LG/CY correspondence between $tt^*$ geometries

The concept of $tt^*$ geometric structure was introduced by physicists (Cecotti-Vafa, BCOV...), and then studied firstly in mathematics by C. Hertling. It is believed that the $tt^*$ geometric structure contains the whole genus $0$ information of the corresponding two dimensional topological field theory. In this talk, a LG/CY correspondence conjecture for $tt^*$ geometry will be given and partial result is given as follows. Let $f\in\mathbb{C}[z_0, \dots, z_{n+2}]$ be a nondegenerate homogeneous polynomial of degree $n+2$, then it defines a Calabi-Yau model represented by a Calabi-Yau hypersurface $X_f$ in $\mathbb{CP}^{n+1}$ or a Landau-Ginzburg model represented by a hypersurface singularity $(\mathbb C^{n+2}, f)$. We build the isomorphism of almost all structures in $tt^*$ geometries between the CY model and the marginal part of the LG model except the isomorphism between real structures. This is a joint work with Lan Tian and Yang Zongrui.