Group cohomology and condensed matter I am mystified by formulas that I find in the condensed matter literature
(see Symmetry protected topological orders and the group cohomology of their symmetry group arXiv:1106.4772v6 (pdf) by Chen, Gu, Liu, and Wen).  These formulas have been used in some very interesting work in condensed matter and I would like to know how to understand them.
I begin with the simplest case.
Let $G$ be a finite group.
One is given an element of $H^2(G,U(1))$ that is represented by an
explicit $U(1)$-valued group cocycle $\nu(a,b,c)$.  This is a homogeneous
cocycle, $\nu(ga,gb,gc)=\nu(a,b,c)$ and obeys the standard cocycle condition
$\nu(a,b,c)\nu^{-1}(a,b,d)\nu(a,c,d)\nu^{-1}(b,c,d)=1$ for $a,b,c,d\in G$.
Let $X=G\times G$ be the Cartesian product of two copies of $G$.  We consider
$G$ acting on $X=G\times G$ by left multiplication on each factor.
   The cocycle $\nu$ is then used to define a twisted
action of $G$ on the complex-valued functions on $X$.  For $g\in G$ and $\Phi:
X\to \mathbb{C}$, the definition (eqn. 27 of the paper) is
$$\hat g(\Phi)=g^*(\Phi) \Lambda(a,b;g)$$
where $g^*(\Phi)$ is the pullback of $\Phi$ by $g$ and (with $a,b\in G$ defining
a point in $X=G\times G$, and $g_*$ an arbitrary element of $G$)
$$\Lambda(a,b;g)=\frac{\nu(a,g^{-1}g_*,g_*)}{\nu(b,g^{-1}g_*,g_*)}.$$
It is shown in appendix F of the paper that this does given an action of $G$
on the functions on $X=G\times G$.
The authors also describe a version in one dimension more.  In this case,
$\nu(a,b,c,d)$ is a homogeneous cocycle representing an element of $H^3(G,U(1))$
and satisfying the usual cocycle relation 
and one takes $X=G\times G\times G\times G$ to be the Cartesian product of four
copies of $G$.  A twisted action of $G$ on the functions on $X$ is now defined by
$$\hat g(\Phi)=g^*(\Phi) \Lambda(a,b,c,d;g)$$
with
$$\Lambda(a,b,c,d;g)=\frac{\nu(a,b,g^{-1}g_*,g_*)\nu(b,c,g^{-1}g_*,g_*)}{\nu(d,c,g^{-1}g_*,g_*)\nu(a,d,g^{-1}g_*,g_*)}.$$
It is shown in appendix G that this does indeed give a twisted action of $G$ on the functions on $X$.
I presume there is supposed to be an analog of this in any dimension though I cannot see this stated explicitly.
Can anyone shed light on these formulas?
 A: The geometric interpretation for $1$-cocyles.
Recall the following construction due to Bisson and Joyal.
Let $p:P\rightarrow B$ be a covering space over the connected manifold $B$. Suppose that the fibres of $p$ are finite. For every topological space $X$, the polynomial functor $p(X)=\{ (u,b),b\in B, u:p^{-1}(b)\rightarrow X\}$ $p(X)$ is a total space of a bundle over $B$ whose fibres are $X^{p^{-1}(b)}$.
Here we suppose $B=BG$ the classifying bundle of $G$ and $p_G:EG\rightarrow BG$ the universal cover. We suppose that $X=U(1)$. The quotient of $EG\times Hom(G,U(1))$ by the diagonal action of $G$, where $G$ acts on $Hom(G,U(1))$ by the pullback.
$\hat g(\Phi)=g^*(\Phi)$
is the polynomial construction $p_G(X)$. It corresponds to $\Lambda=0$. 
Remark that  we can define non zero $\Lambda$ and the definition:
$\hat g(\Phi)(a)=g^*(\Phi)\Lambda(a)$
defines a $U(1)^G$ bundle isomorphic to $p_G(X)$ and we can see these bundle as a deformation of the canonical  flat connection of $p_G(U(1))$.
Interpretation of n-cocycles, n>1
2-cocycles classify gerbes or stacks. There is a notion of classifying space for gerbes. If $G$ is a commutative group, the classifying spaces of a $G$-gerbe is $K(G,2)$. Let $B_2G$ be the classifying space of the $G$-gerbes. The universal gerbe $p_G$ is a functor $:E_2G\rightarrow Ouv(B_2G)$ where $Ouv(B_2G)$ is the category of open subsets of $B_2G$. For every open subset $U$ of $B_2G$, an object of the fibre of $U$ is a $G$-bundle. We can generalize the Bisson Joyal construction here:
If $p_U:T_U\rightarrow U$ is an object of ${E_2G}_U$ the fibre of $U$, we define $p_U(X)$ the polynomial functor associated to $p_U$, we obtain a gerbe $E_2^XG$ such that for every open subset $U$ of $B_2G$, the fibre of $U$ are the bundles $p_U(X)$. Its classifying cocyle is defined by a covering $(U_i)_{i\in I}$ of $B_2G$ and $c_{ijk}: U_{ijk}\rightarrow U(1)^G$. Remark that if $\mu$ is a $U(1)$ valued $2$-cocycle, we can express $\Lambda$ with Cech cohomology and obtain a $2$-boundary $d_{ijk}$.
There exists a notion of connective structure on gerbes, a notion which represents a generalization of the notion of connection. The cocyle $c_{ijk}d_{ijk}$ is a deformation of the canonical flat connective structure defined on $E_2^{U(1)}G$.
For higher dimensional cocyles, there is a notion of $3$-gerbe, but for $n>3$, the notion of $n$-gerbes is not well understood since the notion of $n$-category which must be used to buil such a theory is not well-known also.
Bisson, T.,  Joyal, A. (1995). The Dyer-Lashof algebra in bordism. CR Math. Rep. Acad. Sci. Canada, 17(4), 135-140.
A: Maybe formula (27) is some kind of re-indexed loop transgression map in the case $n=2$. In general, loop transgression is a map
$ \tau : H^n(BG) \rightarrow  H^{n-1}(L(BG))$
where $L(X) = Maps(S^1, X)$ is the free loop space of a topological space $X$. Here and elsewhere, I'll take the coefficient group to be $U(1)$.
There should be a continuous map
$ \psi : B(Y_{G \times G}) \rightarrow L(BG) $.
Here, $Y_{G \times G}$ is the groupoid (the "action groupoid")  corresponding to the diagonal action of $G$ on $G \times G$ from the left, and $B(Y_{G \times G})$ is its classifying space. The map $\psi$ should be
induced by the equivariant map $G \times G \rightarrow G$ given by $(a,b) \mapsto ab^{-1}$. 
Now, suppose you start with a 2-cocycle $\nu \in H^2(BG)$, and transgress it to $\tau(\nu) \in H^1(L(BG))$ and then pull it back to get $\psi^*(\tau(\nu)) \in H^1(B(Y_{G \times G}))$. Maybe one can interpret their $\Lambda (a,b; g)$ function as an element in $H^1(B(Y_{G \times G}))$, and that in fact 
$
 \Lambda = \psi^*(\tau(\nu)).
$
However, I haven't been able to get that to work. If true, it may suggest a similar story for higher $n$.
