Open orbifold Gromov-Witten invariants of [C^3/Z_n]: localization and mirror symmetry

TL;DR

This paper develops a mathematical framework to compute open orbifold Gromov-Witten invariants of [C^3/Z_n], verifying predictions from mirror symmetry through localization techniques and explicit examples.

Contribution

It introduces a localization-based method for calculating open orbifold Gromov-Witten invariants and proves a mirror theorem for orbifold disc invariants.

Findings

Confirmed B-model predictions for [C^3/Z_3]

Proved a mirror theorem for orbifold disc invariants of [C^3/Z_4]

Matched numerous annulus invariants and provided mirror symmetry predictions for genus ≤ 2

Abstract

We develop a mathematical framework for the computation of open orbifold Gromov-Witten invariants of [C^3/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, 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 [C^3/Z_3], where we verify physical predictions of Bouchard, Klemm, Marino and Pasquetti, the main object of our study is the richer case of [C^3/Z_4], 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 \leq 2.