Coherent states and geometry on the Siegel-Jacobi disk

TL;DR

This paper explores the geometric and analytical properties of the Siegel-Jacobi disk using coherent states, including curvature, geodesics, and various kernels, highlighting their significance in the context of the Jacobi group.

Contribution

It provides a detailed geometric analysis of the Siegel-Jacobi disk and connects coherent state representations with fundamental conjectures and kernel functions.

Findings

Ricci form and scalar curvature of the Siegel-Jacobi disk are computed.

Geodesics on the Siegel-Jacobi disk are characterized.

The paper discusses the Berezin kernel, Calabi's diastasis, and other fundamental functions.

Abstract

The coherent state representation of the Jacobi group $G^J_1$ is indexed with two parameters, $\mu (=\frac{1}{\hbar})$, describing the part coming from the Heisenberg group, and $k$, characterizing the positive discrete series representation of $\text{SU}(1,1)$. The Ricci form, the scalar curvature and the geodesics of the Siegel-Jacobi disk $\mathcal{D}^J_1$ are investigated. The significance in the language of coherent states of the transform which realizes the fundamental conjecture on the Siegel-Jacobi disk is emphasized. The Berezin kernel, Calabi's diastasis, the Kobayashi embedding, and the Cauchy formula for the Sigel-Jacobi disk are presented.