On Fukaya categories and prequantization bundles

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.

💡 Research Summary

The paper develops a novel bridge between Lagrangian Floer theory for non‑exact (rational) Lagrangians in an integral symplectic manifold B and exact Lagrangian Floer theory in the filling W of its prequantization bundle V→B. The authors work under monotonicity assumptions on B (minimal Chern number ≠ 1) and exploit the contact geometry of the prequantization bundle to translate Floer data into the language of Chekanov–Eliashberg differential graded algebras (DGAs) associated to Legendrian lifts Λ⊂V.

A key algebraic device is the use of partially ordered groups (POGs) such as (ℝ,≤) and its sub‑POGs ℚ, ℤ. By viewing these as monoidal categories, the authors define module categories over the corresponding monoid rings Z