Fukaya Algebra over $\mathbb{Z}$

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 $A_{\mathrm{\infty }}$-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.