Quantum hyperbolic geometry in loop quantum gravity with cosmological constant

TL;DR

This paper demonstrates that quantum groups can encode hyperbolic geometry in loop quantum gravity with a cosmological constant, providing a novel approach to incorporate $b3$ in the theory.

Contribution

It constructs geometric observables using tensor operators in LQG with quantum groups, revealing a quantized hyperbolic geometry and advancing the understanding of non-zero cosmological constant inclusion.

Findings

Quantum groups encode hyperbolic geometry in LQG.

Tensor operators facilitate the construction of geometric observables.

First explicit demonstration of quantized hyperbolic geometry in this context.

Abstract

Loop Quantum Gravity (LQG) is an attempt to describe the quantum gravity regime. Introducing a non-zero cosmological constant $\Lambda$ in this context has been a withstanding problem. Other approaches, such as Chern-Simons gravity, suggest that quantum groups can be used to introduce $\Lambda$ in the game. Not much is known when defining LQG with a quantum group. Tensor operators can be used to construct observables in any type of discrete quantum gauge theory with a classical/quantum gauge group. We illustrate this by constructing explicitly geometric observables for LQG defined with a quantum group and show for the first time that they encode a quantized hyperbolic geometry. This is a novel argument pointing out the usefulness of quantum groups as encoding a non-zero cosmological constant. We conclude by discussing how tensor operators provide the right formalism to unlock the LQG formulation with a non-zero cosmological constant.