In the context of LQG spin networks are derived based on Ashtekar's formulation implementing local SU(2) gauge symmetry in tangent space. Here SU(2) and 3+1 dim. spacetime are deeply related. Forgetting about this derivation and starting with spin networks, talking about dimensions does no longer makes sense; a spin network or a graph w/o embedding in a manifold does not have a dimension. So all what remains from 3+1 dim. spacetime is the SU(2) spin network. That means we can translate the question "why do we live in 3+1 dim. spacetime?" into "why does spacetime emerge from SU(2) spin networks?". So the basic question is "why SU(2)? Why not SU(N) or something else?". The only reason for SU(2) I know is based on SO(3,1) ~ SU(2) * SU(2), but this argument is no longer valid as soon as we drop 3+1 dim. spacetime as our starting point. Any ideas?