The Hilbert space of 3d gravity: quantum group symmetries and observables

TL;DR

Meusburger and Noui establish a precise link between three-dimensional BF/loop quantum gravity and combinatorial quantisation based on the CS formulation. They show that the kinematical and physical Hilbert spaces of both approaches are equivalent and derive explicit operator relations, clarifying how combinatorial variables map to loop variables, with the cilia playing a key interpretative role. A central result is that quantum group symmetries, realized as the quantum doubles $D(G)$, act naturally on spaces of cylindrical functions and encode the mass and spin observables as well as the constraints, thus unifying the two quantisation frameworks. The work also demonstrates how gauge fixing via maximal-tree contractions in the graph aligns with Fock–Rosly discretisation and provides a pathway to construct physical states through $D(G)$ representations, with implications for broader quantum gravity programs and potential extensions to nonzero cosmological constant.

Abstract

We relate three-dimensional loop quantum gravity to the combinatorial quantisation formalism based on the Chern-Simons formulation for three-dimensional Lorentzian and Euclidean gravity with vanishing cosmological constant. We compare the construction of the kinematical Hilbert space and the implementation of the constraints. This leads to an explicit and very interesting relation between the associated operators in the two approaches and sheds light on their physical interpretation. We demonstrate that the quantum group symmetries arising in the combinatorial formalism, the quantum double of the three-dimensional Lorentz and rotation group, are also present in the loop formalism. We derive explicit expressions for the action of these quantum groups on the space of cylindrical functions associated with graphs. This establishes a direct link between the two quantisation approaches and clarifies the role of quantum group symmetries in three-dimensional gravity.

Figures (13)

• Figure 1: Illustration of the discretisation of a genus two surface $S$ by a graph $\Gamma$. On the right, we focus on a particular part of $\Gamma$ where the structures of the graph have been highlighted: the edges are oriented and the vertices are endowed with a cilium (the short thin lines) which defines a linear ordering of the incident edges. At the vertex $v$, we have we have $S(v)=\{\lambda_1,\lambda_4\}$ and $T(v)=\{\lambda_2,\lambda_3\}$; $O(\lambda_1,s)<O(\lambda_2,t)<O(\lambda_3,s)<O(\lambda_4,t)$.
• Figure 2: Illustrations of the sets $S^\pm$ and $T^\pm$ defined in (\ref{['endedgedef']}): $S^+(t(\lambda_1))=\{\lambda_2,\lambda_4,\lambda_5,\lambda_6\}$, $T^+(t(\lambda_1))=\{\lambda_3,\lambda_4\}$, $T^-(s(\lambda_2))=\{\lambda_1\}$, $T^-(t(\lambda_2))=\{\lambda_7\}$
• Figure 3: The four different configurations for two edges meeting at a vertex.
• Figure 4: Examples of a ciliated graphs with edges that are loops.
• Figure 5: Illustration of a case where the derivative operator ${\hbox{\boldmath $q$}}_\gamma$ has a non-trivial action on a quantum state whose support is a graph $\gamma'$: $\gamma=\lambda_2\lambda_2'\lambda_1$ and $\gamma'=\lambda_4\lambda_3$. The operator ${\hbox{\boldmath $q$}}_\gamma$ acts schematically on the common vertex $\gamma \cap \gamma'$.

Theorems & Definitions (2)

• Example 3.1
• Example 5.1