ABSTRACT. We investigate a multiple mirror phenomenon arising from BerglundHübsh-Krawitz mirror symmetry. We prove that the different mirror Calabi-Yau orbifolds which arise in this context are in fact birational to one another. CONTENTS

ABSTRACT. The celebrated Mirror Theorem states that the genus zero part of the A model (quantum cohomology, rational curves counting) of the Fermat quintic threefold is equivalent to the B model (complex deformation, variation of Hodge structure) of its mirror dual orbifold. In this article, we establish a mirror-dual statement. Namely, the B model of the Fermat quintic threefold is shown to be equivalent to the A model of its mirror, and hence establishes the mirror symmetry as a true duality. CONTENTS

We prove a version of the Landau-Ginzburg/Calabi-Yau correspondence for the mirror quintic. In particular we calculate the genus-zero FJRW theory for the pair (W, G) where W is the Fermat quintic polynomial and G = SL W . We identify it with the Gromov-Witten theory of the mirror quintic three-fold via an explicit analytic continuation and symplectic transformation. In the process we prove a mirror theorem for the corresponding Landau-Ginzburg model (W, G).

We investigate the effect of a general toric wall crossing on genus zero Gromov-Witten theory. Given two complete toric orbifolds X + and X − related by wall crossing under variation of GIT, we prove that their respective I-functions are related by linear transformation and asymptotic expansion. We use this comparison to deduce a similar result for birational complete intersections in X + and X − . This extends the work of the previous authors in [2] to the case of complete intersections in toric varieties, and generalizes some of the results of Coates-Iritani-Jiang [14] on the crepant transformation conjecture to the setting of non-zero discrepancy.

Given Y a non-compact manifold or orbifold, we define a natural subspace of the cohomology of Y called the narrow cohomology. We show that despite Y being non-compact, there is a well-defined and non-degenerate pairing on this subspace. The narrow cohomology proves useful for the study of genus zero Gromov-Witten theory. When Y is a smooth complex variety or Deligne-Mumford stack, one can define a quantum D-module on the narrow cohomology of Y. This yields a new formulation of quantum Serre duality.