Vector bundles for "Matrix algebras converge to the sphere"

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.

46 pages, v3: many small improvements, v4: minor improvements following referee's suggestions

References in corpus (3)

• Noncommutative vector bundles over fuzzy CP^N and their covariant derivatives
• Matricial bridges for "Matrix algebras converge to the sphere"
• Monopoles, Dirac operator and index theory for fuzzy $SU(3)/(U(1)\times U(1))$

Cited by in corpus (3)

• On multimatrix models motivated by random noncommutative geometry II: A Yang-Mills-Higgs matrix model
• Dirac operators for matrix algebras converging to coadjoint orbits
• Quantum metrics on the tensor product of a commutative C*-algebra and an AF C*-algebra