Quantum gravity kinematics from extended TQFTs

Abstract

In this paper, 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. This vacuum state can be thought of as being peaked on connections with homogeneous curvature. 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. These curvature and torsion excitations are classified by the Drinfeld 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 operators which 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 -dimensional gravity with 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 basis which 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 extended TQFTs, as well as techniques from condensed matter, can help design new representations, 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 quantum gravity, and will enable the study of possible phases of spin foam models and group field theories from a new perspective.