Non-abelian analogues of Abel's theorem

Vafa [29] extended a version of mirror symmetry to pairs consisting of a Calabi–Yau manifold and a fixed vector bundle on it. In [30] he considered the mathematical meaning of this extension. In this paper we prove the main facts concerning the geometry of vector bundles on Calabi–Yau manifolds and describe all constructions that enable us to embed them in the general context of modern physical concepts.