LG/CY Correspondence Between $tt^∗$ Geometries (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 byC. Hertling [28]. It is believed that the $tt^∗$ geometric structure contains thewhole genus 0 information of a two dimensional topological field theory. Inthis paper, we propose the LG/CY correspondence conjecture for  $tt^∗$ geometry and obtain the following result. Let $f ∈ \mathbb{C}[z_0,...,z_{n+1}]$ be a nondegeneratehomogeneous polynomial of degree $n$+2, then it defines a Calabi-Yau modelrepresented by a Calabi-Yau hypersurface $X_f$ in $\mathbb{CP}^{n+1}$ or a Landau-Ginzburgmodel represented by a hypersurface singularity ($\mathbb{C}^{n+2}, f$), both can be writtenas a $tt^∗$structure. We proved that there exists a $tt^∗$substructure on Landau-Ginzburg side, which should correspond to the $tt^∗$structure from variation ofHodge structures in Calabi-Yau side. We build the isomorphism of almost allstructures in $tt^∗$ geometries between these two models except the isomorphismbetween real structures.
Keywords: $tt^∗$ geometry; Landau-Ginzburg/Calabi-Yau correspondence; variation of Hodge structures.