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

TL;DR

This paper establishes a precise mathematical correspondence between Fan-Jarvis-Ruan-Witten theory and Gromov-Witten theory for the quintic three-fold, using symplectic transformations and mirror symmetry techniques.

Contribution

It demonstrates the Landau-Ginzburg/Calabi-Yau correspondence at genus zero for quintic three-folds via explicit symplectic transformations and mirror maps.

Findings

Matching of J-function and I-function via mirror map

Validation of physical predictions in enumerative geometry

Prediction of higher genus relations between theories

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.