Differential geometrical method of Landau–Ginzburg models

In this talk, I will survey the differential geometrical method on the study of LG models. LG models in simplest form is a pair $(M,f)$, where $f$ is a holomorphic function defined on a complex manifold. However, due to the LG/CY correspondence conjecture arising in Gauged linear sigma model, which was used by Witten to explain the mirror symmetry phenomena, this simple models contain enough information about the corresponding Calabi–Yau models. Based on the observation of physicists, I initiated a differential geometrical method to study the LG model since 2011. This method leads to the study of Schrödinger equations and its deformation, $tt^*$-geometrical structures and integral systems.