LG/CY correspondence between $t{t}^{\ast }$ geometries

Abstract

The concept of $t{t}^{\ast }$ 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 $t{t}^{\ast }$ 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 $t{t}^{\ast }$ 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 $t{t}^{\ast }$ structure. We proved that there exists a $t{t}^{\ast }$ substructure on Landau-Ginzburg side, which should correspond to the $t{t}^{\ast }$ structure from variation of Hodge structures in Calabi-Yau side. We build the isomorphism of almost all structures in $t{t}^{\ast }$ geometries between these two models except the isomorphism between real structures.

arXiv comment41 pages

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