, Volume 17, Issue 4, pp 879-933

Open orbifold Gromov-Witten invariants of ${[\mathbb{C}^3/\mathbb{Z}_n]}$ : localization and mirror symmetry

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Abstract

We develop a mathematical framework for the computation of open orbifold Gromov-Witten invariants of ${[\mathbb{C}^3/\mathbb{Z}_n]}$ 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 ${[\mathbb{C}^3/\mathbb{Z}_3]}$ , where we verify physical predictions of Bouchard, Klemm, Mariño and Pasquetti [4,5], the main object of our study is the richer case of ${[\mathbb{C}^3/\mathbb{Z}_4]}$ , 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 ≤ 2.