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

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.

43 pages. v2: section 2 updated to match notation of Fan, Jarvis, and Ruan, arXiv:0712.4021. v3: Section 2 focuses only on the Fermat quintic. The final version is available via http://link.springer.com/article/10.1007%2Fs00222-010-0260-0

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