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Article Dans Une Revue Manuscripta mathematica Année : 2014

Quantum differential systems and rational structures

Antoine Douai

Résumé

We consider mirror symmetry (A-side vs B-side, namely singularity side) in the framework of quantum differential systems. We focuse on the logarithmic non-resonant case, which describes the geometric situation and we show that such systems provide a good framework in order to generalize the construction of the rational structure given by Katzarkov, Kontsevich and Pantev for the complex projective space. As an application, we give a closed formula for the rational structure defined by the Lefschetz thimbles on the flat sections of the Gauss-Manin connection associated with the Landau–Ginzburg models of weighted projective spaces (a class of Laurent polynomials). As a by-product, using a mirror theorem, we get a rational structure on the orbifold cohomology of weighted projective spaces. The formula on the B-side is more complicated than the one on the A-side (the latter agrees with one of Iritani’s results), depending on numerous combinatorial data which are rearranged after the mirror transformation.
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Dates et versions

hal-01141566 , version 1 (13-04-2015)

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Antoine Douai. Quantum differential systems and rational structures. Manuscripta mathematica, 2014, pp.285-317. ⟨10.1007/s00229-014-0678-6⟩. ⟨hal-01141566⟩
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