  The U(N) framework and the spinor representation for loop quantum gravity are
two new points of view that can help us deal with the most fundamental problems
of the theory. Here, we review the detailed construction of the U(N) framework
explaining how one can endow the Hilbert space of N-leg intertwiners with a
Fock structure. We then give a description of the classical phase space
corresponding to this system in terms of the spinors, and we will study its
quantization using holomorphic techniques. We take special care in constructing
the usual holonomy operators of LQG in terms of spinors, and in the description
of the Hilbert space of LQG with the different polarization given by these
spinors.
