Operad Structures in Geometric Quantization of the Moduli Space of Spatial Polygons

TL;DR

This paper constructs operad morphisms related to the geometric quantization of the moduli space of spatial polygons and proves their equivalence in dimension, advancing the understanding of quantization in symplectic geometry.

Contribution

It introduces operad morphisms for Kähler and real polarizations and establishes their relationship, generalizing previous recurrence relation methods.

Findings

Constructed operad morphisms for both polarizations.

Proved the equality of quantum Hilbert space dimensions.

Extended recurrence relation methods to a general setting.

Abstract

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.