Path integral representation of spin foam models of 4D gravity

We give a unified description of all recent spin foam models introduced by Engle, Livine, Pereira and Rovelli (ELPR) and by Freidel and Krasnov (FK). We show that the FK models are, for all values of the Immirzi parameter γ, equivalent to path integrals of a discrete theory and we provide an explicit formula for the associated actions. We discuss the relation between the FK and ELPR models and also study the corresponding boundary states. For general Immirzi parameter, these are given by Alexandrov's and Livine's SO(4) projected states. For 0 ⩽ γ < 1, the states can be restricted to SU(2) spin networks.