# rt.representation theory – Spin networks as functionals on the moduli space of connections modulo gauge transformations on a graph

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?