Landau–Ginzburg/Calabi–Yau correspondence

Consider the Lagrangian cones $\mathcal L_{\operatorname{FJRW}}$ and $\mathcal L_{\operatorname{GW}}$ in the symplectic spaces $\mathcal V_{\operatorname{FJRW}}$ and $\mathcal V_{\operatorname{GW}}$, and let $\mathcal D_{\operatorname{FJRW}}$ and $\mathcal D_{\operatorname{GW}}$ denote the corresponding total potential functions. LG–CY correspondence. (1) There is a degree-preserving $\mathbb C[z,z^{-1}]$-valued linear symplectic isomorphism

$$\mathbb U_{\operatorname{LG-CY}}:\mathcal V_{\operatorname{FJRW}}\to\mathcal V_{\operatorname{GW}}$$

and a choice of analytic continuation such that

$$\mathbb U_{\operatorname{LG-CY}}(\mathcal L_{\operatorname{FJRW}})=\mathcal L_{\operatorname{GW}}.$$

(2) Up to an overall constant and a choice of analytic continuation, the total potentials satisfy

$$\mathcal D_{\operatorname{GW}}=\widehat{\mathbb U}_{\operatorname{LG-CY}}(\mathcal D_{\operatorname{FJRW}}).$$

This geometric correspondence generalizes the quintic case to Calabi–Yau orbifold hypersurfaces and finite group quotients; genus-zero special cases are known, while the full statement remains open in the source.