Enumerative meaning of mirror maps for toric Calabi-Yau manifolds

Abstract

We prove that the inverse of a mirror map for a toric Calabi-Yau manifold of the form $K_Y$, where $Y$ is a compact toric Fano manifold, can be expressed in terms of generating functions of genus 0 open Gromov-Witten invariants defined by Fukaya-Oh-Ohta-Ono \cite{FOOO10}. Such a relation between mirror maps and disk counting invariants was first conjectured by Gross and Siebert \cite[Conjecture 0.2 and Remark 5.1]{GS11} as part of their program, and was later formulated in terms of Fukaya-Oh-Ohta-Ono's invariants in the toric Calabi-Yau case in \cite[Conjecture 1.1]{CLL12}.

v1: 13 pages; v2: 19 pages, major revision; v3: 20 pages, small changes

References in corpus (4)

• Mirror symmetry and T-duality in the complement of an anticanonical divisor
• Central charges, symplectic forms, and hypergeometric series in local mirror symmetry
• Seidel elements and mirror transformations
• Some integrality properties in local mirror symmetry

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• Homological mirror symmetry for hypersurfaces in $({\mathbb{C}}^{\ast }{)}^n$
• Quasimap SYZ for toric Calabi-Yau manifolds
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• Relative Gromov-Witten invariants and the enumerative meaning of mirror maps for toric Calabi-Yau orbifolds
• SYZ mirror symmetry for hypertoric varieties
• The Strominger-Yau-Zaslow conjecture and its impact
• SYZ Mirror Symmetry for Dirichlet Branes