Moduli spaces of polygons have been studied since the nineties for their
topological and symplectic properties. Under generic assumptions, these are
symplectic manifolds with natural global action-angle coordinates. This paper
is concerned with the quantization of these manifolds and of their action
coordinates. Applying the geometric quantization procedure, one is lead to
consider invariant subspaces of a tensor product of irreducible representations
of SU(2). These quantum spaces admit natural sets of commuting observables. We
prove that these operators form a semi-classical integrable system, in the
sense that they are Toeplitz operators with principal symbol the square of the
action coordinates. As a consequence, the quantum spaces admit bases whose
vectors concentrate on the Lagrangian submanifolds of constant action. The
coefficients of the change of basis matrices can be estimated in terms of
geometric quantities. We recover this way the already known asymptotics of the
classical 6j-symbols
