[Paper Review] A global mirror symmetry framework for the Landau-Ginzburg/Calabi-Yau correspondence

A Chiodo, Yongbin Ruan|arXiv (Cornell University)|Jul 3, 2013

Algebraic Geometry and Number Theory51 references3 citations

TL;DR

This paper establishes a global mirror symmetry framework for the Landau-Ginzburg/Calabi-Yau (LG-CY) correspondence by unifying FJRW theory on the LG side with Gromov-Witten (GW) theory on the CY side through Orlov's equivalence and analytic continuation. It demonstrates that the LG-CY correspondence can be realized as a symplectic transformation encoded by Orlov's isomorphism, with the key result being a commutative diagram linking flat sections of FJRW and GW theories via analytic continuation and categorical equivalence.

ABSTRACT

We show how the Landau-Ginzburg/Calabi-Yau correspondence for the quintic three-fold can be cast into a global mirror symmetry framework. Then we draw inspiration from Berglund-Hübsch mirror duality construction to provide an analogue conjectural picture featuring all Calabi-Yau hypersurfaces within weighted projective spaces and certain quotients by finite abelian group actions.

Motivation & Objective

• To unify the Landau-Ginzburg/Calabi-Yau correspondence within a global mirror symmetry framework.
• To extend Berglund–Hübsch mirror duality to all Calabi–Yau hypersurfaces in weighted projective spaces and their quotients.
• To establish a mathematical bridge between FJRW theory (LG side) and Gromov–Witten theory (CY side) via categorical and geometric structures.
• To demonstrate that the LG-CY correspondence can be interpreted as a symplectic transformation induced by Orlov’s equivalence.
• To show that analytic continuation in the B-model picture commutes with Orlov’s isomorphism, thereby realizing the correspondence as a global duality.

Proposed method

• Utilizes FJRW theory as the quantum cohomology theory on the LG side, defined via solutions to the Witten equation on Riemann surfaces with orbifold structures.
• Applies the Fan–Jarvis–Ruan–Witten (FJRW) theory to quasihomogeneous singularities, particularly for the quintic threefold and its mirror.
• Employs the B-model connection ∇V on the vector bundle V to define flat sections, linking the B-model to both GW and FJRW theories.
• Uses analytic continuation of flat sections from the large complex structure point (0) to the conifold point (∞) in the B-model moduli space.
• Establishes a commutative diagram linking flat sections of FJRW and GW theories via analytic continuation and Orlov’s equivalence ΦOrlov.
• Introduces symplectic matrices Ua representing Orlov’s isomorphisms, with monodromy T at infinity acting via conjugation to relate different functors.

Experimental results

Research questions

• RQ1 How can the LG-CY correspondence be embedded into a global mirror symmetry framework beyond genus zero?
• RQ2 What is the precise role of Orlov’s equivalence in connecting FJRW theory to Gromov–Witten theory?
• RQ3 How does analytic continuation in the B-model relate to the symplectic structure of the LG-CY correspondence?
• RQ4 Can the LG-CY correspondence be realized as a symplectic transformation induced by categorical equivalences?
• RQ5 How do monodromy actions at infinity affect the realization of the LG-CY correspondence across different branches of the moduli space?

Key findings

• The LG-CY correspondence is realized as a symplectic transformation between the state spaces of FJRW and GW theories, with the transformation matrix U_LG-CY encoding the correspondence.
• A commutative diagram is established showing that analytic continuation in the B-model commutes with Orlov’s equivalence ΦOrlov, linking flat sections of FJRW and GW theories.
• Orlov’s equivalence can be realized as a symplectic matrix Ua, with different choices of a ∈ ℤ parametrizing distinct functors between the derived categories of matrix factorizations and coherent sheaves.
• The monodromy operator T at infinity acts on the symplectic transformation via conjugation, yielding Φa = T^{-a} U_LG-CY T^a, which relates different realizations of the correspondence.
• The framework generalizes to all Calabi–Yau hypersurfaces in weighted projective spaces via Berglund–Hübsch mirror duality and finite abelian group quotients.
• The construction provides a global picture where the LG-CY correspondence short-circuits mirror symmetry by directly relating A-model invariants of the CY to those of the LG model through categorical and geometric data.