LG/CY correspondence between $tt^*$ geometries

Abstract

The concept of $tt^*$ geometric structure was introduced by physicists (see \cite{CV1, BCOV} and references therein) , and then studied firstly in mathematics by C. Hertling \cite{Het1}. It is believed that the $tt^*$ geometric structure contains the whole genus $0$ information of a two dimensional topological field theory. In this paper, we propose the LG/CY correspondence conjecture for $tt^*$ geometry and obtain the following result. Let $f\in\mathbb{C}[z_0, \dots, z_{n+2}]$ be a nondegenerate homogeneous polynomialof 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 $(\mathbb{C}^{n+2}, f)$, both can be written 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 the isomorphism of almost all structures in $tt^*$ geometries between these two models except the isomorphism between real structures.