In Mirror Symmetry, specially in Homological Mirror Symmetry, the mirror of stable holomorphic Vector bundle is the special Lagrangian submanifold.  


In symplectic geometry 

A-brane = Lagrangian submanifold + flat vector bundle • 

In holomorphic geometry

 B-brane = complex submanifold + holomorphic bundle


There is a relation between reflexive Sheaf and Vector bundle which is used in the study of Extension theory for finding canonical metrics

A reflexive sheaf $ F$, on Kähler variety $X$ outside of codimension at least 3 is a holomorphic vector bundle,