The Symplectic Geometry of Cotangent Bundles from a Categorical Viewpoint

Abstract

We describe various approaches to understanding Fukaya categories of cotangent bundles. All of the approaches rely on introducing a suitable class of noncompact Lagrangian submanifolds. We review the work of Nadler-Zaslow [29, 30] and the authors [14], before discussing a new approach using family Floer cohomology [10] and the “wrapped Fukaya category”. The latter, inspired by Viterbo’s symplectic homology, emphasizes the connection to loop spaces, hence seems particularly suitable when trying to extend the existing theory beyond the simply connected case.