Operad structures in geometric quantization of the moduli space of spatial polygons

The moduli space of spatial polygons is known as a symplectic manifold equipped with both Kähler and real polarizations. In this paper, associated to the Kähler and real polarizations, morphisms of operads f K ¥ ah and f re are constructed by using the quantum Hilbert spaces ℋ K ¥ ah and ℋ re , respectively. Moreover, the relationship between the two morphisms of operads f K ¥ ah and f re is studied and then the equality dim ℋ K ¥ ah = dim ℋ re is proved in general setting. This operadic framework is regarded as a development of the recurrence relation method by Kamiyama[6] for proving dim ℋ K ¥ ah = dim ℋ re in a special case.