Classical limit of the d-bar operators on quantum domains

TL;DR

This paper investigates non-commutative analogs of the d-bar operator on quantum domains, demonstrating their convergence to classical operators in the limit as the deformation parameter approaches zero.

Contribution

It introduces a framework for analyzing families of non-commutative d-bar operators on quantum disks and annuli, establishing their classical limit behavior.

Findings

Operators converge to classical d-bar operators as t approaches 0

Inverses of operators form morphisms of continuous fields of Hilbert spaces

Framework for quantum-to-classical transition in non-commutative geometry

Abstract

We study one parameter families $D_t$, $0<t<1$ of non-commutative analogs of the d-bar operator $D_0 = \frac{\d}{\d\bar{z}}$ on disks and annuli in complex plane and show that, under suitable conditions, they converge in the classical limit to their commutative counterpart. More precisely, we endow the corresponding families of Hilbert spaces with the structures of continuous fields over the interval $[0,1)$ and we show that the inverses of the operators $D_t$ subject to APS boundary conditions form morphisms of those continuous fields of Hilbert spaces.