Marc Rieffel
(University of California at Berkeley)
October 29, 2021
Organized by Americas GNCG Seminar.

Dirac Operators for Matrix Algebras Converging to Coadjoint Orbits

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 C*-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 like coadjoint orbits), such as vector bundles, Yang-Mills functionals, Dirac operators, etc., and they want to approximate these by corresponding structures on matrix algebras. 

I will sketch a somewhat unified construction of Dirac operators on coadjoint orbits and on the matrix algebras that converge to them. As Connes showed us, from Dirac operators we may obtain C*-metrics. Our unified construction enables us to prove our main theorem, whose content is that, for the C*-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. This is a long story, but I will sketch how it works.

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