Dirac operators for matrix algebras converging to coadjoint orbits

TL;DR

This paper constructs Dirac operators on coadjoint orbits and their matrix algebra approximations, demonstrating convergence of these noncommutative spaces to classical geometric spaces using quantum Gromov-Hausdorff distance.

Contribution

It provides a unified method to define Dirac operators on coadjoint orbits and matrix algebras, establishing their convergence in a strong quantum metric sense.

Findings

Matrix algebras converge to coadjoint orbits in quantum Gromov-Hausdorff distance

Construction of Dirac operators on both coadjoint orbits and matrix algebras

Verification of convergence for structures involving vector bundles and Yang-Mills functionals

Abstract

In the high-energy 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 provide a somewhat unified construction of Dirac operators on coadjoint orbits and on the matrix algebras that converge to them. This enables us to prove our main theorem, whose content is that, for the quantum metric-space structures determined by the Dirac operators that we construct, the matrix algebras do indeed converge to the coadjoint orbits, for a quite strong version of quantum Gromov-Hausdorff distance.