Symmetries of quantum space-time in 3 dimensions

TL;DR

This paper explores how loop quantum gravity techniques reveal that the local gauge symmetry of 3D quantum gravity with a positive cosmological constant transforms into a quantum group structure, leading to a quantum deformation of space-time symmetries.

Contribution

It demonstrates the emergence of the quantum group $so_q(4)$ and the kappa-Poincaré algebra from 3D quantum gravity using loop quantum gravity methods.

Findings

The gauge symmetry becomes a quantum group $so_q(4)$.

The quantum group structure depends on the cosmological constant.

The flat space-time symmetries are described by the kappa-Poincaré algebra.

Abstract

By applying loop quantum gravity techniques to 3D gravity with a positive cosmological constant $\Lambda$, we show how the local gauge symmetry of the theory, encoded in the constraint algebra, acquires the quantum group structure of $so_q(4)$, with $ q = \exp{(i\hbar \sqrt{\Lambda}/2\kappa)}$. By means of an Inonu-Wigner contraction of the quantum group bi-algebra, keeping $\kappa$ finite, we obtain the kappa-Poincar\'e algebra of the flat quantum space-time symmetries.