In summary: I think this is not possible in odd dimensions, but I am not sure.In summary, the discussion is about the use of SU(2) spin networks as a starting point for a theory of quantum gravity, with the recent entropic and holographic approaches pointing in the same direction. The central question is why SU(2) is chosen as the unique remaining structure, with suggestions that it may be due to simplicity and constraints of the observer. This choice may also be related to the preservation of angular momentum in mechanics and the use of Riemannian curvature. However, there is still a lack of a clear answer as to why SU(2) is the preferred symmetry in this context.
The material [tex]\text{Zn}\text{Cu}_3\text{(OH)}_6\text{Cl}_2[/tex] also known as Herbertsmithite may support a string net ground state (big emphasis on may). If so, then the ground state would be a superpositon of networks of strings each of which basically form a U(1) spin network.
Fractional quantum hall systems definitely realize certain Chern-Simons theories including those based on U(1), SU(N), and Sp(N). These theories may be reformulated in terms of spin networks as Turaev and Viro and others have done.
http://arxiv.org/abs/gr-qc/9408013 Spin Networks, Turaev-Viro Theory and the Loop RepresentationTimothy J. Foxon
(Submitted on 10 Aug 1994)
Abstract: We investigate the Ponzano-Regge and Turaev-Viro topological field theories using spin networks and their $q$-deformed analogues. I propose a new description of the state space for the Turaev-Viro theory in terms of skein space, to which $q$-spin networks belong, and give a similar description of the Ponzano-Regge state space using spin networks.
I give a definition of the inner product on the skein space and show that this corresponds to the topological inner product, defined as the manifold invariant for the union of two 3-manifolds.
Finally, we look at the relation with the loop representation of quantum general relativity, due to Rovelli and Smolin, and suggest that the above inner product may define an inner product on the loop state space.