  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.
