The tetrahedral graph we consider throughout the paper. The orientations are the same as those of the graphical representation of the 6j-symbol.51 The three fat lines form the cycle (126) we will consider in order to explain the action of the new Hamiltonian.
The 3-valent nodes represent 3jm-symbols, where legs carry the spins. A magnetic index is summed when there is a link joining two nodes. The action of is a grasping between e 1 and e 2, denoted by a dashed line which carries the spin 1/2. The final result is proportional to the 3jm-symbol with the spins .
A pictorial representation of (55). The character χ j along the closed loop acts on the left. On the right, we have depicted the situation after re-coupling. A specific 6j-symbol is extracted on each node, and one has to sum over the colorings k 1, … , k n . The dashed lines correspond to the dual 2D triangulation to the plaquette if we think of the latter as embedded in flat 3-space. The vertex s of the 2D triangulation is then dual to the plaquette.
Here, we have displayed the geometric interpretation of the character operator on the plaquette as a tent move. The vertex s is evolved to a new vertex s ′, with an edge of length , the tent pole. Between the initial and the final triangulations we have a piece of 3D triangulation. The character operator then generates the evaluation of the Ponzano-Regge amplitude on this triangulation.
A graphical representation of the action of the new Hamiltonian. The basic idea is that the holonomy around a closed loop in the topological sector only depends on its homotopy type, so that we can deform the grasping on the left to that on the right, picking up this way some holonomy which must be trivial.
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