Quantum gravity kinematics from extended TQFTs

In this paper, we show howextended topological quantum field theories (TQFTs) canbeused toobtaina kinematical setup for quantum gravity, i. e. a kinematical Hilbert space togetherwith a representation of the observable algebra including operators of quantumgeometry. 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 vacuumstate underlying our representation is defined by the Turaev-ViroTQFT. This vacuum state can be thought of as being peaked on connections withhomogeneous curvature. Wetherefore construct here a generalization, ormore precisely a quantumdeformation at root of unity, of the previously introduced SU(2) BF representation. The extended Turaev-ViroTQFT provides a description of the excitations on top of the vacuum. These curvature and torsion excitations are classified by theDrinfeld center category of the quantum deformation of SU(2), and are essential in order to allow for a representation of the holonomies and fluxes. The holonomies and fluxes are generalized to ribbon operatorswhich create and interact with the excitations. 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 gravitywith a cosmological constant. The new representation constructed here presents a number of advantages over the representations which exist so far. In particular, it possesses a very useful finiteness property which guarantees the discreteness of spectra for a wide class of quantum (intrinsic and extrinsic) geometrical operators. Also, the notion of basic excitations leads to a so-called fusion basiswhich offers exciting possibilities for the construction of states with interesting global properties, as well as states with certain stability properties under coarse graining. In addition, the work presented here showcases how the framework of extendedTQFTs, aswell as techniques from condensedmatter, can help design newrepresentations, and construct and understand the associated notion of basic excitations. This is essential in order to find the best starting point for the construction of the dynamics of quantumgravity, andwill enable the study of possible phases of spin foammodels and group field theories from a new perspective.