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On Fukaya categories and prequantization bundles

Tatsuki Kuwagaki, Adrian Petr, Vivek Shende

TL;DR

This work develops a comprehensive bridge between Lagrangian Floer theory in nonexact, compact symplectic manifolds and exact data living on prequantization fillings. By organizing the algebraic and geometric information through pogs, orbit categories, and completion techniques, it expresses Fukaya categories of rational, nonexact Lagrangians in terms of Novikov-augmented data derived from exact fillings and Legendrian invariants. A central achievement is the Fukaya–sheaf correspondence, which relates Fukaya categories to microsheaves and to augmentation categories, and is compatible with wrapped/partially wrapped theories and with Legendrian isotopy of bounding cochains. The paper culminates with concrete computations, notably recovering the quantum cohomology of $\mathbb{C}P^1$ from purely sheaf-theoretic data, highlighting the potential of this framework to compute Floer-theoretic invariants via microlocal and algebraic methods.

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.

On Fukaya categories and prequantization bundles

TL;DR

This work develops a comprehensive bridge between Lagrangian Floer theory in nonexact, compact symplectic manifolds and exact data living on prequantization fillings. By organizing the algebraic and geometric information through pogs, orbit categories, and completion techniques, it expresses Fukaya categories of rational, nonexact Lagrangians in terms of Novikov-augmented data derived from exact fillings and Legendrian invariants. A central achievement is the Fukaya–sheaf correspondence, which relates Fukaya categories to microsheaves and to augmentation categories, and is compatible with wrapped/partially wrapped theories and with Legendrian isotopy of bounding cochains. The paper culminates with concrete computations, notably recovering the quantum cohomology of from purely sheaf-theoretic data, highlighting the potential of this framework to compute Floer-theoretic invariants via microlocal and algebraic methods.

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.
Paper Structure (36 sections, 67 theorems, 137 equations, 8 figures)

This paper contains 36 sections, 67 theorems, 137 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Partially ordered groups
  3. Modules over posets and pogs
  4. Orbit categories for pogs
  5. Change of enrichment and reconstruction
  6. Dense inclusion of pogs
  7. Completion
  8. Almost stuff
  9. Curved $\operatorname{A_{\infty}}$-categories and quotients
  10. Basic definitions
  11. Twisted complexes and bounding cochains
  12. Quotient of $\operatorname{cA_{\infty}}$-categories
  13. Localization
  14. Elementary symplectic geometry of prequantization bundles
  15. Legendrian invariants
  16. ...and 21 more sections

Key Result

Theorem 1.1

Assume that the action spectrum of $\Lambda$ is contained in $\mathbb{Z} \cup (\mathbb{R} \setminus \mathbb{Q})$. Then the Reeb flow contactomorphisms and continuation maps determine an action of $\frac{1}{n} \vec{\mathbb{Z}} / \mathbb{Z}$ on $\operatorname{Mod^{fd}}(\mathcal{A}_{\frac{1}{n} \mathbb

Figures (8)

  • Figure 1: Quilted strip $(w_-, u_+)$ contributing to $\langle \mu_{\mathcal{B}} (\gamma_p, \dots, \gamma_1, \operatorname{in}, x_1, \dots, x_d), t^c \otimes \operatorname{out} \rangle$, where $c = \int w_-^* \Omega + \int u_+^* \omega$.
  • Figure 2: Identification of the curves used to define $\mathcal{B}_G(S\Lambda_j^{b-\eta}(n), -)$ (on the left) with some quilted strips (on the right)
  • Figure 3: Identification of the curves used to define $\mathcal{D}_G^{op}(\Lambda_j^b, -) = \mathcal{D}_G(- ,\Lambda_j^b)$ (on the right) with some quilted strips (on the left)
  • Figure 4: Morse-Bott degeneration (on the right) of the curves used to define the map $\mu_{\mathcal{B}_G} (\gamma_j^n, -) : \mathcal{B}_G(S\Lambda_j^{b-\eta}(n), -) \to \mathcal{B}_G(S\Lambda_j^{b-\eta}(n+1), -)$ (on the left)
  • Figure 5: Curves contributing to $[\mu_{\mathcal{B}_G} (\gamma, -)]$ if $\gamma$ is non-degenerate (on the left) or degenerate (on the right) as an $\alpha^{\circ}$-Reeb chord
  • ...and 3 more figures

Theorems & Definitions (188)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Corollary 1.5
  • Remark 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Remark 1.9: Hitchin fibration
  • Example 2.1
  • ...and 178 more