LG/CY correspondence: The state space isomorphism

We prove the mirror duality 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.