All 3-edge-connected relativistic BC and EPRL spin-networks are integrable

Abstract

We prove statement conjectured in [Baez and Barrett:2001] that every 3-edge-connected SL(2,C) spin-network with invariants of certain class is integrable. It means that the regularized evaluation (defined by a suitable integral) of such a spin-network is finite. Our proof is quite general. It is valid for relativistic spin-networks of Barrett and Crane as well as for spin-networks with the Engle-Pereira-Rovelli-Livine intertwiners and for some generalization of both. The result interesting from the group representation point of view opens also a possibility of defining vertex amplitudes for Spin-Foam models based on non-simplicial decompositions.