Spinfoam 2D quantum gravity and discrete bundles

In four dimensions (4D), general relativity (GR) can be formulated as a constrained BF theory; we show that the same is true in 2D. We describe a spinfoam quantization of this constrained BF formulation of 2D Riemannian GR, obtained using the Barrett-Crane technique of imposing the constraint as a restriction on the representations summed over. We obtain the expected partition function, thus providing support for the viability of the technique. The result requires the nontrivial topology of the bundle where the gravitational connection is defined to be taken into account. For this purpose, we study the definition of a principal bundle over a simplicial base space. The model also sheds light on several other features of spinfoam quantum gravity: the reality of the partition function, the geometrical interpretation of the Newton constant and the issue of possible finiteness of the partition function of quantum GR.