Products of Floer Cohomology of Torus Fibers in Toric Fano Manifolds
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- Cho, CH. Commun. Math. Phys. (2005) 260: 613. doi:10.1007/s00220-005-1421-7
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We compute the ring structure of Floer cohomology groups of Lagrangian torus fibers in some toric Fano manifolds continuing the study of [CO]. Related A∞-formulas hold for a transversal choice of chains. Two different computations are provided: a direct calculation using the classification of holomorphic discs by Oh and the author in [CO], and another method by using an analogue of divisor equation in Gromov-Witten invariants to the case 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 , 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 the A∞(or L∞)-algebra of the Lagrangian submanifold.