Inventiones mathematicae

, Volume 182, Issue 1, pp 117–165 | Cite as

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



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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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Institut Fourier, UMR du CNRS 5582Université de Grenoble 1Saint Martin d’HèresFrance
  2. 2.Department of MathematicsUniversity of MichiganAnn ArborUSA
  3. 3.Yangtze Center of MathematicsSichuan UniversityChengduP.R. China

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