  Spin foam models are the path integral counterparts to loop quantized
canonical theories. In the last few years several spin foam models of gravity
have been proposed, most of which live on finite simplicial lattice spacetime.
The lattice truncates the presumably infinite set of gravitational degrees of
freedom down to a finite set. Models that can accomodate an infinite set of
degrees of freedom and that are independent of any background simplicial
structure, or indeed any a priori spacetime topology, can be obtained from the
lattice models by summing them over all lattice spacetimes. Here we show that
this sum can be realized as the sum over Feynman diagrams of a quantum field
theory living on a suitable group manifold, with each Feynman diagram defining
a particular lattice spacetime. We give an explicit formula for the action of
the field theory corresponding to any given spin foam model in a wide class
which includes several gravity models. Such a field theory was recently found
for a particular gravity model [De Pietri et al, hep-th/9907154]. Our work
generalizes this result as well as Boulatov's and Ooguri's models of three and
four dimensional topological field theories, and ultimately the old matrix
models of two dimensional systems with dynamical topology. A first version of
our result has appeared in a companion paper [gr-qc\0002083]: here we present a
new and more detailed derivation based on the connection formulation of the
spin foam models.
