Wednesday, June 5, 2013

1306.0437 (Kwokwai Chan et al.)

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

Kwokwai Chan, Cheol-Hyun Cho, Siu-Cheong Lau, Hsian-Hua Tseng
Given a toric Calabi-Yau 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 recipe to the Gross fibration of (a toric modification of) $\mathcal{X}$ to construct its instanton-corrected mirror, where the instanton corrections come from genus 0 open orbifold Gromov-Witten 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 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 Gromov-Witten 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) Gromov-Witten invariants for toric Calabi-Yau orbifolds change under toric crepant resolutions - this is an open analogue of Ruan's crepant resolution conjecture.
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