Matrix algebras converge to the sphere for quantum Gromov--Hausdorff distance

TL;DR

This paper rigorously demonstrates that matrix algebras converge to the 2-sphere in the quantum Gromov-Hausdorff sense, using Berezin quantization and coherent states within the framework of compact Lie groups.

Contribution

It formalizes the convergence of matrix algebras to the sphere as quantum metric spaces through Berezin quantization, extending to integral coadjoint orbits.

Findings

Matrix algebras converge to the 2-sphere in quantum Gromov-Hausdorff distance

Uses Berezin quantization with coherent states for rigorous proof

Applicable to integral coadjoint orbits of compact Lie groups

Abstract

On looking at the literature associated with string theory one finds statements that a sequence of matrix algebras converges to the 2-sphere (or to other spaces). There is often careful bookkeeping with lengths, which suggests that one is dealing with ``quantum metric spaces''. We show how to make these ideas precise by means of Berezin quantization using coherent states. We work in the general setting of integral coadjoint orbits for compact Lie groups.