On Fukaya categories and prequantization bundles

TL;DR

This paper establishes a method to compute Floer homology of nonexact rational Lagrangians in symplectic manifolds using exact Lagrangians in their prequantization bundles, linking Floer theory with sheaf theory.

Contribution

It introduces a Fukaya-sheaf correspondence for rational Lagrangians in Weinstein manifolds and demonstrates the natural transformation of bounding cochains under Legendrian isotopy.

Findings

Floer homology of nonexact Lagrangians can be computed via exact fillings.

Fukaya-sheaf correspondence for rational Lagrangians in Weinstein manifolds.

Quantum cohomology of the complex projective line derived from sheaf-theoretic calculations.

Abstract

We show: the Floer homology over the Novikov ring of (nonexact!) rational Lagrangians in an (nonexact!) integral symplectic manifold can be computed in terms of exact Lagrangians in an exact filling of the prequantization bundle. As a consequence, we give a Fukaya-sheaf correspondence for rational (nonexact!) Lagrangians in Weinstein manifolds, as conjectured by Ike and the first-named author. We also show that bounding cochains for immersed rational Lagrangians transform naturally under Legendrian isotopy, as conjectured by Akaho and Joyce. As an illustration, we show that quantum cohomology of the complex projective line -- which requires the counting of one holomorphic sphere -- can be recovered from purely sheaf-theoretic calculations.