Quantum Gravity at the Fifth Root of Unity

We consider quantum transition amplitudes, partition functions and observables for 3D spin foam models within $SU(2)$ quantum group deformation symmetry, where the deformation parameter is a complex fifth root of unity. By considering fermionic cycles through the foam we couple this $SU(2)$ quantum group with the same deformation of $SU(3)$, so that we have quantum numbers linked with spacetime symmetry and charge gauge symmetry in the computation of observables. The generalization to higher-dimensional Lie groups $SU(N)$, $G_2$ and $E_8$ is suggested. On this basis we discuss a unifying framework for quantum gravity. Inside the transition amplitude or partition function for geometries, we have the quantum numbers of particles and fields interacting in the form of a spin foam network $-$ in the framework of state sum models, we have a sum over quantum computations driven by the interplay between aperiodic order and topological order.

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