Vector bundles for Matrix algebras converge to the sphere

JOURNAL OF GEOMETRY AND PHYSICS

卷:132

Article; Proceedings Paper

Rieffel, Marc A.¹ 

[1] Univ Calif Berkeley, Dept Math, Berkeley, CA 94720 USA

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

DOI  :  10.1016/j.geomphys.2018.06.003

来源: Elsevier

【 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.

【 授权许可】

Free