Mathematics – Symplectic Geometry
Scientific paper
2004-12-20
Mathematics
Symplectic Geometry
26 pages, 3 fig; Statements about abstract perturbation and invariance withdrawn. Additional assumption on results about ring
Scientific paper
10.1007/s00220-005-1421-7
We compute the ring structure of Floer cohomology groups of Lagrangian torus fibers in some toric Fano manifolds continuing the study of \cite{CO}. Related $\AI$-formulas hold for transversal choice of chains. Two different computations are provided: a direct calculation using the classification of holomorphic discs by Oh and the author in \cite{CO}, and another method by using an {\it analogue of divisor equation} in Gromov-Witten invariants to thecase of discs. Floer cohomology rings are shown to be isomorphic to Clifford algebras, whose quadratic forms are given by the Hessians of functions $W$, which turn out to be the superpotentials of Landau-Ginzburg mirrors. In the case of $\CP^n$ and $ \CP^1 \times \CP^1$, this proves the prediction made by Hori, Kapustin and Li by B-model calculations via physical arguments. The latter method also provides correspondence between higher derivatives of the superpotential of LG mirror with the higher products of $\AI$(or $\LI$)-algebra of the Lagrangian submanifold.
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