论文标题
$ tt^*$几何形状之间的LG/CY对应关系
LG/CY correspondence between $tt^*$ geometries
论文作者
论文摘要
$ tt^*$几何结构的概念是由物理学家引入的（请参阅\ cite {cv1，bcov}及其中的参考），然后首先由C. Hertling \ cite \ cite {het1}在数学上进行研究。据信，$ tt^*$几何结构包含二维拓扑场理论的整个属$ 0 $信息。在本文中，我们提出了$ tt^*$几何形状的LG/CY对应性猜想，并获得以下结果。令$ f \ in \ mathbb {c} [z_0，\ dots，z_ {n+2}] $为非等级的均质polyenmialof guger $ n+2 $，然后定义了由calabi-yau模型定义了由calabi-yau yaa yau-yau yaa yau yaa yaau yaa ya ya ya ya yaf $ x_f $ in $ x_f $ \ mathbb bb bb bb bb bb bb bb bb bb bb bbb的{cp} Landau-Ginzburg模型由HyperSurface Singularity $（\ Mathbb {C}^{N+2}，F）$表示，两者都可以写成$ tt^*$结构。我们证明了在Landau-Ginzburg方面存在$ TT^*$子结构，该子结构应与Calabi-Yau侧的Hodge结构的变化中的$ TT^*$结构相对应。我们在这两个模型之间建立了几乎所有结构的同构，除了真实结构之间的同构。
The concept of $tt^*$ 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 $tt^*$ 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 $f\in\mathbb{C}[z_0, \dots, z_{n+2}]$ 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 $tt^*$ 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.
