Mirror Symmetry: FJRW-rings and Landau-Ginzburg Orbifolds

For any non-degenerate, quasihomogeneous superpotential $W$ and an admissible group of diagonal symmetries G, Fan, Jarvis and Ruan have constructed a quantum cohomological field theory (FJRW-theory) that gives, among other things, a Frobenius algebra $\mathcal{H}_{W,G}$ ((a,c) ring) and correlators associated with the superpotential. This construction is analogous to a theory of the Gromov-Witten type. The FJRW- theory is a candidate for the mathematical structure behind $\mathcal{N}=$ $2$ superconformal Landau-Ginzburg orbifolds. In this presentation I will give an overview of this theory and discuss the Berglund-H\"{u}bsch-Krawitz mirror symmetry conjecture: For a given invertible superpotential $W$ there exists an invertible superpotential $W^{T}$ such that the Frobenius algebra $\mathcal{H}_{W,G}$ is isomorphic to the (c,c) ring of $W^{T}$, and the Frobenius algebra $\mathcal{H}_ {W^{T},G^{T}}$ is isomorphic to the (c,c) ring of $W$.