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arXiv · 1711.04054

Vector bundles for "Matrix algebras converge to the sphere"

Marc A. Rieffel
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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.

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BibTeXRIS

Marc A. Rieffel. 2017-11-11. Vector bundles for "Matrix algebras converge to the sphere". https://doi.org/10.1016/j.geomphys.2018.06.003

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