## Tohoku Mathematical Journal 2011 December SECOND SERIES VOL. 63, NO. 4

 Tohoku Math. J. 63 (2011), 697-727

Title COUNTING PSEUDO-HOLOMORPHIC DISCS IN CALABI-YAU 3-HOLDS

Author Kenji Fukaya

(Received October 14, 2009, revised August 18, 2010)
Abstract. In this paper we define an invariant of a pair of a 6 dimensional symplectic manifold with vanishing 1st Chern class and its relatively spin Lagrangian submanifold with vanishing Maslov index. This invariant is a function on the set of the path connected components of bounding cochains (solutions of the $A_{\infty}$ version of the Maurer-Cartan equation of the filtered $A_{\infty}$ algebra associated to the Lagrangian submanifold). In the case when the Lagrangian submanifold is a rational homology sphere, it becomes a numerical invariant.
This invariant depends on the choice of almost complex structures. The way how it depends on the almost complex structures is described by a wall crossing formula which involves a moduli space of pseudo-holomorphic spheres.

2000 Mathematics Subject Classification. Primary 57R17; Secondary 81T30.

Key words and phrases. Symplectic geometry, Lagrangian submanifold, Floer homology, Calabi-Yau manifold, $A_{\infty}$ algebra, superpotential, mirror symmetry.