Vector bundles for "Matrix algebras converge to the sphere"

TL;DR

This paper develops a framework for understanding how vector bundles on quantum metric spaces, like matrix algebras, can be approximated by corresponding bundles on classical spaces such as the sphere, with explicit examples.

Contribution

It introduces a method to relate vector bundles on close quantum metric spaces, exemplified by matrix algebras converging to the 2-sphere.

Findings

Established a correspondence between vector bundles on close quantum metric spaces.

Explicitly demonstrated the convergence of vector bundles from matrix algebras to the sphere.

Provided a general framework for approximating geometric structures on quantum spaces.

Abstract

In the high-energy quantum-physics literature one finds statements such as "matrix algebras converge to the sphere". Earlier I provided a general precise setting for understanding such statements, in which the matrix algebras are viewed as quantum metric spaces, and convergence is with respect to a quantum Gromov-Hausdorff-type distance. But physicists want even more to treat structures on spheres (and other spaces), such as vector bundles, Yang-Mills functionals, Dirac operators, etc., and they want to approximate these by corresponding structures on matrix algebras. In the present paper we treat this idea for vector bundles. We develop a general precise way for understanding how, for two compact quantum metric spaces that are close together, to a given vector bundle on one of them there can correspond in a natural way a unique vector bundle on the other. We then show explicitly how this works for the case of matrix algebras converging to the 2-sphere.