Coherent states and geometry on the Siegel–Jacobi disk

Abstract

The coherent state representation of the Jacobi group [Formula: see text] is indexed with two parameters, [Formula: see text], describing the part coming from the Heisenberg group, and k, characterizing the positive discrete series representation of SU(1,1). The Ricci form, the scalar curvature and the geodesics of the Siegel–Jacobi disk [Formula: see text] 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 Siegel–Jacobi disk are presented.