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Fukaya Algebra over $\mathbb{Z}$

Mohamad Rabah

TL;DR

This work develops an integral version of Fukaya theory by constructing global Kuranishi charts and normally complex obstructions to define an integral Fukaya algebra on a Lagrangian $L$. The core strategy combines transversality for orbifold sections with global Kuranishi chart technology to model moduli spaces of genus-zero pseudo-holomorphic curves and bordered disks, enabling the definition of a curved, gapped, filtered $A_{n,K}$-algebra on the Morse complex over the integer Novikov ring and, under Bohr-Sommerfeld hypotheses, its extension to a genuine $A_ abla$-infinity structure. A Quantum Lefschetz principle is proven in Kahler and symplectic settings by relating virtual fundamental classes via equivariant Euler classes of normal bundles, providing integral virtual classes for local Gromov-Witten moduli spaces. The framework yields a robust pathway to integer-valued invariants in Floer theory and furnishes tools with potential applications to mirror symmetry and enumerative geometry.

Abstract

Given a closed, connected, relatively-spin Lagrangian submanifold in a closed symplectic manifold, we associate to it a curved, gapped, filtered, $A_{n, K}$-algebra over the Novikov ring with integer coefficients. Under certain conditions, such an algebra can be extended to an $A_\infty$-algebra. To illustrate our framework, we give a proof of the Quantum Lefschetz Hyperplane Theorem in the K$\ddot a$hler case, and associate virtual fundamental classes to the moduli spaces used in local Gromov-Witten theory, in the symplectic case.

Fukaya Algebra over $\mathbb{Z}$

TL;DR

This work develops an integral version of Fukaya theory by constructing global Kuranishi charts and normally complex obstructions to define an integral Fukaya algebra on a Lagrangian . The core strategy combines transversality for orbifold sections with global Kuranishi chart technology to model moduli spaces of genus-zero pseudo-holomorphic curves and bordered disks, enabling the definition of a curved, gapped, filtered -algebra on the Morse complex over the integer Novikov ring and, under Bohr-Sommerfeld hypotheses, its extension to a genuine -infinity structure. A Quantum Lefschetz principle is proven in Kahler and symplectic settings by relating virtual fundamental classes via equivariant Euler classes of normal bundles, providing integral virtual classes for local Gromov-Witten moduli spaces. The framework yields a robust pathway to integer-valued invariants in Floer theory and furnishes tools with potential applications to mirror symmetry and enumerative geometry.

Abstract

Given a closed, connected, relatively-spin Lagrangian submanifold in a closed symplectic manifold, we associate to it a curved, gapped, filtered, -algebra over the Novikov ring with integer coefficients. Under certain conditions, such an algebra can be extended to an -algebra. To illustrate our framework, we give a proof of the Quantum Lefschetz Hyperplane Theorem in the Khler case, and associate virtual fundamental classes to the moduli spaces used in local Gromov-Witten theory, in the symplectic case.

Paper Structure

This paper contains 38 sections, 73 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Transversality of Orbifold Sections
  3. Global Kuranishi Charts
  4. Organization
  5. An Overview on Global Kuranishi Charts for the Moduli Space of Pseudo-holomorphic Spheres
  6. Construction
  7. Quantum Lefschetz Principle
  8. $L$ is Concave
  9. $L$ is Convex
  10. Global Kuranishi Charts for the Moduli Space of Pseudo-holomorphic Disks
  11. Preliminary Constructions
  12. Space of Framings
  13. Proof of Theorem \ref{['ChartsDisks']}
  14. Using abouzaid2021complex, bai2022arnold
  15. Gromov's Graph Trick
  16. ...and 23 more sections

Key Result

Theorem 1.1

To each closed, connected, relatively-spin Lagrangian submanifold $L$ of $X$ we can associate an $A_\infty$-algebra structure $\{m_k\}_{k\geq 0}$ on $H^*(L;\Lambda^\mathbb{Q}_{0, nov})$, which is well-defined up to isomorphism.

Theorems & Definitions (244)

  • Theorem 1.1: Fukaya2009
  • Theorem 1.2
  • Definition 1.3: Normally Complex Orbibundle
  • Definition 1.4: Normally Complex Polynomial Section
  • Definition 1.5: Global Kuranishi Chart
  • Example 1.6
  • Definition 1.7: Oriented Global Kuranishi Chart
  • Definition 1.8: Normally Complex Global Kuranishi Chart
  • Theorem 2.1
  • Lemma 2.2
  • ...and 234 more