Vector Bundles and Connections in Physics and Mathematics: Some Historical Remarks
The concept of smooth vector bundles on manifolds and the theory of connections on them has played a fundamental role in physics going back to the discovery of Hermann Weyl around 1918 that electromagnet ism can be viewed as a connection on a real line bundle on spacetime. Under the influence of quantum mechanics this idea was modified and the real line bundle was replaced by a hermitian complex line bundle, allowing the discovery, by Dirac, of magnetic monopoles. In mathematics the idea of holomorphic vector bundles goes back to the work of Riemann on multivalued solutions of regular singular differential equations like the hyper geometric equation. This paper examines briefly these historical origins, discusses the evolution of these two streams of thoughts, leading to the surprising climax in which they merge in the theory of moduli spaces.
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