Quantization of compact Riemannian symmetric spaces

TL;DR

This paper investigates the geometric quantization of compact Riemannian symmetric spaces, revealing that projective flatness of the associated quantum Hilbert space field characterizes spaces isometric to compact Lie groups with biinvariant metrics.

Contribution

It establishes a link between projective flatness of quantum Hilbert space fields and the symmetric space being a compact Lie group with a biinvariant metric.

Findings

Projective flatness implies the symmetric space is a compact Lie group with biinvariant metric.

Flatness of the quantum Hilbert space field was previously known for compact Lie groups.

The paper characterizes symmetric spaces based on the flatness property of their quantization fields.

Abstract

The phase space of a compact, irreducible, simply connected, Riemannian symmetric space admits a natural family of K\"ahler polarizations parametrized by the upper half plane $S$. Using this family, geometric quantization, including the half-form correction, produces the field $H^{corr}\rightarrow S$ of quantum Hilbert spaces. We show that projective flatness of $H^{corr}$ implies, that the symmetric space must be isometric to a compact Lie group equipped with a biinvariant metric. In the latter case the flatness of $H^{corr}$ was previously established.