A Convergent Star Product on the Poincar\'e Disc

TL;DR

This paper constructs a convergent star product of Wick type on the Poincaré disc, defining a Fréchet algebra of real-analytic functions with continuous multiplication, and explores its holomorphic dependence and representations.

Contribution

It introduces a topology making the canonical star product on the Poincaré disc convergent and characterizes the resulting algebra explicitly.

Findings

The star product converges as a series on a specific function class.

The algebra is isomorphic to holomorphic functions on an extended doubled disc.

Holomorphic dependence and GNS representations are analyzed.

Abstract

On the Poincar\'e disc and its higher-dimensional analogs one has a canonical formal star product of Wick type. We define a locally convex topology on a certain class of real-analytic functions on the disc for which the star product is continuous and converges as a series. The resulting Fr\'echet algebra is characterized explicitly in terms of the set of all holomorphic functions on an extended and doubled disc of twice the dimension endowed with the natural topology of locally uniform convergence. We discuss the holomorphic dependence on the deformation parameter and the positive functionals and their GNS representations of the resulting Fr\'echet algebra.