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

TL;DR

This paper constructs a non-toric Lagrangian fibration on toric Calabi-Yau orbifolds, applies SYZ mirror symmetry with open Gromov-Witten invariants, and proves conjectures relating these invariants to mirror maps and crepant resolutions.

Contribution

It introduces the Gross fibration for toric CY orbifolds, explicitly computes open GW invariants, and proves the open mirror theorem and crepant resolution conjecture.

Findings

Explicit evaluation of open orbifold Gromov-Witten invariants.

Proof of the open mirror theorem relating invariants to mirror maps.

Demonstration of invariants' behavior under toric crepant resolutions.

Abstract

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 (partial) 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.