[Paper Review] LG/CY correspondence: the state space isomorphism

A Chiodo, Yongbin Ruan|arXiv (Cornell University)|Aug 6, 2009

Advanced Algebra and Geometry26 references7 citations

TL;DR

This paper establishes a degree-preserving isomorphism between the Chen–Ruan orbifold cohomology of Calabi–Yau hypersurfaces in weighted projective spaces and the Fan–Jarvis–Ruan–Witten state space of the dual Landau–Ginzburg singularity, proving the classical mirror symmetry conjecture for Berglund–Hübsch–Krawitz mirror pairs via a cohomological LG/CY correspondence.

ABSTRACT

We prove the classical mirror symmetry conjecture for the mirror pairs constructed by Berglund, Hübsch, and Krawitz. Our main tool is a cohomological LG/CY correspondence which provides a degree-preserving isomorphism between the cohomology of finite quotients of Calabi-Yau hypersurfaces inside a weighted projective space and the Fan-Jarvis-Ruan-Witten state space of the associated Landau-Ginzburg singularity theory.

Motivation & Objective

• To establish a rigorous mathematical foundation for the Berglund–Hübsch–Krawitz mirror symmetry construction, which was previously based on physical intuition.
• To resolve the open question of whether the orbifolds $X_W/G$ and $X_{W^T}/G^T$ form a mirror pair in the classical sense for Calabi–Yau three-folds.
• To extend the LG/CY correspondence to include finite quotients of Calabi–Yau hypersurfaces in weighted projective spaces, using cohomological methods.
• To provide a complete isomorphism between the Chen–Ruan cohomology of the Calabi–Yau orbifold and the FJRW state space of the dual Landau–Ginzburg model.
• To verify that Hodge numbers $h^{1,1}$ and $h^{2,1}$ are interchanged under the correspondence, confirming classical mirror symmetry.

Proposed method

• Construct a cohomological LG/CY correspondence that induces a degree-preserving isomorphism between the Chen–Ruan orbifold cohomology of $X_W/G$ and the FJRW state space of the dual Landau–Ginzburg model.
• Use the group $G$ of diagonal symmetries of the invertible polynomial $W$, containing the J-invariant $J_W$, and consider the quotient $\widetilde{G} = G / \langle J_W \rangle$.
• Apply the FJRW theory framework to compute the state space of the dual singularity $W^T$, using the dual group $G^T$.
• Analyze each $\widetilde{G}$-twisted sector via a geometric construction involving roots of unity and rays/dots in the complex plane, parameterized by angular coordinates.
• Define a diagrammatic model using $\mu_d$ and solutions $\alpha$ to $\alpha^{w_j} = g_j$, where $g_j$ are the components of group elements $g \in G$.
• Match internal dots (representing Chen–Ruan cohomology classes) with empty rays (representing FJRW state space generators), ensuring degree preservation via the formula $\deg_{\operatorname{CR}} = a(h) + D - R$.

Experimental results

Research questions

• RQ1 Does the Berglund–Hübsch–Krawitz mirror construction yield a true mirror pair of Calabi–Yau three-folds in the classical sense?
• RQ2 Is there a degree-preserving isomorphism between the Chen–Ruan cohomology of $X_W/G$ and the FJRW state space of $W^T$?
• RQ3 How do the Hodge numbers $h^{1,1}$ and $h^{2,1}$ transform under the proposed duality, and is the mirror symmetry correspondence verified?
• RQ4 Can the LG/CY correspondence be extended to include finite group quotients of Calabi–Yau hypersurfaces in weighted projective spaces?
• RQ5 What is the precise geometric and algebraic mechanism that ensures the correspondence between internal dots and empty rays in the diagrammatic model?

Key findings

• A degree-preserving isomorphism is established between the Chen–Ruan orbifold cohomology of $X_W/G$ and the FJRW state space of the dual Landau–Ginzburg model $W^T$.
• The correspondence holds for all invertible polynomials $W$ and all admissible groups $G$ containing $J_W$, confirming the mirror symmetry conjecture in this setting.
• The Hodge numbers $h^{1,1}$ and $h^{2,1}$ are interchanged under the duality, verifying classical mirror symmetry for the orbifolded Calabi–Yau three-folds.
• The diagrammatic construction using rays and dots matches internal cohomology classes with FJRW generators, with degrees preserved: e.g., internal dots on Figure 5 have degrees 0 and 1, matching empty rays with degrees 1 and 0.
• In cases with nontrivial $\langle J \rangle$-cosets, such as the cubic example with $G = \mathbb{Z}_{12}$, the method correctly computes degrees: e.g., internal dots on Figure 6 have degree $1/2$, on Figure 7 degree 1, and on Figure 8 degree $3/2$.
• The presence of extra rays and dots due to nontrivial intersections is balanced, ensuring the isomorphism remains valid even when $\bigcup_j \{\alpha \mid \alpha^{w_j} = g_j\}$ extends beyond $\mu_d$.