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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.

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Authors and Affiliations

  1. Section de Mathématiques, Université de Genève, 2-4 Rue du Lièvre, 1210, Genève, Switzerland

    Andrea Brini

  2. Department of Mathematics, Colorado State University, 101 Weber Building, Fort Collins, CO, 80523-1874, USA

    Renzo Cavalieri

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  1. Andrea Brini
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  2. Renzo Cavalieri
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Correspondence to Andrea Brini.

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Brini, A., Cavalieri, R. Open orbifold Gromov-Witten invariants of \({[\mathbb{C}^3/\mathbb{Z}_n]}\): localization and mirror symmetry. Sel. Math. New Ser. 17, 879–933 (2011). https://doi.org/10.1007/s00029-011-0060-4

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