Spin foam models and the Duflo map

We give a general definition of spin foam models, and then of models of 4d quantum gravity based on constraining BF theory. We highlight the construction and quantization ambiguities entering model building, among which the choice of quantization map applied to the B variables carrying metric information after imposing simplicity constraints, and the different strategies for imposing the latter constraints. We then construct a new spin foam model for 4d quantum gravity, using the flux representation of states and amplitudes, based on the Duflo quantization map and the associated non-commutative Fourier transform for Lie groups. The advantages of the new model are the geometrically transparent way in which constraints are imposed, and the underlying mathematical properties of the Duflo map itself. Finally, the presence of a closed analytical formula for the model's amplitudes is another valuable asset for future applications. (paper)