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The small quantum cohomology of a weighted projective space, a mirror D-module and their classical limits
Authors:Antoine Douai  Etienne Mann
Institution:1. Laboratoire J.A Dieudonné, UMR CNRS 6621, Université de Nice, Parc Valrose, 06108, Nice Cedex 2, France
2. Département de Mathématiques, Université de Montpellier 2, Place Eugène Bataillon, 34095, Montpellier Cedex 5, France
Abstract:We first describe a mirror partner (B-model) of the small quantum orbifold cohomology of weighted projective spaces (A-model) in the framework of differential equations: we attach to the A-model (resp. B-model) a quantum differential system (that is a trivial bundle equipped with a suitable flat meromorphic connection and a flat bilinear form) and we give an explicit isomorphism between these two quantum differential systems. On the A-side (resp. on the B-side), the quantum differential system alluded to is naturally produced by the small quantum cohomology (resp. a solution of the Birkhoff problem for the Brieskorn lattice of a Landau–Ginzburg model). Then we study the degenerations of these quantum differential systems and we apply our results to the construction of (classical, limit, logarithmic) Frobenius manifolds.
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