The Open Crepant Transformation Conjecture for Toric Calabi-Yau 3-Orbifolds

TL;DR

This paper proves an open version of Ruan's Crepant Transformation Conjecture for toric Calabi-Yau 3-orbifolds, relating disk invariants via mirror symmetry and wall-crossing techniques.

Contribution

It establishes the open crepant transformation conjecture for toric Calabi-Yau 3-orbifolds using mirror symmetry and secondary fan analysis, extending previous results.

Findings

Identification of disk invariants across crepant transformations

Construction of a global family of mirror curves

Analytic continuation of local coordinates on mirror curves

Abstract

We prove an open version of Ruan's Crepant Transformation Conjecture for toric Calabi-Yau 3-orbifolds, which is an identification of disk invariants of K-equivalent semi-projective toric Calabi-Yau 3-orbifolds relative to corresponding Lagrangian suborbifolds of Aganagic-Vafa type. Our main tool is a mirror theorem of Fang-Liu-Tseng that relates these disk invariants to local coordinates on the B-model mirror curves. Treating toric crepant transformations as wall-crossings in the GKZ secondary fan, we establish the identification of disk invariants through constructing a global family of mirror curves over charts of the secondary variety and understanding analytic continuation on local coordinates. Our work generalizes previous results of Brini-Cavalieri-Ross on disk invariants of threefold type-A singularities and of Ke-Zhou on crepant resolutions with effective outer branes.