Spin-Foams for All Loop Quantum Gravity

The simplicial framework of Engle-Pereira-Rovelli-Livine spin-foam models is generalized to match the diffeomorphism invariant framework of loop quantum gravity. The simplicial spin-foams are generalized to arbitrary linear 2-cell spin-foams. The resulting framework admits all the spin-network states of loop quantum gravity, not only those defined by triangulations (or cubulations). In particular the notion of embedded spin-foam we use allows to consider knotting or linking spin-foam histories. Also the main tools as the vertex structure and the vertex amplitude are naturally generalized to arbitrary valency case. The correspondence between all the SU(2) intertwiners and the SU(2)$\times$SU(2) EPRL intertwiners is proved to be 1-1 in the case of the Barbero-Immirzi parameter $|\gamma|\ge 1$, unless the co-domain of the EPRL map is trivial and the domain is non-trivial.