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Landau–Ginzburg/Calabi–Yau correspondence for quintic three-folds via symplectic transformations

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Abstract

We compute the recently introduced Fan–Jarvis–Ruan–Witten theory of W-curves in genus zero for quintic polynomials in five variables and we show that it matches the Gromov–Witten genus-zero theory of the quintic three-fold via a symplectic transformation. More specifically, we show that the J-function encoding the Fan–Jarvis–Ruan–Witten theory on the A-side equals via a mirror map the I-function embodying the period integrals at the Gepner point on the B-side. This identification inscribes the physical Landau–Ginzburg/Calabi–Yau correspondence within the enumerative geometry of moduli of curves, matches the genus-zero invariants computed by the physicists Huang, Klemm, and Quackenbush at the Gepner point, and yields via Givental’s quantization a prediction on the relation between the full higher genus potential of the quintic three-fold and that of Fan–Jarvis–Ruan–Witten theory.

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Authors and Affiliations

  1. Institut Fourier, UMR du CNRS 5582, Université de Grenoble 1, BP 74, 38402, Saint Martin d’Hères, France

    Alessandro Chiodo

  2. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109-1109, USA

    Yongbin Ruan

  3. Yangtze Center of Mathematics, Sichuan University, Chengdu, 610064, P.R. China

    Yongbin Ruan

Authors
  1. Alessandro Chiodo
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  2. Yongbin Ruan
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Correspondence to Alessandro Chiodo.

Additional information

First author partially supported by the ANR grant ThéorieGW JC09_446540 “Des nouvelles symétries pour la théorie de Gromov-Witten”. Second author partially supported by an NSF grant.

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Chiodo, A., Ruan, Y. Landau–Ginzburg/Calabi–Yau correspondence for quintic three-folds via symplectic transformations. Invent. math. 182, 117–165 (2010). https://doi.org/10.1007/s00222-010-0260-0

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  • DOI: https://doi.org/10.1007/s00222-010-0260-0

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