}(1)$ such that $u(g,h)^{*}\tilde{u}(g,h)=a(g,h)$. Let
$\tilde{c}(g,h,k)\in{\rm U}(1)$ be the phase satisfying
$\tilde{u}(g,h)\tilde{u}(gh,k)=\tilde{c}(g,h,k)\gamma_{g}\left(\tilde{u}(h,k)\right)\tilde{u}(g,hk)$.
Then we have

$\displaystyle a(g,h)a(gh,k)\left(u(g,h)u(gh,k)\right)$ (3.4)
$\displaystyle=\tilde{u}(g,h)\tilde{u}(gh,k)$
$\displaystyle=\tilde{c}(g,h,k)\gamma_{g}\left(\tilde{u}(h,k)\right)\tilde{u}(g,hk)$
$\displaystyle=\tilde{c}(g,h,k)a(h,k)a(g,hk)\gamma_{g}\left(u(h,k)\right)u(g,hk)$
$\displaystyle=\tilde{c}(g,h,k)a(h,k)a(g,hk)\overline{c(g,h,k)}u(g,h)u(gh,k).$

Hence we have
$\tilde{c}(g,h,k)=c(g,h,k)\overline{a(h,k)a(g,hk)}a(g,h)a(gh,k)$. This means
the difference between $\tilde{c}$ and $c$ is just a coboundary, and we get
$[c]_{H^{3}(G,{\mathbb{T}})}=[\tilde{c}]_{H^{3}(G,{\mathbb{T}})}$.

The above argument tells us, if we have automorphisms $\\{\gamma_{g}\\}_{g\in
G}$ on ${\mathcal{A}}$ such that $\gamma_{g}\gamma_{h}\gamma_{gh}^{-1}$ is
inner, then we get some $[c]\in H^{3}(G,{\rm U}(1))$. This can be a hint to
find a $H^{3}(G,{\rm U}(1))$-valued index.

Before closing the subsection, we would like to emphasize the importance of
considering infinite system, not finite quantum spin systems. On a matrix
algebra, any automorphism is inner. Therefore, if we collect some
$\\{\gamma_{g}\\}_{g\in G}$, they are represented as
$\gamma_{g}=\mathop{\mathrm{Ad}}\nolimits(u_{g})$, by some unitary matrix
$u_{g}$. Then we may take $u(g,h)$ in

$\displaystyle\gamma_{g}\gamma_{h}=\mathop{\mathrm{Ad}}\nolimits\left(u(g,h)\right)\gamma_{gh}.$
(3.5)

as $u_{g}u_{h}u_{gh}^{*}$. Substituting this to

$\displaystyle u(g,h)u(gh,k)=c(g,h,k)\gamma_{g}\left(u(h,k)\right)u(g,hk),$

we obtain $c(g,h,k)=1$. Hence we get only a trivial element of $H^{3}(G,{\rm
U}(1))$. This can be seen as a reason that we consider infinite  quantum spin
systems.

###  3.2 Action of $\beta_{g}^{U}$ on automorphisms

From what we saw from the subsection 3.1, our task can be finding
automorphisms $\\{\gamma_{g}\\}_{g\in G}$ on ${\mathcal{A}}$ such that
$\gamma_{g}\gamma_{h}\gamma_{gh}^{-1}$ is inner. But how can we find such a
thing? We have a genuine group action $\beta_{g}$ but a priori, there is no
set of automorphisms which are “nicely obstacled” like the ones in cocycle
action.

A hint comes from former works on gapped phases. The wisdom we can see from
there is if you cut the system, you may obtain some non-trivial index. In
fact, that is the heart of the phenomena called bulk-edge correspondence, in
free Fermions. This correspondence says that if you would like to know about
the bulk(=entire system), cut the system into two and look at the edge. We did
the same thing in section 2 for quantum spin chains as well. We restricted our
attention to the right half of the chain. Also in [BDF], an integer-valued
index for systems with $U(1)$-symmetry was derived by considering
$U(1)$-action on half of the system. From these works, it looks to be a good
idea to consider the action of the $G$-symmetry on the half-plane. Recall that
from the automorphic equivalence, the ground state $\omega_{\Phi}$ of
$\Phi\in{\mathcal{P}}_{U.G.\beta}^{0}$ is given from a product state
$\omega_{\Phi_{0}}$ via some automorphism $\alpha$, as
$\omega_{\Phi}=\omega_{\Phi_{0}}\circ\alpha$. This motivates us to think of an
action (on the half infinite plane) of $G$ on automorphisms. More precisely we
consider the following action.

As we stated above, we consider an action of the $G$-symmetry corresponding to
the upper half-plane.

$\displaystyle\beta_{g}^{U}:=\mathop{\mathrm{id}}\nolimits_{{\mathcal{A}}_{H_{D}}}\otimes\bigotimes_{x\in
H_{U}}\mathop{\mathrm{Ad}}\nolimits\left(U(g)\right),\quad g\in G.$

on the automorphism group
$\mathop{\mathrm{Aut}}\nolimits({\mathcal{A}}_{{\mathbb{Z}}^{2}})$. The action
is given by

$\displaystyle
G\times\mathop{\mathrm{Aut}}\nolimits({\mathcal{A}}_{{\mathbb{Z}}^{2}})\ni(g,\alpha)\to\beta_{g}^{U}\alpha\left(\beta_{g}^{U}\right)^{-1}\in\mathop{\mathrm{Aut}}\nolimits({\mathcal{A}}_{{\mathbb{Z}}^{2}}).$

An idea to allow us to derive $H^{3}(G,{\rm U}(1))$ is cut further. Namely, we
consider the following situation.

######  Lemma 3.1.

Suppose that

$\displaystyle\beta_{g}^{U}\alpha\left(\beta_{g}^{U}\right)^{-1}\alpha^{-1}=({\rm{inner}})\left(\eta_{g,L}\otimes\eta_{g,R}\right).$
(3.6)

Set $\gamma_{g}:=\eta_{g,R}\beta_{g}^{U,R}$. Then we have
$\gamma_{g}\gamma_{h}\gamma_{gh}^{-1}=\mathop{\mathrm{Ad}}\nolimits\left(u(g,h)\right)$,
with some unitary $u(g,h)\in{\mathcal{A}}_{{\mathbb{Z}}^{2}}$. Hence we obtain
some $[c]\in H^{3}(G,{\rm U}(1))$.

  * Proof.

Set $\gamma_{g,R}:=\eta_{g,R}\beta_{g}^{U,R}$ and
$\gamma_{g,L}:=\eta_{g,L}\beta_{g}^{U,L}$. Taking inverse of (3.6), we get

$\alpha\beta_{g}^{U}\alpha^{-1}=({\rm{inner}})\left(\eta_{g,L}^{-1}\beta_{g}^{LU}\otimes\eta_{g,R}^{-1}\beta_{g}^{RU}\right)=({\rm{inner}})\left(\gamma_{g,L}\otimes\gamma_{g,R}\right).$

Because $\beta_{g}^{U}$ is a genuine action of $G$, we get

$\displaystyle\mathop{\mathrm{id}}\nolimits=\alpha\beta_{g}^{U}\alpha^{-1}\circ\alpha\beta_{h}^{U}\alpha^{-1}\circ\left(\alpha\beta_{gh}^{U}\alpha^{-1}\right)^{-1}$
$\displaystyle=({\rm{inner}})\circ\left(\gamma_{g,L}\gamma_{h,L}\gamma_{gh,L}^{-1}\right)\otimes\left(\gamma_{g,R}\gamma_{h,R}\gamma_{gh,R}^{-1}\right).$

This means
$\left(\gamma_{g,L}\gamma_{h,L}\gamma_{gh,L}^{-1}\right)\otimes\left(\gamma_{g,R}\gamma_{h,R}\gamma_{gh,R}^{-1}\right)$
is inner. But as it is of the tensor product form it means the automorphism
$\gamma_{g,R}\gamma_{h,R}\gamma_{gh,R}^{-1}$ on ${\mathcal{A}}_{H_{R}}$ is
inner.  $\square$

In fact, in a model derived out of Dijkgraaf-Witten model in TQFT, the ground
state $\omega_{\Phi}$ is given by some automorphism $\alpha$ and a product
state $\omega_{0}$ as $\omega_{\Phi}=\omega_{0}\circ\alpha$. This $\alpha$
satisfies the property (3.6), and above procedure allows us to recover the
”built in” third cohomology class. (See subsection 3.6.)

We note from the proof that what is happening is heuristically the following:
we are considering a genuine action $\alpha\beta_{g}^{U}\alpha^{-1}$ on the
whole space ${\mathcal{A}}_{{\mathbb{Z}}^{2}}$. This automorphism
$\alpha\beta_{g}^{U}\alpha^{-1}$ can be split into left and right as in (3.6),
modulo inner automorphisms. The fact that the original
$\alpha\beta_{g}^{U}\alpha^{-1}$ is a genuine action result in the fact that
the right half fraction $\gamma_{g}$ of $\alpha\beta_{g}^{U}\alpha^{-1}$ is a
cocycle action, i.e., $\gamma_{g}$ is a genuine action modulo inner
automorphisms.

If the automorphism $\alpha$ in the automorphic equivalence satisfies such a
property, this would have allowed us to define a $H^{3}(G,{\rm U}(1))$-valued
index. Unfortunately, it seems a it too much to ask.

###  3.3 Action of $\beta^{U}$ on $\omega_{\Phi}$

In the previous subsection, we saw that if our automorphism $\alpha$ satisfy
some nice factorization property (3.6) with respect to left-right cut, then we
can obtain some $H^{3}(G,{\rm U}(1))$-valued index. It does not look plausible
(or at least we could not prove) that every $\alpha$ in automorphic
equivalence satisfies such a nice property.

Fortunately, a weak version of (3.6) has turned out to be true. This is the
following Proposition. Recall the definition of $C_{\theta}$ (1.24).

######  Proposition 3.2.

[O7] Let $\Phi\in{\mathcal{P}}_{U.G.\beta}^{0}$. For any
$0<\theta<\frac{\pi}{2}$, there are automorphisms $\eta_{g,\sigma}$,
$\sigma=L,R$ localized in ${C_{\theta},\sigma}$, satisfying

$\displaystyle\omega_{\Phi}=\omega_{\Phi}\circ\left(\eta_{g,L}\beta_{g}^{LU}\otimes\eta_{g,R}\beta_{g}^{RU}\right)\circ({\rm{inner}}).$
(3.7)

If $\alpha$ in the automorphic equivalence
$\omega_{\Phi}=\omega_{\Phi_{0}}\circ\alpha$ satisfies the property (3.6) and
if $\omega_{\Phi_{0}}$ is invariant under $\beta_{g}^{U}$, then
$\omega_{\Phi}$ satisfies the property (3.7) for some
$\tilde{\eta}_{g,\sigma}$, $\sigma=L,R$ on
$\mathop{\mathrm{Aut}}\nolimits\left({\mathcal{A}}_{H_{\sigma}}\right)$.
Indeed, in this case,

$\displaystyle\omega_{\Phi}$
$\displaystyle=\omega_{\Phi_{0}}\circ\alpha=\omega_{\Phi_{0}}\circ\left(\beta_{g}^{U}\right)^{-1}\alpha\beta_{g}^{U}\alpha^{-1}\circ\alpha\left(\beta_{g}^{U}\right)^{-1}$
$\displaystyle=\omega_{\Phi_{0}}\circ({\rm{inner}})\left(\eta_{g^{-1},L}\otimes\eta_{g^{-1},R}\right)\circ\alpha\left(\beta_{g}^{U}\right)^{-1}$
$\displaystyle=\omega_{\Phi}\circ\alpha^{-1}\left(\eta_{g^{-1},L}\otimes\eta_{g^{-1},R}\right)\circ\alpha\left(\beta_{g}^{U}\right)^{-1}\circ({\rm{inner}}).$

Now, by the quasi-locality of $\alpha$, there are automorphisms
$\tilde{\eta}_{g,\sigma}$, $\sigma=L,R$ on
$\mathop{\mathrm{Aut}}\nolimits\left({\mathcal{A}}_{H_{\sigma}}\right)$ such
that

$\displaystyle\alpha^{-1}\left(\eta_{g^{-1},L}\otimes\eta_{g^{-1},R}\right)\circ\alpha=\left(\tilde{\eta}_{g^{-1},L}\otimes\tilde{\eta}_{g^{-1},R}\right)\circ({\rm{inner}}).$

(Lemma 1.29.) Substituting this to above, we obtain (3.7) with
$\eta_{g,\sigma}$ replaced by $\tilde{\eta}_{g,\sigma}$.

Intuitively, Proposition 3.2 can be understood as follows. Let us rewrite
(3.7) as

$\displaystyle\omega_{\Phi}\circ\beta_{g}^{U}=\omega_{\Phi}\circ\left(\eta_{g^{-1},L}\otimes\eta_{g^{-1},R}\right)\circ({\rm{inner}}).$

On the left-hand side, we are considering the action of $\beta_{g}^{U}$ on
$\omega_{\Phi}$. Recall that $\omega_{\Phi}$ is $\beta_{g}$-invariant,

$\displaystyle\omega_{\Phi}\circ\beta_{g}=\omega_{\Phi},$

because it is a unique ground state of a $\beta$-invariant interaction
$\Phi\in{\mathcal{P}}_{U.G.\beta}^{0}$. Clearly, on the upper half-plane, the
action of $\beta_{g}^{U}$ and $\beta_{g}$ are the same, while on the lower
half-plane, the action of $\beta_{g}^{U}$ and $\mathop{\mathrm{id}}\nolimits$
are the same, i.e.,

$\displaystyle\left.\beta_{g}^{U}\right|_{{\mathcal{A}}_{H_{U}}}=\left.\beta_{g}\right|_{{\mathcal{A}}_{H_{U}}},\quad\left.\beta_{g}^{U}\right|_{{\mathcal{A}}_{H_{D}}}=\mathop{\mathrm{id}}\nolimits_{{\mathcal{A}}_{H_{D}}}.$

Now recall that as a unique gapped ground state, $\omega_{\Phi}$ shows
exponential decay of correlation functions, namely, it is “almost” a product
state. (Theorem 1.11.) From this “low-correlation” and the fact that
$\omega_{\Phi}$ is $\beta_{g}$-invariant, we intuitively expect that the
effect of $\beta_{g}^{U}$ on $\omega_{\Phi}$ should be localized around the
$x$-axis. The Proposition 3.2 guarantees that it is the case.

Let us now prove Proposition 3.2.

  * Proof of Proposition 3.2.

Let $\Phi\in{\mathcal{P}}_{U.G.\beta}^{0}$. Then by the automorphic
equivalence Theorem 1.26, there is an automorphism $\alpha$ such that
$\omega_{\Phi}=\omega_{\Phi_{0}}\circ\alpha$. Using the factorization property
of $\alpha$, for any $0<\theta^{\prime}<\frac{\pi}{2}$, we obtain the
following decompositions

$\displaystyle\begin{split}&\alpha\circ\beta_{g}^{U}\circ\alpha^{-1}=({\rm{inner}})\circ\left(\mathop{\mathrm{id}}\nolimits_{{\mathcal{A}}_{H_{D}}}\otimes\xi_{U}\right)\circ\left(\Xi_{H_{U},L}\otimes\Xi_{H_{U},R}\right),\\\
&\alpha\circ\beta_{g}\circ\alpha^{-1}=({\rm{inner}})\circ\left(\xi_{D}\otimes\xi_{U}\right)\circ\left(\Xi_{{\mathbb{Z}}^{2},L}\otimes\Xi_{{\mathbb{Z}}^{2},R}\right).\end{split}$
(3.8)

Here, $\xi_{U}$ (resp. $\xi_{D}$) is an automorphism localized on the upper
(resp. lower) half-plane $H_{U}$ (resp. $H_{D}$), while $\Xi_{H_{U},L}$ and
$\Xi_{{\mathbb{Z}}^{2},L}$ (resp. $\Xi_{H_{U},R}$ and
$\Xi_{{\mathbb{Z}}^{2},R}$) are automorphisms localized on
$C_{\theta^{\prime},L}$ (resp. $C_{\theta^{\prime},R}$).

Because $\omega_{\Phi}$ is a unique ground state of a $\beta$-invariant
interaction, it is $\beta$-invariant. Therefore, we have

$\displaystyle\omega_{\Phi_{0}}\circ\alpha\circ\beta_{g}\circ\alpha^{-1}=\omega_{\Phi_{0}}.$
(3.9)

From the decomposition (3.8), we then get

$\displaystyle\omega_{D}\circ\xi_{D}\otimes\omega_{U}\circ\xi_{U}=\omega_{\Phi_{0}}\circ\left(\xi_{D}\otimes\xi_{U}\right)=\omega_{\Phi_{0}}\circ\left(\Xi_{{\mathbb{Z}}^{2},L}^{-1}\otimes\Xi_{{\mathbb{Z}}^{2},R}^{-1}\right)\circ({\rm{inner}})$
(3.10)

Note that the left-hand side is of the tensor product for cut $H_{D}-H_{U}$,
while the right-hand side is almost tensor product form for cut
$\left(H_{D}\cup C_{\theta^{\prime}}\right)-\left(H_{D}\cup
C_{\theta^{\prime}}\right)^{c}$. From this, the pure state
$\omega_{U}\circ\xi_{U}$ on ${\mathcal{A}}_{H_{U}}$ satisfies the split
property with respect to ${\mathcal{A}}_{\left(C_{\theta^{\prime},U}\right)}$
and ${\mathcal{A}}_{\left(C_{\theta^{\prime}}\right)^{c}\cap H_{U}}$.
Furthermore, the restriction of $\omega_{U}\circ\xi_{U}$ onto
${\mathcal{A}}_{\left(C_{\theta^{\prime}}\right)^{c}\cap H_{U}}$ is quasi-
equivalent to the restriction of the pure product state $\omega_{\Phi_{0}}$
onto the same algebra. As a result (Lemma 1.19), there is an automorphism $S$
on ${\mathcal{A}}_{\left(C_{\theta,U}\right)}$ such that

$\displaystyle\begin{split}&\omega_{U}\circ\xi_{U}=\omega_{\Phi_{0}}\circ\left(S\otimes\mathop{\mathrm{id}}\nolimits_{{\mathcal{A}}_{\left(\left(C_{\theta^{\prime}}\right)^{c}\cap
H_{U}\right)}}\right)\circ({\rm{inner}})\\\
&=\left(\left(\omega_{\Phi_{0}}|_{{\mathcal{A}}_{\left(C_{\theta^{\prime},U}\right)}}\circ
S\right)\otimes\left(\omega_{\Phi_{0}}|_{{\mathcal{A}}_{\left(\left(C_{\theta^{\prime}}\right)^{c}\cap
H_{U}\right)}}\right)\right)\circ({\rm{inner}}).\end{split}$ (3.11)

On the other hand, from (3.10), $\omega_{U}\circ\xi_{U}$ also satisfies the
split property with respect to ${H_{U}\cap H_{L}}$-${H_{U}\cap H_{R}}$ cut. As
a result (Lemma 1.19), there are automorphisms
$\tilde{\eta}_{\sigma,g}\in\mathop{\mathrm{Aut}}\nolimits\left({\mathcal{A}}_{\left(C_{\theta^{\prime}}\right)_{\sigma}}\right)$,
$g\in G$, $\sigma=L,R$ such that

$\displaystyle\omega_{\Phi_{0}}|_{{\mathcal{A}}_{\left(C_{\theta^{\prime},U}\right)}}\circ
S=\left(\left(\omega_{\Phi_{0}}|_{{\mathcal{A}}_{\left(C_{\theta^{\prime},U}\cap
H_{L}\right)}}\circ\tilde{\eta}_{L,g}\right)\otimes\left(\omega_{\Phi_{0}}|_{{\mathcal{A}}_{\left(C_{\theta^{\prime},U}\cap
H_{R}\right)}}\circ\tilde{\eta}_{R,g}\right)\right)\circ({\rm{inner}}).$
(3.12)

Substituting this to (3.11), we obtain

$\displaystyle\omega_{U}\circ\xi_{U}=\left(\left(\omega|_{{\mathcal{A}}_{H_{U}\cap
H_{L}}}\circ\tilde{\eta}_{L,g}\right)\otimes\left(\omega|_{{\mathcal{A}}_{H_{U}\cap
H_{R}}}\circ\tilde{\eta}_{R,g}\right)\right)\circ({\rm{inner}})$

Now, from this and (3.8), we obtain a decomposition of
$\omega_{\Phi}\circ\beta_{g}^{U}$:

$\displaystyle\begin{split}&\omega_{\Phi}\circ\beta_{g}^{U}=\omega_{\Phi_{0}}\circ\alpha\circ\beta_{g}^{U}\circ\alpha^{-1}\circ\alpha=\left(\omega_{D}\otimes\omega_{U}\xi_{U}\right)\circ\left(\Xi_{H_{U},L}\otimes\Xi_{H_{U},R}\right)\circ({\rm{inner}})\circ\alpha\\\
&=\omega_{\Phi_{0}}\circ\left(\hat{\eta}_{g,L}\otimes\hat{\eta}_{g,R}\right)\circ\alpha\circ({\rm{inner}}),\end{split}$
(3.13)

where $\hat{\eta}_{g,\sigma}$ is an automorphism localized in
${C_{\theta^{\prime}},\sigma}$. Because of the factorization property of
$\alpha$ for any $\theta$ with $\theta^{\prime}<\theta<\frac{\pi}{2}$, we may
find (Lemma 1.29) automorphisms $\eta_{g^{-1},\sigma}$, $\sigma=L,R$ localized
in ${C_{\theta},\sigma}$, satisfying

$\displaystyle\begin{split}&\omega_{\Phi}\circ\beta_{g}^{U}=\omega_{\Phi_{0}}\circ\left(\hat{\eta}_{g,L}\otimes\hat{\eta}_{g,R}\right)\circ\alpha\circ({\rm{inner}})=\omega_{\Phi}\circ\alpha^{-1}\circ\left(\hat{\eta}_{g,L}\otimes\hat{\eta}_{g,R}\right)\circ\alpha\circ({\rm{inner}})\\\
&=\omega_{\Phi}\circ\left(\eta_{g^{-1},L}\otimes\eta_{g^{-1},R}\right)\circ({\rm{inner}}).\end{split}$
(3.14)

Hence we have obtained the Proposition 3.2.  $\square$

###  3.4 Definition of $H^{3}(G,{\rm U}(1))$-valued index

Recall when an automorphism satisfies a nice condition (3.6), we could obtain
a $H^{3}(G,{\rm U}(1))$-valued index by considering $u(g,h)\in{\mathcal{A}}$
implementing $\gamma_{g}\gamma_{h}\gamma_{gh}^{-1}$ , for
$\gamma_{g}:=\eta_{g,R}\beta_{g}^{U,R}$. From our weaker version Proposition
3.2, we cannot gurantee the existence of such $u(g,h)$ inside of
${\mathcal{A}}$. However, we can get some unitary $u(g,h)$ on the GNS-Hilbert
space ${\mathcal{H}}_{R}$ of $\omega_{R}$, implementing

$\displaystyle\gamma_{g}\gamma_{h}\gamma_{gh}^{-1}=\eta_{g,R}\beta_{g}^{RU}\eta_{h,R}\left(\beta_{g}^{RU}\right)^{-1}\left(\eta_{gh,R}\right)^{-1}$

with respect to a representation $\pi_{R}\alpha_{R}$ of ${\mathcal{A}}_{R}$.
This is the following proposition.

######  Proposition 3.3.

[O7] For $\Phi\in{\mathcal{P}}_{U.G.\beta}^{0}$, and $0<\theta<\frac{\pi}{2}$,
let $\eta_{g,\sigma}$, $\sigma=L,R$ be automorphisms given by Proposition 3.2.
Let $\alpha$ be an automorphism on ${\mathcal{A}}_{{\mathbb{Z}}^{2}}$
satisfying the factorization property
$\alpha=\left(\alpha_{L}\otimes\alpha_{R}\right)\circ\Theta\circ({\rm{inner}})$
for
$\alpha_{\sigma}\in\mathop{\mathrm{Aut}}\nolimits({\mathcal{A}}_{H_{\sigma}})$,
$\sigma=L,R$ and
$\Theta\in\mathop{\mathrm{Aut}}\nolimits\left({\mathcal{A}}_{C_{\theta}^{c}}\right)$.
Suppose that $\omega_{\Phi}=\omega_{\Phi_{0}}\circ\alpha$.

Then there is a unitary $u(g,h)$ on ${\mathcal{H}}_{R}$ such that

$\displaystyle\mathop{\mathrm{Ad}}\nolimits\left(u(g,h)\right)\pi_{R}\circ\alpha_{R}(A)=\pi_{R}\circ\alpha_{R}\eta_{g,R}\beta_{g}^{RU}\eta_{h,R}\left(\beta_{g}^{RU}\right)^{-1}\left(\eta_{gh,R}\right)^{-1}(A),\quad
A\in{\mathcal{A}}_{R}.$ (3.15)

######  Remark 3.4.

Setting $\gamma_{g}:=\eta_{g,R}\beta_{g}^{U,R}$, we can regard this as a weak
version of Lemma 3.1.

  * Proof.

With
$\tilde{\beta}_{g}:=\alpha\left(\eta_{g,L}\otimes\eta_{g,R}\right)\beta_{g}^{U}\alpha^{-1}$,
we get
$\omega_{\Phi_{0}}\circ\tilde{\beta}_{g}:=\omega_{\Phi_{0}}\circ({\rm{inner}})$.
Substituting the definition of $\tilde{\beta}_{g}$ and the factorization
property
$\alpha=\left(\alpha_{L}\otimes\alpha_{R}\right)\circ\Theta\circ({\rm{inner}})$,
we get

$\displaystyle\omega_{\Phi_{0}}\circ({\rm{inner}})=\omega_{\Phi_{0}}\circ\tilde{\beta}_{g}\tilde{\beta}_{h}\tilde{\beta}_{gh}^{-1}$
$\displaystyle=\omega_{\Phi_{0}}\circ\alpha\left(\eta_{g,L}\beta_{g}^{LU}\eta_{h,L}\left(\beta_{g}^{LU}\right)^{-1}\left(\eta_{gh,L}\right)^{-1}\otimes\eta_{g,R}\beta_{g}^{RU}\eta_{h,R}\left(\beta_{g}^{RU}\right)^{-1}\left(\eta_{gh,R}\right)^{-1}\right)\alpha^{-1}$
$\displaystyle=\omega_{\Phi_{0}}\circ\left(\alpha_{L}\otimes\alpha_{R}\right)\Theta$
$\displaystyle\quad\left(\eta_{g,L}\beta_{g}^{LU}\eta_{h,L}\left(\beta_{g}^{LU}\right)^{-1}\left(\eta_{gh,L}\right)^{-1}\otimes\eta_{g,R}\beta_{g}^{RU}\eta_{h,R}\left(\beta_{g}^{RU}\right)^{-1}\left(\eta_{gh,R}\right)^{-1}\right)$
$\displaystyle\quad\Theta^{-1}\left(\alpha_{L}\otimes\alpha_{R}\right)^{-1}\circ({\rm{inner}}).$

However, $\Theta$ is localized at $\left(C_{\theta}\right)^{c}$, while
$\eta_{g,R}\beta_{g}^{RU}\eta_{h,R}\left(\beta_{g}^{RU}\right)^{-1}\left(\eta_{gh,R}\right)^{-1}$
is localized at $C_{\theta}$. Therefore, they commute, and we obtain

$\displaystyle\left(\omega_{L}\otimes\omega_{R}\right)\circ({\rm{inner}})=\omega_{\Phi_{0}}\circ({\rm{inner}})$
$\displaystyle=\omega_{\Phi_{0}}\circ\left(\alpha_{L}\eta_{g,L}\beta_{g}^{LU}\eta_{h,L}\left(\beta_{g}^{LU}\right)^{-1}\left(\eta_{gh,L}\right)^{-1}\alpha_{L}^{-1}\otimes\alpha_{R}\eta_{g,R}\beta_{g}^{RU}\eta_{h,R}\left(\beta_{g}^{RU}\right)^{-1}\left(\eta_{gh,R}\right)^{-1}\alpha_{R}^{-1}\right)$
$\displaystyle=\omega_{L}\circ\left(\alpha_{L}\eta_{g,L}\beta_{g}^{LU}\eta_{h,L}\left(\beta_{g}^{LU}\right)^{-1}\left(\eta_{gh,L}\right)^{-1}\alpha_{L}^{-1}\right)$
$\displaystyle\quad\otimes\omega_{R}\circ\left(\alpha_{R}\eta_{g,R}\beta_{g}^{RU}\eta_{h,R}\left(\beta_{g}^{RU}\right)^{-1}\left(\eta_{gh,R}\right)^{-1}\alpha_{R}^{-1}\right).$

Because $\omega_{L}$ and $\omega_{R}$ are pure, we may apply Lemma 1.7. As a
result, there is a unitary $u(g,h)$ on ${\mathcal{H}}_{R}$ such that

$\displaystyle\begin{split}&\mathop{\mathrm{Ad}}\nolimits\left(u(g,h)\right)\pi_{R}(A)=\pi_{R}\circ\alpha_{R}\eta_{g,R}\beta_{g}^{RU}\eta_{h,R}\left(\beta_{g}^{RU}\right)^{-1}\left(\eta_{gh,R}\right)^{-1}\alpha_{R}^{-1}(A)\end{split}$

This proves the Proposition.  $\square$

Note that in general, $\gamma_{g}=\eta_{g,R}\beta_{g}^{RU}$ does not need to
be implemented by a unitary for the
representation$({\mathcal{H}}_{R},\pi_{R}\circ\alpha_{R})$. In general,
representations $\pi_{R}\circ\alpha_{R}$ and
$\pi_{R}\circ\alpha_{R}\circ\gamma_{g}$ are mutually singular. In fact, if
they are quasi-equivalent the third cohomology class obtained below will be
$0$. Intuitively, $\pi_{R}\circ\alpha_{R}$ and
$\pi_{R}\circ\alpha_{R}\circ\gamma_{g}$ are macroscopically different.
However, from the above Proposition 3.3, if we consider the combination
$\gamma_{g}\gamma_{h}\gamma_{gh}^{-1}$, it is implementable in
$({\mathcal{H}}_{R},\pi_{R}\circ\alpha_{R})$, namely, $\pi_{R}\circ\alpha_{R}$
and $\pi_{R}\circ\alpha_{R}\circ\gamma_{g}\gamma_{h}\gamma_{gh}^{-1}$ are
quasi-equivalent. Physically, they are macroscopically the same.

This situation is parallel to the situation in subsection 3.2. Recall in
subsection 3.2, the fact that the original $\alpha\beta_{g}^{U}\alpha^{-1}$ is
a genuine action result in the fact that the right half fraction $\gamma_{g}$
is a genuine action modulo inner automorphisms of ${\mathcal{A}}$. Here, in
Proposition 3.3, the fact that the original $\alpha\beta_{g}^{U}\alpha^{-1}$
is a genuine action result in the fact that $\gamma_{g}$ is a genuine action
modulo unitaries on ${\mathcal{H}}_{R}$.

Having this $u(g,h)$, we expect something like (3.2)

$\displaystyle
u(g,h)u(gh,k)=c(g,h,k)\gamma_{g}\left(u(h,k)\right)u(g,hk),\quad g,h,k\in G.$

should holds. One question occurs : our
$\gamma_{g}:=\eta_{g,R}\beta_{g}^{U,R}$ is an automorphism on
${\mathcal{A}}_{R}$, and it does not act on a bounded opearators on
${\mathcal{H}}_{R}$, in general. In particular, we do not know an action of
our $\gamma_{g}:=\eta_{g,R}\beta_{g}^{U,R}$ on $u(g,h)$. Fortunately, there is
an alternative.

######  Proposition 3.5.

[O7] Let $\Phi\in{\mathcal{P}}_{U.G.\beta}^{0}$, and $0<\theta<\frac{\pi}{2}$,
and $\eta_{g,\sigma}$, $\sigma=L,R$ be automorphisms given by Proposition 3.2.
Let $\alpha$ be an automorphism on ${\mathcal{A}}_{{\mathbb{Z}}^{2}}$
satisfying the factorization property
$\alpha=\left(\alpha_{L}\otimes\alpha_{R}\right)\circ\Theta\circ({\rm{inner}})$
with
$\alpha_{\sigma}\in\mathop{\mathrm{Aut}}\nolimits({\mathcal{A}}_{H_{\sigma}})$,
$\sigma=L,R$ and
$\Theta\in\mathop{\mathrm{Aut}}\nolimits\left({\mathcal{A}}_{C_{\theta}^{c}}\right)$.
Suppose that $\omega_{\Phi}=\omega_{\Phi_{0}}\circ\alpha$. Then there are
unitaries $W_{g}$, $g\in G$ on ${\mathcal{H}}_{0}$ such that

$\displaystyle\mathop{\mathrm{Ad}}\nolimits\left(W_{g}\right)\circ\pi_{0}=\pi_{0}\circ\left(\alpha_{0}\right)\circ\Theta\circ\eta_{g}\beta_{g}^{U}\circ\Theta^{-1}\circ\alpha_{0}^{-1},\quad
g\in G,$ (3.16)

where $\alpha_{0}:=\alpha_{L}\otimes\alpha_{R}$ and
$\eta_{g}:=\eta_{g,L}\otimes\eta_{g,R}$. Furthermore, there is a $c\in
Z^{3}(G,{\mathbb{T}})$ such that

$\displaystyle\mathbb{I}_{{\mathcal{H}}_{L}}\otimes
u(g,h)u(gh,k)=c(g,h,k)\left(W_{g}\left(\mathbb{I}_{{\mathcal{H}}_{L}}\otimes
u(h,k)\right)W_{g}^{*}\right)\left(\mathbb{I}_{{\mathcal{H}}_{L}}\otimes
u(g,hk)\right),$ (3.17)

for all $g,h,k\in G$.

  * Proof.

As in Proposition 3.3, an automorphism

$\tilde{\beta}_{g}:=\alpha\left(\eta_{g,L}\otimes\eta_{g,R}\right)\beta_{g}^{U}\alpha^{-1}=\alpha_{0}\Theta\circ\left(\eta_{g,L}\otimes\eta_{g,R}\right)\beta_{g}^{U}\Theta^{-1}\alpha_{0}^{-1}\circ({\rm{inner}})$

satisfies
$\omega_{\Phi_{0}}\circ\tilde{\beta}_{g}:=\omega_{\Phi_{0}}\circ({\rm{inner}})$.
It means $\omega_{\Phi_{0}}$ is invariant under an automorphism

$\alpha_{0}\Theta\circ\left(\eta_{g,L}\otimes\eta_{g,R}\right)\beta_{g}^{U}\Theta^{-1}\alpha_{0}^{-1}\circ\mathop{\mathrm{Ad}}\nolimits(V_{g})$

with some unitary $V_{g}\in{\mathcal{A}}$. Applying Theorem 1.2, we obtain
$\tilde{W}_{g}$ satisfying

$\displaystyle\mathop{\mathrm{Ad}}\nolimits(\tilde{W}_{g})\circ\pi_{0}=\pi_{0}\circ\alpha_{0}\Theta\circ\left(\eta_{g,L}\otimes\eta_{g,R}\right)\beta_{g}^{U}\Theta^{-1}\alpha_{0}^{-1}\circ\mathop{\mathrm{Ad}}\nolimits(V_{g}).$

Absorbing $V_{g}$ part, we obtain $W_{g}$ satisfying (3.16).

Next we prove that there is a number $c(g,h,k)\in{\mathbb{T}}$ satisfying
(3.17). From Proposition 3.3, we have

$\displaystyle\begin{split}&\mathop{\mathrm{Ad}}\nolimits\left(\mathbb{I}_{{\mathcal{H}}_{L}}\otimes
u(g,h)u(gh,k)\right)\pi_{0}\\\
&=\pi_{L}\otimes\pi_{R}\circ\alpha_{R}\circ\left(\eta_{g,R}\beta_{g}^{RU}\right)\left(\eta_{h,R}\beta_{h}^{RU}\right)\left(\eta_{k}^{R}\beta_{k}^{RU}\right)\left(\eta_{ghk}^{R}\beta_{ghk}^{RU}\right)^{-1}\alpha_{R}^{-1}.\end{split}$
(3.18)

On the other hand, from (3.16) and (3.15), using the fact that the support of
$\Theta$ and $\eta$s are disjoint (as in the proof of Proposition 3.2), we
obtain

$\displaystyle\mathop{\mathrm{Ad}}\nolimits\left(W_{g}\left(\mathbb{I}_{{\mathcal{H}}_{L}}\otimes
u(h,k)\right)W_{g}^{*}\right)\circ\pi_{0}$
$\displaystyle=\pi_{L}\otimes\pi_{R}\left(\alpha_{R}\eta_{g,R}\beta_{g}^{RU}\left(\eta_{h,R}\beta_{h}^{RU}\eta_{k}^{R}\left(\beta_{h}^{RU}\right)^{-1}\left(\eta_{hk}^{R}\right)^{-1}\right)\left(\eta_{g,R}\beta_{g}^{RU}\right)^{-1}\alpha_{R}^{-1}\right)$

for all $g,h,k\in G$. Using this and Proposition 3.2, we see that

$\displaystyle\mathop{\mathrm{Ad}}\nolimits\left(\left(W_{g}\left(\mathbb{I}_{{\mathcal{H}}_{L}}\otimes
u(h,k)\right)W_{g}^{*}\right)\left(\mathbb{I}_{{\mathcal{H}}_{L}}\otimes
u(g,hk)\right)\right)\pi_{0}$ (3.19)

is also equal to the right-hand side of (3.18). Because $\pi_{0}$ is
irreducible, this means that there is a number $c(g,h,k)\in{\mathbb{T}}$
satisfying (3.17).

The proof that $c$ is a $3$-cocycle is analogous to that in subsectoin 3.1. In
order to carry the argument there out, we just need the following:

$\displaystyle\mathop{\mathrm{Ad}}\nolimits\left(W_{g}W_{h}\right)\left(\mathbb{I}_{{\mathcal{H}}_{L}}\otimes
u(k,f)\right)=\left(\mathop{\mathrm{Ad}}\nolimits\left(\left(\mathbb{I}_{{\mathcal{H}}_{L}}\otimes
u(g,h)\right)W_{gh}\right)\right)\left(\mathbb{I}_{{\mathcal{H}}_{L}}\otimes
u(k,f)\right).$ (3.20)

To see this, first we note from (3.17) that

$\displaystyle\mathop{\mathrm{Ad}}\nolimits\left(W_{gh}\right)\left(\mathbb{I}_{{\mathcal{H}}_{L}}\otimes
u(k,f)\right)\in\mathbb{I}_{{\mathcal{H}}_{L}}\otimes{\mathcal{B}}({\mathcal{H}}_{R}),$
(3.21)

for any $g,h,k,f\in G$. We get

$\displaystyle\begin{split}&\mathop{\mathrm{Ad}}\nolimits\left(W_{g}W_{h}W_{gh}^{*}\right)\left(\mathbb{I}_{{\mathcal{H}}_{L}}\otimes\pi_{R}(A)\right)\\\
&=\mathbb{I}_{{\mathcal{H}}_{L}}\otimes\pi_{R}\circ\alpha_{R}\circ\eta_{g,R}\beta_{g}^{RU}\eta_{h,R}\left(\beta_{g}^{RU}\right)^{-1}\eta_{gh,R}^{-1}\circ\alpha_{R}^{-1}(A)\\\
&=\mathbb{I}_{{\mathcal{H}}_{L}}\otimes\mathop{\mathrm{Ad}}\nolimits
u(g,h)\circ\pi_{R}(A),\end{split}$

for all $A\in{\mathcal{A}}_{R}$, again, using the commutativity of $\Theta$
and $\eta$s. Because $\pi_{R}$ is irreducible, this means

$\displaystyle W_{g}W_{h}W_{gh}^{*}\left(\mathbb{I}_{{\mathcal{H}}_{L}}\otimes
u(g,h)\right)^{*}\in{\mathcal{B}}({\mathcal{H}}_{L})\otimes\mathbb{I}_{{\mathcal{H}}_{R}}.$

Hence using (3.21), we obtain

$\displaystyle\begin{split}&\mathop{\mathrm{Ad}}\nolimits\left(W_{g}W_{h}W_{gh}^{*}\left(\mathbb{I}_{{\mathcal{H}}_{L}}\otimes
u(g,h)\right)^{*}\right)\left(\mathop{\mathrm{Ad}}\nolimits\left(\left(\mathbb{I}_{{\mathcal{H}}_{L}}\otimes
u(g,h)\right)W_{gh}\right)\right)\left(\mathbb{I}_{{\mathcal{H}}_{L}}\otimes
u(k,f)\right)\\\
&=\left(\mathop{\mathrm{Ad}}\nolimits\left(\left(\mathbb{I}_{{\mathcal{H}}_{L}}\otimes
u(g,h)\right)W_{gh}\right)\right)\left(\mathbb{I}_{{\mathcal{H}}_{L}}\otimes
u(k,f)\right).\end{split}$

This implies the required property (3.20).  $\square$

Hence we have obtained some third group cohomology. In order to define it, we
made a lot of choices. First of all, there can be many path to connect $\Phi$
to $\Phi_{0}$. According to that, there can be many choices for $\alpha$. The
factorization property was key to define the index, but there is a freedom of
the direction $\theta$ to cut the plane, as well as the choice of
$\alpha_{R}$, $\alpha_{L}$, $\Theta$. Furthermore, the choice of
$\eta_{g,L},\eta_{g,R}$ of Proposition 3.2 is neither unique. Does resulting
third cohomology class obtained in Proposition 3.5 depend on such choices?
Fortunately, the answer turns out to be no.

######  Theorem 3.6.

[O7] Let $\Phi\in{\mathcal{P}}_{U.G.\beta}^{0}$. The third cohomology class
$[c]_{H^{3}(G,{\rm U}(1))}$ obtained from Proposition 3.5 does not depend on
the choice of $\alpha$, $0<\theta<\frac{\pi}{2}$,
$\alpha_{\sigma}\in\mathop{\mathrm{Aut}}\nolimits({\mathcal{A}}_{H_{\sigma}})$,
$\sigma=L,R$,
$\Theta\in\mathop{\mathrm{Aut}}\nolimits\left({\mathcal{A}}_{C_{\theta}^{c}}\right)$,
$W_{g}$, $u(g,h)$, and $\eta_{g,\sigma}$, $\sigma=L,R$.

######  Definition 3.7.

[O7] We denote by $h(\Phi)$, the third cohomology class $[c]_{H^{3}(G,{\rm
U}(1))}$ obtained in Theorem 3.6.

###  3.5 What $h(\Phi)$ represents

Now let us investigate what $h(\Phi)$ represents. In $1$-dimension, the
$H^{2}(G,{\rm U}(1))$ represents how the left-half and right-half of the
chains are correlated, in terms of group action. We can have an analogous
observation in $2$-dimensional systems. Let us denote $u(g,h)$ $c(g,h,k)$
given in the previous subsection by $u_{R}(g,h)$ and $c_{R}(g,h,k)$. The same
argument gives us $u_{L}(g,h)$ and $c_{L}(g,h,k)$, for left half-plane. Hence
we get

$\displaystyle\mathop{\mathrm{Ad}}\nolimits\left(u_{\sigma}(g,h)\right)\pi_{\sigma}\circ\alpha_{\sigma}(A)=\pi_{\sigma}\circ\alpha_{\sigma}\eta_{g,\sigma}\beta_{g}^{\sigma
U}\eta_{h,\sigma}\left(\beta_{g}^{\sigma
U}\right)^{-1}\left(\eta_{gh,\sigma}\right)^{-1}(A),\quad
A\in{\mathcal{A}}_{\sigma}.$ (3.22)

for $\sigma=L,R$. By the definition of $W_{g}$, and the commutativity of
$\Theta$ and $\eta$s, we have

$\displaystyle\mathop{\mathrm{Ad}}\nolimits\left({W_{g}W_{h}W_{gh}^{*}}\right)\circ\bigotimes_{\sigma=L,R}\pi_{\sigma}=\bigotimes_{\sigma=L,R}\pi_{\sigma}\circ\alpha_{\sigma}\eta_{g,\sigma}\beta_{g}^{\sigma
U}\eta_{h,\sigma}\left(\beta_{g}^{\sigma
U}\right)^{-1}\left(\eta_{gh,\sigma}\right)^{-1}\alpha_{\sigma}^{-1}.$

Comparing these equations, from the irreducibility of $\pi_{0}$, we have

$\displaystyle\mathop{\mathrm{Ad}}\nolimits\left({W_{g}W_{h}W_{gh}^{*}}\right)(x)=\mathop{\mathrm{Ad}}\nolimits\left({u_{L}(g,h)}\otimes{u_{R}(g,h)}\right)(x),\quad
x\in{\mathcal{B}}({\mathcal{H}}_{0}).$

This means there is some $\sigma(g,h)\in{\rm U}(1)$ such that

$\displaystyle W_{g}W_{h}W_{gh}^{*}=\sigma(g,h)u_{L}(g,h)\otimes u_{R}(g,h).$

Absorbing the ${\rm U}(1)$-phase to $u_{L}(g,h)$, we may set $\sigma(g,h)=1$.
Hence we get

$\displaystyle W_{g}W_{h}W_{gh}^{*}=u_{L}(g,h)\otimes u_{R}(g,h).$

By the same argument in the previous section for the left half-plane, there
are $c_{R},c_{L}\in Z^{3}(G,{\rm U}(1))$ such that

$\displaystyle\begin{split}&\mathbb{I}_{{\mathcal{H}}_{L}}\otimes
u_{R}(g,h)u_{R}(gh,k)={c_{R}(g,h,k)}\left(W_{g}\left(\mathbb{I}_{{\mathcal{H}}_{L}}\otimes
u_{R}(h,k)\right)W_{g}^{*}\right)\left(\mathbb{I}_{{\mathcal{H}}_{L}}\otimes
u_{R}(g,hk)\right)\\\
&u_{L}(g,h)u_{L}(gh,k)\otimes\mathbb{I}_{{\mathcal{H}}_{R}}={c_{L}(g,h,k)}\left(W_{g}\left(u_{L}(h,k)\otimes\mathbb{I}_{{\mathcal{H}}_{R}}\right)W_{g}^{*}\right)\left(u_{L}(g,hk)\otimes\mathbb{I}_{{\mathcal{H}}_{L}}\right).\end{split}$
(3.23)

Now we set

$V(g,h):=W_{g}W_{h}W_{gh}^{*}.$

Clearly, it satisfies the $2$-cocycle relation

${V(g,h)}{V(gh,k)}=\mathop{\mathrm{Ad}}\nolimits(W_{g})\left({V(h,k)}\right)\cdot{V(g,hk)},$

because

$\left({W_{g}W_{h}W_{gh}^{*}}\right)\left({W_{gh}W_{k}W_{ghk}^{*}}\right)=W_{g}\left({W_{h}W_{k}W_{hk}^{*}}\right)W_{g}^{*}\left({W_{g}W_{hk}W_{ghk}^{*}}\right).$

On the other hand, $u_{L},u_{R}$ satisfies the obstructed versions, (3.23).
Now, substituting $V(g,h)=W_{g}W_{h}W_{gh}^{*}=u_{L}(g,h)\otimes u_{R}(g,h)$,
we get

$\displaystyle[c_{L}]_{H^{3}(G,{\rm U}(1))}\cdot[c_{R}]_{H^{3}(G,{\rm
U}(1))}=1.$

From this, we can interpret that what we see via the third cohomology valued
index $h_{\Phi}$ is how $V(g,h)$ satisfying a genuine $2$-cocycle relation
split into left-right $u^{L}$, $u^{R}$, satisfying “obstacled $2$-cocycle
relations”.

Suppose that $\omega_{\Phi}$ is of product form with respect to
$H_{D}$-$H_{U}$ cut

$\omega_{\Phi}=\omega_{\Phi}|_{{\mathcal{A}}_{H_{D}}}\otimes\omega_{\Phi}|_{{\mathcal{A}}_{H_{U}}}.$

Because $\omega_{\Phi}$ is $\beta$-invariant, we have

$\omega_{\Phi}|_{{\mathcal{A}}_{H_{D}}}\circ\beta_{g}^{D}\otimes\omega_{\Phi}|_{{\mathcal{A}}_{H_{U}}}\circ\beta_{g}^{U}=\omega_{\Phi}\circ\beta_{g}=\omega_{\Phi}=\omega_{\Phi}|_{{\mathcal{A}}_{H_{D}}}\otimes\omega_{\Phi}|_{{\mathcal{A}}_{H_{U}}}.$

Hence we have

$\displaystyle\omega_{\Phi}|_{{\mathcal{A}}_{H_{U}}}\circ\beta_{g}^{U}=\omega_{\Phi}|_{{\mathcal{A}}_{H_{U}}},$

and

$\displaystyle\omega_{\Phi}\circ\beta_{g}^{U}=\omega_{\Phi}.$

It means that in Proposition 3.2, we may take $\eta_{g,R}$, $\eta_{g,L}$ as
identities. In this case, we may take $u_{R}$, $u_{L}$ as identities. As a
result, we get

$\displaystyle[c_{L}]_{H^{3}(G,{\rm U}(1))}=[c_{R}]_{H^{3}(G,{\rm U}(1))}=1.$

Hence $1$ is split into left and right in a trivial manner.

Suppose that $\omega_{\Phi}$ is of product form with respect to
$H_{L}$-$H_{R}$ cut

$\omega_{\Phi}=\omega_{\Phi}|_{{\mathcal{A}}_{H_{L}}}\otimes\omega_{\Phi}|_{{\mathcal{A}}_{H_{R}}}.$

In this case, in order to define the $H^{3}(G,{\rm U}(1))$-valued index, we
may take $\alpha$ of the form $\alpha=\alpha_{L}\otimes\alpha_{R}$, with
$\alpha_{L}\in\mathop{\mathrm{Aut}}\nolimits\left({\mathcal{A}}_{H_{L}}\right)$,
and
$\alpha_{R}\in\mathop{\mathrm{Aut}}\nolimits\left({\mathcal{A}}_{H_{R}}\right)$.
(Actually, this is allowed because our $H^{3}(G,{\rm U}(1))$-valued index can
actually be defined on the wider class of systems. See the original paper
[O7].) As a result, we may set

$\displaystyle\omega_{\Phi}|_{{\mathcal{A}}_{H_{L}}}=\omega_{L}\circ\alpha_{L},\quad\omega_{\Phi}|_{{\mathcal{A}}_{H_{R}}}=\omega_{R}\circ\alpha_{R},$
$\displaystyle\Theta=\mathop{\mathrm{id}}\nolimits.$

Now, from Proposition 3.2, there are automorphisms $\eta_{g,\sigma}$,
$\sigma=L,R$ localized in $C_{\theta,\sigma}$, satisfying

$\displaystyle\omega_{\Phi}=\omega_{\Phi}\circ\left(\eta_{g,L}\beta_{g}^{LU}\otimes\eta_{g,R}\beta_{g}^{RU}\right)\circ({\rm{inner}}).$

From this, we see that

$\displaystyle\begin{split}&\omega_{L}\circ\alpha_{L}\circ\left(\beta_{g}^{LU}\right)^{-1}\otimes\omega_{R}\circ\alpha_{R}\circ\left(\beta_{g}^{RU}\right)^{-1}\\\
&=\omega_{\Phi}\circ\left(\beta_{g}^{U}\right)^{-1}\sim_{q.e.}\omega_{\Phi}\circ\left(\eta_{g,L}\otimes\eta_{g,R}\right)\\\
&=\omega_{L}\circ\alpha_{L}\circ\eta_{g,L}\otimes\omega_{R}\circ\alpha_{R}\circ\eta_{g,R}.\end{split}$

This implies that

$\displaystyle\omega_{\sigma}\circ\alpha_{\sigma}\circ\left(\beta_{g}^{\sigma
U}\right)^{-1}\sim_{q.e.}\omega_{\sigma}\circ\alpha_{\sigma}\circ\eta_{g,\sigma},$

for $\sigma=L,R$. Because $\omega_{\sigma}$ is pure, we see that there is a
unitary $v_{g,\sigma}$ on ${\mathcal{H}}_{\sigma}$ such that

$\displaystyle\mathop{\mathrm{Ad}}\nolimits\left(v_{g,\sigma}\right)\circ\pi_{\sigma}(A)=\pi_{\sigma}\circ\alpha_{\sigma}\circ\eta_{g,\sigma}\beta_{g}^{\sigma
U}\circ\alpha_{\sigma}^{-1}(A),\quad A\in{\mathcal{A}}_{H_{\sigma}},$

for each $\sigma=L,R$. Because $\Theta=\mathop{\mathrm{id}}\nolimits$ now, we
have

$\displaystyle\begin{split}\mathop{\mathrm{Ad}}\nolimits\left(W_{g}\right)\circ\pi_{0}&=\pi_{0}\circ\left(\alpha_{0}\right)\circ\Theta\circ\eta_{g}\beta_{g}^{U}\circ\Theta^{-1}\circ\alpha_{0}^{-1}\\\
&=\pi_{0}\circ\left(\alpha_{0}\right)\circ\eta_{g}\beta_{g}^{U}\circ\alpha_{0}^{-1}\\\
&=\mathop{\mathrm{Ad}}\nolimits\left(v_{g,L}\otimes
v_{g,R}\right)\circ\pi_{0}.\end{split}$

Because $\pi_{0}$ is irreducible, (absorbing possible ${\rm U}(1)$-phase in
$v_{g,L}$) we get

$W_{g}=v_{g,L}\otimes v_{g,R}.$

Now, we may choose $u_{\sigma}(g,h)$ in (3.22) as

$u_{\sigma}(g,h)=v_{g,\sigma}v_{h,\sigma}v_{gh,\sigma}^{*}.$

With this choice, we immediately see

$\displaystyle[c_{L}]_{H^{3}(G,{\rm U}(1))}=[c_{R}]_{H^{3}(G,{\rm U}(1))}=1.$

Hence $1$ is split into left and right in a trivial manner.

As we have seen, if either $H_{D}$-$H_{U}$ cut or $H_{L}$-$H_{R}$ cut is
trivial, namely, product, then our $H^{3}(G,{\rm U}(1))$-valued index become
trivial. In this sense, $[{c_{R}}]_{H^{3}(G,{\rm U}(1))}$ indicates how left-
right/ up-down planes are correlated, in terms of the group action.

###  3.6 Two-dimensional Dijkgraaf-Witten model

When the unique ground state is given by an automorphism $\alpha$ satisfying
the factorization property (3.6), the $H^{3}(G,{\rm U}(1))$-valued index can
be calculated without going through the GNS representation. In this section,
we give such an example of $2$-dimensional SPT.

######  Definition 3.8.

For
$\sigma^{(0)}\in\mathop{\mathrm{Aut}}\nolimits\left({\mathcal{A}}_{{\mathbb{Z}}^{2}}\right)$
and $\Gamma\subset{\mathbb{Z}}^{2}$, we set

$\displaystyle\left(d^{0}_{\Gamma}\sigma^{(0)}\right)(g):=\left(\sigma^{(0)}\right)^{-1}\beta_{g}^{\Gamma}\circ\sigma^{(0)}\circ\left(\beta_{g}^{\Gamma}\right)^{-1},\quad
g\in G.$ (3.24)

For
$\sigma^{(1)}:G\to\mathop{\mathrm{Aut}}\nolimits\left({\mathcal{A}}_{{\mathbb{Z}}^{2}}\right)$
and $\Gamma\subset{\mathbb{Z}}^{2}$, we set

$\displaystyle\left(d^{1}_{\Gamma}\sigma^{(1)}\right)(g,h):=\sigma^{(1)}(g)\beta_{g}^{\Gamma}\sigma^{(1)}(h)\left(\beta_{g}^{\Gamma}\right)^{-1}\left(\sigma^{(1)}(gh)\right),$
(3.25)

for $g,h\in G$.

######  Definition 3.9.

For
$\alpha\in\mathop{\mathrm{Aut}}\nolimits\left({\mathcal{A}}_{{\mathbb{Z}}^{2}}\right)$,
we say that $d^{0}_{H_{U}}\alpha$ is factorized into left and right if there
are automorphisms
$\gamma_{g,\sigma}\in\mathop{\mathrm{Aut}}\nolimits\left({\mathcal{A}}_{H_{\sigma}}\right)$,
$g\in G$, $\sigma=L,R$ such that

$\displaystyle\left(d^{0}_{H_{U}}\alpha\right)(g)=({\rm{inner}})\circ\left(\gamma_{g,L}\otimes\gamma_{g,R}\right),\quad
g\in G.$ (3.26)

Note that this situation is the same as that of Lemma 3.1. Hence there are
unitaries
$v_{\sigma}(g,h)\in{\mathcal{U}}\left({\mathcal{A}}_{H_{\sigma}}\right)$,
$g,h\in G$, $\sigma=L,R$ such that

$\displaystyle\gamma_{g,{\sigma}}\beta_{g}^{{\sigma}U}\gamma_{h,{\sigma}}\beta_{h}^{{\sigma}U}\left(\gamma_{gh,{\sigma}}\beta_{gh}^{{\sigma}U}\right)^{-1}=\mathop{\mathrm{Ad}}\nolimits\left(v_{\sigma}(g,h)\right).$
(3.27)

From Lemma 3.1 and the basic fact given in section 3.1 on cocycle actions, we
have

$\displaystyle
v_{R}(g,h)v_{R}(gh,k)=c_{R}(g,h,k)\gamma_{g,{R}}\beta_{g}^{{R}U}\left(v_{R}(h,k)\right)v_{R}(g,hk),\quad
g,h,k\in G,$ (3.28)

for some $3$-cocycle $c_{R}$. The index $[c_{R}]_{H^{3}(G,{\rm U}(1))}$
coincides with our index obtained in subsection 3.4, under the following
conditions.

######  Theorem 3.10.

[O7] Suppose that $\omega_{\Phi}$ has a form $\omega_{\Phi}=\omega_{0}\alpha$,
where $\omega_{0}$ is a $\beta$-invariant infinite tensor product state and
$\alpha$ is an automorphism. Suppose that $\alpha$ allows a decomposition like
(1.25) for an arbitrary direction $0<\theta<\frac{\pi}{2}$ and
$d^{0}_{H_{U}}\alpha$ is factorized into left and right as in (3.26) with some
$\gamma_{g,\sigma}\in\mathop{\mathrm{Aut}}\nolimits\left({\mathcal{A}}_{C_{\theta_{0}},\sigma}\right)$
and $0<\theta_{0}<\frac{\pi}{2}$, for $\sigma=L,R$. Then we have
$h(\Phi)=[c_{R}]_{H^{3}(G,{\mathbb{T}})}$.

(See Theorem 6.4 of [O7] for a more general setting.)

  * Proof.

We factorize $\alpha$ with respect to the angle $\theta_{0}$ as in (1.25). Let
$v_{R}(g,h)$ $c_{R}(g,h,k)$ be the unitary and $3$-cocycle given in (3.27) and
(3.28).

Because $\omega_{0}$ is infinite tensor product state and
$\omega_{0}\beta_{g}=\omega_{0}$, we have
$\omega_{0}\beta_{g}^{U}=\omega_{0}$. From this and (1.25), (3.26), we get

$\displaystyle\begin{split}\omega_{\Phi}\beta_{g}^{U}=\omega_{0}\alpha\beta_{g}^{U}=\omega_{0}\left(\beta_{g}^{U}\right)^{-1}\alpha\beta_{g}^{U}=\omega_{0}\alpha\left(\gamma_{g^{-1},L}\otimes\gamma_{g^{-1},R}\right)\circ({\rm{inner}})\\\
=\omega_{\Phi}\left(\gamma_{g^{-1},L}\otimes\gamma_{g^{-1},R}\right)\circ({\rm{inner}}).\end{split}$

This means we may take $\eta_{g,\sigma}:=\gamma_{g,\sigma}$ in Proposition
3.2. With this choice, because of (3.27), we may take $u(g,h)$ in Proposition
3.3 as $u(g,h):=\pi_{R}\alpha_{R}\left(v_{R}(g,h)\right)$. From (3.27) and the
fact that our $\gamma_{g,R}$s are localized in $C_{\theta_{0},R}$, we see that
$v_{R}(g,h)$s are elements in ${\mathcal{A}}_{C_{\theta_{0},R}}$. For $W_{g}$
given in Proposition 3.5, we have

$\displaystyle\begin{split}&\mathop{\mathrm{Ad}}\nolimits(W_{g})\left(\mathbb{I}_{{\mathcal{H}}_{L}}\otimes
u(h,k)\right)=\mathop{\mathrm{Ad}}\nolimits(W_{g})\left(\mathbb{I}_{{\mathcal{H}}_{L}}\otimes\pi_{R}\alpha_{R}\left(v_{R}(h,k)\right)\right)\\\
&=\pi_{0}\left(\alpha_{0}\right)\circ\Theta\circ\eta_{g}\beta_{g}^{U}\circ\Theta^{-1}\circ\alpha_{0}^{-1}\left(\mathbb{I}_{{\mathcal{A}}_{H_{L}}}\otimes\alpha_{R}\left(v_{R}(h,k)\right)\right)\\\
&=\pi_{0}\left(\mathbb{I}_{{\mathcal{A}}_{H_{L}}}\otimes\left(\alpha_{R}\gamma_{g,R}\beta_{g}^{RU}v_{R}(h,k)\right)\right),\end{split}$

because $\Theta$ is localized in $C_{\theta_{0}}^{c}$. (We substituted
$\eta_{g,\sigma}:=\gamma_{g,\sigma}$ at the last line.) Substituting this, we
get

$\displaystyle\begin{split}&\left(W_{g}\left(\mathbb{I}_{{\mathcal{H}}_{L}}\otimes
u(h,k)\right)W_{g}^{*}\right)\left(\mathbb{I}_{{\mathcal{H}}_{L}}\otimes
u(g,hk)\right)\\\
&=\mathbb{I}_{{\mathcal{A}}_{H_{L}}}\otimes\pi_{R}\alpha_{R}\left(\gamma_{g,R}\beta_{g}^{RU}\left(v_{R}(h,k)\right)v_{R}(g,hk)\right)\\\
&=\overline{c_{R}(g,h,k)}\cdot\mathbb{I}_{{\mathcal{A}}_{H_{L}}}\otimes\pi_{R}\alpha_{R}\left(v_{R}(g,h)v_{R}(gh,k)\right)\\\
&=\overline{c_{R}(g,h,k)}\cdot\mathbb{I}_{{\mathcal{A}}_{H_{L}}}\otimes
u(g,h)u(gh,k).\end{split}$

This proves the theorem.  $\square$

Now we apply this to the analysis of an example of two-dimensional model
related to Dijkgraaf-Witten model [CGLW] [Y] [MM]. We learned about this model
from Hal Tasaki. The calculation below is carried out by him. Analogous
analysis can be carried out for general $d$-dimensional Dijkgraaf-Witten model
(or more generally, for models with some abelian structures) to derive a
$H^{d+1}(G,{\rm U}(1))$-valued index.

The standard basis of ${\mathbb{R}}^{2}$ is denoted by
$\\{\bm{e}_{j}\\}_{j=1,2}$. Let us recall the canonical triangulation of the
unit hypercube $[0,1]^{2}$. We denote by $\mathop{\mathrm{Sym}}\nolimits(2)$
the symmetric group of order $2$. For
$\pi\in\mathop{\mathrm{Sym}}\nolimits(2)$,
$\mathop{\mathrm{sgn}}\nolimits(\pi)$ denotes the signature of the
permutation. For each $\pi\in\mathop{\mathrm{Sym}}\nolimits(2)$, we set

$\displaystyle T_{\pi}:=\left\\{\bm{x}=(x_{i})_{i=1}^{2}\in[0,1]^{2}\mid
x_{\pi(2)}\leq x_{\pi(1)}\right\\}.$ (3.29)

The unique integral corners of $T_{\pi}$ are the following $3$ points:

$\displaystyle\begin{array}[]{cccc}&&x_{\pi(2)}&x_{\pi(1)}\\\
{\bm{v}}_{0}^{(\pi)}&:=&0&0\\\ {\bm{v}}_{1}^{(\pi)}&:=&0&1\\\
{\bm{v}}_{2}^{(\pi)}&:=&1&1\\\ \end{array}$ (3.34)

The simplex $T_{\pi}$ is equal to the convex combination
$\mathop{\mathrm{co}}\nolimits\\{{\bm{v}}_{k}^{(\pi)}\\}_{k=1}^{2}$ of vectors
$\\{{\bm{v}}_{k}^{(\pi)}\\}_{k=1}^{2}$. The simplices give a simplicization
$[0,1]^{2}:=\cup_{\pi\in\mathop{\mathrm{Sym}}\nolimits(2)}T_{\pi}$.

For each $\pi\in\mathop{\mathrm{Sym}}\nolimits(2)$ and $j=0,1,2$, we denote by
$S_{j}^{(\pi)}$ the convex combination
$\mathop{\mathrm{co}}\nolimits\\{{\bm{v}}_{k}^{(\pi)}\mid k\neq j\\}$ of
vectors $\\{{\bm{v}}_{k}^{(\pi)}\\}_{k\neq j}$. By (3.34), we have

$\displaystyle S_{0}^{(\pi)}:=\left\\{\bm{x}=(x_{i})_{i=1}^{2}\in[0,1]^{2}\mid
x_{\pi(1)}=1\right\\},$ (3.35) $\displaystyle
S_{1}^{(\pi)}=\left\\{\bm{x}=(x_{i})_{i=1}^{2}\in[0,1]^{2}\mid
x_{1}=x_{2}\right\\},$ (3.36) $\displaystyle
S_{2}^{(\pi)}:=\left\\{\bm{x}=(x_{i})_{i=1}^{2}\in[0,1]^{2}\mid
x_{\pi(2)}=0\right\\}.$ (3.37)

For $\pi,\pi^{\prime}\in\mathop{\mathrm{Sym}}\nolimits(2)$ with
$\pi\neq\pi^{\prime}$, we have $T_{\pi}\cap
T_{\pi^{\prime}}=S_{1}^{(\pi)}=S_{1}^{(\pi^{\prime})}$. For
$\pi,\pi^{\prime}\in\mathop{\mathrm{Sym}}\nolimits(2)$ and $j=1,2$,
$T_{\pi}\cap\left(T_{\pi^{\prime}}+\bm{e}_{j}\right)$ is $1$-dimensional if
and only if $\pi^{\prime}\neq\pi$ and $\pi(1)=\pi^{\prime}(2)=j$. In this
case, we have
$T_{\pi}\cap\left(T_{\pi^{\prime}}+\bm{e}_{j}\right)=S_{0}^{(\pi)}=S_{2}^{(\pi^{\prime})}+\bm{e}_{j}$.
For $\bm{x}\in{\mathbb{Z}}^{2}$, $\pi\in\mathop{\mathrm{Sym}}\nolimits(2)$,
and $k=0,1,2$, we set
$\bm{v}_{k,\bm{x}}^{(\pi)}:=\bm{v}_{k}^{(\pi)}+\bm{x}\in{\mathbb{Z}}^{2}$, and
$S_{k,\bm{x}}^{(\pi)}:=S_{k}^{(\pi)}+\bm{x}=\mathop{\mathrm{co}}\nolimits\\{\bm{v}_{i,\bm{x}}^{(\pi)}\mid
i\neq k\\}$.

For any
$(\bm{x},\pi,k)\in{\mathbb{Z}}^{2}\times\mathop{\mathrm{Sym}}\nolimits(2)\times\\{0,1,2\\}$,
there exists a unique
$(\bm{x}^{\prime},\pi^{\prime},k^{\prime})\in{\mathbb{Z}}^{2}\times\mathop{\mathrm{Sym}}\nolimits(2)\times\\{0,1,2\\}$
with $(\bm{x},\pi,k)\neq(\bm{x}^{\prime},\pi^{\prime},k^{\prime})$ such that
$S_{k,\bm{x}}^{(\pi)}=S_{k^{\prime},\bm{x}^{\prime}}^{(\pi^{\prime})}$.
Furthermore, for this $(\bm{x}^{\prime},\pi^{\prime},k^{\prime})$, one of the
following occurs.

(i)

We have $\bm{x}=\bm{x}^{\prime}$, $k=k^{\prime}=1$,
$\pi^{\prime}=\pi\circ(1,2)$, and
$\bm{v}_{j,\bm{x}}^{(\pi)}=\bm{v}_{j,\bm{x}^{\prime}}^{(\pi^{\prime})}$, for
any $j=0,2$.

(ii)

We have $\bm{x}^{\prime}=\bm{x}+\bm{e}_{\pi(1)}$, $k=0$, $k^{\prime}=2$,
$\pi^{\prime}=\pi\circ(1,2)$ and
$\bm{v}_{j,\bm{x}}^{(\pi)}=\bm{v}_{j-1,\bm{x}^{\prime}}^{(\pi^{\prime})}$, for
$j=1,2$.

(iii)

We have $\bm{x}=\bm{x}^{\prime}+\bm{e}_{\pi^{\prime}(1)}$, $k=2$,
$k^{\prime}=0$, $\pi=\pi^{\prime}\circ(1,2)$ and
$\bm{v}_{j-1,\bm{x}}^{(\pi)}=\bm{v}_{j,\bm{x}^{\prime}}^{(\pi^{\prime})}$, for
$j=1,2$.

For such
$(\bm{x},\pi,k)\in{\mathbb{Z}}^{2}\times\mathop{\mathrm{Sym}}\nolimits(2)\times\\{0,1,2\\}$,
and
$(\bm{x}^{\prime},\pi^{\prime},k^{\prime})\in{\mathbb{Z}}^{2}\times\mathop{\mathrm{Sym}}\nolimits(2)\times\\{0,1,2\\}$
we write $(\bm{x},\pi,k)\sim(\bm{x}^{\prime},\pi^{\prime},k^{\prime})$. Each
$(\bm{x},\pi,k)\in{\mathbb{Z}}^{2}\times\mathop{\mathrm{Sym}}\nolimits(2)\times\\{0,1,2\\}$
has exactly one
$(\bm{x}^{\prime},\pi^{\prime},k^{\prime})\in{\mathbb{Z}}^{2}\times\mathop{\mathrm{Sym}}\nolimits(2)\times\\{0,1,2\\}$
satisfying $(\bm{x},\pi,k)\sim(\bm{x}^{\prime},\pi^{\prime},k^{\prime})$.

Now let us introduce the model related to the Dijkgraaf-Witten model. Let $G$
be a fixed finite group. Let $\delta_{g}$, $g\in G$ be the orthogonal basis of
$l^{2}(G)$, and $\\{e_{g,h}\in B(l^{2}(G))\mid g,h\in G\\}$ the system of
matrix units associated to the basis $\\{\delta_{g}\mid g\in G\\}$. We denote
by $U$ the left action of $G$ on $l^{2}(G)$, i.e.,
$U_{g}\delta_{h}:=\delta_{gh}$, for any $g,h\in G$. Note that the algebra of
diagonal matrices $\mathop{\mathrm{D}}\nolimits_{d}$ with respect to the basis
$\\{\delta_{g}\mid g\in G\\}$ is invariant under
$\mathop{\mathrm{Ad}}\nolimits(U_{g})$, $g\in G$. For any finite
$\Lambda\subset{\mathbb{Z}}^{2}$ and
$\bm{s}=(s\left(\bm{x}\right))_{\bm{x}\in\Lambda},\bm{s}^{\prime}=(s^{\prime}\left({\bm{x}}\right))_{\bm{x}\in\Lambda}\in
G^{\times\Lambda}$, we set
$e^{(\Lambda)}_{\bm{s},\bm{s}^{\prime}}:=\bigotimes_{\bm{x}\in\Lambda}e_{s\left({\bm{x}}\right),{\bm{s}^{\prime}\left(\bm{x}\right)}}$.
For each $\Gamma\subset{\mathbb{Z}}^{2}$, we denote by
${\mathcal{D}}_{\Gamma}$ the abelian subalgebra of ${\mathcal{A}}_{\Gamma}$
generated by the tensor product of the diagonal matrices. For each
$L\in{\mathbb{N}}$, set

$\displaystyle\begin{split}&\Lambda_{L}^{(1)}:=[-L,L]\quad\partial\Lambda_{L}^{(1)}:=\\{-L,L\\},\\\
&\Lambda_{L}^{(2)}:=[-L,L]^{2}\quad\partial\Lambda_{L}^{(2)}:=\left(\\{-L,L\\}\times[-L,L]\right)\cup\left([-L,L]\times\\{-L,L\\}\right).\end{split}$

Here, with a bit abuse of notation, we denote $[-L,L]^{k}\cap{\mathbb{Z}}^{k}$
by $[-L,L]^{k}$. Analogous notation will be used for $[0,L]$, $[-L,-1]$ etc
below.

Let us recall a basic fact about group cohomology. The group ring
${\mathbb{Z}}[G^{\times n+1}]$ is a $G$-module with a $G$-action

$g\cdot(g_{0},\cdots,g_{n})=(gg_{0},\cdots,gg_{n}).$

We associate a trivial action of $G$ on ${\rm U}(1)$. For a homomorphism
$\varphi\in\mathop{\mathrm{Hom}}\nolimits_{{\mathbb{Z}}[G]}\left({\mathbb{Z}}[G^{\times
i+1}],{\rm U}(1)\right)$ from a $G$-module ${\mathbb{Z}}[G^{\times i+1}]$ and
to a $G$-module ${\rm U}(1)$ we have

$\displaystyle\varphi(gg_{0},gg_{1},\ldots,gg_{i})=\varphi(g_{0},g_{1},\ldots,g_{i}).$
(3.38)

Define
$D^{i}:\mathop{\mathrm{Hom}}\nolimits_{{\mathbb{Z}}[G]}\left({\mathbb{Z}}[G^{\times(i+1)}],{\rm
U}(1)\right)\to\mathop{\mathrm{Hom}}\nolimits_{{\mathbb{Z}}[G]}\left({\mathbb{Z}}[G^{\times
i+2}],{\rm U}(1)\right)$ by

$\displaystyle\begin{split}&\left(D^{i}\varphi\right)\left(g_{0},g_{1},\cdots,g_{i},g_{i+1}\right):=\prod_{j=0}^{i+1}\varphi\left(g_{0},\cdots,g_{j-1},\hat{g}_{j},g_{j+1},\cdots
g_{i+1}\right)^{(-1)^{j}},\end{split}$

where $\hat{g}_{j}$ means that $g_{j}$ is excluded. The group of $i$-cochains
of $G$ with coefficients in ${\rm U}(1)$ is the set of functions from
$G^{\times i}$ to ${\rm U}(1)$. The $i$-th differential
$d^{i}:C^{i}\left(G,{\rm U}(1)\right)\to C^{i+1}\left(G,{\rm U}(1)\right)$ is
given by

$\displaystyle\begin{split}&\left(d^{i}_{\Gamma_{0}}\sigma\right)\left(g_{0},\ldots,g_{i}\right)\\\
&=g_{0}\cdot\sigma\left(g_{1},\ldots,g_{i}\right)\left(\prod_{k=1}^{i}\sigma(g_{0},\ldots,g_{k-1}g_{k},\ldots,g_{i})^{(-1)^{k}}\right)\sigma\left(g_{0},\ldots,g_{i-1}\right)^{(-1)^{i+1}}.\end{split}$

There are isomorphisms
$\Psi^{i}:\mathop{\mathrm{Hom}}\nolimits_{{\mathbb{Z}}[G]}\left({\mathbb{Z}}[G^{\times
i+1}],{\rm U}(1)\right)\to C^{i}\left(G,{\rm U}(1)\right)$, $i=0,1,2,\ldots$
such that

$\displaystyle\Psi^{i}(\varphi)(g_{0},\ldots,g_{i-1})=\varphi\left(e,g_{0},g_{0}g_{1},g_{0}g_{1}g_{2},\cdots,g_{0}g_{1}g_{2}\cdots
g_{i-1}\right).$

With this isomorphism, we have

$\Psi^{i+1}\circ D^{i}=d^{i}\circ\Psi^{i}$

for all $i=0,1,2,\ldots$.

The two-dimensional Dijkgraaf-Witten model is defined by fixing an arbitrary
$\nu\in\ker D^{3}=\left(\Psi^{3}\right)^{-1}\left(Z^{3}\left(G,{\rm
U}(1)\right)\right)$. For the rest of this section, we fix such $\nu$. By
definition, we have

$\displaystyle\frac{\nu(g_{1},g_{2},g_{3},g_{4})\nu(g_{0},g_{1},g_{3},g_{4})\nu(g_{0},g_{1},g_{2},g_{3})}{\nu(g_{0},g_{2},g_{3},g_{4})\nu(g_{0},g_{1},g_{2},g_{4})}=1,$
(3.39)

for any $g_{0},g_{1},g_{2},g_{3},g_{4},g_{5}\in G$.

For $\bm{s}:{\mathbb{Z}}^{2}\to G$,
$\bm{s}=(s(\bm{v}))_{\bm{v}\in{\mathbb{Z}}^{2}}$, set

$\displaystyle\begin{split}&{\mathfrak{q}}\left(\bm{s},\bm{x}\right):=\prod_{\pi\in\mathop{\mathrm{Sym}}\nolimits(2)}\nu\left(e,s\left({{\bm{v}}_{0,\bm{x}}^{\left(\pi\right)}}\right),s\left({{\bm{v}}_{1,\bm{x}}^{\left(\pi\right)}}\right),s\left({{\bm{v}}_{2,\bm{x}}^{\left(\pi\right)}}\right)\right)^{\mathop{\mathrm{sgn}}\nolimits\pi},\;\bm{x}\in{\mathbb{Z}}^{2}\quad\text{and}\\\
&{\mathfrak{p}}\left(g,\bm{s},{y}\right):=\nu\left(e,g,s\left(({y},0)\right),s\left(({y}+1,0)\right)\right),\quad
g\in G,\quad y\in{\mathbb{Z}}.\end{split}$

For each $L\in{\mathbb{N}}$, set

$\displaystyle\begin{split}&V_{L}^{(0)}:=\sum_{\bm{s}\in
G^{\times\left(\Lambda_{L+1}^{(2)}\right)}}\;\left(\prod_{\bm{x}\in\Lambda_{L}^{(2)}}\;\left({\mathfrak{q}}\left(\bm{s},\bm{x}\right)\right)\right)\cdot
e_{\bm{s},\bm{s}}^{\left(\Lambda_{L+1}^{(2)}\right)},\\\
&V_{L}^{(1)}\left(g\right)=\sum_{\bm{s}\in
G^{\times\left(\Lambda_{L+1}^{(1)}\times\\{\bm{0}\\}\right)}}\;\left(\prod_{y\in\Lambda_{L}^{(1)}}\left({\mathfrak{p}}\left(g,\bm{s},{y}\right)\right)^{-1}\right)\cdot
e_{\bm{s},\bm{s}}^{\left(\Lambda_{L+1}^{(1)}\times\\{\bm{0}\\}\right)}.\end{split}$

We also introduce

$\displaystyle\begin{split}&V_{+,L}^{(0)}:=\sum_{\bm{s}\in
G^{\times\left(\Lambda_{L+1}^{(1)}\times[0,L+1]\right)}}\;\;\left(\prod_{\bm{x}\in\Lambda_{L}^{(1)}\times[0,L]}\;\;\left({\mathfrak{q}}\left(\bm{s},\bm{x}\right)\right)\right)\cdot
e_{\bm{s},\bm{s}}^{\left(\Lambda_{L+1}^{(1)}\times[0,L+1]\right)},\\\
&V_{+,L}^{(1)}\left(g\right):=\sum_{\bm{s}\in
G^{\times\left([0,L+1]\times\\{\bm{0}\\}\right)}}\;\;\left(\prod_{y\in[0,L]}\;\;\left({\mathfrak{p}}\left(g,\bm{s},{y}\right)\right)^{-1}\right)\cdot
e_{\bm{s},\bm{s}}^{\left([0,L+1]\times\\{\bm{0}\\}\right)},\\\ \end{split}$

and

$\displaystyle\begin{split}&V_{-,L}^{(0)}:=\sum_{\bm{s}\in
G^{\times\left(\Lambda_{L+1}^{(1)}\times[-L,-1]\right)}}\;\;\left(\prod_{\bm{x}\in\Lambda_{L}^{(1)}\times[-L,-2]}\;\;\left({\mathfrak{q}}\left(\bm{s},\bm{x}\right)\right)\right)\cdot
e_{\bm{s},\bm{s}}^{\left(\Lambda_{L+1}^{(1)}\times[-L,-1]\right)},\\\
&V_{-,L}^{(1)}\left(g\right):=\sum_{\bm{s}\in
G^{\times\left([-L,-1]\times\\{\bm{0}\\}\right)}}\;\;\left(\prod_{y\in[-L,-2]}\quad\left({\mathfrak{p}}\left(g,\bm{s},{y}\right)\right)^{-1}\right)\cdot
e_{\bm{s},\bm{s}}^{\left([-L,-1]\times\\{\bm{0}\\}\right)}.\end{split}$

Furthermore, set

$\displaystyle\begin{split}&V_{\partial,L}^{(0)}:=\sum_{\bm{s}\in
G^{\times\left(\Lambda_{L+1}^{(1)}\times\\{-1,0\\}\right)}}\left(\prod_{\bm{x}\in\Lambda_{L}^{(1)}\times\\{-1\\}}\;{\mathfrak{q}}\left(\bm{s},\bm{x}\right)\right)e_{\bm{s},\bm{s}}^{\left(\Lambda_{L+1}^{(1)}\times\\{-1,0\\}\right)},\\\
&V_{\partial,L,+}^{(0)}:=\sum_{\bm{s}\in
G^{\times\left([0,L+1]\times\\{-1,0\\}\right)}}\left(\prod_{\bm{x}\in[0,L]\times\\{-1\\}}\;\left({\mathfrak{q}}\left(\bm{s},\bm{x}\right)\right)\right)e_{\bm{s},\bm{s}}^{\left([0,L+1]\times\\{-1,0\\}\right)},\\\
&V_{\partial,L,-}^{(0)}:=\sum_{\bm{s}\in
G^{\times\left([-L,-1]]\times\\{-1,0\\}\right)}}\left(\prod_{\bm{x}\in[-L,-2]\times\\{-1\\}}\;\left({\mathfrak{q}}\left(\bm{s},\bm{x}\right)\right)\right)e_{\bm{s},\bm{s}}^{\left([-L,-2]\times\\{-1,0\\}\right)},\\\
\\\ &V_{\partial,L,0}^{(0)}:=\sum_{\bm{s}\in
G^{\times\left(\\{-1,0\\}\times\\{-1,0\\}\right)}}\left(\prod_{\bm{x}=(-1,-1)}\left({\mathfrak{q}}\left(\bm{s},\bm{x}\right)\right)\right)e_{\bm{s},\bm{s}}^{\left(\\{-1,0\\}\times\\{-1,0\\}\right)},\\\
&V_{\partial,L}^{(1)}\left(g\right):=\sum_{\bm{s}\in
G^{\times\left(\\{-1,0\\}\times\\{\bm{0}\\}\right)}}\left({\mathfrak{p}}\left(g,\bm{s},{-1}\right)\right)^{-1}e_{\bm{s},\bm{s}}^{\left(\\{-1,0\\}\times\\{\bm{0}\\}\right)}.\end{split}$

Finally, for a unitary $u\in{\mathcal{U}}({\mathcal{A}}_{{\mathbb{Z}}^{2}})$,
we set

$\displaystyle\left(\widetilde{d^{0}}{u}\right)\left(g\right)={u}^{-1}\beta_{g}^{U}\left({u}\right),\quad
g\in G.$

For these objects, the following Lemma holds.

######  Lemma 3.11.

In the setting above,

(i)

For any $5\leq L\in\mathbb{N}$ and $g\in G$,

$\displaystyle\begin{split}&V_{L}^{(0)}=V_{\partial,L}^{(0)}\left(V_{-,L}^{(0)}\otimes
V_{+,L}^{(0)}\right),\\\
&V_{L}^{(1)}(g)=V_{\partial,L}^{(1)}(g)\left(V_{-,L}^{(1)}(g)\otimes
V_{+,L}^{(1)}(g)\right),\\\
&V_{\partial,L}^{(0)}=V_{\partial,L,0}^{(0)}\left(V_{\partial,L,-}^{(0)}\otimes
V_{\partial,L,+}^{(0)}\right)\end{split}$ (3.40)

(ii)

For any $5\leq L\in\mathbb{N}$ and $g\in G$,

$\displaystyle\left(\tilde{d}^{0}V_{+,L}^{(0)}\right)(g)=\left(\text{unitary
in }{\mathcal{D}}_{\partial\Lambda_{L}}\right)\cdot V_{L}^{(1)}(g)$ (3.41)
$\displaystyle\left(\tilde{d}^{0}V_{L}^{(0)}\right)(g)=\left(\text{unitary in
}{\mathcal{D}}_{\partial\Lambda_{L}}\right).$ (3.42)

(iii)

For any $5\leq L\in\mathbb{N}$ and $(g,h)\in G^{\times 2}$,

$\displaystyle
V_{+,L}^{(1)}(g)\cdot\left(\beta_{g}^{U}\left(V_{+,L}^{(1)}(h)\right)\right)\cdot\left(V_{+,L}^{(1)}(gh)\right)^{-1}=\left(\text{unitary
in }{\mathcal{D}}_{\partial\Lambda_{L}^{(1)}}\right)\cdot V\left(g,h\right),$
(3.43)

where

$\displaystyle V\left(g,h\right):=\sum_{s\in G}\nu\left(e,g,gh,s\right)\cdot
e_{s,s}^{\\{\bm{0}\\}}.$ (3.44)

(iv)

For the unitary $V$ in (3.44), we have

$\displaystyle\overline{\nu(e,g,gh,ghk)}V(g,h)V(gh,k)=\beta_{g}^{{\sigma}U}\left(V(h,k)\right)V\left(g,hk\right),\quad
g,h,k\in G.$ (3.45)

  * Proof.

(i) is trivial from the definitions.
To see (ii), by definition, we first have

$\displaystyle\begin{split}\tilde{d}^{0}V_{+,L}^{(0)}(g)=&{V_{+,L}^{(0)}}^{-1}\beta_{g}^{U}\left({V_{+,L}^{(0)}}\right)\\\
=&\sum_{\bm{s}\in
G^{\times\left(\Lambda_{L+1}^{(1)}\times[0,L+1]\right)}}\;\;\prod_{\bm{x}\in\Lambda_{L}^{(1)}\times[0,L]}\;\;\prod_{\pi\in\mathop{\mathrm{Sym}}\nolimits(2)}\\\
&\nu\left(g,s\left({{\bm{v}}_{0,\bm{x}}^{\left(\pi\right)}}\right),s\left({{\bm{v}}_{1,\bm{x}}^{\left(\pi\right)}}\right),s\left({{\bm{v}}_{2,\bm{x}}^{\left(\pi\right)}}\right)\right)^{\mathop{\mathrm{sgn}}\nolimits{\pi}}\\\
&\nu\left(e,s\left({{\bm{v}}_{0,\bm{x}}^{\left(\pi\right)}}\right),s\left({{\bm{v}}_{1,\bm{x}}^{\left(\pi\right)}}\right),s\left({{\bm{v}}_{2,\bm{x}}^{\left(\pi\right)}}\right)\right)^{-{\mathop{\mathrm{sgn}}\nolimits{\pi}}}e_{\bm{s},\bm{s}}^{\left(\Lambda_{L+1}^{(1)}\times[0,L+1]\right)}.\end{split}$
(3.46)

The first $\nu$ comes from $\beta_{g}^{U}\left({V_{+,L}^{(0)}}\right)$ and we
used change of varianles $g^{-1}s\to s$ and (3.38). Because $\nu\in\ker
D^{3}$, using (3.39), we obtain

$\displaystyle\begin{split}&\tilde{d}^{0}V_{+,L}^{(0)}\left(g\right)\\\
&=\sum_{\bm{s}\in
G^{\times\left(\Lambda_{L+1}^{(1)}\times[0,L+1]\right)}}\;\;\prod_{\bm{x}\in\Lambda_{L}^{(1)}\times[0,L]}\;\;\prod_{\pi\in\mathop{\mathrm{Sym}}\nolimits(2)}\\\
&\quad\left(\begin{gathered}\nu\left(e,g,s\left({\bm{v}}_{1,\bm{x}}^{\left(\pi\right)}\right),s\left({\bm{v}}_{2,\bm{x}}^{\left(\pi\right)}\right)\right)^{-1}\nu\left(e,g,s\left({\bm{v}}_{0,\bm{x}}^{\left(\pi\right)}\right),s\left({\bm{v}}_{2,\bm{x}}^{\left(\pi\right)}\right)\right)\\\
\nu\left(e,g,s\left({\bm{v}}_{0,\bm{x}}^{\left(\pi\right)}\right),s\left({\bm{v}}_{1,\bm{x}}^{\left(\pi\right)}\right)\right)^{-1}\end{gathered}\right)^{\mathop{\mathrm{sgn}}\nolimits{\pi}}\quad\quad
e_{\bm{s},\bm{s}}^{\left(\Lambda_{L+1}^{(1)}\times[0,L+1]\right)}.\end{split}$
(3.47)

Now for each $\bm{s}:{\mathbb{Z}}^{2}\to G$, $g\in G$ and
$(\bm{x},\pi,j)\in\left(\Lambda_{L}^{(1)}\times[0,L]\right)\times\mathop{\mathrm{Sym}}\nolimits(2)\times\\{0,1,2\\}$,
we set

$\displaystyle\psi_{\bm{s},g}\left(\bm{x},\pi,j\right):=\left\\{\begin{gathered}\nu\left(e,g,s\left({\bm{v}}_{1,\bm{x}}^{\left(\pi\right)}\right),s\left({\bm{v}}_{2,\bm{x}}^{\left(\pi\right)}\right)\right)^{-\mathop{\mathrm{sgn}}\nolimits{\pi}},\quad\text{if}\quad
j=0\\\
\nu\left(e,g,s\left({\bm{v}}_{0,\bm{x}}^{\left(\pi\right)}\right),s\left({\bm{v}}_{2,\bm{x}}^{\left(\pi\right)}\right)\right)^{\mathop{\mathrm{sgn}}\nolimits{\pi}}\quad\text{if}\quad
j=1\\\
\nu\left(e,g,s\left({\bm{v}}_{0,\bm{x}}^{\left(\pi\right)}\right),s\left({\bm{v}}_{1,\bm{x}}^{\left(\pi\right)}\right)\right)^{-\mathop{\mathrm{sgn}}\nolimits{\pi}}\quad\text{if}\quad
j=2\end{gathered}\right..$

With this notation, we have

$\displaystyle\begin{split}&\tilde{d}^{0}V_{+,L}^{(0)}\left(g\right)\\\
&=\sum_{\bm{s}\in
G^{\times\left(\Lambda_{L+1}^{(1)}\times[0,L+1]\right)}}\;\;\left(\prod_{\bm{x}\in\Lambda_{L}^{(1)}\times[0,L]}\;\;\prod_{\pi\in\mathop{\mathrm{Sym}}\nolimits(2)}\prod_{j=0}^{2}\psi_{\bm{s},g}\left(\bm{x},\pi,j\right)\right)\cdot
e_{\bm{s},\bm{s}}^{\left(\Lambda_{L+1}^{(1)}\times[0,L+1]\right)}\end{split}$
(3.48)

By the observation about
$(\bm{x},\pi,j)\sim(\bm{x^{\prime}},{\pi^{\prime}},j^{\prime})$, we see that
if $(\bm{x},\pi,j)\sim(\bm{x^{\prime}},{\pi^{\prime}},j^{\prime})$, then
$\psi_{\bm{s},g}\left(\bm{x},\pi,j\right)\psi_{\bm{s},g}\left(\bm{x^{\prime}},{\pi^{\prime}},j^{\prime}\right)=1$.
Therefore, a term $\psi_{\bm{s},g}(\bm{x},\pi,j)$ corresponding to
$(\bm{x},\pi,j)$ cancels out with that of
$(\bm{x^{\prime}},{\pi^{\prime}},j^{\prime})$ , if the term
$\psi_{\bm{s},g}(\bm{x^{\prime}},{\pi^{\prime}},j^{\prime})$ also appears in
(3.48). Recall that there is exactly one
$(\bm{x^{\prime}},{\pi^{\prime}},j^{\prime})$ such that
$(\bm{x},\pi,j)\sim(\bm{x^{\prime}},{\pi^{\prime}},j^{\prime})$. The terms
which may not disappear is that of either $\bm{x}\in\partial\Lambda_{L}^{(2)}$
or

$\displaystyle\bm{x}\in\Lambda_{L}^{(1)}\times\\{0\\},\quad\text{and}\quad
j=2,\quad\text{and}\quad\pi=\mathop{\mathrm{id}}\nolimits.$ (3.49)

The former terms form some unitary in
${\mathcal{D}}_{\partial\Lambda_{L}^{(2)}}$. For the latter case, setting
$y\in\Lambda_{L}^{(1)}$ so that $(y,0)=\bm{x}$, we obtain

$\displaystyle{\bm{v}}_{0,\bm{x}}^{\left(\pi\right)}=\left(y,0\right),\quad{\bm{v}}_{1,\bm{x}}^{\left(\pi\right)}=\left(y+1,0\right).$
(3.50)

Hence we have

$\displaystyle\begin{split}&\tilde{d}^{0}V_{+,L}^{(0)}\left(g\right)\\\
&=\left(\text{unitary in
}{\mathcal{D}}_{\partial\Lambda_{L}^{(2)}}\right)\cdot\sum_{\bm{s}\in
G^{\times\left(\Lambda_{L+1}^{(1)}\times\\{\bm{0}\\}\right)}}\\\
&\quad\prod_{y\in\Lambda_{L}^{(1)}}\nu\left(\begin{gathered}e,g,s\left((y,0)\right),s\left((y+1,0)\right)\end{gathered}\right)^{-1}\cdot
e_{\bm{s},\bm{s}}^{\left(\Lambda_{L+1}^{(1)}\times\\{\bm{0}\\}\right)}\\\
&=\left(\text{unitary in
}{\mathcal{D}}_{\partial\Lambda_{L}^{(2)}}\right)\cdot
V_{L}^{(1)}(g).\end{split}$

This completes the proof of the first part of (ii). The proof for the second
part of (ii) is the same, except for this time, $[0,L]$ is replaced by
$[-L,L]$. Therefore, there is no contribution from out side of
$\partial\Lambda_{L}^{(2)}$.

The proof of (iii) is basically the same. We have

$\displaystyle\begin{split}&V_{+,L}^{(1)}(g)\cdot\left(\beta_{g}^{U}\left(V_{+,L}^{(1)}(h)\right)\right)\cdot\left(V_{+,L}^{(1)}(gh)\right)^{-1}\\\
&=\sum_{\bm{s}\in
G^{\times\left([0,L+1]\times\\{0\\}\right)}}\;\;\prod_{y\in[0,L]}\\\
&\quad\left(\begin{gathered}\nu\left(g,gh,s(y,0),s(y+1,0)\right)^{-1}\\\
\nu\left(e,gh,s(y,0),s(y+1,0)\right)\nu\left(e,g,s(y,0),s(y+1,0)\right)^{-1}\end{gathered}\right)e_{\bm{s},\bm{s}}^{[0,L+1]\times\\{0\\}}\\\
&=\sum_{\bm{s}\in
G^{\times\left([0,L+1]\times\\{\bm{0}\\}\right)}}\;\;\prod_{y\in[0,L]}\left(\nu\left(e,g,gh,s(y+1,0)\right)^{-1}\nu\left(e,g,gh,s(y,0)\right)\right)e_{\bm{s},\bm{s}}^{[0,L+1]\times\\{0\\}}\\\
&=\left(\text{unitary in
}{\mathcal{D}}_{\partial\Lambda_{L}^{(1)}}\right)\sum_{s\in
G}\nu\left(e,g,gh,s\right)\cdot e_{s,s}^{\\{\bm{0}\\}}.\end{split}$

For the second equality, we used (3.39). Third equality is just a simple
cancellation.

(iv) can be checked directly, using (3.39).  $\square$

We can easily take the thermodynamic limit.

######  Lemma 3.12.

There are automorphisms

$\displaystyle\begin{split}&\sigma^{(0)}\in\mathop{\mathrm{Aut}}\nolimits\left({{\mathcal{A}}_{{\mathbb{Z}}^{2}}}\right),\quad\sigma_{+}^{(0)}\in\mathop{\mathrm{Aut}}\nolimits\left({\mathcal{A}}_{{\mathbb{Z}}\times[0,\infty)}\right),\quad\sigma_{-}^{(0)}\in\mathop{\mathrm{Aut}}\nolimits\left({\mathcal{A}}_{{\mathbb{Z}}\times(-\infty,-1]}\right)\\\
&\Xi^{(0)}\in\mathop{\mathrm{Aut}}\nolimits\left({\mathcal{A}}_{{\mathbb{Z}}^{(1)}\times\\{-1,0\\}}\right),\quad\Xi_{+}^{(0)}\in\mathop{\mathrm{Aut}}\nolimits\left({\mathcal{A}}_{{[0,\infty)}\times\\{-1,0\\}}\right),\Xi_{-}^{(0)}\in\mathop{\mathrm{Aut}}\nolimits\left({\mathcal{A}}_{(-\infty,-1]\times\\{-1,0\\}}\right),\quad\\\
&\sigma_{+}^{(1)}(g)\in\mathop{\mathrm{Aut}}\nolimits\left({\mathcal{A}}_{[0,\infty)\times\\{0\\}}\right),\quad\sigma_{-}^{(1)}(g)\in\mathop{\mathrm{Aut}}\nolimits\left({\mathcal{A}}_{(-\infty,-1]\times\\{0\\}}\right)\end{split}$

such that

$\displaystyle\begin{split}&\sigma^{(0)}(A)=\lim_{L\to\infty}\mathop{\mathrm{Ad}}\nolimits\left(V_{L}^{(0)}\right)(A),\quad\sigma_{+}^{(0)}(A)=\lim_{L\to\infty}\mathop{\mathrm{Ad}}\nolimits\left(V_{+,L}^{(0)}\right)(A),\\\
&\sigma_{-}^{(0)}(A)=\lim_{L\to\infty}\mathop{\mathrm{Ad}}\nolimits\left(V_{-,L}^{(0)}\right)(A),\quad\Xi^{(0)}(A)=\lim_{L\to\infty}\mathop{\mathrm{Ad}}\nolimits\left(V_{\partial
L}^{(0)}\right)(A),\\\
&\Xi_{+}^{(0)}(A)=\lim_{L\to\infty}\mathop{\mathrm{Ad}}\nolimits\left(V_{\partial,L,+}^{(0)}\right)(A),\quad\Xi_{-}^{(0)}(A)=\lim_{L\to\infty}\mathop{\mathrm{Ad}}\nolimits\left(V_{\partial,L,-}^{(0)}\right)(A),\\\
&\sigma^{(1)}(g)(A)=\lim_{L\to\infty}\mathop{\mathrm{Ad}}\nolimits\left(V_{L}^{(1)}\left(g\right)\right)(A),\quad\sigma_{+}^{(1)}\left(g\right)(A)=\lim_{L\to\infty}\mathop{\mathrm{Ad}}\nolimits\left(V_{+,L}^{(1)}\left(g\right)\right)(A),\\\
&\sigma_{-}^{(1)}\left(g\right)(A)=\lim_{L\to\infty}\mathop{\mathrm{Ad}}\nolimits\left(V_{-,L}^{(1)}\left(g\right)\right)(A),\quad\Xi^{(1)}\left(g\right)(A)=\lim_{L\to\infty}\mathop{\mathrm{Ad}}\nolimits\left(V_{\partial,L}^{(1)}\left(g\right)\right)(A),\end{split}$
(3.51)

for any $g\in G$, $L\in{\mathbb{N}}$, and
$A\in{\mathcal{A}}_{\Lambda_{L-1}^{(2)}}$.
Furthermore, the following holds:

(i)

For each

$\displaystyle\begin{split}&\sigma^{(0)}=\Xi^{(0)}\left(\sigma_{-}^{(0)}\otimes\sigma_{+}^{(0)}\right),\quad\Xi^{(0)}=({\rm{inner}})\left(\Xi_{-}^{(0)}\otimes\Xi_{+}^{(0)}\right),\\\
&\sigma^{(1)}\left(g\right)=\Xi^{(1)}\left(g\right)\circ\left(\sigma_{-}^{(1)}\left(g\right)\otimes\sigma_{+}^{(1)}\left(g\right)\right).\end{split}$
(3.52)

(ii)

For each $g\in G$,

$\displaystyle
d^{1}_{H_{U}}\sigma_{+}^{(0)}\left(g\right)=\sigma^{(1)}\left(g\right).$
(3.53)

(iii)

$\displaystyle\sigma_{+}^{(1)}(g)\beta_{g}^{U}\sigma_{+}^{(1)}\left(h\right)\left(\beta_{g}^{U}\right)^{-1}\left(\sigma_{+}^{(1)}(gh)\right)^{-1}=\mathop{\mathrm{Ad}}\nolimits\left(V\left(g,h\right)\right)$
(3.54)

  * Proof.

For any $M,L\in{\mathbb{N}}$ with $M+1\geq L$, $V_{L}^{(m)}$ is of the form

$\displaystyle V_{L}^{(m)}=V_{M+1}^{(m)}\left(\text{unitary in
}\quad{\mathcal{D}}_{\Lambda_{L}^{(d)}\setminus\Lambda_{M}^{(d)}}\right).$
(3.55)

From this and the fact that $V_{L}^{(m)},V_{M+1}^{(m)}$ belong to a common
abelian subalgebra, we have

$\displaystyle\mathop{\mathrm{Ad}}\nolimits\left({V_{L}^{(m)}}\right)(A)=\mathop{\mathrm{Ad}}\nolimits\left({V_{M+1}^{(m)}}\right)(A),\quad\mathop{\mathrm{Ad}}\nolimits\left({\left(V_{L}^{(m)}\right)^{*}}\right)(A)=\mathop{\mathrm{Ad}}\nolimits\left({\left(V_{M+1}^{(m)}\right)^{*}}\right)(A),$
(3.56)

for all $A\in{\mathcal{A}}_{\Lambda_{M}^{(2)}}$. Therefore, taking
$L\to\infty$ limit, we obtain an automorphism satisfying (3.51). The existence
of other automorphisms are the same. The properties (i), (ii), (iii)
corresponds to the (i), (ii), (iii) of Lemma 3.11.  $\square$

Let

$\displaystyle\xi:=\frac{1}{\sqrt{|G|}}\sum_{g\in G}\delta_{g},$ (3.57)

and set a state $\rho$ on ${\mathcal{B}}(l^{2}(G))$ such that

$\displaystyle\rho(A)=\left\langle\xi,A\xi\right\rangle,\quad
A\in{\mathcal{B}}(l^{2}(G)).$ (3.58)

Combining Lemma 3.11 and Lemma 3.12, we obtain the following:

######  Theorem 3.13.

Let $\sigma^{(0)}$ be an automorphism given by Lemma 3.12. Let $\omega_{0}$ be
an infinite tensor product state on
${\mathcal{A}}_{{\mathbb{Z}}^{2}}:=\bigotimes_{\bm{x}\in{\mathbb{Z}}^{2}}{\mathcal{B}}(l^{2}(G))$
defined by

$\displaystyle\omega_{0}:=\bigotimes_{\bm{x}\in{\mathbb{Z}}^{2}}\rho$ (3.59)

Then the $H^{3}(G,{\rm U}(1))$-valued index of the model is
$\left[\Psi^{3}\left(\nu\right)\right]_{H^{3}(G,{\rm U}(1))}$.

  * Proof.

Note that all the automorphisms in Lemma 3.12 and their conjugation with
respect to $\beta_{g}^{U}$ mutually commute. Using Lemma 3.12, we obtain

$\displaystyle\begin{split}&d^{0}_{H_{U}}\left(\sigma^{(0)}\right)(g)=d^{0}_{H_{U}}\left(\Xi^{(0)}\left(\sigma_{-}^{(0)}\otimes\sigma_{+}^{(0)}\right)\right)(g)\\\
&=({\rm{inner}})\circ\left(d^{0}_{H_{U}\cap H_{L}}\Xi^{(0)}_{-}(g)\otimes
d^{0}_{H_{U}\cap H_{R}}\Xi^{(0)}_{+}(g)\right)\circ d^{0}_{H_{U}\cap
H_{R}}\sigma_{+}^{(0)}\left(g\right)\\\
&=({\rm{inner}})\circ\left(d^{0}_{H_{U}\cap
H_{L}}\Xi^{(0)}_{-}(g)\circ\sigma_{-}^{(1)}(g)\otimes d^{0}_{H_{U}\cap
H_{R}}\Xi^{(0)}_{+}(g)\circ\sigma_{+}^{(1)}(g)\right)\end{split}$

Hence $d^{0}_{H_{U}}\sigma^{(0)}$ is factorized into left and right with
respect to automorphisms

$\displaystyle\begin{split}\gamma_{g,L}=d^{0}_{H_{U}\cap
H_{L}}\Xi^{(0)}_{-}(g)\circ\sigma_{-}^{(1)}(g)\\\
\gamma_{g,R}=d^{0}_{H_{U}\cap
H_{R}}\Xi^{(0)}_{+}(g)\circ\sigma_{+}^{(1)}(g).\end{split}$ (3.60)

For this $\gamma_{g,R}$, by a straight forward calculation, we get

$\displaystyle\begin{split}&d^{1}_{H_{U}\cap
H_{R}}\gamma_{R}(g,h)=d^{1}_{H_{U}\cap H_{R}}d^{0}_{H_{U}\cap
H_{R}}\Xi^{(0)}_{+}(g,h)\circ
d^{1}_{H_{U}}\sigma_{+}^{(1)}(g,h)=\mathop{\mathrm{Ad}}\nolimits\left(V\left(g,h\right)\right).\end{split}$
(3.61)

Hence we may set $v(g,h)=V(g,h)$ in Theorem 3.10. Then the corresponding
$3$-cocycle $c_{R}$ of Theorem 3.10 is $c(g,h,k)=\Psi_{3}(\nu)(g,h,k)$ from
(iv) of Lemma 3.11. This proves the theorem.  $\square$

## Acknowledgment.

The author is grateful to all the 2020 Current Development of Mathematics
Conference organizers of, for giving her the opportunity to give a talk there
and write this lecture note. The author is thankful to Hal Tasaki for his help
on this lecture note, particularly for authorizing her to write down his
calculation on Dijkgraaf-Witten model in this lecture note. This work is
supported by JSPS KAKENHI Grant Number 16K05171 and 19K03534 and JST CREST
Grant Number JPMJCR19T2.

## References

  * [ALLNY] H. Abdul-Rahman, M. Lemm, A. Lucia, B. Nachtergaele and A. Young  A class of two-dimensional AKLT models with a gap, Analytic trends in mathematical physics,  Contemp. Math. 741, Amer. Math. Soc., Providence, RI 2020
  * [AKLT] I. Affleck, T. Kennedy, E.H. Lieb, and H. Tasaki.  Valence bond ground states in isotropic quantum antiferromagnets.  Comm. Math. Phys., 115, 477–528, 1988.
  * [AL] I. Affleck and E. H. Lieb,  A proof of part of Haldane’s conjecture on spin chains.  Lett. Math. Phys. 12, 57–69 (1986).
  * [A] W.B. Arveson.  Continuous analogues of Fock space I.  Mem. Amer. Math. Soc., 409, 1989.
  * [BMNS] S. Bachmann, S. Michalakis, B. Nachtergaele, and R. Sims.  Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems.  Communications in Mathematical Physics 309, 835–871, 2012.
  * [BN] S. Bachmann and B. Nachtergaele.  On gapped phases with a continuous symmetry and boundary operators  J. Stat. Phy. 154, 91–112, 2014.
  * [BDF] S. Bachmann, W. De Roeck, M. Fraas,  The adiabatic theorem and linear response theory for extended quantum systems, Commun. Math. Phys. 361, 997–1027, 2018.
  * [B] O. Bratteli.  Inductive limits of finite-dimensional $C^{*}$-algebras.  Trans. Amer. Math. Soc. 171 195–234 (1972).
  * [BJP] O. Bratteli P. Jorgensen, G. Price.  Endomorphisms of $B({\mathcal{H}})$.  Quantization, nonlinear partial differential equations, and operator algebra. 93–138, Proc. Sympos. Pure Math., 59, 1996.
  * [BJ] O. Bratteli, P. E. T. Jorgensen.  Endomorphisms of $B(H)$ II. Finitely Correlated States on $O_{n}$.  Journal of functional analysis. 145, 323–373 1997.
  * [BR1] O. Bratteli, D. W. Robinson.  Operator Algebras and Quntum Statistical Mechanics 1. Springer-Verlag, 1986.
  * [BR2] O. Bratteli, D. W. Robinson.  Operator Algebras and Quantum Statistical Mechanics 2. Springer-Verlag, 1996.
  * [CGLW] X. Chen, Z. C. Gu, Z. X. Liu, and X. G. Wen,  Symmetry protected topological orders and the group cohomology of their symmetry group. Phys. Rev. B 87, 155114 (2013)
  * [CGW] X. Chen, Z.-C. Gu, and X.-G. Wen,  Classification of gapped symmetric phases in one-dimensional spin systems, Phys. Rev. B 83, 035107 2011.
  * [C] A. Connes,  Periodic automorphisms of the hyperfinite factor of type $II_{1}$.  Acta Sci. Math. (Szeged) 39 (1977), no. 1–2, 39–66.
  * [DL] S. Doplicher, R. Longo.  Standard and split inclusions of von Neumann algebras. Invent. Math. 75 493–536. 1984.
  * [FNW] M. Fannes, B. Nachtergaele, and R.F. Werner.  Finitely correlated states on quantum spin chains.  Comm. Math. Phys., 144, 443–490, 1992.
  * [FKK] H. Futamura, N. Kataoka and A. Kishimoto.  Homogeneity of the pure state space for separable $C^{*}$algebras.  Internat. J. Math. 12 813–845 (2001).
  * [FNW2] M. Fannes, B. Nachtergaele, and R.F. Werner.  Finitely correlated pure states.  Journal of Functional Analysis., 120, 511–534, 1994.
  * [Fu] Y. Fuji  Effective field theory for one-dimensional valence-bond-solid phases and their symmetry protection,  Physical Review B, 10 1410.4211,2016.
  * [GW] Z.-C. Gu, and X.-G. Wen,  Tensor-entanglement-filtering renormalization approach and symmetry-protected topological order, Phys. Rev. B, 80, 155131 2009.
  * [GCW] X. Chen, Z.-C. Gu, and X.-G. Wen,  Classification of gapped symmetric phases in one-dimensional spin systems, Phys. Rev. B, 3, 1008.3745 2011
  * [Hal1] F.D.M. Haldane,  Continuum dynamics of the 1-D Heisenberg antiferromagnet: identification with the $O(3)$ nonlinear sigma model, Phys. Lett. 93A, 464–468 1983.
  * [Hal2] F.D.M. Haldane,  Nonlinear field theory of large-spin Heisenberg antiferromagnets: semiclassically quantized solitons of the one-dimensional easy-axis Néel state, Phys. Rev. Lett. 50 1153–1156 1983.
  * [H1] M. Hastings.  An area law for one-dimensional quantum systems.  Journal of Statistical Mechanics. P08024, 2007.
  * [H2] M. Hastings.  Quasi-adiabatic Continuation for Disordered Systems: Applications to Correlations, Lieb-Schultz-Mattis, and Hall Conductance. http://arxiv/org/abs/1001.5280v2 [math-ph], 2010.
  * [H3] M.B. Hastings,  Lieb-Schultz-Mattis in higher dimensions,  Phys. Rev. B 69, 104431 (2004).
  * [HK] M. B. Hastings, T.= Koma,  Spectral Gap and Exponential Decay of Correlations.  Commun. Math. Phys. 265, 781–804 (2006).
  * [HW] M. B. Hastings, X. G. Wen,  Quasi-adiabatic continuation of quantum states: The stability of topological ground-state degeneracy and emergent gauge invariance, Phys. Rev. B 72 045141 2005 .
  * [J] V. Jones.  Actions of finite groups on the hyperfinite type $II_{1}$ factor. Mem. Amer. Math. Soc. 28 (1980), no. 237.
  * [K] T. Kennedy,  Exact diagonalization of open spin 1 chains, J. Phys.: Cond. Matt. 2, 5737–5745, 1990.
  * [KT1] T. Kennedy and H. Tasaki.  Hidden ${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$-symmetry breaking in Haldane-gap antiferromagnets.  Phys. Rev. B, 45 304–307, 1992.
  * [KT2] T. Kennedy and H. Tasaki.  Hidden symmetry breaking and the Haldane phase in $S=1$ quantum spin chains.  Communications in Mathematical Physics, 147 431–484, 1992.
  * [Ki] A. Kitaev.  Fault-tolerant quantum computation by anyons.  Ann. Physics,303 (1) 2-30,2003.
  * [KOS] A. Kishimoto, N. Ozawa, and S. Sakai.  Homogeneity of the pure state space of a separable C*-algebra.  Canad. Math. Bull. 46 365–37 (2003).
  * [LN] M. Lemm and B. Nachtergaele  Gapped PVBS models for all species numbers and dimensions, Rev. Math. Phys. 31 (2019), no. 9, 1950028
  * [LSW] M. Lemm A. W. Sandvik and L. Wang  Existence of a Spectral Gap in the Affleck-Kennedy-Lieb-Tasaki Model on the Hexagonal Lattice, Phys. Rev. Lett. 124 (2020), 177204
  * [LSM] E. Lieb, T. Schultz, and D. Mattis.  Two soluble models of an antiferromagnetic chain,  Ann. Phys.16,407–466 (1961).
  * [M1] T. Matsui.  A characterization of matrix product pure states.  Infinite dimensional analysis and quantum probability. 1 647–661. 1998.
  * [M2] T. Matsui.  The split property and the symmetry breaking of the quantum spin chain.  Communications in Mathematical Physics, 218 393–416, 2001.
  * [M3] T. Matsui.  Boundedness of entanglement entropy and split property of quantum spin chains.  Reviews in Mathematical Physics, 1350017, 2013.
  * [MM] J. Miller and A. Miyake,  Hierarchy of universal entanglement in 2D measurement-based quantum computation,  Quantum Information 2, 16036 (2016).
  * [MO] A. Moon and Y. Ogata  Automorphic equivalence within gapped phases in the bulk  Journal of Functional Analysis :Volume 278, Issue 8, 1 May 2020, 108422
  * [NO] P. Naaijkens and Y. Ogata  The split and approximate split property in 2D systems: stability and absence of superselection sectors  arXiv:2102.07707
  * [N] B. Nachtergaele.  The spectral gap for some spin chains with discrete symmetry breaking.  Comm. Math. Phys., 175(3):565–606, 1996.
  * [NOS] B. Nachtergaele, Y. Ogata, and R. Sims.  Propogation of correlations in quantum lattice systems.  J. Stat. Phys, 124, 1–13, 2006.
  * [NR] M. den Nijs and K. Rommelse,  Preroughening transitions in crystal surfaces and valence-bond phases in quantum spin chains, Phys. Rev. B 40, 4709, 1989.
  * [NS06] B. Nachtergaele and R. Sims.  Lieb-Robinson Bounds and the Exponential Clustering Theorem.  Commun. Math. Phys. 265, 119–130 (2006)
  * [NS07] B. Nachtergaele and R. Sims,  A multi-dimensional Lieb-Schultz-Mattis theorem, Comm. Math. Phys. 276, 437–472 (2007).
  * [NS09] B. Nachtergaele and R. Sims. Locality Estimates for Quantum Spin Systems. In: Sidoravicius V. (eds) New Trends in Mathematical Physics. Springer (2009)
  * [NSY] B. Nachtergaele, R. Sims, and A. Young.  Quasi-locality bounds for quantum lattice systems. I. Lieb-Robinson bounds, quasi-local maps, and spectral flow automorphisms. J. Math. Phys. 60, 061101 (2019)
  * [O1] Y. Ogata.  A class of asymmetric gapped Hamiltonians on quantum spin chains and its classification I.  Communications in Mathematical Physics, 348, 847–895, 2016.
  * [O2] Y. Ogata.  A class of asymmetric gapped Hamiltonians on quantum spin chains and its classification II.  Communications in Mathematical Physics, 348, 897–957, 2016.
  * [O3] Y. Ogata.  A class of asymmetric gapped Hamiltonians on quantum spin chains and its classification III.  Communications in Mathematical Physics, 352, 1205–1263, 2017.
  * [O4] Y. Ogata  A $\mathbb{Z}_{2}$-Index of Symmetry Protected Topological Phases with Time Reversal Symmetry for Quantum Spin Chains.  Communications in Mathematical Physics, 374, 705–734, 2020.
  * [O5] Y. Ogata  A classification of pure states on quantum spin chains satisfying the split property with on-site finite group symmetries.  Transactions of the American Mathematical Society, Series B 8(2) 39 - 65 2021\.
  * [O6] Y. Ogata  A ${\mathbb{Z}}_{2}$-index of symmetry protected topological phases with reflection symmetry for quantum spin chains  To appear in Communications in Mathematical Physics.
  * [O7] Y. Ogata  A $H^{3}(G,{\mathbb{T}})$-valued index of symmetry protected topological phases with on-site finite group symmetry for two-dimensional quantum spin systems arXiv:2101.00426
  * [OTT] Y. Ogata and Y. Tachikawa and H. Tasaki  General Lieb-Schultz-Mattis type theorems for quantum spin chains, arXiv:2004.06458
  * [OT] Y. Ogata and H. Tasaki  Lieb-Schultz-Mattis type theorems for quantum spin chains without continuous symmetry.  Communications in Mathematical Physics 372(3) 951–962 , 2019.
  * [Os] M. Oshikawa,  Commensurability, excitation gap, and topology in quantum many-particle systems on a periodic lattice,  Phys. Rev. Lett. 84, 1535 (2000).
  * [P] R. T. Powers.  Representations of uniformly hyperfinite algebras and their associated von Neumann rings.  Ann. of Math. 86 138–171 (1967).
  * [PTBO1] F. Pollmann, A. Turner, E. Berg, and M. Oshikawa  Entanglement spectrum of a topological phase in one dimension. Phys. Rev. B 81, 064439, 2010.
  * [PTBO2] F. Pollmann, A. Turner, E. Berg, and M. Oshikawa  Symmetry protection of topological phases in one-dimensional quantum spin systems. Phys. Rev. B 81, 075125, 2012.
  * [PWSVC] D. Perez-Garcia, M.M. Wolf, M. Sanz, F. Verstraete, and J.I. Cirac, String order and symmetries in quantum spin lattices, Phys. Rev. Lett. 100, 167202 2008.
  * [T] M. Takesaki,  Theory of operator algebras. I.  Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin, 2002.
  * [Tas1] H. Tasaki  Topological phase transition and ${\mathbb{Z}}_{2}$ index for $S=1$ quantum spin chains  Phys. Rev. Lett. 121, 140604 (2018).
  * [Tas2] H. Tasaki,  Physics and mathematics of quantum many-body systems,  Graduate Texts in Physics, Springer 2020
  * [Y] B. Yoshida,  Topological phases with generalized global symmetries, Phys. Rev. B 93, 155131 (2016).
  * [WPVZ] H. Watanabe, H.C. Po, A. Vishwanath, and M.P. Zaletel,  Filling constraints for spin-orbit coupled insulators in symmorphic and nonsymmorphic crystals,  Proc. Natl. Acad. Sci. U.S.A. 112, 14551–14556 (2015).
  * [No] “Topological Phase Transitions and Topological Phases of Matter: Scientific Background on the Nobel Prize in Physics 2016” (Class for Physics of the Royal Swedish Academy of Sciences, 2016)
https://www.nobelprize.org/prizes/physics/2016/advanced-information/