Fukaya Algebra over $\mathbb{Z}$

TL;DR

This paper constructs a Fukaya algebra over the integers for Lagrangian submanifolds in symplectic manifolds, extending to $A_$-algebras and applying it to prove the Quantum Lefschetz Hyperplane Theorem and define virtual classes.

Contribution

It introduces a new integral coefficient Fukaya algebra framework and demonstrates its applications in symplectic geometry and Gromov-Witten theory.

Findings

Constructed a curved, filtered $A_{n,K}$-algebra over $Z$

Extended algebra to an $A_$-algebra under certain conditions

Applied framework to prove Quantum Lefschetz Hyperplane Theorem

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.