Constructing the LG/CY isomorphism between $tt^*$ geometries

Abstract

For a nondegenerate homogeneous polynomial $f\in\mathbb{C}[z_0, \dots, z_{n+1}]$ with degree $n+2$, we can obtain a $tt^*$ structure from the Landau-Ginzburg model $(\C^{n+2}, f)$ and a (new) $tt^*$ structure on the Calabi-Yau hypersurface defined by the zero locus of $f$ in $\C P^{n+1}$. We can prove that the big residue map considered by Steenbrink gives an isomorphism between the two $tt^*$ structures. We also build the correspondence for non-Calabi-Yau cases, and it turns out that only partial structure can be preserved. As an application, we show that the $tt^*$ geometry structure of Landau-Ginzburg model on relavant deformation space uniquely determines the $tt^*$ geometry structure on Calabi-Yau side. This explains the folklore conclusion in physical literature. This result is based on our early work \cite{FLY}.