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U(N) framework and simplicity constraints for spin foam models

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In the context of loop quantum gravity and spin foam models, the simplicity constraints are essential in that they allow to write general relativity as a constrained topological theory.
I will first recall the spin foam quantization procedure and focus more particularly on the step consisting in implementing the simplicity constraints.
Then, I will present the U(N) framework initially developed for SU(2) intertwiners. Finally, I will show how we can apply this new framework to impose the simplicity constraints in the context of 4d Euclidean gravity and present new solutions defined in term of U(N) coherent states.