LAUSR

We revise the classical continuum formulation behind the Spin Foam approach to the quantization of gravity. Based on the recent applications of the current EPRL-FK model beyond triangulations, we identify the tension with the implementation of the `volume' part of simplicity constraints, required to finish the reduction from the topological BF theory to gravity. The crucial role played by 4d normals and the condition of their closure, in the linear version of constraints, necessitates the extension of the configuration space, which we supplement with an additional condition of vanishing torsion. We characterize fully the extended Poincar\'{e} BF theory both at the Lagrangian and Hamiltonian levels. The simplicity constraints are introduced naturally, in the spirit of Pleba\'{n}ski formulation, and we give their tetradic dual version to that of using 3-forms. This brings us much closer to the metric theory of General Relativity.