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

Kwokwai Chan The Chinese University of Hong Kong Cheol-Hyun Cho Seoul National University Siu-Cheong Lau Boston University Hsian-Hua Tseng Ohio State Unviersity

Differential Geometry mathscidoc:1610.10014

Journal of Differential Geometry, 103, 207-288, 2016
For a toric Calabi-Yau (CY) orbifold X whose underlying toric variety is semi-projective, we construct and study a non-toric Lagrangian torus fibration on X, which we call the Gross fibration. We apply the Strominger-Yau-Zaslow (SYZ) recipe to the Gross fibration of 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 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 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 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.
SYZ, toric, Calabi-Yau, orbifold, open Gromov-Witten, mirror symmetry
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@inproceedings{kwokwai2016gross,
  title={Gross fibrations, SYZ mirror symmetry, and open Gromov-Witten invariants for toric Calabi-Yau orbifolds},
  author={Kwokwai Chan, Cheol-Hyun Cho, Siu-Cheong Lau, and Hsian-Hua Tseng},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161004023312450099079},
  booktitle={Journal of Differential Geometry},
  volume={103},
  pages={207-288},
  year={2016},
}
Kwokwai Chan, Cheol-Hyun Cho, Siu-Cheong Lau, and Hsian-Hua Tseng. Gross fibrations, SYZ mirror symmetry, and open Gromov-Witten invariants for toric Calabi-Yau orbifolds. 2016. Vol. 103. In Journal of Differential Geometry. pp.207-288. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161004023312450099079.
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