Journal of Differential Geometry

Gross fibrations, SYZ mirror symmetry, and open Gromov–Witten invariants for toric Calabi–Yau orbifolds

Kwokwai Chan, Cheol-Hyun Cho, Siu-Cheong Lau, and Hsian-Hua Tseng

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For a toric Calabi–Yau (CY) orbifold $\mathcal{X}$ whose underlying toric variety is semi-projective, we construct and study a non-toric Lagrangian torus fibration on $\mathcal{X}$, which we call the Gross fibration. We apply the Strominger–Yau–Zaslow (SYZ) recipe to the Gross fibration of $\mathcal{X}$ to construct its mirror with the instanton corrections coming from genus $0$ open orbifold Gromov–Witten (GW) invariants, which are virtual counts of holomorphic orbi-disks in $\mathcal{X}$ bounded by fibers of the Gross fibration. We explicitly evaluate all these invariants by first proving an open/closed equality and then employing the toric mirror theorem for suitable toric (parital) compactifications of $\mathcal{X}$. Our calculations are then applied to

(1) prove a conjecture of Gross-Siebert on a relation between genus $0$ open orbifold GW invariants and mirror maps of $\mathcal{X}$—this is called the open mirror theorem, which leads to an enumerative meaning of mirror maps, and

(2) demonstrate how open (orbifold) GW invariants for toric CY orbifolds change under toric crepant resolutions—an open analogue of Ruan’s crepant resolution conjecture.

Article information

J. Differential Geom., Volume 103, Number 2 (2016), 207-288.

First available in Project Euclid: 16 May 2016

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Chan, Kwokwai; Cho, Cheol-Hyun; Lau, Siu-Cheong; Tseng, Hsian-Hua. Gross fibrations, SYZ mirror symmetry, and open Gromov–Witten invariants for toric Calabi–Yau orbifolds. J. Differential Geom. 103 (2016), no. 2, 207--288. doi:10.4310/jdg/1463404118.

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