(untitled) (Communications in Mathematical Research, 2021)
The concept of tt^∗ geometric structure was introduced by physicists (see [4, 10] and references therein), and then studied firstly in mathematics by C. Hertling [28]. 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 ∈ C[z_0,...,z_n+1] 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 CP^n+1 or a Landau-Ginzburg model represented by a hypersurface singularity (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.