A mirror theorem for the mirror quintic

Abstract

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

A Landau–Ginzburg/Calabi–Yau correspondence for the mirror quintic

“…We will now describe the Chen-Ruan cohomology of the mirror quintic W . For more detail, refer to [18]. For an element g ∈ G, denote by [g] the corresponding element inḠ and I(g) := {k ∈ {1, 2, 3, 4, 5} | Θ k (g) = 0}.…”

“…Thus one must first check convergence in order to prove the LG/CY correspondence. For the purposes of this paper however, the necessary convergence will follow from the mirror theorem of [18] . This is no longer true for the mirror quintic.…”

“…In analogy to the original mirror theorem, it was proven in [18] that the orbifold GW theory of W may be related to the Picard-Fuchs equations of M ψ around ψ = ∞. To be more precise, it was shown that after restricting to the degree two part of the untwisted subspace of the state space, H 2 un (W ) ⊂ H * CR (W ), the components of the mirror quintic J-function, J W (s, z), give solutions to the Picard-Fuchs equations of a holomorphic (3, 0)-form on M ψ around ψ = ∞.…”

The Crepant Transformation Conjecture for toric complete intersections

“…A mirror Y to the quintic 3-fold arises [7,22,47] as a crepant resolution of an anticanonical hypersurface in X = P 4 /(Z/5Z) 3 . A mirror theorem for Y has been proved by Lee-Shoemaker [61]. The variety Y is a Calabi-Yau 3-fold with h 1,1 (Y ) = 101.…”

A brief survey of FJRW theory

“…They further showed that the narrow sectors of the FJRW state space correspond to the ambient cohomology of H * CR ([X W /G]). The LG/CY correspondence has been verified in genus zero for the Fermat Quintic withG = {1} [CR10]; for the narrow/ambient part of a general hypersurface in Gorenstein weighted projective space withG = {1} [CIR14]; for the mirror quintic (with groupG = (Z/5) 3 ) [PS13,LS12]; and for arbitrary Fermat polynomials with arbitrary admissible groups [PLS14]. This last paper [PLS14] is also interesting because it connects the LG/CY correspondence to the better-understood Crepant Transformation Conjecture.…”