Gauge theory on a lattice: intertwiners, gauge potentials...

[Heidi]

Hi Pfs
i am interested in spin networks (a pecular lattices) and i found two ways to define them. they both take G = SU(2) as the Lie group.
in the both ways the L oriented edges are colored with G representations (elements of G^L
the difference is about the N nodes.
1) in the first way the coloring of the nodes is like the links: elements of G^N
2) in the second the nodes are colored with intertwiners between the ingoing links and the outgoing links from the node
How to see that they are equivalent
i would appreciate an example with one or two nodes
I have doubts because when the nodes are trivalent there is only one intertwiner and in that case is the coloring of the nodes still a choince,
thanks

 

[azntoon]

The two ways of defining spin networks are equivalent because they use the same Lie group (G = SU(2)) and the same representations for the edges (elements of G^L). In the first way, the nodes are colored with elements of G^N, while in the second way the nodes are colored with intertwiners between the incoming and outgoing links from the node.To illustrate the equivalence, let's consider a simple example with one node. In the first approach, the node is colored with an element of G^N, say, g. In the second approach, the node is colored with the intertwiner between the incoming and outgoing links, which is also g. Thus, the two approaches yield the same coloring for the node and are therefore equivalent.When the nodes are trivalent, there is only one intertwiner, so the choice of a specific element of G^N is not necessary. However, in this case the two approaches are still equivalent since the intertwiner is the same as the element of G^N.