# Quantum Gravity at the Fifth Root of Unity

## TL;DR

This paper explores quantum gravity models using $SU(2)$ quantum groups at a fifth root of unity, coupling spacetime and charge symmetries, and suggests a unifying framework involving higher Lie groups and quantum computations.

## Contribution

It introduces a novel approach to 3D spin foam models with complex root of unity deformation, coupling spacetime and gauge symmetries, and proposes generalizations to higher-dimensional Lie groups.

## Key findings

- Coupling of $SU(2)$ and $SU(3)$ quantum groups at fifth root of unity.
- Framework for quantum gravity involving particle and field quantum numbers.
- Potential extension to higher Lie groups like $SU(N)$, $G_2$, and $E_8$.

## 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 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.