Abstract. We give a geometric definition of smooth toric Deligne-Mumford stacks using the action of a"torus". We show that our definition is equivalent to the one of Borisov, Chen and Smith in terms of stacky fans. In particular, we give a geometric interpretation of the combinatorial data contained in a stacky fan. We also give a bottom up classification in terms of simplicial toric varieties and fiber products of root stacks.

We give an interpretation of quantum Serre theorem of Coates and Givental as a duality of twisted quantum D-modules. This interpretation admits a non-equivariant limit, and we obtain a precise relationship among (1) the quantum D-module of X twisted by a convex vector bundle E and the Euler class, (2) the quantum D-module of the total space of the dual bundle E ∨ → X, and (3) the quantum D-module of a submanifold Z ⊂ X cut out by a regular section of E.When E is the anticanonical line bundle K −1 X , we identify these twisted quantum D-modules with second structure connections with different parameters, which arise as Fourier-Laplace transforms of the quantum D-module of X. In this case, we show that the duality pairing is identified with Dubrovin's second metric (intersection form).

In this article, we prove the following results.• We show a mirror theorem: the Frobenius manifold associated to the orbifold quantum cohomology of weighted projective spaces is isomorphic to the one attached to a specific Laurent polynomial. • We show a reconstruction theorem; that is, we can reconstruct in an algorithmic way the full genus 0 Gromov-Witten potential from the 3-point invariants. Licensed to Univ of Mississippi. Prepared on Sat Jul 4 02:26:12 EDT 2015 for download from IP 130.74.92.202. License or copyright restrictions may apply to redistribution; see http://www.ams.org/license/jour-dist-license.pdfNote that, in a more general and algebraic context, Borisov, Chen and Smith [BCS05] computed the orbifold cohomology ring for toric Deligne-Mumford stacks. We will not use these results because, firstly we will use the techniques developed by Chen and Ruan, and secondly the author did not find in the literature a complete and explicit description of weighted projective spaces as toric Deligne-Mumford stacks.Afterward, using [CCLT06], we prove two propositions 1 (cf. Propositions 4.14 and 4.17) on the value of some orbifold Gromov-Witten invariants with 3 marked points, and we show in Section 6.c that these propositions imply an isomorphism between the Frobenius manifolds coming from the A side and 1 In a previous version of this article, these propositions were conjectures. Licensed to Univ of Mississippi. Prepared on Sat Jul 4 02:26:12 EDT 2015 for download from IP 130.74.92.202. License or copyright restrictions may apply to redistribution; see http://www.ams.org/license/jour-dist-license.pdf ORBIFOLD QUANTUM COHOMOLOGY Licensed to Univ of Mississippi. Prepared on Sat Jul 4 02:26:12 EDT 2015 for download from IP 130.74.92.202.License or copyright restrictions may apply to redistribution; see http://www.ams.org/license/jour-dist-license.pdf

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.