I have just read a big part of John Baez's nice article Spin network states in Gauge theory. The definitions are quite clear in that article. However, there is a part which is not explained explicitly there, in my humble opinion. Let us say you have a spin network corresponding to a compact connected Lie group $G$ (the gauge group), which is a finite oriented graph with each edge labeling an irreducible representation of $G$ and each vertex $v$ labeling an intertwining operator (for the action of $G$) from the tensor product of the (irreducible) representations labeled by the set of all incoming edges at $v$ to the tensor product of the reps labeled by the set of all outgoing edges at $v$.

What I would like to understand though, is how to regard a spin network as an element of $\mathrm{L}^2(\mathcal{A}/\mathcal{G})$, where $\mathcal{A}$ denotes the space of connections on the graph (here regarded as parallel transport maps associated to each edge, please see the article for more detail) and $\mathcal{G}$ is the group of gauge transformations which acts on $\mathcal{A}$.

Given a spin network and a "connection" $A$ on the graph, how do we "evaluate" the spin network on the connection $A$, or rather on the equivalence class of $A$ under the group $\mathcal{G}$ of gauge transformations? The description in that article is via identifications and the Peter-Weyl theorem. Could someone perhaps spell it out for me in more concrete terms?


In Spin Networks in Nonperturbative Quantum Gravity, John Baez explains what I am trying to understand in section $2$. Essentially, you form a big tensor by tensoring out the intertwining operators at each vertex, and you form another big tensor by tensoring out the "parallel transport maps" associated to each edge when given a connection $A$ on the graph, and then you just contract each upper (resp. lower) index of the first big tensor with a lower (resp. upper) index of the second big tensor. The result is the "evaluation" of the spin network at the connection $A$. It is explained in more detail in the paper linked to in this answer. I thank the author for sending me a link to this paper.

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    $\begingroup$ Short comment concerning the motivation for this evaluation map: The simplest case is if your graph has only one vertex. Then edges are closed loops, and the evaluation of the network on a connection is the trace (with respect to a given representation) of the holonomy around these loops (so-called Wilson loops). The evaluation for more complex networks is what you get if you try to extend the holonomy picture to a prescription that behaves nicely with respect to path concatenation. $\endgroup$ – Tobias Diez Nov 17 '20 at 21:15

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