Ruan's LG/CY correspondence conjecture for elliptic orbifold projective lines

The Landau–Ginzburg side is the Fan–Jarvis–Ruan–Witten theory of a singularity $(W,G)$, with state space ${\mathcal H}_{FJRW}$ and total potential ${\mathcal A}_{FJRW}$, while the Calabi–Yau side is the Gromov–Witten theory of $X_W/\widetilde{G}$, with Chen–Ruan cohomology $H^*_{CR}(X_W/\widetilde{G})$, Givental cone ${\mathcal L}_{GW}$, and total potential ${\mathcal A}_{GW}$. Let ${\mathcal L}_{FJRW}$ denote the FJRW Givental cone, and let ${\mathbb U}_{\rm LG/CY}$ be a degree-preserving $\mathbb{C}[z,z^{-1}]$-valued linear symplectic map between the corresponding Givental symplectic vector spaces.

Ruan's LG/CY correspondence conjecture. (1) There is a graded vector-space isomorphism

$${\mathcal H}_{FJRW}\longrightarrow H^*_{CR}(X_W/\widetilde{G}).$$

(2) There is a choice of analytic continuation in the Kähler parameter such that

$${\mathbb U}_{\rm LG/CY}({\mathcal L}_{FJRW})={\mathcal L}_{GW}.$$

(3) Up to an overall constant and a choice of analytic continuation, the total potentials are related by quantization:

$${\mathcal A}_{GW}=\widehat{{\mathbb U}_{\rm LG/CY}}({\mathcal A}_{FJRW}).$$

This conjecture proposes that the FJRW and Gromov–Witten theories agree after identifying their state spaces, Givental cones, and total potentials through a symplectic transformation. The paper states that its main goal is to prove the conjecture for elliptic orbifold $\mathbb{P}^1$; the supplied text does not specify the resulting resolution status beyond that aim.