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

Abstract

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_𝖪ä𝗁 and 𝖿_𝗋𝖾 are constructed by using the quantum Hilbert spaces ℋ_Käh and ℋ_re, respectively. Moreover, the relationship between the two morphisms of operads 𝖿_𝖪ä𝗁 and 𝖿_𝗋𝖾 is studied and then the equality dim ℋ_Käh = dim ℋ_re is proved in general setting. This operadic framework is regarded as a development of the recurrence relation method by Kamiyama (2000) for proving dim ℋ_Käh = dim ℋ_re in a special case.