Spacetime as a Feynman diagram: the connection formulation

Spin foam models are the path integral counterparts to loop quantizedcanonical theories. In the last few years several spin foam models of gravityhave been proposed, most of which live on finite simplicial lattice spacetime.The lattice truncates the presumably infinite set of gravitational degrees offreedom down to a finite set. Models that can accomodate an infinite set ofdegrees of freedom and that are independent of any background simplicialstructure, or indeed any a priori spacetime topology, can be obtained from thelattice models by summing them over all lattice spacetimes. Here we show thatthis sum can be realized as the sum over Feynmann diagrams of a quantum fieldtheory living on a suitable group manifold, with each Feynmann diagram defininga particular lattice spacetime. We give an explicit formula for the action ofthe field theory corresponding to any given spin foam model in a wide classwhich includes several gravity models. Such a field theory was recently foundfor a particular gravity model De Pietri et al, hep-th\\9907154. Our workgeneralizes this result as well as Boulatov's and Ooguri's models of three andfour dimensional topological field theories, and ultimately the old matrixmodels of two dimensional systems with dynamical topology. A first version ofour result has appeared in a companion paper gr-qc\\0002083: here we present anew and more detailed derivation based on the connection formulation of thespin foam models.