The group cohomology can be useful to describe **the many-body quantum Condensed Matter systems with emergent underlying intrinsic Topological Orders**. These classes of systems at low energy and long distance behave like **certain TQFT theories**. In particular, one can write explicit exact solvable Lattice quantum Hamiltonian operator $\hat{H}$ on the space discretized lattice, and its ground states give rise to **discrete gauge theory TQFT with finite group $G$ (quantum double models in 2+1d or its 3+1d, etc analogous, $D(G)$, with Drinfield and Hopf algebra), twisted discrete gauge theories (twisted quantum double models $D^\omega(G)$)**.

See:

Twisted Quantum Double Model of Topological Phases in Two-Dimension arxiv:1211.3695 PhysRevB.87.125114

Twisted Gauge Theory Model of Topological Phases in Three Dimensions arxiv:1409.3216
PhysRevB.92.045101

Non-Abelian String and Particle Braiding in Topological Order: Modular SL(3,Z) Representation and 3+1D Twisted Gauge Theory arxiv:1404.7854 PhysRevB.91.035134

**The low energy and long distance physics of the theories are the same as the Dijkgraaf-Witten topological gauge theories, Kitaev toric code and quantum double models, and some of Levin-Wen string-net models, and many of twisted discrete gauge theory.**

This is how the quantum Hamiltonian operator (acting on the Hilbert space of quantum states) looks like:
$$
\hat{H}=-\sum_v A_v-\sum_f B_f,
$$
where $B_f$ is the face operator defined at each triangular face $f$, and $A_v$ is the vertex operator defined on each vertex $v$. As in the TQD model in $(2+1)$-d, each operator $A_v$ behaves as a gauge transformation on the group elements respectively on the edges meeting at $v$, and a $B_f$ detects whether the flux through face $f$ is zero. This kind of Hamiltonians generically feature ground states that are gauge invariant and bear zero flux everywhere.

A normalized
$$\omega_d\in H^d(G,U(1)),$$
as a function
$\omega:G^4\rightarrow U(1)$, satisfies the $d$-cocycle
condition. The cocycle $\omega$ will fill into the spacetime lattice $d$-simplex ($d+1$-cell). Let us take $D=4$-dim spacetime as an example below.

The operator $A_v$ is a summation
$$ A_v=\frac{1}{|G|}\sum_{[vv']=g\in G}A_v^g.
$$
The value $|G|$ is the order of the group $G$. The operator $A_v^g$ acts on
a vertex $v$ with a group element $g\in G$ by replacing $v$ by a new enumeration $v'$ that is ``slightly" less than $v$ but greater than all the enumerations that are less than $v$ in the original set of enumerations before the action of the operator, such that $v'v=g$. In a dynamical picture of Hamiltonian evolution, $v'$ is understood as on the next \textquotedblleft time" slice, and there is an edge $v'v\in G$ in the $(3+1)$ dimensional \textquotedblleft spacetime" picture. That is, the new vertex $v'$ and the original vertices before the action of $A_v$ delineate a $4$-dimensional picture. Let us consider as follows the simplest subgraph---namely a single tetrahedron---of some large $\Gamma$ to illustrate how an $A_v$ acts:
where $v'_4v_4=g$.

The action of $B_f$ on a basis vector is
The discrete
delta function $\delta_{v_1v_2\cdot v_2v_3\cdot v_3v_1}$ is unity if ${v_1v_2\cdot v_2v_3\cdot v_3v_1=1 }$, where $1$ is the identity element in $G$, and 0 otherwise. Note again that here, the ordering of $v_1,v_2$, and $v_3$ does not matter because of the identities
$\delta_{v_1v_2\cdot v_2v_3\cdot v_3v_1}
=\delta_{v_3v_1\cdot v_1v_2\cdot v_2v_3}$ and
$\delta_{v_1v_2\cdot v_2v_3\cdot v_3v_1}
=\delta_{\overline{v_1v_2\cdot v_2v_3\cdot v_3v_1}}
=\delta_{\overline{v_3v_1}\cdot \overline{v_2v_3}\cdot \overline{v_1v_2}}
=\delta_{v_1v_3\cdot v_3v_2\cdot v_2v_1}$.
In other words,
in any state on which $B_f=1$ on a triangular face $f$, the three group degrees of freedom
around $v$ is related by a chain
rule:
$$
v_1v_3=v_1v_2\cdot v_2v_3
$$
for any enumeration
$v_1,v_2,v_3$ of the three vertices of the face $f$.

Here are how the space lattice and the triangulation of spacetime lattices looks like:

Here are how the **$SL(N,\mathbb{Z})$'s modular $S$ and $T$-transformations** look like:

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