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Tohoku Mathematical Journal
2011
December
SECOND SERIES VOL. 63, NO. 4
Tohoku Math. J.
63 (2011), 697727

Title
COUNTING PSEUDOHOLOMORPHIC DISCS IN CALABIYAU 3HOLDS
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 MaurerCartan 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 pseudoholomorphic spheres.
2000 Mathematics Subject Classification.
Primary 57R17; Secondary 81T30.
Key words and phrases.
Symplectic geometry, Lagrangian submanifold, Floer homology, CalabiYau manifold, $A_{\infty}$ algebra, superpotential, mirror symmetry.


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