Quantum gravity at the fifth root of unity

Abstract

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 postulated to be 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. In this case, the amplitudes have a polytope interpretation. The generalization to higher-dimensional Lie groups SU(N), G2 and E8 is suggested.