Landau--Ginzburg mirror symmetry for ADE singularities

Abstract

We prove Landau-Ginzburg mirror symmetry conjecture for ADE singularities with the nontrivial symmetry groups. In particular, we consider $(f,G)$ with $f\in \mathbb{C}[z_1,z_2,z_3]$ defining ADE singularity and $G$ --- nontrivial group of diagonal symmetries of $f$. We show that orbifold Saito theory of $(f,G)$ is isomorphic to the Fan-Jarvis-Ruan-Witten theory of its Berglund-Hübsch-Henningson dual pair $(\tilde{f},\tilde{G})$.