Cohomological Landau–Ginzburg/Calabi–Yau correspondence

Let $(W,G)$ be of Calabi–Yau type: $W$ is nondegenerate, not necessarily invertible, with weights $q_j$ satisfying

$$\sum_j q_j=1,$$

and $G$ contains $\langle j_W\rangle$ and lies in $SL_W$. Set $\widetilde G=G/\langle j_W\rangle$. Cohomological LG–CY correspondence. There is a bigraded vector-space isomorphism

$$\mathcal H_{W,G}^{*,*}\cong H^{*,*}_{\operatorname{CR}}\left([X_W/\widetilde G];\mathbb C\right),$$

where the right-hand side is the Chen–Ruan orbifold cohomology of $[X_W/\widetilde G]$. This is the cohomological form of the LG–CY correspondence, identifying FJRW state space with Chen–Ruan cohomology; the source supplies no general resolution.