Quantum Cohomology and Periods

Iritani, Hiroshi

Annales de l'Institut Fourier, Tome 61 (2011) no. 7, pp. 2909-2958.

Dans un précédent article, l’auteur a défini une structure entière sur la cohomologie quantique à l’aide de la K-théorie et d’une classe Gamma. Cette structure est compatible avec la symétrie miroir pour les orbifolds toriques. Le principe de Lefschetz quantique appliqué aux résultats précédents, nous donne une relation explicite entre les solutions du module différentiel quantique pour une intersection complète torique et les périodes (ou les intégrales oscillantes) de leur miroir. Nous expliquons en détail l’isomorphisme miroir pour une variation de structure de Hodge entière pour une paire miroir (au sens de Batyrev) d’hypersurfaces de Calabi-Yau.

In a previous paper, the author introduced an integral structure in quantum cohomology defined by the $K$-theory and the Gamma class and showed that it is compatible with mirror symmetry for toric orbifolds. Applying the quantum Lefschetz principle to the previous results, we find an explicit relationship between solutions to the quantum differential equation of toric complete intersections and the periods (or oscillatory integrals) of their mirrors. We describe in detail the mirror isomorphism of variations of integral Hodge structure for a mirror pair of Calabi-Yau hypersurfaces (Batyrev’s mirror).

Classification : 14N35, 14D05, 14D07, 14J33, 32G20, 53D37
Keywords: quantum cohomology, mirror symmetry, Gamma class, $K$-theory, period, oscillatory integral, variation of Hodge structure, GKZ system, toric variety, orbifold
Mot clés : cohomologie quantique, symétrie miroir, $K$-théorie, période, intégrale oscillante, variation de structure de Hodge, système GKZ, variété torique, orbifold

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