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The concept of $tt^*$ geometric structure was introduced by physicists (see\cite{CV1, BCOV} and references therein) , and then studied firstly inmathematics by C. Hertling \cite{Het1}. It is believed that the $tt^*$geometric structure contains the whole genus $0$ information of a twodimensional topological field theory. In this paper, we propose the LG/CYcorrespondence conjecture for $tt^*$ geometry and obtain the following result.Let $f\in\mathbb{C}[z_0, \dots, z_{n+2}]$ be a nondegenerate homogeneouspolynomialof degree $n+2$, then it defines a Calabi-Yau model represented by aCalabi-Yau hypersurface $X_f$ in $\mathbb{CP}^{n+1}$ or a Landau-Ginzburg modelrepresented by a hypersurface singularity $(\mathbb{C}^{n+2}, f)$, both can bewritten as a $tt^*$ structure. We proved that there exists a $tt^*$substructure on Landau-Ginzburg side, which should correspond to the $tt^*$structure from variation of Hodge structures in Calabi-Yau side. We build theisomorphism of almost all structures in $tt^*$ geometries between these twomodels except the isomorphism between real structures.

LG/CY correspondence between $tt^*$ geometries