Open orbifold Gromov-Witten invariants of [ℂ³ /ℤ_n]: localization and mirror symmetry

We develop a mathematical framework for the computation of open orbifold Gromov-Witten invariants of [ℂ³ /ℤ_n] and provide extensive checks with predictions from open string mirror symmetry. To this aim, we set up a computation of open string invariants in the spirit of Katz-Liu [23], defining them by localization. The orbifold is viewed as an open chart of a global quotient of the resolved conifold, and the Lagrangian as the fixed locus of an appropriate anti-holomorphic involution. We consider two main applications of the formalism. After warming up with the simpler example of [ℂ³ /ℤ₃], where we verify physical predictions of Bouchard, Klemm, Mariño and Pasquetti [4,5], the main object of our study is the richer case of [ℂ³ /ℤ₄], where two different choices are allowed for the Lagrangian. For one choice, we make numerical checks to confirm the B-model predictions; for the other, we prove a mirror theorem for orbifold disc invariants, match a large number of annulus invariants, and give mirror symmetry predictions for open string invariants of genus ≤ 2.