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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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