What is the role of averaging holonomies in gauge invariance within GR?

[naima]

In this paper the author shows that holonomies on Wilson loops are useful tools in GR. I have no problem with the gauge invariance on these loops which comes from the cyclicity of thr trace.
Bonzom writes then:
We are moreover interested in SU(2) gauge invariant states. Gauge transformations act on holonomies only on their endpoints. If h is a map from Σ to SU(2), then the holonomy transforms as
$U_e (A^h ) = h(t(e))U_e (A)h(s(e))^{−1}$, with t(e), s(e) being respectively the source and target points of the path e. When focusing on a single graph Γ, this reduces gauge transformations
to an action of SU(2)^V on the set of cylindrical functions over Γ. So from any function f over SU(2)^E , one gets an invariant function by averaging over the SU(2)^V action.

Here we have edges and vertices.What is this averaging over the action ar the nodes?
Thanks for your help.

 

[Ben Niehoff]

This is not really a GR paper, I'm not sure why you would expect classical GR people to be able to answer a detailed question about LQG.

If the authors haven't made it clear what they mean, then you could ask them, or perhaps try to dig in the literature.

 

[naima]

Could this question be sent to a better place?
thanks

 

[PeterDonis]

Moved to Quantum Physics.

 

[naima]

I can describe now why i had a problem with this averaging.
In LQG we have oriented edges which are colored by a representation if SU(2).
Each edge may be considered as an open path with a starting point and a target point. we call them the nodes.
let us consider the simpler spin network with only one edge. it may be a sub network of a bigger one. suppose that it is colored by h a given matrix of SU(2). A gauge transformation acts like this: a pair of elements of the group g1 and g2 transforms h
in $h' = g1.h.(g2)^{-1}$
We will say that h and h' belong to the same class of equivalence if one can find such a pair to map h to h'. it is obvious to see that the identity is equivalent to any element of SU(2). If you have a function f on the set of the representation you do not need averaging or something else to get a gauge invariant function. the constant function f(id) works!
This is not so easy with more complex cases. take the Mercedes Benz network. It has
6 edges and 4 nodes. and there are several classes of equivalence.
if each class "i" was equipped with a pecular element $m_i$ we could build a gauge invariant function f' from any f by taking for any matrix m f'(m) = #f(m_i)## .
The author gives a more elegant way to build such a gauge invariant function. it is also constant on each equivalence class but its value is the mean value of f on the class.
This is why they are talking about the average of the holonomies.