Quantum hyperbolic geometry in loop quantum gravity with cosmological constant

Loop quantum gravity (LQG) is an attempt to describe the quantum gravity regime. Introducing a nonzero cosmological constant $\Lambda$ in this context has been a standing problem. Other approaches, such as Chern-Simons gravity, suggest that quantum groups can be used to introduce $\Lambda$ into 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 nonzero cosmological constant. We conclude by discussing how tensor operators provide the right formalism to unlock the LQG formulation with a nonzero cosmological constant.

Figure 1

Hyperbolic triangle in Poincaré disc. Normals are such that ${\overrightarrow{n}}_i=\sinh \frac{l_i}{R}{\widehat{n}}_i$. ${\widehat{u}}_i$ are the normalized tangent vectors and ${\widehat{u}}_b·{\widehat{u}}_c=\cos {\theta }_a$.