Higher rank vector bundles in Fukaya category

Abstract

In this thesis, we study higher rank vector bundles in Fukaya category and its relation with twisted complex. We enlarge the class of objects of Fukaya category so that they contain not only flat line bundles, but also any finite rank flat vector bundles. For this purpose we use the de Rham version of Fukaya category and verify that it has some nice features, which other versions do not share.

We prove that every flat vector bundle on a Lagrangian submanifold can be realized as a twisted complex of flat line bundles when the fundamental group of the Lagrangian submanifold is abelian. Futhermore, we also prove that every flat vector bundle whose holonomy reprsentation is triangularizable is quasi-isomorphic to a certain twisted complex of flat line bundles.