Quantum gravity kinematics from extended TQFTs

TL;DR

<3-5 sentence high-level summary>This paper constructs a kinematical framework for (2+1)-dimensional quantum gravity with a positive cosmological constant by grounding the holonomy–flux algebra in an extended TQFT, specifically the Turaev–Viro model based on SU(2)$_k$. It introduces a finite-dimensional, puncture-based graph Hilbert space whose vacuum is provided by the TV state sum, and whose excitations are labeled by the Drinfeld center, realized through ribbon operators and the tube algebra. The work develops a self-contained graphical calculus for SU(2)$_k$ and defines left/open and closed ribbon operators that generate and measure curvature and torsion excitations, forming a fusion basis that is stable under coarse-graining. This framework links kinematics to extended TQFTs, enabling studies of phases in spin foams and group field theories and offering a natural path to coupling matter (mass and spin) to gravity with a cosmological constant in 2+1 dimensions.

Abstract

We show how extended topological quantum field theories (TQFTs) can be used to obtain a kinematical setup for quantum gravity, i.e. a kinematical Hilbert space together with a representation of the observable algebra including operators of quantum geometry. In particular, we consider the holonomy-flux algebra of (2+1)-dimensional Euclidean loop quantum gravity, and construct a new representation of this algebra that incorporates a positive cosmological constant. The vacuum state underlying our representation is defined by the Turaev-Viro TQFT. We therefore construct here a generalization, or more precisely a quantum deformation at root of unity, of the previously-introduced SU(2) BF representation. The extended Turaev-Viro TQFT provides a description of the excitations on top of the vacuum, which are essential to allow for a representation of the holonomies and fluxes. These excitations agree with the ones induced by massive and spinning particles, and therefore the framework presented here allows automatically for a description of the coupling of such matter to (2+1)-dimensional gravity with a cosmological constant. The new representation presents a number of advantages over the representations which exist so far. It possesses a very useful finiteness property which guarantees the discreteness of spectra for a wide class of quantum (intrinsic and extrinsic) geometrical operators. The notion of basic excitations leads to a fusion basis which offers exciting possibilities for constructing states with interesting global properties. The work presented here showcases how the framework of extended TQFTs can help design new representations and understand the associated notion of basic excitations. This is essential for the construction of the dynamics of quantum gravity, and will enable the study of possible phases of spin foam models and group field theories from a new perspective.

Figures (7)

• Figure 1: Example of a state on the sphere with one puncture. Using the rules of graphical calculus, one can show that any such state is proportional to the empty graph.
• Figure 2: Minimal graph on the sphere with two punctures. The basis states for $\mathcal{H}_{\mathbb{S}_2}$ are given by the admissible spin colorings of this minimal graph.
• Figure 3: Basis of states for the sphere with two punctures and level $\mathrm{k}=1$.
• Figure 4: Minimal triangulation (in pink) of the two-punctured sphere and its dual graph $\mathcal{Q}$. One can see that the circular edge tangent to a marked point is the triangulation boundary of the hole defining the puncture and on which the marked point has been placed.
• Figure 5: Triangulation of a cylinder and its dual graph. For each of the two circle boundaries of the cylinder, the dual graph has an open strand ending at a marked point. These circle boundaries and their marked point are the punctures of the two-punctured sphere.

Theorems & Definitions (16)

• Definition 4.1: Space $\mathcal{V}_{\Sigma_p}$ of graphs
• proof : Proof of \ref{['orthonormality of p-punctured Q']}
• proof : Proof of the sliding property \ref{['strand sliding']}
• proof : Proof of \ref{['Bp on face']} in the case $s=6$
• proof : Proof of \ref{['OjtoQ']}
• proof : Proof of \ref{['pulling over O']}
• proof : Proof of \ref{['QAlgebra']}
• proof : Proof of \ref{['OtoQ']}
• proof : Proof of \ref{['O times O']}
• proof : Proof of \ref{['orthonormality of Os']}