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Global differential geometry --- Géométrie différentielle globale --- Symplectic manifolds --- Variétés symplectiques

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Symplectic manifolds. --- Topology. --- Topologie. --- Variétés symplectiques. --- Topology --- Géometrie différentielle globale --- Géometrie symplectique

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Global differential geometry --- Géométrie différentielle globale --- Symplectic manifolds --- Variétés symplectiques --- Géométrie algébrique --- Géométrie algébrique --- Géométrie différentielle globale --- Variétés symplectiques

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"We construct a global B-model for any quasi-homogeneous polynomial f that has properties similar to the properties of the physic's B-model on a Calabi-Yau manifold. The main ingredients in our construction are K. Saito's theory of primitive forms and Givental's higher genus reconstruction. More precisely, we consider the moduli space M[unfilled bullet]mar of the so-called marginal deformations of f. For each point [sigma] [is an element of] M[unfilled bullet]mar we introduce the notion of an opposite subspace in the twisted de Rham cohomology of the corresponding singularity f[sigma] and prove that opposite subspaces are in one-to-one correspondence with the splittings of the Hodge structure in the vanishing cohomology of f[sigma]. Therefore, according to M. Saito, an opposite subspace gives rise to a semi-simple Frobenius structure on the space of miniversal deformations of f[sigma]. Using Givental's higher genus reconstruction we define a total ancestor potential A[sigma](h,q) wh ose properties can be described quite elegantly in terms of the properties of the corresponding opposite subspace. For example, if the opposite subspace corresponds to the splitting of the Hodge structure given by complex conjugation, then the total ancestor potential is monodromy invariant and it satisfies the BCOV holomorphic anomaly equations. The coefficients of the total ancestor potential could be viewed as quasi-modular forms on M[unfilled bullet]mar in a certain generalized sense. As an application of our construction, we consider the case of a Fermat polynomial W that defines a Calabi-Yau hypersurface XW in a weighted-projective space. We have constructed two opposite subspaces and proved that the corresponding total ancestor potentials can be identified with respectively the total ancestor potential of the orbifold quotient XW/G̃W and the total ancestor potential of FJRW invariants corresponding to (W,GW). Here GW is the maximal group of diagonal symmetries of W and GW is a quotient of GW by the subgroup of those elements that act trivially on XW . In particular, our result establishes the so-called Landau-Ginzburg/Calabi-Yau correspondence for the pair (W,GW)"--

Gromov-Witten invariants. --- Calabi-Yau manifolds. --- Symplectic manifolds. --- Homology theory. --- Invariants de Gromov-Witten --- Calabi-Yau, Variétés de --- Variétés symplectiques --- Homologie

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Differential geometry. Global analysis --- Symplectic manifolds --- Homology theory --- Moduli theory --- Variétés symplectiques --- Homologie --- Variétés topologiques à 4 dimensions --- Symplectic manifolds. --- Moduli theory. --- Homology theory. --- Variétés symplectiques --- Variétés topologiques à 4 dimensions