Let \((M,\omega)\) be a symplectic manifold of real dimension \(2\times 3\) with \(c^1(M)=0\) in \(H^2(M;\mathbb Q)\). Let \(L\subset M\) be a relatively spin Lagrangian submanifold with its zero Maslov index homomorphism \(\mu_L:H_2(M,L;\mathbb Z)\to2\,\mathbb Z\).
In this paper, the author considers such a pair \((M,L)\) whose example is a Calabi-Yau \(3\)-fold \(M\) and its special Lagrangian submanifold \(L\). The main purpose is to define and study a special invariant of \((M,L)\) that is independent of the various choices involved in the construction and that depends on the almost contact structure \(J\) on \(M\). The way how it depends on \(J\) is described by a wall crossing formula which involves a moduli space of pseudo-holomorphic spheres. This invariant is a function on the set of the path connected components of bounding cochains that are solutions of the \(A_{\infty}\) version of the Maurer-Cartan equation of the filtered \(A_{\infty}\) algebra associated to the Lagrangian submanifold \(L\). In the case when \(L\) is a rational homology sphere, it becomes a numerical invariant.