LG/CY Correspondence Between tt* Geometries

• 作者: Huijun Fan Tian Lan Zongrui Yang
• 来源期刊: 数学研究通讯
• 年，卷（期）: 2021,（3）
• 页码范围: 297-348
• 页数: 52页
• 语种: 英文
• DOI: 10.4208/cmr.2020-0050

摘要

The concept of tt* geometric structure was introduced by physicists(see[4,9]and references therein),and then studied firstly in mathematics by C.Hertling[26].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* geome-try and obtain the following result.Let f ∈?[z0,…,zn+1]be a nondegenerate homogeneous polynomial of degree n+2,then it defines a Calabi-Yau model represented by a Calabi-Yau hypersurface Xf in(CP)n+1 or a Landau-Ginzburg model represented by a hypersurface singularity( ?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.