Vector bundles for Matrix algebras converge to the sphere

[摘要] 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. (C) 2018 Elsevier B.V. All rights reserved.

[发布日期] 2018-10-01

[效力级别]  Proceedings Paper

[关键词] C*-metric space;Quantum Gromov-Hausdorff distance;Vector bundles;Projective modules;Sphere;Matrix algebras