FJRW-Rings and Landau-Ginzburg Mirror Symmetry in Two Dimensions

For any non-degenerate, quasi-homogeneous hypersurface singularity W and an admissible group of diagonal symmetries G, Fan, Jarvis, and Ruan have constructed a cohomological field theory which is a candidate for the mathematical structure behind the Landau-Ginzburg A-model. When using the orbifold Milnor ring of a singularity W as a B-model, and the Frobenius algebra H_{W,G} constructed by Fan, Jarvis, and Ruan, as an A-model, the following conjecture is obtained: For a quasi-homogeneous singularity W and a group G of symmetries of W, there is a dual singularity W^T such that the orbifold A-model of W/G is isomorphic to the B-model of W^T. I will show that this conjecture holds for a two-dimensional invertible loop potential $W$ with its maximal group of diagonal symmetries $G_{W}$.