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 K\"ahler and real polarizations. In this paper, associated to the K\"ahler and real polarizations, morphisms of operads $\mathsf{f}_{\mathsf{K}\ddot{\mathsf{a}}\mathsf{h}}$ and $\mathsf{f}_{\mathsf{re}}$ are constructed by using the quantum Hilbert spaces $\mathscr{H}_{\mathrm{K}\ddot{\mathrm{a}}\mathrm{h}}$ and $\mathscr{H}_\mathrm{re}$, respectively. Moreover, the relationship between the two morphisms of operads $\mathsf{f}_{\mathsf{K}\ddot{\mathsf{a}}\mathsf{h}}$ and $\mathsf{f}_{\mathsf{re}}$ is studied and then the equality $\dim \mathscr{H} _{\mathrm{K}\ddot{\mathrm{a}}\mathrm{h}}=\dim \mathscr{H}_\mathrm{re}$ is proved in general setting. This operadic framework is regarded as a development of the recurrence relation method by Kamiyama for proving $\dim \mathscr{H}_{\mathrm{K}\ddot{\mathrm{a}}\mathrm{h}}=\dim \mathscr{H}_\mathrm{re}$ in a special case.