Landau-Ginzburg/Calabi-Yau correspondence, global mirror symmetry and Orlov equivalence

TL;DR

This paper establishes a deep correspondence between the Gromov-Witten theory of Calabi-Yau hypersurfaces and FJRW theory, revealing connections through integral local systems and Orlov equivalence, thus advancing understanding of mirror symmetry and derived categories.

Contribution

It demonstrates the equivalence of Gromov-Witten and FJRW theories in genus zero after analytic continuation, linking them via Gamma-integral structures and Orlov equivalence.

Findings

Matching of Gromov-Witten and FJRW theories after analytic continuation

Identification of integral local systems from Gamma-structures

Proof of Orlov equivalence between derived categories

Abstract

We show that the Gromov-Witten theory of Calabi-Yau hypersurfaces matches, in genus zero and after an analytic continuation, the quantum singularity theory (FJRW theory) recently introduced by Fan, Jarvis and Ruan following ideas of Witten. Moreover, on both sides, we highlight two remarkable integral local systems arising from the common formalism of Gamma-integral structures applied to the derived category of the hypersurface {W=0} and to the category of graded matrix factorizations of W. In this setup, we prove that the analytic continuation matches Orlov equivalence between the two above categories.